Properties

Label 177.10.a.b.1.2
Level $177$
Weight $10$
Character 177.1
Self dual yes
Analytic conductor $91.161$
Analytic rank $1$
Dimension $21$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

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

Embedding invariants

Embedding label 1.2
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-39.7720 q^{2} -81.0000 q^{3} +1069.81 q^{4} +1152.50 q^{5} +3221.53 q^{6} -1754.07 q^{7} -22185.4 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q-39.7720 q^{2} -81.0000 q^{3} +1069.81 q^{4} +1152.50 q^{5} +3221.53 q^{6} -1754.07 q^{7} -22185.4 q^{8} +6561.00 q^{9} -45837.2 q^{10} +74114.4 q^{11} -86654.9 q^{12} +124475. q^{13} +69762.8 q^{14} -93352.4 q^{15} +334612. q^{16} -282894. q^{17} -260944. q^{18} +498014. q^{19} +1.23296e6 q^{20} +142079. q^{21} -2.94768e6 q^{22} -1.32368e6 q^{23} +1.79701e6 q^{24} -624872. q^{25} -4.95061e6 q^{26} -531441. q^{27} -1.87652e6 q^{28} -2.58997e6 q^{29} +3.71281e6 q^{30} -8.69842e6 q^{31} -1.94930e6 q^{32} -6.00327e6 q^{33} +1.12512e7 q^{34} -2.02156e6 q^{35} +7.01905e6 q^{36} -1.67010e7 q^{37} -1.98070e7 q^{38} -1.00824e7 q^{39} -2.55686e7 q^{40} +3.96049e6 q^{41} -5.65078e6 q^{42} +3.80945e7 q^{43} +7.92886e7 q^{44} +7.56154e6 q^{45} +5.26455e7 q^{46} +8.57711e6 q^{47} -2.71036e7 q^{48} -3.72769e7 q^{49} +2.48524e7 q^{50} +2.29144e7 q^{51} +1.33165e8 q^{52} -2.60493e7 q^{53} +2.11365e7 q^{54} +8.54168e7 q^{55} +3.89146e7 q^{56} -4.03391e7 q^{57} +1.03008e8 q^{58} -1.21174e7 q^{59} -9.98696e7 q^{60} +1.24109e8 q^{61} +3.45954e8 q^{62} -1.15084e7 q^{63} -9.37939e7 q^{64} +1.43457e8 q^{65} +2.38762e8 q^{66} -2.73027e8 q^{67} -3.02643e8 q^{68} +1.07218e8 q^{69} +8.04015e7 q^{70} +9.64618e7 q^{71} -1.45558e8 q^{72} -4.37807e8 q^{73} +6.64231e8 q^{74} +5.06146e7 q^{75} +5.32782e8 q^{76} -1.30002e8 q^{77} +4.00999e8 q^{78} -3.71631e8 q^{79} +3.85640e8 q^{80} +4.30467e7 q^{81} -1.57517e8 q^{82} +7.07879e8 q^{83} +1.51998e8 q^{84} -3.26034e8 q^{85} -1.51509e9 q^{86} +2.09787e8 q^{87} -1.64426e9 q^{88} +8.00284e8 q^{89} -3.00738e8 q^{90} -2.18337e8 q^{91} -1.41609e9 q^{92} +7.04572e8 q^{93} -3.41129e8 q^{94} +5.73960e8 q^{95} +1.57893e8 q^{96} +8.70957e8 q^{97} +1.48258e9 q^{98} +4.86265e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q + 20q^{2} - 1701q^{3} + 4950q^{4} + 2058q^{5} - 1620q^{6} - 17167q^{7} - 2853q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q + 20q^{2} - 1701q^{3} + 4950q^{4} + 2058q^{5} - 1620q^{6} - 17167q^{7} - 2853q^{8} + 137781q^{9} - 31559q^{10} - 38751q^{11} - 400950q^{12} - 58915q^{13} + 3453q^{14} - 166698q^{15} + 1655714q^{16} - 64233q^{17} + 131220q^{18} - 1937236q^{19} - 1065507q^{20} + 1390527q^{21} - 5386882q^{22} - 1838574q^{23} + 231093q^{24} + 4565755q^{25} - 839702q^{26} - 11160261q^{27} - 4471034q^{28} + 15658544q^{29} + 2556279q^{30} - 14282802q^{31} - 2205286q^{32} + 3138831q^{33} + 19005532q^{34} - 8633300q^{35} + 32476950q^{36} + 7531195q^{37} + 26649773q^{38} + 4772115q^{39} + 17775672q^{40} + 18338245q^{41} - 279693q^{42} - 22480305q^{43} - 80230922q^{44} + 13502538q^{45} - 83894107q^{46} - 110397260q^{47} - 134112834q^{48} + 130653638q^{49} + 65575693q^{50} + 5202873q^{51} + 177908014q^{52} + 145498338q^{53} - 10628820q^{54} + 86448944q^{55} + 354387888q^{56} + 156916116q^{57} + 115508368q^{58} - 254464581q^{59} + 86306067q^{60} + 287595506q^{61} + 819899030q^{62} - 112632687q^{63} + 822446413q^{64} + 77238206q^{65} + 436337442q^{66} - 392860610q^{67} + 167325073q^{68} + 148924494q^{69} - 424902116q^{70} - 248960491q^{71} - 18718533q^{72} - 758406074q^{73} - 923266846q^{74} - 369826155q^{75} - 2312747568q^{76} - 878126795q^{77} + 68015862q^{78} - 1925801029q^{79} - 1898919861q^{80} + 903981141q^{81} - 3249102191q^{82} - 1650336307q^{83} + 362153754q^{84} - 2342480762q^{85} - 3609864952q^{86} - 1268342064q^{87} - 5987792887q^{88} - 574997526q^{89} - 207058599q^{90} - 4481387117q^{91} - 5317166770q^{92} + 1156906962q^{93} - 5360726568q^{94} - 2789231462q^{95} + 178628166q^{96} - 4651540898q^{97} - 5566652976q^{98} - 254245311q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −39.7720 −1.75769 −0.878846 0.477106i \(-0.841686\pi\)
−0.878846 + 0.477106i \(0.841686\pi\)
\(3\) −81.0000 −0.577350
\(4\) 1069.81 2.08948
\(5\) 1152.50 0.824661 0.412330 0.911034i \(-0.364715\pi\)
0.412330 + 0.911034i \(0.364715\pi\)
\(6\) 3221.53 1.01480
\(7\) −1754.07 −0.276124 −0.138062 0.990424i \(-0.544087\pi\)
−0.138062 + 0.990424i \(0.544087\pi\)
\(8\) −22185.4 −1.91497
\(9\) 6561.00 0.333333
\(10\) −45837.2 −1.44950
\(11\) 74114.4 1.52628 0.763142 0.646230i \(-0.223655\pi\)
0.763142 + 0.646230i \(0.223655\pi\)
\(12\) −86654.9 −1.20636
\(13\) 124475. 1.20875 0.604374 0.796701i \(-0.293423\pi\)
0.604374 + 0.796701i \(0.293423\pi\)
\(14\) 69762.8 0.485342
\(15\) −93352.4 −0.476118
\(16\) 334612. 1.27644
\(17\) −282894. −0.821491 −0.410746 0.911750i \(-0.634732\pi\)
−0.410746 + 0.911750i \(0.634732\pi\)
\(18\) −260944. −0.585897
\(19\) 498014. 0.876698 0.438349 0.898805i \(-0.355563\pi\)
0.438349 + 0.898805i \(0.355563\pi\)
\(20\) 1.23296e6 1.72311
\(21\) 142079. 0.159421
\(22\) −2.94768e6 −2.68274
\(23\) −1.32368e6 −0.986299 −0.493150 0.869945i \(-0.664154\pi\)
−0.493150 + 0.869945i \(0.664154\pi\)
\(24\) 1.79701e6 1.10561
\(25\) −624872. −0.319934
\(26\) −4.95061e6 −2.12461
\(27\) −531441. −0.192450
\(28\) −1.87652e6 −0.576956
\(29\) −2.58997e6 −0.679991 −0.339996 0.940427i \(-0.610426\pi\)
−0.339996 + 0.940427i \(0.610426\pi\)
\(30\) 3.71281e6 0.836869
\(31\) −8.69842e6 −1.69166 −0.845830 0.533453i \(-0.820894\pi\)
−0.845830 + 0.533453i \(0.820894\pi\)
\(32\) −1.94930e6 −0.328627
\(33\) −6.00327e6 −0.881201
\(34\) 1.12512e7 1.44393
\(35\) −2.02156e6 −0.227709
\(36\) 7.01905e6 0.696493
\(37\) −1.67010e7 −1.46499 −0.732493 0.680775i \(-0.761643\pi\)
−0.732493 + 0.680775i \(0.761643\pi\)
\(38\) −1.98070e7 −1.54097
\(39\) −1.00824e7 −0.697871
\(40\) −2.55686e7 −1.57920
\(41\) 3.96049e6 0.218888 0.109444 0.993993i \(-0.465093\pi\)
0.109444 + 0.993993i \(0.465093\pi\)
\(42\) −5.65078e6 −0.280212
\(43\) 3.80945e7 1.69924 0.849618 0.527399i \(-0.176833\pi\)
0.849618 + 0.527399i \(0.176833\pi\)
\(44\) 7.92886e7 3.18914
\(45\) 7.56154e6 0.274887
\(46\) 5.26455e7 1.73361
\(47\) 8.57711e6 0.256390 0.128195 0.991749i \(-0.459082\pi\)
0.128195 + 0.991749i \(0.459082\pi\)
\(48\) −2.71036e7 −0.736956
\(49\) −3.72769e7 −0.923755
\(50\) 2.48524e7 0.562346
\(51\) 2.29144e7 0.474288
\(52\) 1.33165e8 2.52565
\(53\) −2.60493e7 −0.453476 −0.226738 0.973956i \(-0.572806\pi\)
−0.226738 + 0.973956i \(0.572806\pi\)
\(54\) 2.11365e7 0.338268
\(55\) 8.54168e7 1.25867
\(56\) 3.89146e7 0.528770
\(57\) −4.03391e7 −0.506162
\(58\) 1.03008e8 1.19521
\(59\) −1.21174e7 −0.130189
\(60\) −9.98696e7 −0.994839
\(61\) 1.24109e8 1.14767 0.573835 0.818971i \(-0.305455\pi\)
0.573835 + 0.818971i \(0.305455\pi\)
\(62\) 3.45954e8 2.97342
\(63\) −1.15084e7 −0.0920415
\(64\) −9.37939e7 −0.698819
\(65\) 1.43457e8 0.996807
\(66\) 2.38762e8 1.54888
\(67\) −2.73027e8 −1.65527 −0.827634 0.561269i \(-0.810313\pi\)
−0.827634 + 0.561269i \(0.810313\pi\)
\(68\) −3.02643e8 −1.71649
\(69\) 1.07218e8 0.569440
\(70\) 8.04015e7 0.400242
\(71\) 9.64618e7 0.450498 0.225249 0.974301i \(-0.427680\pi\)
0.225249 + 0.974301i \(0.427680\pi\)
\(72\) −1.45558e8 −0.638323
\(73\) −4.37807e8 −1.80439 −0.902194 0.431330i \(-0.858045\pi\)
−0.902194 + 0.431330i \(0.858045\pi\)
\(74\) 6.64231e8 2.57499
\(75\) 5.06146e7 0.184714
\(76\) 5.32782e8 1.83184
\(77\) −1.30002e8 −0.421445
\(78\) 4.00999e8 1.22664
\(79\) −3.71631e8 −1.07347 −0.536736 0.843751i \(-0.680343\pi\)
−0.536736 + 0.843751i \(0.680343\pi\)
\(80\) 3.85640e8 1.05263
\(81\) 4.30467e7 0.111111
\(82\) −1.57517e8 −0.384737
\(83\) 7.07879e8 1.63722 0.818611 0.574348i \(-0.194744\pi\)
0.818611 + 0.574348i \(0.194744\pi\)
\(84\) 1.51998e8 0.333106
\(85\) −3.26034e8 −0.677452
\(86\) −1.51509e9 −2.98673
\(87\) 2.09787e8 0.392593
\(88\) −1.64426e9 −2.92279
\(89\) 8.00284e8 1.35204 0.676019 0.736884i \(-0.263704\pi\)
0.676019 + 0.736884i \(0.263704\pi\)
\(90\) −3.00738e8 −0.483166
\(91\) −2.18337e8 −0.333765
\(92\) −1.41609e9 −2.06085
\(93\) 7.04572e8 0.976680
\(94\) −3.41129e8 −0.450654
\(95\) 5.73960e8 0.722979
\(96\) 1.57893e8 0.189733
\(97\) 8.70957e8 0.998904 0.499452 0.866341i \(-0.333535\pi\)
0.499452 + 0.866341i \(0.333535\pi\)
\(98\) 1.48258e9 1.62368
\(99\) 4.86265e8 0.508762
\(100\) −6.68496e8 −0.668496
\(101\) 1.96316e9 1.87719 0.938597 0.345014i \(-0.112126\pi\)
0.938597 + 0.345014i \(0.112126\pi\)
\(102\) −9.11351e8 −0.833652
\(103\) −3.06917e8 −0.268691 −0.134345 0.990935i \(-0.542893\pi\)
−0.134345 + 0.990935i \(0.542893\pi\)
\(104\) −2.76152e9 −2.31471
\(105\) 1.63746e8 0.131468
\(106\) 1.03603e9 0.797071
\(107\) −1.66746e9 −1.22978 −0.614890 0.788613i \(-0.710800\pi\)
−0.614890 + 0.788613i \(0.710800\pi\)
\(108\) −5.68543e8 −0.402120
\(109\) −1.86533e9 −1.26572 −0.632858 0.774268i \(-0.718118\pi\)
−0.632858 + 0.774268i \(0.718118\pi\)
\(110\) −3.39720e9 −2.21235
\(111\) 1.35278e9 0.845810
\(112\) −5.86932e8 −0.352457
\(113\) −1.52291e9 −0.878662 −0.439331 0.898325i \(-0.644784\pi\)
−0.439331 + 0.898325i \(0.644784\pi\)
\(114\) 1.60437e9 0.889677
\(115\) −1.52554e9 −0.813362
\(116\) −2.77078e9 −1.42083
\(117\) 8.16678e8 0.402916
\(118\) 4.81932e8 0.228832
\(119\) 4.96214e8 0.226834
\(120\) 2.07106e9 0.911751
\(121\) 3.13500e9 1.32955
\(122\) −4.93605e9 −2.01725
\(123\) −3.20799e8 −0.126375
\(124\) −9.30569e9 −3.53469
\(125\) −2.97114e9 −1.08850
\(126\) 4.57713e8 0.161781
\(127\) −4.56037e9 −1.55555 −0.777773 0.628545i \(-0.783651\pi\)
−0.777773 + 0.628545i \(0.783651\pi\)
\(128\) 4.72842e9 1.55694
\(129\) −3.08565e9 −0.981054
\(130\) −5.70557e9 −1.75208
\(131\) 1.59631e9 0.473584 0.236792 0.971560i \(-0.423904\pi\)
0.236792 + 0.971560i \(0.423904\pi\)
\(132\) −6.42238e9 −1.84125
\(133\) −8.73549e8 −0.242078
\(134\) 1.08588e10 2.90945
\(135\) −6.12485e8 −0.158706
\(136\) 6.27610e9 1.57313
\(137\) −3.08714e9 −0.748711 −0.374355 0.927285i \(-0.622136\pi\)
−0.374355 + 0.927285i \(0.622136\pi\)
\(138\) −4.26429e9 −1.00090
\(139\) 1.68833e9 0.383610 0.191805 0.981433i \(-0.438566\pi\)
0.191805 + 0.981433i \(0.438566\pi\)
\(140\) −2.16269e9 −0.475793
\(141\) −6.94746e8 −0.148027
\(142\) −3.83648e9 −0.791836
\(143\) 9.22537e9 1.84489
\(144\) 2.19539e9 0.425482
\(145\) −2.98493e9 −0.560762
\(146\) 1.74125e10 3.17156
\(147\) 3.01943e9 0.533330
\(148\) −1.78669e10 −3.06106
\(149\) 3.27902e9 0.545012 0.272506 0.962154i \(-0.412148\pi\)
0.272506 + 0.962154i \(0.412148\pi\)
\(150\) −2.01305e9 −0.324671
\(151\) −3.26329e9 −0.510809 −0.255405 0.966834i \(-0.582209\pi\)
−0.255405 + 0.966834i \(0.582209\pi\)
\(152\) −1.10486e10 −1.67885
\(153\) −1.85606e9 −0.273830
\(154\) 5.17043e9 0.740769
\(155\) −1.00249e10 −1.39505
\(156\) −1.07863e10 −1.45819
\(157\) −1.01602e10 −1.33460 −0.667301 0.744788i \(-0.732551\pi\)
−0.667301 + 0.744788i \(0.732551\pi\)
\(158\) 1.47805e10 1.88683
\(159\) 2.10999e9 0.261815
\(160\) −2.24656e9 −0.271006
\(161\) 2.32183e9 0.272341
\(162\) −1.71205e9 −0.195299
\(163\) 4.63462e9 0.514245 0.257123 0.966379i \(-0.417226\pi\)
0.257123 + 0.966379i \(0.417226\pi\)
\(164\) 4.23698e9 0.457361
\(165\) −6.91876e9 −0.726692
\(166\) −2.81538e10 −2.87773
\(167\) 5.62382e8 0.0559509 0.0279754 0.999609i \(-0.491094\pi\)
0.0279754 + 0.999609i \(0.491094\pi\)
\(168\) −3.15208e9 −0.305285
\(169\) 4.88944e9 0.461072
\(170\) 1.29670e10 1.19075
\(171\) 3.26747e9 0.292233
\(172\) 4.07540e10 3.55052
\(173\) −1.09709e10 −0.931186 −0.465593 0.884999i \(-0.654159\pi\)
−0.465593 + 0.884999i \(0.654159\pi\)
\(174\) −8.34366e9 −0.690057
\(175\) 1.09607e9 0.0883417
\(176\) 2.47996e10 1.94822
\(177\) 9.81506e8 0.0751646
\(178\) −3.18289e10 −2.37647
\(179\) −1.54049e10 −1.12155 −0.560777 0.827967i \(-0.689498\pi\)
−0.560777 + 0.827967i \(0.689498\pi\)
\(180\) 8.08944e9 0.574371
\(181\) 7.95236e9 0.550735 0.275367 0.961339i \(-0.411201\pi\)
0.275367 + 0.961339i \(0.411201\pi\)
\(182\) 8.68369e9 0.586656
\(183\) −1.00528e10 −0.662608
\(184\) 2.93664e10 1.88873
\(185\) −1.92478e10 −1.20812
\(186\) −2.80223e10 −1.71670
\(187\) −2.09665e10 −1.25383
\(188\) 9.17590e9 0.535721
\(189\) 9.32183e8 0.0531402
\(190\) −2.28276e10 −1.27077
\(191\) −4.91634e9 −0.267296 −0.133648 0.991029i \(-0.542669\pi\)
−0.133648 + 0.991029i \(0.542669\pi\)
\(192\) 7.59731e9 0.403464
\(193\) 1.64699e10 0.854441 0.427221 0.904147i \(-0.359493\pi\)
0.427221 + 0.904147i \(0.359493\pi\)
\(194\) −3.46397e10 −1.75577
\(195\) −1.16200e10 −0.575507
\(196\) −3.98793e10 −1.93017
\(197\) 1.15190e10 0.544898 0.272449 0.962170i \(-0.412166\pi\)
0.272449 + 0.962170i \(0.412166\pi\)
\(198\) −1.93397e10 −0.894246
\(199\) 2.25046e9 0.101726 0.0508630 0.998706i \(-0.483803\pi\)
0.0508630 + 0.998706i \(0.483803\pi\)
\(200\) 1.38630e10 0.612664
\(201\) 2.21151e10 0.955669
\(202\) −7.80788e10 −3.29953
\(203\) 4.54297e9 0.187762
\(204\) 2.45141e10 0.991015
\(205\) 4.56445e9 0.180508
\(206\) 1.22067e10 0.472276
\(207\) −8.68468e9 −0.328766
\(208\) 4.16508e10 1.54290
\(209\) 3.69100e10 1.33809
\(210\) −6.51252e9 −0.231080
\(211\) 1.52648e10 0.530175 0.265088 0.964224i \(-0.414599\pi\)
0.265088 + 0.964224i \(0.414599\pi\)
\(212\) −2.78679e10 −0.947529
\(213\) −7.81340e9 −0.260095
\(214\) 6.63181e10 2.16157
\(215\) 4.39038e10 1.40129
\(216\) 1.17902e10 0.368536
\(217\) 1.52576e10 0.467108
\(218\) 7.41879e10 2.22474
\(219\) 3.54624e10 1.04176
\(220\) 9.13800e10 2.62996
\(221\) −3.52131e10 −0.992976
\(222\) −5.38027e10 −1.48667
\(223\) −5.65463e10 −1.53120 −0.765601 0.643316i \(-0.777558\pi\)
−0.765601 + 0.643316i \(0.777558\pi\)
\(224\) 3.41920e9 0.0907420
\(225\) −4.09978e9 −0.106645
\(226\) 6.05693e10 1.54442
\(227\) −2.12964e10 −0.532341 −0.266170 0.963926i \(-0.585758\pi\)
−0.266170 + 0.963926i \(0.585758\pi\)
\(228\) −4.31553e10 −1.05762
\(229\) −7.88715e9 −0.189522 −0.0947612 0.995500i \(-0.530209\pi\)
−0.0947612 + 0.995500i \(0.530209\pi\)
\(230\) 6.06739e10 1.42964
\(231\) 1.05301e10 0.243321
\(232\) 5.74594e10 1.30216
\(233\) −1.78278e10 −0.396275 −0.198137 0.980174i \(-0.563489\pi\)
−0.198137 + 0.980174i \(0.563489\pi\)
\(234\) −3.24809e10 −0.708202
\(235\) 9.88510e9 0.211435
\(236\) −1.29633e10 −0.272027
\(237\) 3.01021e10 0.619769
\(238\) −1.97354e10 −0.398704
\(239\) −1.96048e10 −0.388662 −0.194331 0.980936i \(-0.562254\pi\)
−0.194331 + 0.980936i \(0.562254\pi\)
\(240\) −3.12369e10 −0.607738
\(241\) 5.90159e10 1.12692 0.563459 0.826144i \(-0.309470\pi\)
0.563459 + 0.826144i \(0.309470\pi\)
\(242\) −1.24685e11 −2.33693
\(243\) −3.48678e9 −0.0641500
\(244\) 1.32773e11 2.39803
\(245\) −4.29615e10 −0.761785
\(246\) 1.27588e10 0.222128
\(247\) 6.19901e10 1.05971
\(248\) 1.92978e11 3.23947
\(249\) −5.73382e10 −0.945251
\(250\) 1.18168e11 1.91324
\(251\) 5.30307e10 0.843326 0.421663 0.906753i \(-0.361447\pi\)
0.421663 + 0.906753i \(0.361447\pi\)
\(252\) −1.23119e10 −0.192319
\(253\) −9.81040e10 −1.50537
\(254\) 1.81375e11 2.73417
\(255\) 2.64088e10 0.391127
\(256\) −1.40036e11 −2.03779
\(257\) 9.08290e10 1.29875 0.649375 0.760468i \(-0.275031\pi\)
0.649375 + 0.760468i \(0.275031\pi\)
\(258\) 1.22723e11 1.72439
\(259\) 2.92946e10 0.404518
\(260\) 1.53472e11 2.08281
\(261\) −1.69928e10 −0.226664
\(262\) −6.34886e10 −0.832415
\(263\) −1.48249e11 −1.91070 −0.955348 0.295484i \(-0.904519\pi\)
−0.955348 + 0.295484i \(0.904519\pi\)
\(264\) 1.33185e11 1.68747
\(265\) −3.00218e10 −0.373964
\(266\) 3.47428e10 0.425498
\(267\) −6.48230e10 −0.780600
\(268\) −2.92087e11 −3.45865
\(269\) 3.71116e10 0.432140 0.216070 0.976378i \(-0.430676\pi\)
0.216070 + 0.976378i \(0.430676\pi\)
\(270\) 2.43598e10 0.278956
\(271\) −2.65662e10 −0.299204 −0.149602 0.988746i \(-0.547799\pi\)
−0.149602 + 0.988746i \(0.547799\pi\)
\(272\) −9.46597e10 −1.04859
\(273\) 1.76853e10 0.192699
\(274\) 1.22782e11 1.31600
\(275\) −4.63120e10 −0.488311
\(276\) 1.14704e11 1.18983
\(277\) −1.48091e11 −1.51137 −0.755686 0.654935i \(-0.772696\pi\)
−0.755686 + 0.654935i \(0.772696\pi\)
\(278\) −6.71481e10 −0.674268
\(279\) −5.70704e10 −0.563887
\(280\) 4.48490e10 0.436056
\(281\) 1.98809e10 0.190221 0.0951105 0.995467i \(-0.469680\pi\)
0.0951105 + 0.995467i \(0.469680\pi\)
\(282\) 2.76314e10 0.260185
\(283\) −9.03809e10 −0.837602 −0.418801 0.908078i \(-0.637550\pi\)
−0.418801 + 0.908078i \(0.637550\pi\)
\(284\) 1.03196e11 0.941306
\(285\) −4.64908e10 −0.417412
\(286\) −3.66911e11 −3.24275
\(287\) −6.94695e9 −0.0604402
\(288\) −1.27893e10 −0.109542
\(289\) −3.85591e10 −0.325152
\(290\) 1.18717e11 0.985647
\(291\) −7.05475e10 −0.576718
\(292\) −4.68372e11 −3.77023
\(293\) 9.32623e10 0.739268 0.369634 0.929178i \(-0.379483\pi\)
0.369634 + 0.929178i \(0.379483\pi\)
\(294\) −1.20089e11 −0.937430
\(295\) −1.39652e10 −0.107362
\(296\) 3.70517e11 2.80540
\(297\) −3.93874e10 −0.293734
\(298\) −1.30413e11 −0.957962
\(299\) −1.64765e11 −1.19219
\(300\) 5.41482e10 0.385957
\(301\) −6.68202e10 −0.469201
\(302\) 1.29787e11 0.897845
\(303\) −1.59016e11 −1.08380
\(304\) 1.66642e11 1.11906
\(305\) 1.43035e11 0.946439
\(306\) 7.38194e10 0.481309
\(307\) 1.60471e11 1.03104 0.515519 0.856878i \(-0.327599\pi\)
0.515519 + 0.856878i \(0.327599\pi\)
\(308\) −1.39077e11 −0.880600
\(309\) 2.48603e10 0.155129
\(310\) 3.98711e11 2.45206
\(311\) −8.93285e10 −0.541462 −0.270731 0.962655i \(-0.587265\pi\)
−0.270731 + 0.962655i \(0.587265\pi\)
\(312\) 2.23683e11 1.33640
\(313\) 1.18187e11 0.696018 0.348009 0.937491i \(-0.386858\pi\)
0.348009 + 0.937491i \(0.386858\pi\)
\(314\) 4.04090e11 2.34582
\(315\) −1.32634e10 −0.0759030
\(316\) −3.97576e11 −2.24300
\(317\) −1.60397e11 −0.892131 −0.446066 0.895000i \(-0.647175\pi\)
−0.446066 + 0.895000i \(0.647175\pi\)
\(318\) −8.39186e10 −0.460189
\(319\) −1.91954e11 −1.03786
\(320\) −1.08097e11 −0.576289
\(321\) 1.35064e11 0.710014
\(322\) −9.23438e10 −0.478692
\(323\) −1.40885e11 −0.720200
\(324\) 4.60520e10 0.232164
\(325\) −7.77807e10 −0.386720
\(326\) −1.84328e11 −0.903884
\(327\) 1.51092e11 0.730761
\(328\) −8.78648e10 −0.419163
\(329\) −1.50448e10 −0.0707954
\(330\) 2.75173e11 1.27730
\(331\) −1.93237e11 −0.884837 −0.442419 0.896809i \(-0.645879\pi\)
−0.442419 + 0.896809i \(0.645879\pi\)
\(332\) 7.57299e11 3.42094
\(333\) −1.09575e11 −0.488329
\(334\) −2.23670e10 −0.0983444
\(335\) −3.14663e11 −1.36503
\(336\) 4.75415e10 0.203491
\(337\) −1.27963e11 −0.540441 −0.270221 0.962798i \(-0.587097\pi\)
−0.270221 + 0.962798i \(0.587097\pi\)
\(338\) −1.94463e11 −0.810423
\(339\) 1.23356e11 0.507296
\(340\) −3.48796e11 −1.41552
\(341\) −6.44679e11 −2.58195
\(342\) −1.29954e11 −0.513655
\(343\) 1.36169e11 0.531196
\(344\) −8.45139e11 −3.25398
\(345\) 1.23569e11 0.469595
\(346\) 4.36337e11 1.63674
\(347\) 1.92497e11 0.712755 0.356378 0.934342i \(-0.384012\pi\)
0.356378 + 0.934342i \(0.384012\pi\)
\(348\) 2.24433e11 0.820315
\(349\) −4.61360e11 −1.66466 −0.832329 0.554282i \(-0.812993\pi\)
−0.832329 + 0.554282i \(0.812993\pi\)
\(350\) −4.35928e10 −0.155277
\(351\) −6.61509e10 −0.232624
\(352\) −1.44471e11 −0.501579
\(353\) 2.93844e11 1.00724 0.503618 0.863926i \(-0.332002\pi\)
0.503618 + 0.863926i \(0.332002\pi\)
\(354\) −3.90365e10 −0.132116
\(355\) 1.11172e11 0.371508
\(356\) 8.56154e11 2.82506
\(357\) −4.01933e10 −0.130963
\(358\) 6.12684e11 1.97135
\(359\) 7.15931e10 0.227482 0.113741 0.993510i \(-0.463717\pi\)
0.113741 + 0.993510i \(0.463717\pi\)
\(360\) −1.67756e11 −0.526400
\(361\) −7.46700e10 −0.231400
\(362\) −3.16281e11 −0.968021
\(363\) −2.53935e11 −0.767613
\(364\) −2.33580e11 −0.697395
\(365\) −5.04572e11 −1.48801
\(366\) 3.99820e11 1.16466
\(367\) 2.29259e11 0.659672 0.329836 0.944038i \(-0.393007\pi\)
0.329836 + 0.944038i \(0.393007\pi\)
\(368\) −4.42921e11 −1.25896
\(369\) 2.59847e10 0.0729625
\(370\) 7.65525e11 2.12350
\(371\) 4.56922e10 0.125216
\(372\) 7.53761e11 2.04075
\(373\) 6.54074e11 1.74959 0.874797 0.484489i \(-0.160994\pi\)
0.874797 + 0.484489i \(0.160994\pi\)
\(374\) 8.33880e11 2.20385
\(375\) 2.40662e11 0.628445
\(376\) −1.90286e11 −0.490978
\(377\) −3.22385e11 −0.821938
\(378\) −3.70748e10 −0.0934040
\(379\) −4.97719e11 −1.23910 −0.619552 0.784955i \(-0.712686\pi\)
−0.619552 + 0.784955i \(0.712686\pi\)
\(380\) 6.14030e11 1.51065
\(381\) 3.69390e11 0.898095
\(382\) 1.95533e11 0.469824
\(383\) 1.45916e11 0.346505 0.173253 0.984877i \(-0.444572\pi\)
0.173253 + 0.984877i \(0.444572\pi\)
\(384\) −3.83002e11 −0.898897
\(385\) −1.49827e11 −0.347549
\(386\) −6.55040e11 −1.50184
\(387\) 2.49938e11 0.566412
\(388\) 9.31761e11 2.08719
\(389\) −1.67014e11 −0.369812 −0.184906 0.982756i \(-0.559198\pi\)
−0.184906 + 0.982756i \(0.559198\pi\)
\(390\) 4.62151e11 1.01156
\(391\) 3.74461e11 0.810236
\(392\) 8.27001e11 1.76896
\(393\) −1.29301e11 −0.273424
\(394\) −4.58132e11 −0.957763
\(395\) −4.28305e11 −0.885250
\(396\) 5.20212e11 1.06305
\(397\) 9.16620e11 1.85196 0.925981 0.377570i \(-0.123240\pi\)
0.925981 + 0.377570i \(0.123240\pi\)
\(398\) −8.95053e10 −0.178803
\(399\) 7.07575e10 0.139764
\(400\) −2.09090e11 −0.408378
\(401\) 7.07689e11 1.36676 0.683381 0.730062i \(-0.260509\pi\)
0.683381 + 0.730062i \(0.260509\pi\)
\(402\) −8.79564e11 −1.67977
\(403\) −1.08273e12 −2.04479
\(404\) 2.10021e12 3.92236
\(405\) 4.96113e10 0.0916290
\(406\) −1.80683e11 −0.330028
\(407\) −1.23778e12 −2.23599
\(408\) −5.08364e11 −0.908247
\(409\) 7.48832e11 1.32321 0.661606 0.749852i \(-0.269875\pi\)
0.661606 + 0.749852i \(0.269875\pi\)
\(410\) −1.81538e11 −0.317277
\(411\) 2.50059e11 0.432268
\(412\) −3.28344e11 −0.561424
\(413\) 2.12547e10 0.0359483
\(414\) 3.45407e11 0.577870
\(415\) 8.15830e11 1.35015
\(416\) −2.42638e11 −0.397228
\(417\) −1.36754e11 −0.221477
\(418\) −1.46799e12 −2.35195
\(419\) −8.42398e11 −1.33522 −0.667612 0.744510i \(-0.732683\pi\)
−0.667612 + 0.744510i \(0.732683\pi\)
\(420\) 1.75178e11 0.274699
\(421\) −1.16454e11 −0.180670 −0.0903351 0.995911i \(-0.528794\pi\)
−0.0903351 + 0.995911i \(0.528794\pi\)
\(422\) −6.07111e11 −0.931885
\(423\) 5.62744e10 0.0854632
\(424\) 5.77913e11 0.868392
\(425\) 1.76772e11 0.262823
\(426\) 3.10755e11 0.457167
\(427\) −2.17695e11 −0.316900
\(428\) −1.78387e12 −2.56960
\(429\) −7.47255e11 −1.06515
\(430\) −1.74614e12 −2.46304
\(431\) 1.58044e11 0.220613 0.110306 0.993898i \(-0.464817\pi\)
0.110306 + 0.993898i \(0.464817\pi\)
\(432\) −1.77827e11 −0.245652
\(433\) −1.02201e12 −1.39721 −0.698604 0.715508i \(-0.746195\pi\)
−0.698604 + 0.715508i \(0.746195\pi\)
\(434\) −6.06826e11 −0.821033
\(435\) 2.41780e11 0.323756
\(436\) −1.99555e12 −2.64469
\(437\) −6.59212e11 −0.864687
\(438\) −1.41041e12 −1.83110
\(439\) −1.14406e12 −1.47014 −0.735072 0.677989i \(-0.762852\pi\)
−0.735072 + 0.677989i \(0.762852\pi\)
\(440\) −1.89500e12 −2.41031
\(441\) −2.44573e11 −0.307918
\(442\) 1.40050e12 1.74535
\(443\) −5.61863e11 −0.693129 −0.346564 0.938026i \(-0.612652\pi\)
−0.346564 + 0.938026i \(0.612652\pi\)
\(444\) 1.44722e12 1.76730
\(445\) 9.22326e11 1.11497
\(446\) 2.24896e12 2.69138
\(447\) −2.65601e11 −0.314663
\(448\) 1.64521e11 0.192961
\(449\) −3.36126e11 −0.390296 −0.195148 0.980774i \(-0.562519\pi\)
−0.195148 + 0.980774i \(0.562519\pi\)
\(450\) 1.63057e11 0.187449
\(451\) 2.93529e11 0.334085
\(452\) −1.62923e12 −1.83595
\(453\) 2.64326e11 0.294916
\(454\) 8.47000e11 0.935691
\(455\) −2.51633e11 −0.275243
\(456\) 8.94938e11 0.969284
\(457\) −3.44761e11 −0.369739 −0.184869 0.982763i \(-0.559186\pi\)
−0.184869 + 0.982763i \(0.559186\pi\)
\(458\) 3.13688e11 0.333122
\(459\) 1.50341e11 0.158096
\(460\) −1.63205e12 −1.69950
\(461\) 1.37614e12 1.41909 0.709543 0.704662i \(-0.248902\pi\)
0.709543 + 0.704662i \(0.248902\pi\)
\(462\) −4.18804e11 −0.427683
\(463\) −5.46956e11 −0.553144 −0.276572 0.960993i \(-0.589198\pi\)
−0.276572 + 0.960993i \(0.589198\pi\)
\(464\) −8.66635e11 −0.867971
\(465\) 8.12019e11 0.805430
\(466\) 7.09048e11 0.696528
\(467\) −1.43232e12 −1.39352 −0.696762 0.717302i \(-0.745377\pi\)
−0.696762 + 0.717302i \(0.745377\pi\)
\(468\) 8.73693e11 0.841885
\(469\) 4.78907e11 0.457060
\(470\) −3.93151e11 −0.371637
\(471\) 8.22973e11 0.770533
\(472\) 2.68828e11 0.249308
\(473\) 2.82335e12 2.59352
\(474\) −1.19722e12 −1.08936
\(475\) −3.11195e11 −0.280486
\(476\) 5.30856e11 0.473964
\(477\) −1.70909e11 −0.151159
\(478\) 7.79722e11 0.683147
\(479\) 1.06013e12 0.920126 0.460063 0.887886i \(-0.347827\pi\)
0.460063 + 0.887886i \(0.347827\pi\)
\(480\) 1.81972e11 0.156465
\(481\) −2.07885e12 −1.77080
\(482\) −2.34718e12 −1.98077
\(483\) −1.88068e11 −0.157236
\(484\) 3.35386e12 2.77806
\(485\) 1.00378e12 0.823757
\(486\) 1.38676e11 0.112756
\(487\) 8.36469e11 0.673860 0.336930 0.941530i \(-0.390611\pi\)
0.336930 + 0.941530i \(0.390611\pi\)
\(488\) −2.75339e12 −2.19775
\(489\) −3.75404e11 −0.296899
\(490\) 1.70867e12 1.33898
\(491\) −1.26919e12 −0.985504 −0.492752 0.870170i \(-0.664009\pi\)
−0.492752 + 0.870170i \(0.664009\pi\)
\(492\) −3.43195e11 −0.264058
\(493\) 7.32685e11 0.558607
\(494\) −2.46547e12 −1.86264
\(495\) 5.60419e11 0.419556
\(496\) −2.91060e12 −2.15931
\(497\) −1.69200e11 −0.124393
\(498\) 2.28046e12 1.66146
\(499\) 6.69271e11 0.483225 0.241613 0.970373i \(-0.422324\pi\)
0.241613 + 0.970373i \(0.422324\pi\)
\(500\) −3.17856e12 −2.27439
\(501\) −4.55529e10 −0.0323033
\(502\) −2.10914e12 −1.48231
\(503\) −1.58991e12 −1.10743 −0.553717 0.832705i \(-0.686791\pi\)
−0.553717 + 0.832705i \(0.686791\pi\)
\(504\) 2.55319e11 0.176257
\(505\) 2.26254e12 1.54805
\(506\) 3.90179e12 2.64598
\(507\) −3.96045e11 −0.266200
\(508\) −4.87874e12 −3.25028
\(509\) −2.11937e12 −1.39951 −0.699755 0.714382i \(-0.746708\pi\)
−0.699755 + 0.714382i \(0.746708\pi\)
\(510\) −1.05033e12 −0.687480
\(511\) 7.67943e11 0.498236
\(512\) 3.14857e12 2.02488
\(513\) −2.64665e11 −0.168721
\(514\) −3.61245e12 −2.28280
\(515\) −3.53721e11 −0.221579
\(516\) −3.30107e12 −2.04989
\(517\) 6.35687e11 0.391324
\(518\) −1.16510e12 −0.711019
\(519\) 8.88646e11 0.537621
\(520\) −3.18264e12 −1.90885
\(521\) 1.08323e12 0.644097 0.322049 0.946723i \(-0.395629\pi\)
0.322049 + 0.946723i \(0.395629\pi\)
\(522\) 6.75837e11 0.398405
\(523\) 2.01209e10 0.0117596 0.00587978 0.999983i \(-0.498128\pi\)
0.00587978 + 0.999983i \(0.498128\pi\)
\(524\) 1.70776e12 0.989545
\(525\) −8.87814e10 −0.0510041
\(526\) 5.89617e12 3.35841
\(527\) 2.46073e12 1.38968
\(528\) −2.00877e12 −1.12480
\(529\) −4.90165e10 −0.0272140
\(530\) 1.19403e12 0.657313
\(531\) −7.95020e10 −0.0433963
\(532\) −9.34535e11 −0.505817
\(533\) 4.92980e11 0.264580
\(534\) 2.57814e12 1.37205
\(535\) −1.92174e12 −1.01415
\(536\) 6.05719e12 3.16979
\(537\) 1.24780e12 0.647530
\(538\) −1.47600e12 −0.759568
\(539\) −2.76275e12 −1.40991
\(540\) −6.55245e11 −0.331613
\(541\) 8.36978e11 0.420075 0.210037 0.977693i \(-0.432641\pi\)
0.210037 + 0.977693i \(0.432641\pi\)
\(542\) 1.05659e12 0.525909
\(543\) −6.44141e11 −0.317967
\(544\) 5.51444e11 0.269964
\(545\) −2.14979e12 −1.04379
\(546\) −7.03379e11 −0.338706
\(547\) 6.32552e10 0.0302102 0.0151051 0.999886i \(-0.495192\pi\)
0.0151051 + 0.999886i \(0.495192\pi\)
\(548\) −3.30267e12 −1.56442
\(549\) 8.14276e11 0.382557
\(550\) 1.84192e12 0.858300
\(551\) −1.28984e12 −0.596147
\(552\) −2.37868e12 −1.09046
\(553\) 6.51866e11 0.296412
\(554\) 5.88990e12 2.65652
\(555\) 1.55907e12 0.697507
\(556\) 1.80619e12 0.801545
\(557\) −3.22149e12 −1.41810 −0.709052 0.705156i \(-0.750877\pi\)
−0.709052 + 0.705156i \(0.750877\pi\)
\(558\) 2.26980e12 0.991139
\(559\) 4.74179e12 2.05395
\(560\) −6.76438e11 −0.290658
\(561\) 1.69829e12 0.723899
\(562\) −7.90705e11 −0.334350
\(563\) 2.97740e11 0.124896 0.0624481 0.998048i \(-0.480109\pi\)
0.0624481 + 0.998048i \(0.480109\pi\)
\(564\) −7.43248e11 −0.309299
\(565\) −1.75515e12 −0.724598
\(566\) 3.59463e12 1.47225
\(567\) −7.55068e10 −0.0306805
\(568\) −2.14004e12 −0.862689
\(569\) 3.07736e12 1.23076 0.615380 0.788231i \(-0.289003\pi\)
0.615380 + 0.788231i \(0.289003\pi\)
\(570\) 1.84903e12 0.733682
\(571\) −1.52734e12 −0.601277 −0.300638 0.953738i \(-0.597200\pi\)
−0.300638 + 0.953738i \(0.597200\pi\)
\(572\) 9.86942e12 3.85487
\(573\) 3.98224e11 0.154323
\(574\) 2.76294e11 0.106235
\(575\) 8.27132e11 0.315551
\(576\) −6.15382e11 −0.232940
\(577\) 2.28483e12 0.858148 0.429074 0.903269i \(-0.358840\pi\)
0.429074 + 0.903269i \(0.358840\pi\)
\(578\) 1.53357e12 0.571517
\(579\) −1.33406e12 −0.493312
\(580\) −3.19332e12 −1.17170
\(581\) −1.24167e12 −0.452077
\(582\) 2.80582e12 1.01369
\(583\) −1.93063e12 −0.692134
\(584\) 9.71291e12 3.45535
\(585\) 9.41221e11 0.332269
\(586\) −3.70923e12 −1.29940
\(587\) −3.97506e10 −0.0138189 −0.00690943 0.999976i \(-0.502199\pi\)
−0.00690943 + 0.999976i \(0.502199\pi\)
\(588\) 3.23022e12 1.11438
\(589\) −4.33194e12 −1.48308
\(590\) 5.55426e11 0.188709
\(591\) −9.33036e11 −0.314597
\(592\) −5.58834e12 −1.86997
\(593\) 5.18902e12 1.72321 0.861607 0.507576i \(-0.169458\pi\)
0.861607 + 0.507576i \(0.169458\pi\)
\(594\) 1.56652e12 0.516293
\(595\) 5.71886e11 0.187061
\(596\) 3.50794e12 1.13879
\(597\) −1.82287e11 −0.0587316
\(598\) 6.55304e12 2.09550
\(599\) −1.37203e12 −0.435454 −0.217727 0.976010i \(-0.569864\pi\)
−0.217727 + 0.976010i \(0.569864\pi\)
\(600\) −1.12290e12 −0.353722
\(601\) 1.19015e12 0.372105 0.186053 0.982540i \(-0.440431\pi\)
0.186053 + 0.982540i \(0.440431\pi\)
\(602\) 2.65757e12 0.824710
\(603\) −1.79133e12 −0.551756
\(604\) −3.49111e12 −1.06733
\(605\) 3.61308e12 1.09642
\(606\) 6.32438e12 1.90498
\(607\) −2.77137e12 −0.828601 −0.414300 0.910140i \(-0.635974\pi\)
−0.414300 + 0.910140i \(0.635974\pi\)
\(608\) −9.70778e11 −0.288107
\(609\) −3.67981e11 −0.108405
\(610\) −5.68879e12 −1.66355
\(611\) 1.06763e12 0.309911
\(612\) −1.98564e12 −0.572163
\(613\) −6.07233e12 −1.73693 −0.868467 0.495746i \(-0.834895\pi\)
−0.868467 + 0.495746i \(0.834895\pi\)
\(614\) −6.38227e12 −1.81225
\(615\) −3.69721e11 −0.104216
\(616\) 2.88413e12 0.807053
\(617\) 2.19148e12 0.608772 0.304386 0.952549i \(-0.401549\pi\)
0.304386 + 0.952549i \(0.401549\pi\)
\(618\) −9.88743e11 −0.272669
\(619\) 4.79703e12 1.31330 0.656651 0.754195i \(-0.271972\pi\)
0.656651 + 0.754195i \(0.271972\pi\)
\(620\) −1.07248e13 −2.91492
\(621\) 7.03459e11 0.189813
\(622\) 3.55277e12 0.951724
\(623\) −1.40375e12 −0.373331
\(624\) −3.37371e12 −0.890794
\(625\) −2.20378e12 −0.577708
\(626\) −4.70054e12 −1.22338
\(627\) −2.98971e12 −0.772547
\(628\) −1.08695e13 −2.78862
\(629\) 4.72459e12 1.20347
\(630\) 5.27514e11 0.133414
\(631\) 6.52235e12 1.63784 0.818921 0.573907i \(-0.194573\pi\)
0.818921 + 0.573907i \(0.194573\pi\)
\(632\) 8.24478e12 2.05566
\(633\) −1.23645e12 −0.306097
\(634\) 6.37930e12 1.56809
\(635\) −5.25582e12 −1.28280
\(636\) 2.25730e12 0.547056
\(637\) −4.64002e12 −1.11659
\(638\) 7.63439e12 1.82424
\(639\) 6.32886e11 0.150166
\(640\) 5.44949e12 1.28394
\(641\) 6.39535e12 1.49625 0.748123 0.663560i \(-0.230955\pi\)
0.748123 + 0.663560i \(0.230955\pi\)
\(642\) −5.37177e12 −1.24799
\(643\) 1.91656e12 0.442154 0.221077 0.975256i \(-0.429043\pi\)
0.221077 + 0.975256i \(0.429043\pi\)
\(644\) 2.48392e12 0.569051
\(645\) −3.55621e12 −0.809037
\(646\) 5.60328e12 1.26589
\(647\) −1.05257e12 −0.236147 −0.118073 0.993005i \(-0.537672\pi\)
−0.118073 + 0.993005i \(0.537672\pi\)
\(648\) −9.55007e11 −0.212774
\(649\) −8.98071e11 −0.198705
\(650\) 3.09350e12 0.679735
\(651\) −1.23587e12 −0.269685
\(652\) 4.95818e12 1.07450
\(653\) −2.09151e12 −0.450143 −0.225072 0.974342i \(-0.572262\pi\)
−0.225072 + 0.974342i \(0.572262\pi\)
\(654\) −6.00922e12 −1.28445
\(655\) 1.83975e12 0.390547
\(656\) 1.32523e12 0.279398
\(657\) −2.87245e12 −0.601463
\(658\) 5.98363e11 0.124437
\(659\) −2.76756e12 −0.571626 −0.285813 0.958285i \(-0.592264\pi\)
−0.285813 + 0.958285i \(0.592264\pi\)
\(660\) −7.40178e12 −1.51841
\(661\) −4.96249e12 −1.01110 −0.505549 0.862798i \(-0.668710\pi\)
−0.505549 + 0.862798i \(0.668710\pi\)
\(662\) 7.68541e12 1.55527
\(663\) 2.85226e12 0.573295
\(664\) −1.57046e13 −3.13523
\(665\) −1.00676e12 −0.199632
\(666\) 4.35802e12 0.858331
\(667\) 3.42829e12 0.670675
\(668\) 6.01643e11 0.116908
\(669\) 4.58025e12 0.884040
\(670\) 1.25148e13 2.39931
\(671\) 9.19823e12 1.75167
\(672\) −2.76955e11 −0.0523899
\(673\) −7.88631e12 −1.48186 −0.740928 0.671584i \(-0.765614\pi\)
−0.740928 + 0.671584i \(0.765614\pi\)
\(674\) 5.08933e12 0.949929
\(675\) 3.32083e11 0.0615714
\(676\) 5.23079e12 0.963401
\(677\) −8.80468e12 −1.61089 −0.805443 0.592673i \(-0.798073\pi\)
−0.805443 + 0.592673i \(0.798073\pi\)
\(678\) −4.90611e12 −0.891669
\(679\) −1.52772e12 −0.275822
\(680\) 7.23319e12 1.29730
\(681\) 1.72501e12 0.307347
\(682\) 2.56402e13 4.53828
\(683\) −5.13195e11 −0.0902380 −0.0451190 0.998982i \(-0.514367\pi\)
−0.0451190 + 0.998982i \(0.514367\pi\)
\(684\) 3.49558e12 0.610614
\(685\) −3.55793e12 −0.617433
\(686\) −5.41571e12 −0.933678
\(687\) 6.38859e11 0.109421
\(688\) 1.27469e13 2.16898
\(689\) −3.24248e12 −0.548138
\(690\) −4.91459e12 −0.825403
\(691\) 7.47766e12 1.24771 0.623856 0.781539i \(-0.285565\pi\)
0.623856 + 0.781539i \(0.285565\pi\)
\(692\) −1.17369e13 −1.94569
\(693\) −8.52940e11 −0.140482
\(694\) −7.65598e12 −1.25280
\(695\) 1.94579e12 0.316348
\(696\) −4.65421e12 −0.751803
\(697\) −1.12040e12 −0.179814
\(698\) 1.83492e13 2.92596
\(699\) 1.44405e12 0.228789
\(700\) 1.17259e12 0.184588
\(701\) −7.62118e11 −0.119204 −0.0596020 0.998222i \(-0.518983\pi\)
−0.0596020 + 0.998222i \(0.518983\pi\)
\(702\) 2.63096e12 0.408881
\(703\) −8.31730e12 −1.28435
\(704\) −6.95148e12 −1.06660
\(705\) −8.00693e11 −0.122072
\(706\) −1.16868e13 −1.77041
\(707\) −3.44351e12 −0.518339
\(708\) 1.05003e12 0.157055
\(709\) −5.96209e12 −0.886116 −0.443058 0.896493i \(-0.646106\pi\)
−0.443058 + 0.896493i \(0.646106\pi\)
\(710\) −4.42154e12 −0.652996
\(711\) −2.43827e12 −0.357824
\(712\) −1.77546e13 −2.58911
\(713\) 1.15140e13 1.66848
\(714\) 1.59857e12 0.230192
\(715\) 1.06322e13 1.52141
\(716\) −1.64804e13 −2.34347
\(717\) 1.58799e12 0.224394
\(718\) −2.84740e12 −0.399843
\(719\) −5.49982e12 −0.767483 −0.383741 0.923441i \(-0.625365\pi\)
−0.383741 + 0.923441i \(0.625365\pi\)
\(720\) 2.53019e12 0.350878
\(721\) 5.38352e11 0.0741921
\(722\) 2.96977e12 0.406730
\(723\) −4.78029e12 −0.650627
\(724\) 8.50754e12 1.15075
\(725\) 1.61840e12 0.217552
\(726\) 1.00995e13 1.34923
\(727\) −9.18960e12 −1.22009 −0.610045 0.792367i \(-0.708849\pi\)
−0.610045 + 0.792367i \(0.708849\pi\)
\(728\) 4.84388e12 0.639149
\(729\) 2.82430e11 0.0370370
\(730\) 2.00679e13 2.61546
\(731\) −1.07767e13 −1.39591
\(732\) −1.07546e13 −1.38451
\(733\) 4.38244e12 0.560722 0.280361 0.959895i \(-0.409546\pi\)
0.280361 + 0.959895i \(0.409546\pi\)
\(734\) −9.11807e12 −1.15950
\(735\) 3.47988e12 0.439817
\(736\) 2.58025e12 0.324125
\(737\) −2.02352e13 −2.52641
\(738\) −1.03347e12 −0.128246
\(739\) −9.16996e12 −1.13101 −0.565506 0.824744i \(-0.691319\pi\)
−0.565506 + 0.824744i \(0.691319\pi\)
\(740\) −2.05916e13 −2.52433
\(741\) −5.02120e12 −0.611822
\(742\) −1.81727e12 −0.220091
\(743\) −6.46289e12 −0.777996 −0.388998 0.921239i \(-0.627179\pi\)
−0.388998 + 0.921239i \(0.627179\pi\)
\(744\) −1.56312e13 −1.87031
\(745\) 3.77906e12 0.449450
\(746\) −2.60139e13 −3.07525
\(747\) 4.64440e12 0.545741
\(748\) −2.24302e13 −2.61985
\(749\) 2.92483e12 0.339572
\(750\) −9.57162e12 −1.10461
\(751\) 5.46521e12 0.626942 0.313471 0.949598i \(-0.398508\pi\)
0.313471 + 0.949598i \(0.398508\pi\)
\(752\) 2.87001e12 0.327267
\(753\) −4.29549e12 −0.486895
\(754\) 1.28219e13 1.44471
\(755\) −3.76093e12 −0.421244
\(756\) 9.97262e11 0.111035
\(757\) −1.05794e13 −1.17092 −0.585462 0.810700i \(-0.699087\pi\)
−0.585462 + 0.810700i \(0.699087\pi\)
\(758\) 1.97953e13 2.17796
\(759\) 7.94642e12 0.869128
\(760\) −1.27335e13 −1.38448
\(761\) −1.77850e13 −1.92231 −0.961156 0.276006i \(-0.910989\pi\)
−0.961156 + 0.276006i \(0.910989\pi\)
\(762\) −1.46914e13 −1.57857
\(763\) 3.27191e12 0.349495
\(764\) −5.25957e12 −0.558509
\(765\) −2.13911e12 −0.225817
\(766\) −5.80339e12 −0.609049
\(767\) −1.50830e12 −0.157366
\(768\) 1.13429e13 1.17652
\(769\) 1.89029e13 1.94922 0.974610 0.223910i \(-0.0718823\pi\)
0.974610 + 0.223910i \(0.0718823\pi\)
\(770\) 5.95891e12 0.610884
\(771\) −7.35715e12 −0.749834
\(772\) 1.76197e13 1.78534
\(773\) 1.69443e13 1.70693 0.853465 0.521150i \(-0.174497\pi\)
0.853465 + 0.521150i \(0.174497\pi\)
\(774\) −9.94053e12 −0.995577
\(775\) 5.43540e12 0.541220
\(776\) −1.93225e13 −1.91287
\(777\) −2.37286e12 −0.233549
\(778\) 6.64250e12 0.650015
\(779\) 1.97238e12 0.191898
\(780\) −1.24312e13 −1.20251
\(781\) 7.14921e12 0.687588
\(782\) −1.48931e13 −1.42415
\(783\) 1.37641e12 0.130864
\(784\) −1.24733e13 −1.17912
\(785\) −1.17096e13 −1.10059
\(786\) 5.14258e12 0.480595
\(787\) −4.43370e12 −0.411983 −0.205992 0.978554i \(-0.566042\pi\)
−0.205992 + 0.978554i \(0.566042\pi\)
\(788\) 1.23231e13 1.13855
\(789\) 1.20082e13 1.10314
\(790\) 1.70345e13 1.55600
\(791\) 2.67129e12 0.242620
\(792\) −1.07880e13 −0.974263
\(793\) 1.54484e13 1.38725
\(794\) −3.64558e13 −3.25518
\(795\) 2.43176e12 0.215908
\(796\) 2.40757e12 0.212555
\(797\) −1.36101e13 −1.19481 −0.597406 0.801939i \(-0.703802\pi\)
−0.597406 + 0.801939i \(0.703802\pi\)
\(798\) −2.81417e12 −0.245661
\(799\) −2.42641e12 −0.210622
\(800\) 1.21806e12 0.105139
\(801\) 5.25066e12 0.450679
\(802\) −2.81462e13 −2.40235
\(803\) −3.24478e13 −2.75401
\(804\) 2.36591e13 1.99685
\(805\) 2.67590e12 0.224589
\(806\) 4.30625e13 3.59411
\(807\) −3.00604e12 −0.249496
\(808\) −4.35534e13 −3.59477
\(809\) −1.44698e13 −1.18766 −0.593832 0.804589i \(-0.702386\pi\)
−0.593832 + 0.804589i \(0.702386\pi\)
\(810\) −1.97314e12 −0.161055
\(811\) −2.01968e12 −0.163942 −0.0819708 0.996635i \(-0.526121\pi\)
−0.0819708 + 0.996635i \(0.526121\pi\)
\(812\) 4.86013e12 0.392325
\(813\) 2.15186e12 0.172746
\(814\) 4.92291e13 3.93017
\(815\) 5.34140e12 0.424078
\(816\) 7.66743e12 0.605403
\(817\) 1.89716e13 1.48972
\(818\) −2.97825e13 −2.32580
\(819\) −1.43251e12 −0.111255
\(820\) 4.88311e12 0.377168
\(821\) −2.01085e12 −0.154467 −0.0772335 0.997013i \(-0.524609\pi\)
−0.0772335 + 0.997013i \(0.524609\pi\)
\(822\) −9.94533e12 −0.759795
\(823\) 1.20717e13 0.917212 0.458606 0.888640i \(-0.348349\pi\)
0.458606 + 0.888640i \(0.348349\pi\)
\(824\) 6.80906e12 0.514535
\(825\) 3.75127e12 0.281926
\(826\) −8.45340e11 −0.0631861
\(827\) −9.24762e12 −0.687472 −0.343736 0.939066i \(-0.611693\pi\)
−0.343736 + 0.939066i \(0.611693\pi\)
\(828\) −9.29099e12 −0.686951
\(829\) 8.27922e12 0.608827 0.304414 0.952540i \(-0.401540\pi\)
0.304414 + 0.952540i \(0.401540\pi\)
\(830\) −3.24472e13 −2.37315
\(831\) 1.19954e13 0.872590
\(832\) −1.16750e13 −0.844697
\(833\) 1.05454e13 0.758857
\(834\) 5.43900e12 0.389289
\(835\) 6.48144e11 0.0461405
\(836\) 3.94868e13 2.79591
\(837\) 4.62270e12 0.325560
\(838\) 3.35039e13 2.34691
\(839\) −1.50143e13 −1.04611 −0.523053 0.852300i \(-0.675207\pi\)
−0.523053 + 0.852300i \(0.675207\pi\)
\(840\) −3.63277e12 −0.251757
\(841\) −7.79922e12 −0.537612
\(842\) 4.63163e12 0.317563
\(843\) −1.61036e12 −0.109824
\(844\) 1.63305e13 1.10779
\(845\) 5.63508e12 0.380228
\(846\) −2.23815e12 −0.150218
\(847\) −5.49900e12 −0.367120
\(848\) −8.71641e12 −0.578837
\(849\) 7.32086e12 0.483590
\(850\) −7.03059e12 −0.461962
\(851\) 2.21068e13 1.44491
\(852\) −8.35888e12 −0.543463
\(853\) −2.93288e13 −1.89681 −0.948403 0.317066i \(-0.897302\pi\)
−0.948403 + 0.317066i \(0.897302\pi\)
\(854\) 8.65815e12 0.557012
\(855\) 3.76575e12 0.240993
\(856\) 3.69931e13 2.35499
\(857\) 1.11635e13 0.706946 0.353473 0.935445i \(-0.385001\pi\)
0.353473 + 0.935445i \(0.385001\pi\)
\(858\) 2.97198e13 1.87221
\(859\) −6.02793e12 −0.377745 −0.188873 0.982002i \(-0.560483\pi\)
−0.188873 + 0.982002i \(0.560483\pi\)
\(860\) 4.69689e13 2.92797
\(861\) 5.62703e11 0.0348952
\(862\) −6.28573e12 −0.387769
\(863\) −6.78324e12 −0.416283 −0.208142 0.978099i \(-0.566741\pi\)
−0.208142 + 0.978099i \(0.566741\pi\)
\(864\) 1.03594e12 0.0632443
\(865\) −1.26440e13 −0.767913
\(866\) 4.06475e13 2.45586
\(867\) 3.12329e12 0.187727
\(868\) 1.63228e13 0.976014
\(869\) −2.75432e13 −1.63842
\(870\) −9.61606e12 −0.569063
\(871\) −3.39849e13 −2.00080
\(872\) 4.13830e13 2.42380
\(873\) 5.71435e12 0.332968
\(874\) 2.62182e13 1.51985
\(875\) 5.21157e12 0.300561
\(876\) 3.79381e13 2.17674
\(877\) 2.63908e13 1.50645 0.753225 0.657763i \(-0.228497\pi\)
0.753225 + 0.657763i \(0.228497\pi\)
\(878\) 4.55017e13 2.58406
\(879\) −7.55424e12 −0.426816
\(880\) 2.85815e13 1.60662
\(881\) −1.55550e12 −0.0869920 −0.0434960 0.999054i \(-0.513850\pi\)
−0.0434960 + 0.999054i \(0.513850\pi\)
\(882\) 9.72718e12 0.541226
\(883\) 1.11147e13 0.615283 0.307642 0.951502i \(-0.400460\pi\)
0.307642 + 0.951502i \(0.400460\pi\)
\(884\) −3.76714e13 −2.07480
\(885\) 1.13118e12 0.0619853
\(886\) 2.23464e13 1.21831
\(887\) −1.87813e13 −1.01876 −0.509378 0.860543i \(-0.670125\pi\)
−0.509378 + 0.860543i \(0.670125\pi\)
\(888\) −3.00119e13 −1.61970
\(889\) 7.99918e12 0.429524
\(890\) −3.66828e13 −1.95978
\(891\) 3.19038e12 0.169587
\(892\) −6.04940e13 −3.19941
\(893\) 4.27152e12 0.224776
\(894\) 1.05635e13 0.553080
\(895\) −1.77541e13 −0.924902
\(896\) −8.29395e12 −0.429908
\(897\) 1.33460e13 0.688310
\(898\) 1.33684e13 0.686020
\(899\) 2.25286e13 1.15031
\(900\) −4.38600e12 −0.222832
\(901\) 7.36917e12 0.372527
\(902\) −1.16742e13 −0.587218
\(903\) 5.41244e12 0.270893
\(904\) 3.37863e13 1.68261
\(905\) 9.16508e12 0.454169
\(906\) −1.05128e13 −0.518371
\(907\) −2.28696e13 −1.12208 −0.561041 0.827788i \(-0.689599\pi\)
−0.561041 + 0.827788i \(0.689599\pi\)
\(908\) −2.27832e13 −1.11231
\(909\) 1.28803e13 0.625732
\(910\) 1.00079e13 0.483792
\(911\) 2.28776e13 1.10047 0.550234 0.835011i \(-0.314539\pi\)
0.550234 + 0.835011i \(0.314539\pi\)
\(912\) −1.34980e13 −0.646088
\(913\) 5.24641e13 2.49887
\(914\) 1.37118e13 0.649887
\(915\) −1.15858e13 −0.546427
\(916\) −8.43778e12 −0.396003
\(917\) −2.80004e12 −0.130768
\(918\) −5.97937e12 −0.277884
\(919\) −3.93536e13 −1.81997 −0.909985 0.414640i \(-0.863907\pi\)
−0.909985 + 0.414640i \(0.863907\pi\)
\(920\) 3.38447e13 1.55756
\(921\) −1.29982e13 −0.595270
\(922\) −5.47319e13 −2.49431
\(923\) 1.20070e13 0.544538
\(924\) 1.12653e13 0.508414
\(925\) 1.04360e13 0.468699
\(926\) 2.17536e13 0.972257
\(927\) −2.01368e12 −0.0895637
\(928\) 5.04862e12 0.223464
\(929\) −1.33437e13 −0.587770 −0.293885 0.955841i \(-0.594948\pi\)
−0.293885 + 0.955841i \(0.594948\pi\)
\(930\) −3.22956e13 −1.41570
\(931\) −1.85644e13 −0.809855
\(932\) −1.90724e13 −0.828007
\(933\) 7.23561e12 0.312613
\(934\) 5.69663e13 2.44938
\(935\) −2.41639e13 −1.03398
\(936\) −1.81183e13 −0.771572
\(937\) −2.48681e13 −1.05394 −0.526969 0.849885i \(-0.676672\pi\)
−0.526969 + 0.849885i \(0.676672\pi\)
\(938\) −1.90471e13 −0.803370
\(939\) −9.57315e12 −0.401846
\(940\) 1.05752e13 0.441788
\(941\) −1.95665e13 −0.813503 −0.406751 0.913539i \(-0.633338\pi\)
−0.406751 + 0.913539i \(0.633338\pi\)
\(942\) −3.27313e13 −1.35436
\(943\) −5.24243e12 −0.215889
\(944\) −4.05462e12 −0.166179
\(945\) 1.07434e12 0.0438226
\(946\) −1.12290e14 −4.55860
\(947\) −1.14035e13 −0.460748 −0.230374 0.973102i \(-0.573995\pi\)
−0.230374 + 0.973102i \(0.573995\pi\)
\(948\) 3.22037e13 1.29499
\(949\) −5.44959e13 −2.18105
\(950\) 1.23768e13 0.493008
\(951\) 1.29921e13 0.515072
\(952\) −1.10087e13 −0.434380
\(953\) −1.44702e13 −0.568272 −0.284136 0.958784i \(-0.591707\pi\)
−0.284136 + 0.958784i \(0.591707\pi\)
\(954\) 6.79741e12 0.265690
\(955\) −5.66608e12 −0.220428
\(956\) −2.09735e13 −0.812100
\(957\) 1.55483e13 0.599209
\(958\) −4.21633e13 −1.61730
\(959\) 5.41505e12 0.206737
\(960\) 8.75589e12 0.332721
\(961\) 4.92230e13 1.86171
\(962\) 8.26799e13 3.11252
\(963\) −1.09402e13 −0.409927
\(964\) 6.31360e13 2.35467
\(965\) 1.89815e13 0.704624
\(966\) 7.47984e12 0.276373
\(967\) −2.49135e13 −0.916252 −0.458126 0.888887i \(-0.651479\pi\)
−0.458126 + 0.888887i \(0.651479\pi\)
\(968\) −6.95511e13 −2.54604
\(969\) 1.14117e13 0.415808
\(970\) −3.99222e13 −1.44791
\(971\) −2.58118e13 −0.931819 −0.465910 0.884832i \(-0.654273\pi\)
−0.465910 + 0.884832i \(0.654273\pi\)
\(972\) −3.73021e12 −0.134040
\(973\) −2.96144e12 −0.105924
\(974\) −3.32681e13 −1.18444
\(975\) 6.30024e12 0.223273
\(976\) 4.15282e13 1.46494
\(977\) 1.21393e13 0.426255 0.213127 0.977024i \(-0.431635\pi\)
0.213127 + 0.977024i \(0.431635\pi\)
\(978\) 1.49306e13 0.521858
\(979\) 5.93126e13 2.06360
\(980\) −4.59608e13 −1.59173
\(981\) −1.22384e13 −0.421905
\(982\) 5.04781e13 1.73221
\(983\) −1.04154e13 −0.355782 −0.177891 0.984050i \(-0.556927\pi\)
−0.177891 + 0.984050i \(0.556927\pi\)
\(984\) 7.11705e12 0.242004
\(985\) 1.32756e13 0.449356
\(986\) −2.91404e13 −0.981858
\(987\) 1.21863e12 0.0408738
\(988\) 6.63178e13 2.21424
\(989\) −5.04250e13 −1.67595
\(990\) −2.22890e13 −0.737450
\(991\) 1.65937e13 0.546527 0.273264 0.961939i \(-0.411897\pi\)
0.273264 + 0.961939i \(0.411897\pi\)
\(992\) 1.69558e13 0.555925
\(993\) 1.56522e13 0.510861
\(994\) 6.72944e12 0.218645
\(995\) 2.59365e12 0.0838895
\(996\) −6.13412e13 −1.97508
\(997\) 2.39331e13 0.767134 0.383567 0.923513i \(-0.374695\pi\)
0.383567 + 0.923513i \(0.374695\pi\)
\(998\) −2.66183e13 −0.849361
\(999\) 8.87557e12 0.281937
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.10.a.b.1.2 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.b.1.2 21 1.1 even 1 trivial