Properties

Label 177.10.a.a.1.17
Level $177$
Weight $10$
Character 177.1
Self dual yes
Analytic conductor $91.161$
Analytic rank $1$
Dimension $21$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

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

Embedding invariants

Embedding label 1.17
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+27.0996 q^{2} +81.0000 q^{3} +222.388 q^{4} -1517.97 q^{5} +2195.07 q^{6} +2210.22 q^{7} -7848.37 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q+27.0996 q^{2} +81.0000 q^{3} +222.388 q^{4} -1517.97 q^{5} +2195.07 q^{6} +2210.22 q^{7} -7848.37 q^{8} +6561.00 q^{9} -41136.3 q^{10} +20491.3 q^{11} +18013.4 q^{12} +146616. q^{13} +59896.0 q^{14} -122955. q^{15} -326550. q^{16} +10640.9 q^{17} +177800. q^{18} -363427. q^{19} -337578. q^{20} +179028. q^{21} +555306. q^{22} +1.47841e6 q^{23} -635718. q^{24} +351104. q^{25} +3.97324e6 q^{26} +531441. q^{27} +491526. q^{28} -7.12759e6 q^{29} -3.33204e6 q^{30} -8.92102e6 q^{31} -4.83102e6 q^{32} +1.65979e6 q^{33} +288365. q^{34} -3.35504e6 q^{35} +1.45909e6 q^{36} -3.34760e6 q^{37} -9.84874e6 q^{38} +1.18759e7 q^{39} +1.19136e7 q^{40} +1.53989e7 q^{41} +4.85158e6 q^{42} -2.67443e7 q^{43} +4.55702e6 q^{44} -9.95939e6 q^{45} +4.00644e7 q^{46} -5.05532e7 q^{47} -2.64506e7 q^{48} -3.54685e7 q^{49} +9.51477e6 q^{50} +861916. q^{51} +3.26057e7 q^{52} -5.26983e7 q^{53} +1.44018e7 q^{54} -3.11051e7 q^{55} -1.73466e7 q^{56} -2.94376e7 q^{57} -1.93155e8 q^{58} +1.21174e7 q^{59} -2.73438e7 q^{60} +3.72572e7 q^{61} -2.41756e8 q^{62} +1.45012e7 q^{63} +3.62751e7 q^{64} -2.22559e8 q^{65} +4.49798e7 q^{66} -1.12028e8 q^{67} +2.36642e6 q^{68} +1.19751e8 q^{69} -9.09203e7 q^{70} -2.42362e8 q^{71} -5.14931e7 q^{72} -1.68777e7 q^{73} -9.07187e7 q^{74} +2.84394e7 q^{75} -8.08219e7 q^{76} +4.52902e7 q^{77} +3.21832e8 q^{78} +5.59348e8 q^{79} +4.95693e8 q^{80} +4.30467e7 q^{81} +4.17303e8 q^{82} -7.22195e8 q^{83} +3.98136e7 q^{84} -1.61526e7 q^{85} -7.24760e8 q^{86} -5.77335e8 q^{87} -1.60823e8 q^{88} +5.52358e8 q^{89} -2.69896e8 q^{90} +3.24054e8 q^{91} +3.28781e8 q^{92} -7.22602e8 q^{93} -1.36997e9 q^{94} +5.51671e8 q^{95} -3.91312e8 q^{96} +3.40681e8 q^{97} -9.61183e8 q^{98} +1.34443e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} - 54663q^{10} - 151769q^{11} + 421686q^{12} - 153611q^{13} - 286771q^{14} - 240084q^{15} + 805530q^{16} - 723621q^{17} - 433026q^{18} - 549388q^{19} - 527311q^{20} - 2492775q^{21} + 2973158q^{22} + 169962q^{23} - 1994301q^{24} + 8035779q^{25} - 2337392q^{26} + 11160261q^{27} - 22659054q^{28} - 16845442q^{29} - 4427703q^{30} - 19307976q^{31} - 44923568q^{32} - 12293289q^{33} - 35547496q^{34} - 34882596q^{35} + 34156566q^{36} - 41561129q^{37} - 52335371q^{38} - 12442491q^{39} - 125735038q^{40} - 68169291q^{41} - 23228451q^{42} - 25719587q^{43} - 126277032q^{44} - 19446804q^{45} - 292814271q^{46} - 174095332q^{47} + 65247930q^{48} + 7479350q^{49} - 227877439q^{50} - 58613301q^{51} - 232397708q^{52} - 228390500q^{53} - 35075106q^{54} - 29426208q^{55} + 326778474q^{56} - 44500428q^{57} + 480343762q^{58} + 254464581q^{59} - 42712191q^{60} - 183928964q^{61} - 21753862q^{62} - 201914775q^{63} + 310571245q^{64} + 5308466q^{65} + 240825798q^{66} - 82724114q^{67} - 138336205q^{68} + 13766922q^{69} + 1030274876q^{70} - 404721965q^{71} - 161538381q^{72} + 154162574q^{73} + 36352054q^{74} + 650898099q^{75} + 1068940636q^{76} - 448535481q^{77} - 189328752q^{78} + 272529635q^{79} - 345587859q^{80} + 903981141q^{81} - 38412637q^{82} + 432518643q^{83} - 1835383374q^{84} - 126211490q^{85} - 3699273072q^{86} - 1364480802q^{87} + 170111045q^{88} - 1255621070q^{89} - 358643943q^{90} + 1448885849q^{91} + 1568933320q^{92} - 1563946056q^{93} - 1908445164q^{94} - 2896546490q^{95} - 3638809008q^{96} + 1007235486q^{97} - 9506868248q^{98} - 995756409q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 27.0996 1.19764 0.598822 0.800882i \(-0.295636\pi\)
0.598822 + 0.800882i \(0.295636\pi\)
\(3\) 81.0000 0.577350
\(4\) 222.388 0.434352
\(5\) −1517.97 −1.08617 −0.543085 0.839678i \(-0.682744\pi\)
−0.543085 + 0.839678i \(0.682744\pi\)
\(6\) 2195.07 0.691460
\(7\) 2210.22 0.347932 0.173966 0.984752i \(-0.444342\pi\)
0.173966 + 0.984752i \(0.444342\pi\)
\(8\) −7848.37 −0.677445
\(9\) 6561.00 0.333333
\(10\) −41136.3 −1.30085
\(11\) 20491.3 0.421990 0.210995 0.977487i \(-0.432330\pi\)
0.210995 + 0.977487i \(0.432330\pi\)
\(12\) 18013.4 0.250773
\(13\) 146616. 1.42376 0.711880 0.702301i \(-0.247844\pi\)
0.711880 + 0.702301i \(0.247844\pi\)
\(14\) 59896.0 0.416698
\(15\) −122955. −0.627101
\(16\) −326550. −1.24569
\(17\) 10640.9 0.0309001 0.0154500 0.999881i \(-0.495082\pi\)
0.0154500 + 0.999881i \(0.495082\pi\)
\(18\) 177800. 0.399215
\(19\) −363427. −0.639774 −0.319887 0.947456i \(-0.603645\pi\)
−0.319887 + 0.947456i \(0.603645\pi\)
\(20\) −337578. −0.471780
\(21\) 179028. 0.200878
\(22\) 555306. 0.505394
\(23\) 1.47841e6 1.10159 0.550796 0.834640i \(-0.314324\pi\)
0.550796 + 0.834640i \(0.314324\pi\)
\(24\) −635718. −0.391123
\(25\) 351104. 0.179765
\(26\) 3.97324e6 1.70516
\(27\) 531441. 0.192450
\(28\) 491526. 0.151125
\(29\) −7.12759e6 −1.87134 −0.935668 0.352882i \(-0.885202\pi\)
−0.935668 + 0.352882i \(0.885202\pi\)
\(30\) −3.33204e6 −0.751043
\(31\) −8.92102e6 −1.73495 −0.867474 0.497482i \(-0.834258\pi\)
−0.867474 + 0.497482i \(0.834258\pi\)
\(32\) −4.83102e6 −0.814448
\(33\) 1.65979e6 0.243636
\(34\) 288365. 0.0370073
\(35\) −3.35504e6 −0.377913
\(36\) 1.45909e6 0.144784
\(37\) −3.34760e6 −0.293647 −0.146824 0.989163i \(-0.546905\pi\)
−0.146824 + 0.989163i \(0.546905\pi\)
\(38\) −9.84874e6 −0.766222
\(39\) 1.18759e7 0.822008
\(40\) 1.19136e7 0.735821
\(41\) 1.53989e7 0.851061 0.425531 0.904944i \(-0.360087\pi\)
0.425531 + 0.904944i \(0.360087\pi\)
\(42\) 4.85158e6 0.240581
\(43\) −2.67443e7 −1.19295 −0.596477 0.802630i \(-0.703433\pi\)
−0.596477 + 0.802630i \(0.703433\pi\)
\(44\) 4.55702e6 0.183292
\(45\) −9.95939e6 −0.362057
\(46\) 4.00644e7 1.31931
\(47\) −5.05532e7 −1.51115 −0.755576 0.655061i \(-0.772643\pi\)
−0.755576 + 0.655061i \(0.772643\pi\)
\(48\) −2.64506e7 −0.719200
\(49\) −3.54685e7 −0.878944
\(50\) 9.51477e6 0.215295
\(51\) 861916. 0.0178402
\(52\) 3.26057e7 0.618413
\(53\) −5.26983e7 −0.917393 −0.458696 0.888593i \(-0.651683\pi\)
−0.458696 + 0.888593i \(0.651683\pi\)
\(54\) 1.44018e7 0.230487
\(55\) −3.11051e7 −0.458353
\(56\) −1.73466e7 −0.235705
\(57\) −2.94376e7 −0.369374
\(58\) −1.93155e8 −2.24119
\(59\) 1.21174e7 0.130189
\(60\) −2.73438e7 −0.272382
\(61\) 3.72572e7 0.344529 0.172265 0.985051i \(-0.444892\pi\)
0.172265 + 0.985051i \(0.444892\pi\)
\(62\) −2.41756e8 −2.07785
\(63\) 1.45012e7 0.115977
\(64\) 3.62751e7 0.270271
\(65\) −2.22559e8 −1.54645
\(66\) 4.49798e7 0.291789
\(67\) −1.12028e8 −0.679186 −0.339593 0.940573i \(-0.610289\pi\)
−0.339593 + 0.940573i \(0.610289\pi\)
\(68\) 2.36642e6 0.0134215
\(69\) 1.19751e8 0.636004
\(70\) −9.09203e7 −0.452605
\(71\) −2.42362e8 −1.13188 −0.565942 0.824445i \(-0.691487\pi\)
−0.565942 + 0.824445i \(0.691487\pi\)
\(72\) −5.14931e7 −0.225815
\(73\) −1.68777e7 −0.0695603 −0.0347801 0.999395i \(-0.511073\pi\)
−0.0347801 + 0.999395i \(0.511073\pi\)
\(74\) −9.07187e7 −0.351685
\(75\) 2.84394e7 0.103787
\(76\) −8.08219e7 −0.277887
\(77\) 4.52902e7 0.146824
\(78\) 3.21832e8 0.984474
\(79\) 5.59348e8 1.61570 0.807850 0.589389i \(-0.200631\pi\)
0.807850 + 0.589389i \(0.200631\pi\)
\(80\) 4.95693e8 1.35303
\(81\) 4.30467e7 0.111111
\(82\) 4.17303e8 1.01927
\(83\) −7.22195e8 −1.67033 −0.835167 0.549997i \(-0.814629\pi\)
−0.835167 + 0.549997i \(0.814629\pi\)
\(84\) 3.98136e7 0.0872519
\(85\) −1.61526e7 −0.0335627
\(86\) −7.24760e8 −1.42873
\(87\) −5.77335e8 −1.08042
\(88\) −1.60823e8 −0.285875
\(89\) 5.52358e8 0.933181 0.466590 0.884474i \(-0.345482\pi\)
0.466590 + 0.884474i \(0.345482\pi\)
\(90\) −2.69896e8 −0.433615
\(91\) 3.24054e8 0.495371
\(92\) 3.28781e8 0.478478
\(93\) −7.22602e8 −1.00167
\(94\) −1.36997e9 −1.80982
\(95\) 5.51671e8 0.694903
\(96\) −3.91312e8 −0.470222
\(97\) 3.40681e8 0.390728 0.195364 0.980731i \(-0.437411\pi\)
0.195364 + 0.980731i \(0.437411\pi\)
\(98\) −9.61183e8 −1.05266
\(99\) 1.34443e8 0.140663
\(100\) 7.80813e7 0.0780813
\(101\) −1.32540e7 −0.0126736 −0.00633680 0.999980i \(-0.502017\pi\)
−0.00633680 + 0.999980i \(0.502017\pi\)
\(102\) 2.33576e7 0.0213662
\(103\) 1.81417e9 1.58822 0.794111 0.607773i \(-0.207937\pi\)
0.794111 + 0.607773i \(0.207937\pi\)
\(104\) −1.15070e9 −0.964520
\(105\) −2.71758e8 −0.218188
\(106\) −1.42810e9 −1.09871
\(107\) 8.38202e8 0.618190 0.309095 0.951031i \(-0.399974\pi\)
0.309095 + 0.951031i \(0.399974\pi\)
\(108\) 1.18186e8 0.0835910
\(109\) 2.20518e9 1.49632 0.748159 0.663519i \(-0.230938\pi\)
0.748159 + 0.663519i \(0.230938\pi\)
\(110\) −8.42937e8 −0.548944
\(111\) −2.71156e8 −0.169537
\(112\) −7.21747e8 −0.433415
\(113\) −1.14724e9 −0.661915 −0.330958 0.943646i \(-0.607372\pi\)
−0.330958 + 0.943646i \(0.607372\pi\)
\(114\) −7.97748e8 −0.442378
\(115\) −2.24418e9 −1.19652
\(116\) −1.58509e9 −0.812818
\(117\) 9.61949e8 0.474587
\(118\) 3.28376e8 0.155920
\(119\) 2.35188e7 0.0107511
\(120\) 9.64999e8 0.424826
\(121\) −1.93805e9 −0.821924
\(122\) 1.00966e9 0.412623
\(123\) 1.24731e9 0.491361
\(124\) −1.98393e9 −0.753578
\(125\) 2.43182e9 0.890914
\(126\) 3.92978e8 0.138899
\(127\) −1.27289e9 −0.434183 −0.217091 0.976151i \(-0.569657\pi\)
−0.217091 + 0.976151i \(0.569657\pi\)
\(128\) 3.45652e9 1.13814
\(129\) −2.16629e9 −0.688752
\(130\) −6.03125e9 −1.85209
\(131\) −1.54326e9 −0.457845 −0.228923 0.973445i \(-0.573520\pi\)
−0.228923 + 0.973445i \(0.573520\pi\)
\(132\) 3.69119e8 0.105824
\(133\) −8.03254e8 −0.222598
\(134\) −3.03590e9 −0.813423
\(135\) −8.06711e8 −0.209034
\(136\) −8.35140e7 −0.0209331
\(137\) −7.41578e8 −0.179851 −0.0899257 0.995948i \(-0.528663\pi\)
−0.0899257 + 0.995948i \(0.528663\pi\)
\(138\) 3.24522e9 0.761707
\(139\) −3.30588e7 −0.00751140 −0.00375570 0.999993i \(-0.501195\pi\)
−0.00375570 + 0.999993i \(0.501195\pi\)
\(140\) −7.46121e8 −0.164147
\(141\) −4.09481e9 −0.872464
\(142\) −6.56791e9 −1.35559
\(143\) 3.00436e9 0.600813
\(144\) −2.14250e9 −0.415230
\(145\) 1.08195e10 2.03259
\(146\) −4.57380e8 −0.0833085
\(147\) −2.87295e9 −0.507458
\(148\) −7.44467e8 −0.127546
\(149\) 2.41685e9 0.401708 0.200854 0.979621i \(-0.435628\pi\)
0.200854 + 0.979621i \(0.435628\pi\)
\(150\) 7.70696e8 0.124300
\(151\) −1.13268e10 −1.77301 −0.886506 0.462718i \(-0.846874\pi\)
−0.886506 + 0.462718i \(0.846874\pi\)
\(152\) 2.85231e9 0.433412
\(153\) 6.98152e7 0.0103000
\(154\) 1.22735e9 0.175843
\(155\) 1.35418e10 1.88445
\(156\) 2.64106e9 0.357041
\(157\) 1.27072e10 1.66917 0.834587 0.550876i \(-0.185706\pi\)
0.834587 + 0.550876i \(0.185706\pi\)
\(158\) 1.51581e10 1.93503
\(159\) −4.26856e9 −0.529657
\(160\) 7.33333e9 0.884629
\(161\) 3.26762e9 0.383279
\(162\) 1.16655e9 0.133072
\(163\) −1.61227e9 −0.178893 −0.0894464 0.995992i \(-0.528510\pi\)
−0.0894464 + 0.995992i \(0.528510\pi\)
\(164\) 3.42452e9 0.369660
\(165\) −2.51952e9 −0.264630
\(166\) −1.95712e10 −2.00047
\(167\) −1.92747e10 −1.91763 −0.958814 0.284035i \(-0.908327\pi\)
−0.958814 + 0.284035i \(0.908327\pi\)
\(168\) −1.40507e9 −0.136084
\(169\) 1.08918e10 1.02709
\(170\) −4.37729e8 −0.0401962
\(171\) −2.38445e9 −0.213258
\(172\) −5.94762e9 −0.518161
\(173\) −1.21602e10 −1.03213 −0.516063 0.856550i \(-0.672603\pi\)
−0.516063 + 0.856550i \(0.672603\pi\)
\(174\) −1.56455e10 −1.29395
\(175\) 7.76016e8 0.0625460
\(176\) −6.69144e9 −0.525669
\(177\) 9.81506e8 0.0751646
\(178\) 1.49687e10 1.11762
\(179\) −1.73653e10 −1.26428 −0.632141 0.774853i \(-0.717824\pi\)
−0.632141 + 0.774853i \(0.717824\pi\)
\(180\) −2.21485e9 −0.157260
\(181\) −6.72148e9 −0.465491 −0.232746 0.972538i \(-0.574771\pi\)
−0.232746 + 0.972538i \(0.574771\pi\)
\(182\) 8.78173e9 0.593279
\(183\) 3.01783e9 0.198914
\(184\) −1.16031e10 −0.746268
\(185\) 5.08156e9 0.318951
\(186\) −1.95822e10 −1.19965
\(187\) 2.18047e8 0.0130395
\(188\) −1.12424e10 −0.656372
\(189\) 1.17460e9 0.0669595
\(190\) 1.49501e10 0.832247
\(191\) 1.29637e10 0.704820 0.352410 0.935846i \(-0.385362\pi\)
0.352410 + 0.935846i \(0.385362\pi\)
\(192\) 2.93829e9 0.156041
\(193\) −2.92326e9 −0.151656 −0.0758280 0.997121i \(-0.524160\pi\)
−0.0758280 + 0.997121i \(0.524160\pi\)
\(194\) 9.23231e9 0.467953
\(195\) −1.80273e10 −0.892841
\(196\) −7.88778e9 −0.381771
\(197\) −1.57311e10 −0.744151 −0.372076 0.928202i \(-0.621354\pi\)
−0.372076 + 0.928202i \(0.621354\pi\)
\(198\) 3.64336e9 0.168465
\(199\) −1.47294e10 −0.665803 −0.332902 0.942962i \(-0.608028\pi\)
−0.332902 + 0.942962i \(0.608028\pi\)
\(200\) −2.75559e9 −0.121781
\(201\) −9.07424e9 −0.392128
\(202\) −3.59177e8 −0.0151785
\(203\) −1.57535e10 −0.651097
\(204\) 1.91680e8 0.00774891
\(205\) −2.33750e10 −0.924397
\(206\) 4.91634e10 1.90213
\(207\) 9.69987e9 0.367197
\(208\) −4.78776e10 −1.77356
\(209\) −7.44710e9 −0.269978
\(210\) −7.36454e9 −0.261312
\(211\) −2.45527e10 −0.852762 −0.426381 0.904544i \(-0.640212\pi\)
−0.426381 + 0.904544i \(0.640212\pi\)
\(212\) −1.17195e10 −0.398471
\(213\) −1.96313e10 −0.653494
\(214\) 2.27149e10 0.740371
\(215\) 4.05970e10 1.29575
\(216\) −4.17094e9 −0.130374
\(217\) −1.97174e10 −0.603644
\(218\) 5.97594e10 1.79206
\(219\) −1.36710e9 −0.0401606
\(220\) −6.91741e9 −0.199086
\(221\) 1.56013e9 0.0439943
\(222\) −7.34821e9 −0.203046
\(223\) −6.08234e10 −1.64702 −0.823510 0.567301i \(-0.807987\pi\)
−0.823510 + 0.567301i \(0.807987\pi\)
\(224\) −1.06776e10 −0.283372
\(225\) 2.30359e9 0.0599217
\(226\) −3.10898e10 −0.792739
\(227\) 5.45777e10 1.36427 0.682133 0.731228i \(-0.261053\pi\)
0.682133 + 0.731228i \(0.261053\pi\)
\(228\) −6.54658e9 −0.160438
\(229\) 2.14644e9 0.0515774 0.0257887 0.999667i \(-0.491790\pi\)
0.0257887 + 0.999667i \(0.491790\pi\)
\(230\) −6.08165e10 −1.43300
\(231\) 3.66851e9 0.0847687
\(232\) 5.59399e10 1.26773
\(233\) 5.86692e10 1.30409 0.652047 0.758179i \(-0.273911\pi\)
0.652047 + 0.758179i \(0.273911\pi\)
\(234\) 2.60684e10 0.568386
\(235\) 7.67382e10 1.64137
\(236\) 2.69476e9 0.0565478
\(237\) 4.53072e10 0.932824
\(238\) 6.37350e8 0.0128760
\(239\) 1.00854e9 0.0199941 0.00999707 0.999950i \(-0.496818\pi\)
0.00999707 + 0.999950i \(0.496818\pi\)
\(240\) 4.01511e10 0.781173
\(241\) 1.96232e9 0.0374707 0.0187354 0.999824i \(-0.494036\pi\)
0.0187354 + 0.999824i \(0.494036\pi\)
\(242\) −5.25205e10 −0.984373
\(243\) 3.48678e9 0.0641500
\(244\) 8.28556e9 0.149647
\(245\) 5.38401e10 0.954682
\(246\) 3.38015e10 0.588475
\(247\) −5.32844e10 −0.910885
\(248\) 7.00154e10 1.17533
\(249\) −5.84978e10 −0.964368
\(250\) 6.59013e10 1.06700
\(251\) 1.01815e11 1.61912 0.809560 0.587037i \(-0.199706\pi\)
0.809560 + 0.587037i \(0.199706\pi\)
\(252\) 3.22490e9 0.0503749
\(253\) 3.02946e10 0.464861
\(254\) −3.44947e10 −0.519996
\(255\) −1.30836e9 −0.0193775
\(256\) 7.50975e10 1.09281
\(257\) −1.38531e11 −1.98084 −0.990420 0.138089i \(-0.955904\pi\)
−0.990420 + 0.138089i \(0.955904\pi\)
\(258\) −5.87056e10 −0.824880
\(259\) −7.39893e9 −0.102169
\(260\) −4.94944e10 −0.671701
\(261\) −4.67641e10 −0.623779
\(262\) −4.18218e10 −0.548336
\(263\) 6.09592e10 0.785667 0.392833 0.919610i \(-0.371495\pi\)
0.392833 + 0.919610i \(0.371495\pi\)
\(264\) −1.30267e10 −0.165050
\(265\) 7.99944e10 0.996444
\(266\) −2.17679e10 −0.266593
\(267\) 4.47410e10 0.538772
\(268\) −2.49136e10 −0.295005
\(269\) −1.27333e10 −0.148270 −0.0741352 0.997248i \(-0.523620\pi\)
−0.0741352 + 0.997248i \(0.523620\pi\)
\(270\) −2.18615e10 −0.250348
\(271\) 6.61671e10 0.745213 0.372607 0.927989i \(-0.378464\pi\)
0.372607 + 0.927989i \(0.378464\pi\)
\(272\) −3.47480e9 −0.0384919
\(273\) 2.62484e10 0.286003
\(274\) −2.00965e10 −0.215398
\(275\) 7.19457e9 0.0758591
\(276\) 2.66313e10 0.276249
\(277\) 1.43076e11 1.46018 0.730092 0.683349i \(-0.239477\pi\)
0.730092 + 0.683349i \(0.239477\pi\)
\(278\) −8.95881e8 −0.00899599
\(279\) −5.85308e10 −0.578316
\(280\) 2.63316e10 0.256015
\(281\) −2.20811e10 −0.211272 −0.105636 0.994405i \(-0.533688\pi\)
−0.105636 + 0.994405i \(0.533688\pi\)
\(282\) −1.10968e11 −1.04490
\(283\) 2.05484e11 1.90431 0.952156 0.305611i \(-0.0988608\pi\)
0.952156 + 0.305611i \(0.0988608\pi\)
\(284\) −5.38984e10 −0.491636
\(285\) 4.46854e10 0.401203
\(286\) 8.14168e10 0.719560
\(287\) 3.40348e10 0.296111
\(288\) −3.16963e10 −0.271483
\(289\) −1.18475e11 −0.999045
\(290\) 2.93203e11 2.43432
\(291\) 2.75951e10 0.225587
\(292\) −3.75341e9 −0.0302136
\(293\) −1.49334e11 −1.18374 −0.591869 0.806034i \(-0.701610\pi\)
−0.591869 + 0.806034i \(0.701610\pi\)
\(294\) −7.78558e10 −0.607755
\(295\) −1.83938e10 −0.141407
\(296\) 2.62732e10 0.198930
\(297\) 1.08899e10 0.0812120
\(298\) 6.54955e10 0.481103
\(299\) 2.16759e11 1.56840
\(300\) 6.32459e9 0.0450803
\(301\) −5.91108e10 −0.415066
\(302\) −3.06952e11 −2.12344
\(303\) −1.07357e9 −0.00731711
\(304\) 1.18677e11 0.796960
\(305\) −5.65553e10 −0.374217
\(306\) 1.89196e9 0.0123358
\(307\) 8.55676e10 0.549777 0.274889 0.961476i \(-0.411359\pi\)
0.274889 + 0.961476i \(0.411359\pi\)
\(308\) 1.00720e10 0.0637731
\(309\) 1.46948e11 0.916961
\(310\) 3.66978e11 2.25690
\(311\) −5.01393e10 −0.303918 −0.151959 0.988387i \(-0.548558\pi\)
−0.151959 + 0.988387i \(0.548558\pi\)
\(312\) −9.32065e10 −0.556866
\(313\) 3.25435e11 1.91652 0.958262 0.285892i \(-0.0922897\pi\)
0.958262 + 0.285892i \(0.0922897\pi\)
\(314\) 3.44360e11 1.99908
\(315\) −2.20124e10 −0.125971
\(316\) 1.24392e11 0.701782
\(317\) −1.76459e11 −0.981472 −0.490736 0.871308i \(-0.663272\pi\)
−0.490736 + 0.871308i \(0.663272\pi\)
\(318\) −1.15676e11 −0.634341
\(319\) −1.46054e11 −0.789685
\(320\) −5.50645e10 −0.293560
\(321\) 6.78944e10 0.356912
\(322\) 8.85511e10 0.459031
\(323\) −3.86721e9 −0.0197691
\(324\) 9.57308e9 0.0482613
\(325\) 5.14775e10 0.255943
\(326\) −4.36918e10 −0.214250
\(327\) 1.78619e11 0.863900
\(328\) −1.20856e11 −0.576548
\(329\) −1.11734e11 −0.525778
\(330\) −6.82779e10 −0.316933
\(331\) 2.19730e11 1.00615 0.503075 0.864243i \(-0.332202\pi\)
0.503075 + 0.864243i \(0.332202\pi\)
\(332\) −1.60608e11 −0.725512
\(333\) −2.19636e10 −0.0978825
\(334\) −5.22337e11 −2.29664
\(335\) 1.70054e11 0.737711
\(336\) −5.84615e10 −0.250232
\(337\) 4.62471e11 1.95321 0.976607 0.215030i \(-0.0689850\pi\)
0.976607 + 0.215030i \(0.0689850\pi\)
\(338\) 2.95164e11 1.23009
\(339\) −9.29266e10 −0.382157
\(340\) −3.59215e9 −0.0145780
\(341\) −1.82803e11 −0.732131
\(342\) −6.46176e10 −0.255407
\(343\) −1.67583e11 −0.653744
\(344\) 2.09899e11 0.808161
\(345\) −1.81779e11 −0.690809
\(346\) −3.29536e11 −1.23612
\(347\) 5.15153e11 1.90745 0.953727 0.300674i \(-0.0972117\pi\)
0.953727 + 0.300674i \(0.0972117\pi\)
\(348\) −1.28392e11 −0.469281
\(349\) 3.01493e10 0.108783 0.0543917 0.998520i \(-0.482678\pi\)
0.0543917 + 0.998520i \(0.482678\pi\)
\(350\) 2.10297e10 0.0749078
\(351\) 7.79179e10 0.274003
\(352\) −9.89938e10 −0.343689
\(353\) −1.03856e11 −0.355997 −0.177999 0.984031i \(-0.556962\pi\)
−0.177999 + 0.984031i \(0.556962\pi\)
\(354\) 2.65984e10 0.0900205
\(355\) 3.67898e11 1.22942
\(356\) 1.22838e11 0.405329
\(357\) 1.90502e9 0.00620716
\(358\) −4.70593e11 −1.51416
\(359\) −1.87883e11 −0.596982 −0.298491 0.954412i \(-0.596483\pi\)
−0.298491 + 0.954412i \(0.596483\pi\)
\(360\) 7.81650e10 0.245274
\(361\) −1.90608e11 −0.590689
\(362\) −1.82149e11 −0.557493
\(363\) −1.56982e11 −0.474538
\(364\) 7.20657e10 0.215165
\(365\) 2.56199e10 0.0755543
\(366\) 8.17821e10 0.238228
\(367\) −4.28989e9 −0.0123438 −0.00617190 0.999981i \(-0.501965\pi\)
−0.00617190 + 0.999981i \(0.501965\pi\)
\(368\) −4.82776e11 −1.37224
\(369\) 1.01032e11 0.283687
\(370\) 1.37708e11 0.381990
\(371\) −1.16475e11 −0.319190
\(372\) −1.60698e11 −0.435079
\(373\) 1.70398e11 0.455800 0.227900 0.973685i \(-0.426814\pi\)
0.227900 + 0.973685i \(0.426814\pi\)
\(374\) 5.90897e9 0.0156167
\(375\) 1.96977e11 0.514370
\(376\) 3.96760e11 1.02372
\(377\) −1.04502e12 −2.66433
\(378\) 3.18312e10 0.0801936
\(379\) 9.02573e10 0.224702 0.112351 0.993669i \(-0.464162\pi\)
0.112351 + 0.993669i \(0.464162\pi\)
\(380\) 1.22685e11 0.301832
\(381\) −1.03104e11 −0.250676
\(382\) 3.51310e11 0.844123
\(383\) 3.19534e11 0.758792 0.379396 0.925234i \(-0.376132\pi\)
0.379396 + 0.925234i \(0.376132\pi\)
\(384\) 2.79978e11 0.657104
\(385\) −6.87491e10 −0.159476
\(386\) −7.92192e10 −0.181630
\(387\) −1.75469e11 −0.397651
\(388\) 7.57633e10 0.169713
\(389\) −2.31291e11 −0.512136 −0.256068 0.966659i \(-0.582427\pi\)
−0.256068 + 0.966659i \(0.582427\pi\)
\(390\) −4.88532e11 −1.06931
\(391\) 1.57317e10 0.0340393
\(392\) 2.78370e11 0.595436
\(393\) −1.25004e11 −0.264337
\(394\) −4.26307e11 −0.891229
\(395\) −8.49073e11 −1.75492
\(396\) 2.98986e10 0.0610974
\(397\) 5.25050e11 1.06082 0.530412 0.847740i \(-0.322037\pi\)
0.530412 + 0.847740i \(0.322037\pi\)
\(398\) −3.99160e11 −0.797395
\(399\) −6.50636e10 −0.128517
\(400\) −1.14653e11 −0.223932
\(401\) −4.99786e11 −0.965238 −0.482619 0.875830i \(-0.660314\pi\)
−0.482619 + 0.875830i \(0.660314\pi\)
\(402\) −2.45908e11 −0.469630
\(403\) −1.30797e12 −2.47015
\(404\) −2.94753e9 −0.00550480
\(405\) −6.53436e10 −0.120686
\(406\) −4.26914e11 −0.779783
\(407\) −6.85967e10 −0.123916
\(408\) −6.76463e9 −0.0120857
\(409\) −4.76147e10 −0.0841368 −0.0420684 0.999115i \(-0.513395\pi\)
−0.0420684 + 0.999115i \(0.513395\pi\)
\(410\) −6.33452e11 −1.10710
\(411\) −6.00678e10 −0.103837
\(412\) 4.03451e11 0.689847
\(413\) 2.67820e10 0.0452968
\(414\) 2.62863e11 0.439772
\(415\) 1.09627e12 1.81427
\(416\) −7.08305e11 −1.15958
\(417\) −2.67777e9 −0.00433671
\(418\) −2.01813e11 −0.323338
\(419\) −1.07510e12 −1.70406 −0.852029 0.523495i \(-0.824628\pi\)
−0.852029 + 0.523495i \(0.824628\pi\)
\(420\) −6.04358e10 −0.0947704
\(421\) −8.47296e11 −1.31452 −0.657258 0.753666i \(-0.728284\pi\)
−0.657258 + 0.753666i \(0.728284\pi\)
\(422\) −6.65368e11 −1.02131
\(423\) −3.31680e11 −0.503718
\(424\) 4.13596e11 0.621483
\(425\) 3.73607e9 0.00555476
\(426\) −5.32001e11 −0.782653
\(427\) 8.23466e10 0.119873
\(428\) 1.86406e11 0.268512
\(429\) 2.43353e11 0.346879
\(430\) 1.10016e12 1.55185
\(431\) −7.52112e11 −1.04987 −0.524934 0.851143i \(-0.675910\pi\)
−0.524934 + 0.851143i \(0.675910\pi\)
\(432\) −1.73542e11 −0.239733
\(433\) −4.65766e11 −0.636755 −0.318377 0.947964i \(-0.603138\pi\)
−0.318377 + 0.947964i \(0.603138\pi\)
\(434\) −5.34333e11 −0.722950
\(435\) 8.76376e11 1.17352
\(436\) 4.90405e11 0.649929
\(437\) −5.37296e11 −0.704770
\(438\) −3.70478e10 −0.0480982
\(439\) 4.37413e11 0.562085 0.281042 0.959695i \(-0.409320\pi\)
0.281042 + 0.959695i \(0.409320\pi\)
\(440\) 2.44125e11 0.310509
\(441\) −2.32709e11 −0.292981
\(442\) 4.22790e10 0.0526896
\(443\) −1.51523e12 −1.86922 −0.934611 0.355671i \(-0.884252\pi\)
−0.934611 + 0.355671i \(0.884252\pi\)
\(444\) −6.03018e10 −0.0736389
\(445\) −8.38462e11 −1.01359
\(446\) −1.64829e12 −1.97254
\(447\) 1.95764e11 0.231926
\(448\) 8.01760e10 0.0940358
\(449\) 7.17098e11 0.832664 0.416332 0.909213i \(-0.363315\pi\)
0.416332 + 0.909213i \(0.363315\pi\)
\(450\) 6.24264e10 0.0717649
\(451\) 3.15542e11 0.359140
\(452\) −2.55133e11 −0.287504
\(453\) −9.17472e11 −1.02365
\(454\) 1.47903e12 1.63391
\(455\) −4.91904e11 −0.538057
\(456\) 2.31037e11 0.250230
\(457\) 4.12562e10 0.0442452 0.0221226 0.999755i \(-0.492958\pi\)
0.0221226 + 0.999755i \(0.492958\pi\)
\(458\) 5.81678e10 0.0617714
\(459\) 5.65503e9 0.00594673
\(460\) −4.99080e11 −0.519709
\(461\) 1.07089e12 1.10431 0.552154 0.833742i \(-0.313806\pi\)
0.552154 + 0.833742i \(0.313806\pi\)
\(462\) 9.94151e10 0.101523
\(463\) −3.76411e11 −0.380669 −0.190334 0.981719i \(-0.560957\pi\)
−0.190334 + 0.981719i \(0.560957\pi\)
\(464\) 2.32752e12 2.33110
\(465\) 1.09689e12 1.08799
\(466\) 1.58991e12 1.56184
\(467\) −7.84332e11 −0.763087 −0.381543 0.924351i \(-0.624607\pi\)
−0.381543 + 0.924351i \(0.624607\pi\)
\(468\) 2.13926e11 0.206138
\(469\) −2.47605e11 −0.236310
\(470\) 2.07957e12 1.96578
\(471\) 1.02928e12 0.963699
\(472\) −9.51015e10 −0.0881959
\(473\) −5.48025e11 −0.503414
\(474\) 1.22781e12 1.11719
\(475\) −1.27601e11 −0.115009
\(476\) 5.23030e9 0.00466977
\(477\) −3.45754e11 −0.305798
\(478\) 2.73310e10 0.0239459
\(479\) 1.69258e12 1.46906 0.734529 0.678577i \(-0.237403\pi\)
0.734529 + 0.678577i \(0.237403\pi\)
\(480\) 5.94000e11 0.510741
\(481\) −4.90813e11 −0.418084
\(482\) 5.31779e10 0.0448766
\(483\) 2.64677e11 0.221286
\(484\) −4.31000e11 −0.357004
\(485\) −5.17143e11 −0.424397
\(486\) 9.44904e10 0.0768289
\(487\) 1.93469e12 1.55858 0.779292 0.626661i \(-0.215579\pi\)
0.779292 + 0.626661i \(0.215579\pi\)
\(488\) −2.92408e11 −0.233400
\(489\) −1.30594e11 −0.103284
\(490\) 1.45905e12 1.14337
\(491\) 1.30146e12 1.01056 0.505282 0.862954i \(-0.331388\pi\)
0.505282 + 0.862954i \(0.331388\pi\)
\(492\) 2.77386e11 0.213423
\(493\) −7.58442e10 −0.0578244
\(494\) −1.44398e12 −1.09092
\(495\) −2.04081e11 −0.152784
\(496\) 2.91316e12 2.16121
\(497\) −5.35673e11 −0.393818
\(498\) −1.58527e12 −1.15497
\(499\) 2.28177e12 1.64748 0.823738 0.566971i \(-0.191885\pi\)
0.823738 + 0.566971i \(0.191885\pi\)
\(500\) 5.40807e11 0.386970
\(501\) −1.56125e12 −1.10714
\(502\) 2.75914e12 1.93913
\(503\) −9.08924e11 −0.633099 −0.316550 0.948576i \(-0.602524\pi\)
−0.316550 + 0.948576i \(0.602524\pi\)
\(504\) −1.13811e11 −0.0785682
\(505\) 2.01191e10 0.0137657
\(506\) 8.20971e11 0.556738
\(507\) 8.82237e11 0.592993
\(508\) −2.83075e11 −0.188588
\(509\) −1.79179e12 −1.18320 −0.591600 0.806232i \(-0.701503\pi\)
−0.591600 + 0.806232i \(0.701503\pi\)
\(510\) −3.54561e10 −0.0232073
\(511\) −3.73035e10 −0.0242022
\(512\) 2.65372e11 0.170663
\(513\) −1.93140e11 −0.123125
\(514\) −3.75415e12 −2.37234
\(515\) −2.75386e12 −1.72508
\(516\) −4.81757e11 −0.299161
\(517\) −1.03590e12 −0.637691
\(518\) −2.00508e11 −0.122362
\(519\) −9.84975e11 −0.595899
\(520\) 1.74672e12 1.04763
\(521\) 2.11097e11 0.125520 0.0627600 0.998029i \(-0.480010\pi\)
0.0627600 + 0.998029i \(0.480010\pi\)
\(522\) −1.26729e12 −0.747065
\(523\) −1.05833e12 −0.618532 −0.309266 0.950976i \(-0.600083\pi\)
−0.309266 + 0.950976i \(0.600083\pi\)
\(524\) −3.43203e11 −0.198866
\(525\) 6.28573e10 0.0361109
\(526\) 1.65197e12 0.940949
\(527\) −9.49280e10 −0.0536101
\(528\) −5.42006e11 −0.303495
\(529\) 3.84553e11 0.213504
\(530\) 2.16782e12 1.19339
\(531\) 7.95020e10 0.0433963
\(532\) −1.78634e11 −0.0966857
\(533\) 2.25772e12 1.21171
\(534\) 1.21246e12 0.645257
\(535\) −1.27236e12 −0.671459
\(536\) 8.79234e11 0.460111
\(537\) −1.40659e12 −0.729934
\(538\) −3.45066e11 −0.177575
\(539\) −7.26796e11 −0.370905
\(540\) −1.79403e11 −0.0907941
\(541\) −4.87448e10 −0.0244647 −0.0122324 0.999925i \(-0.503894\pi\)
−0.0122324 + 0.999925i \(0.503894\pi\)
\(542\) 1.79310e12 0.892500
\(543\) −5.44440e11 −0.268751
\(544\) −5.14065e10 −0.0251665
\(545\) −3.34739e12 −1.62526
\(546\) 7.11320e11 0.342530
\(547\) −1.10428e12 −0.527395 −0.263698 0.964605i \(-0.584942\pi\)
−0.263698 + 0.964605i \(0.584942\pi\)
\(548\) −1.64918e11 −0.0781188
\(549\) 2.44445e11 0.114843
\(550\) 1.94970e11 0.0908522
\(551\) 2.59036e12 1.19723
\(552\) −9.39853e11 −0.430858
\(553\) 1.23628e12 0.562153
\(554\) 3.87730e12 1.74878
\(555\) 4.11606e11 0.184146
\(556\) −7.35189e9 −0.00326259
\(557\) −2.95668e12 −1.30154 −0.650768 0.759277i \(-0.725553\pi\)
−0.650768 + 0.759277i \(0.725553\pi\)
\(558\) −1.58616e12 −0.692617
\(559\) −3.92115e12 −1.69848
\(560\) 1.09559e12 0.470762
\(561\) 1.76618e10 0.00752838
\(562\) −5.98388e11 −0.253029
\(563\) 3.12555e12 1.31111 0.655554 0.755148i \(-0.272435\pi\)
0.655554 + 0.755148i \(0.272435\pi\)
\(564\) −9.10637e11 −0.378956
\(565\) 1.74148e12 0.718952
\(566\) 5.56852e12 2.28069
\(567\) 9.51426e10 0.0386591
\(568\) 1.90215e12 0.766790
\(569\) −4.28087e12 −1.71209 −0.856046 0.516900i \(-0.827086\pi\)
−0.856046 + 0.516900i \(0.827086\pi\)
\(570\) 1.21096e12 0.480498
\(571\) −2.27517e12 −0.895678 −0.447839 0.894114i \(-0.647806\pi\)
−0.447839 + 0.894114i \(0.647806\pi\)
\(572\) 6.68133e11 0.260964
\(573\) 1.05006e12 0.406928
\(574\) 9.22330e11 0.354636
\(575\) 5.19077e11 0.198028
\(576\) 2.38001e11 0.0900903
\(577\) 4.94509e12 1.85730 0.928651 0.370954i \(-0.120969\pi\)
0.928651 + 0.370954i \(0.120969\pi\)
\(578\) −3.21062e12 −1.19650
\(579\) −2.36784e11 −0.0875586
\(580\) 2.40612e12 0.882858
\(581\) −1.59621e12 −0.581162
\(582\) 7.47817e11 0.270173
\(583\) −1.07986e12 −0.387131
\(584\) 1.32463e11 0.0471233
\(585\) −1.46021e12 −0.515482
\(586\) −4.04690e12 −1.41770
\(587\) 3.65831e12 1.27177 0.635886 0.771783i \(-0.280635\pi\)
0.635886 + 0.771783i \(0.280635\pi\)
\(588\) −6.38910e11 −0.220415
\(589\) 3.24214e12 1.10998
\(590\) −4.98464e11 −0.169356
\(591\) −1.27422e12 −0.429636
\(592\) 1.09316e12 0.365794
\(593\) −1.48698e12 −0.493810 −0.246905 0.969040i \(-0.579414\pi\)
−0.246905 + 0.969040i \(0.579414\pi\)
\(594\) 2.95112e11 0.0972631
\(595\) −3.57008e10 −0.0116775
\(596\) 5.37478e11 0.174483
\(597\) −1.19308e12 −0.384402
\(598\) 5.87409e12 1.87839
\(599\) −7.33466e11 −0.232787 −0.116394 0.993203i \(-0.537133\pi\)
−0.116394 + 0.993203i \(0.537133\pi\)
\(600\) −2.23203e11 −0.0703103
\(601\) −5.59924e12 −1.75063 −0.875315 0.483553i \(-0.839346\pi\)
−0.875315 + 0.483553i \(0.839346\pi\)
\(602\) −1.60188e12 −0.497102
\(603\) −7.35013e11 −0.226395
\(604\) −2.51895e12 −0.770110
\(605\) 2.94191e12 0.892750
\(606\) −2.90934e10 −0.00876329
\(607\) 3.51612e12 1.05127 0.525635 0.850710i \(-0.323828\pi\)
0.525635 + 0.850710i \(0.323828\pi\)
\(608\) 1.75572e12 0.521063
\(609\) −1.27604e12 −0.375911
\(610\) −1.53263e12 −0.448179
\(611\) −7.41192e12 −2.15152
\(612\) 1.55261e10 0.00447384
\(613\) −9.62938e10 −0.0275439 −0.0137720 0.999905i \(-0.504384\pi\)
−0.0137720 + 0.999905i \(0.504384\pi\)
\(614\) 2.31885e12 0.658437
\(615\) −1.89337e12 −0.533701
\(616\) −3.55454e11 −0.0994650
\(617\) 3.01324e12 0.837048 0.418524 0.908206i \(-0.362548\pi\)
0.418524 + 0.908206i \(0.362548\pi\)
\(618\) 3.98223e12 1.09819
\(619\) −2.35963e12 −0.646006 −0.323003 0.946398i \(-0.604692\pi\)
−0.323003 + 0.946398i \(0.604692\pi\)
\(620\) 3.01154e12 0.818514
\(621\) 7.85689e11 0.212001
\(622\) −1.35875e12 −0.363985
\(623\) 1.22083e12 0.324683
\(624\) −3.87808e12 −1.02397
\(625\) −4.37717e12 −1.14745
\(626\) 8.81915e12 2.29531
\(627\) −6.03215e11 −0.155872
\(628\) 2.82593e12 0.725009
\(629\) −3.56216e10 −0.00907373
\(630\) −5.96528e11 −0.150868
\(631\) 1.16636e12 0.292887 0.146444 0.989219i \(-0.453217\pi\)
0.146444 + 0.989219i \(0.453217\pi\)
\(632\) −4.38997e12 −1.09455
\(633\) −1.98877e12 −0.492342
\(634\) −4.78198e12 −1.17545
\(635\) 1.93220e12 0.471596
\(636\) −9.49278e11 −0.230057
\(637\) −5.20026e12 −1.25141
\(638\) −3.95799e12 −0.945762
\(639\) −1.59014e12 −0.377295
\(640\) −5.24689e12 −1.23621
\(641\) 3.45037e12 0.807245 0.403622 0.914926i \(-0.367751\pi\)
0.403622 + 0.914926i \(0.367751\pi\)
\(642\) 1.83991e12 0.427454
\(643\) 2.83775e12 0.654674 0.327337 0.944908i \(-0.393849\pi\)
0.327337 + 0.944908i \(0.393849\pi\)
\(644\) 7.26679e11 0.166478
\(645\) 3.28836e12 0.748102
\(646\) −1.04800e11 −0.0236763
\(647\) 8.39979e12 1.88451 0.942256 0.334892i \(-0.108700\pi\)
0.942256 + 0.334892i \(0.108700\pi\)
\(648\) −3.37846e11 −0.0752717
\(649\) 2.48300e11 0.0549384
\(650\) 1.39502e12 0.306528
\(651\) −1.59711e12 −0.348514
\(652\) −3.58549e11 −0.0777024
\(653\) −1.28646e12 −0.276877 −0.138439 0.990371i \(-0.544208\pi\)
−0.138439 + 0.990371i \(0.544208\pi\)
\(654\) 4.84051e12 1.03464
\(655\) 2.34262e12 0.497298
\(656\) −5.02850e12 −1.06016
\(657\) −1.10735e11 −0.0231868
\(658\) −3.02794e12 −0.629695
\(659\) −1.00991e12 −0.208592 −0.104296 0.994546i \(-0.533259\pi\)
−0.104296 + 0.994546i \(0.533259\pi\)
\(660\) −5.60310e11 −0.114943
\(661\) −2.30424e12 −0.469484 −0.234742 0.972058i \(-0.575425\pi\)
−0.234742 + 0.972058i \(0.575425\pi\)
\(662\) 5.95458e12 1.20501
\(663\) 1.26371e11 0.0254001
\(664\) 5.66805e12 1.13156
\(665\) 1.21931e12 0.241779
\(666\) −5.95205e11 −0.117228
\(667\) −1.05375e13 −2.06145
\(668\) −4.28647e12 −0.832925
\(669\) −4.92670e12 −0.950908
\(670\) 4.60841e12 0.883515
\(671\) 7.63448e11 0.145388
\(672\) −8.64886e11 −0.163605
\(673\) 6.41400e12 1.20521 0.602603 0.798041i \(-0.294130\pi\)
0.602603 + 0.798041i \(0.294130\pi\)
\(674\) 1.25328e13 2.33926
\(675\) 1.86591e11 0.0345958
\(676\) 2.42221e12 0.446120
\(677\) −4.42017e12 −0.808704 −0.404352 0.914603i \(-0.632503\pi\)
−0.404352 + 0.914603i \(0.632503\pi\)
\(678\) −2.51827e12 −0.457688
\(679\) 7.52979e11 0.135947
\(680\) 1.26772e11 0.0227369
\(681\) 4.42079e12 0.787659
\(682\) −4.95389e12 −0.876833
\(683\) 1.21590e12 0.213799 0.106900 0.994270i \(-0.465908\pi\)
0.106900 + 0.994270i \(0.465908\pi\)
\(684\) −5.30273e11 −0.0926290
\(685\) 1.12569e12 0.195349
\(686\) −4.54144e12 −0.782953
\(687\) 1.73862e11 0.0297783
\(688\) 8.73336e12 1.48605
\(689\) −7.72643e12 −1.30615
\(690\) −4.92614e12 −0.827343
\(691\) −6.17385e12 −1.03016 −0.515080 0.857142i \(-0.672238\pi\)
−0.515080 + 0.857142i \(0.672238\pi\)
\(692\) −2.70428e12 −0.448306
\(693\) 2.97149e11 0.0489412
\(694\) 1.39605e13 2.28445
\(695\) 5.01823e10 0.00815866
\(696\) 4.53113e12 0.731923
\(697\) 1.63858e11 0.0262979
\(698\) 8.17034e11 0.130284
\(699\) 4.75221e12 0.752918
\(700\) 1.72577e11 0.0271670
\(701\) 1.04454e13 1.63378 0.816890 0.576794i \(-0.195696\pi\)
0.816890 + 0.576794i \(0.195696\pi\)
\(702\) 2.11154e12 0.328158
\(703\) 1.21661e12 0.187868
\(704\) 7.43325e11 0.114052
\(705\) 6.21579e12 0.947645
\(706\) −2.81446e12 −0.426358
\(707\) −2.92942e10 −0.00440955
\(708\) 2.18275e11 0.0326479
\(709\) −1.00404e13 −1.49226 −0.746131 0.665800i \(-0.768091\pi\)
−0.746131 + 0.665800i \(0.768091\pi\)
\(710\) 9.96989e12 1.47241
\(711\) 3.66989e12 0.538566
\(712\) −4.33511e12 −0.632179
\(713\) −1.31889e13 −1.91121
\(714\) 5.16253e10 0.00743397
\(715\) −4.56052e12 −0.652585
\(716\) −3.86184e12 −0.549143
\(717\) 8.16918e10 0.0115436
\(718\) −5.09154e12 −0.714972
\(719\) 2.36265e12 0.329701 0.164850 0.986319i \(-0.447286\pi\)
0.164850 + 0.986319i \(0.447286\pi\)
\(720\) 3.25224e12 0.451010
\(721\) 4.00972e12 0.552593
\(722\) −5.16540e12 −0.707436
\(723\) 1.58948e11 0.0216337
\(724\) −1.49478e12 −0.202187
\(725\) −2.50252e12 −0.336401
\(726\) −4.25416e12 −0.568328
\(727\) 1.99053e12 0.264279 0.132140 0.991231i \(-0.457815\pi\)
0.132140 + 0.991231i \(0.457815\pi\)
\(728\) −2.54329e12 −0.335587
\(729\) 2.82430e11 0.0370370
\(730\) 6.94288e11 0.0904872
\(731\) −2.84585e11 −0.0368624
\(732\) 6.71130e11 0.0863987
\(733\) −9.77749e11 −0.125101 −0.0625503 0.998042i \(-0.519923\pi\)
−0.0625503 + 0.998042i \(0.519923\pi\)
\(734\) −1.16254e11 −0.0147835
\(735\) 4.36105e12 0.551186
\(736\) −7.14224e12 −0.897189
\(737\) −2.29559e12 −0.286610
\(738\) 2.73792e12 0.339756
\(739\) −2.84759e12 −0.351219 −0.175609 0.984460i \(-0.556190\pi\)
−0.175609 + 0.984460i \(0.556190\pi\)
\(740\) 1.13008e12 0.138537
\(741\) −4.31603e12 −0.525900
\(742\) −3.15642e12 −0.382276
\(743\) −6.47321e12 −0.779238 −0.389619 0.920976i \(-0.627393\pi\)
−0.389619 + 0.920976i \(0.627393\pi\)
\(744\) 5.67125e12 0.678579
\(745\) −3.66870e12 −0.436323
\(746\) 4.61771e12 0.545886
\(747\) −4.73832e12 −0.556778
\(748\) 4.84910e10 0.00566374
\(749\) 1.85261e12 0.215088
\(750\) 5.33800e12 0.616032
\(751\) 8.14699e12 0.934582 0.467291 0.884103i \(-0.345230\pi\)
0.467291 + 0.884103i \(0.345230\pi\)
\(752\) 1.65082e13 1.88243
\(753\) 8.24700e12 0.934799
\(754\) −2.83196e13 −3.19092
\(755\) 1.71937e13 1.92579
\(756\) 2.61217e11 0.0290840
\(757\) 4.96495e12 0.549520 0.274760 0.961513i \(-0.411402\pi\)
0.274760 + 0.961513i \(0.411402\pi\)
\(758\) 2.44594e12 0.269113
\(759\) 2.45386e12 0.268387
\(760\) −4.32972e12 −0.470759
\(761\) −3.30511e12 −0.357236 −0.178618 0.983919i \(-0.557163\pi\)
−0.178618 + 0.983919i \(0.557163\pi\)
\(762\) −2.79407e12 −0.300220
\(763\) 4.87392e12 0.520617
\(764\) 2.88297e12 0.306140
\(765\) −1.05977e11 −0.0111876
\(766\) 8.65925e12 0.908763
\(767\) 1.77660e12 0.185358
\(768\) 6.08290e12 0.630935
\(769\) −5.13105e12 −0.529100 −0.264550 0.964372i \(-0.585223\pi\)
−0.264550 + 0.964372i \(0.585223\pi\)
\(770\) −1.86307e12 −0.190995
\(771\) −1.12210e13 −1.14364
\(772\) −6.50098e11 −0.0658720
\(773\) −4.46041e12 −0.449332 −0.224666 0.974436i \(-0.572129\pi\)
−0.224666 + 0.974436i \(0.572129\pi\)
\(774\) −4.75515e12 −0.476244
\(775\) −3.13220e12 −0.311883
\(776\) −2.67379e12 −0.264697
\(777\) −5.99314e11 −0.0589874
\(778\) −6.26789e12 −0.613357
\(779\) −5.59637e12 −0.544487
\(780\) −4.00905e12 −0.387807
\(781\) −4.96631e12 −0.477644
\(782\) 4.26323e11 0.0407669
\(783\) −3.78789e12 −0.360139
\(784\) 1.15823e13 1.09489
\(785\) −1.92891e13 −1.81301
\(786\) −3.38756e12 −0.316582
\(787\) 1.49260e13 1.38694 0.693469 0.720486i \(-0.256081\pi\)
0.693469 + 0.720486i \(0.256081\pi\)
\(788\) −3.49841e12 −0.323223
\(789\) 4.93769e12 0.453605
\(790\) −2.30095e13 −2.10177
\(791\) −2.53566e12 −0.230301
\(792\) −1.05516e12 −0.0952917
\(793\) 5.46251e12 0.490527
\(794\) 1.42287e13 1.27049
\(795\) 6.47954e12 0.575297
\(796\) −3.27564e12 −0.289193
\(797\) 1.20121e12 0.105452 0.0527260 0.998609i \(-0.483209\pi\)
0.0527260 + 0.998609i \(0.483209\pi\)
\(798\) −1.76320e12 −0.153917
\(799\) −5.37933e11 −0.0466948
\(800\) −1.69619e12 −0.146409
\(801\) 3.62402e12 0.311060
\(802\) −1.35440e13 −1.15601
\(803\) −3.45847e11 −0.0293537
\(804\) −2.01800e12 −0.170321
\(805\) −4.96014e12 −0.416306
\(806\) −3.54453e13 −2.95836
\(807\) −1.03139e12 −0.0856039
\(808\) 1.04022e11 0.00858567
\(809\) 1.01524e13 0.833298 0.416649 0.909067i \(-0.363204\pi\)
0.416649 + 0.909067i \(0.363204\pi\)
\(810\) −1.77078e12 −0.144538
\(811\) −6.37555e12 −0.517516 −0.258758 0.965942i \(-0.583313\pi\)
−0.258758 + 0.965942i \(0.583313\pi\)
\(812\) −3.50340e12 −0.282805
\(813\) 5.35954e12 0.430249
\(814\) −1.85894e12 −0.148408
\(815\) 2.44737e12 0.194308
\(816\) −2.81459e11 −0.0222233
\(817\) 9.71962e12 0.763220
\(818\) −1.29034e12 −0.100766
\(819\) 2.12612e12 0.165124
\(820\) −5.19832e12 −0.401514
\(821\) 7.50250e12 0.576317 0.288159 0.957583i \(-0.406957\pi\)
0.288159 + 0.957583i \(0.406957\pi\)
\(822\) −1.62781e12 −0.124360
\(823\) −7.71506e12 −0.586192 −0.293096 0.956083i \(-0.594686\pi\)
−0.293096 + 0.956083i \(0.594686\pi\)
\(824\) −1.42383e13 −1.07593
\(825\) 5.82760e11 0.0437973
\(826\) 7.25782e11 0.0542495
\(827\) −1.52511e13 −1.13377 −0.566886 0.823796i \(-0.691852\pi\)
−0.566886 + 0.823796i \(0.691852\pi\)
\(828\) 2.15714e12 0.159493
\(829\) −1.77884e13 −1.30810 −0.654050 0.756451i \(-0.726931\pi\)
−0.654050 + 0.756451i \(0.726931\pi\)
\(830\) 2.97085e13 2.17285
\(831\) 1.15891e13 0.843038
\(832\) 5.31852e12 0.384801
\(833\) −3.77419e11 −0.0271594
\(834\) −7.25664e10 −0.00519384
\(835\) 2.92584e13 2.08287
\(836\) −1.65615e12 −0.117266
\(837\) −4.74099e12 −0.333891
\(838\) −2.91347e13 −2.04086
\(839\) −2.77513e13 −1.93355 −0.966773 0.255637i \(-0.917715\pi\)
−0.966773 + 0.255637i \(0.917715\pi\)
\(840\) 2.13286e12 0.147811
\(841\) 3.62954e13 2.50190
\(842\) −2.29614e13 −1.57432
\(843\) −1.78857e12 −0.121978
\(844\) −5.46022e12 −0.370399
\(845\) −1.65334e13 −1.11560
\(846\) −8.98838e12 −0.603274
\(847\) −4.28352e12 −0.285974
\(848\) 1.72086e13 1.14279
\(849\) 1.66442e13 1.09946
\(850\) 1.01246e11 0.00665263
\(851\) −4.94914e12 −0.323480
\(852\) −4.36577e12 −0.283846
\(853\) 6.16511e11 0.0398722 0.0199361 0.999801i \(-0.493654\pi\)
0.0199361 + 0.999801i \(0.493654\pi\)
\(854\) 2.23156e12 0.143565
\(855\) 3.61952e12 0.231634
\(856\) −6.57852e12 −0.418790
\(857\) 1.39335e13 0.882364 0.441182 0.897418i \(-0.354559\pi\)
0.441182 + 0.897418i \(0.354559\pi\)
\(858\) 6.59476e12 0.415438
\(859\) −2.84121e13 −1.78047 −0.890233 0.455505i \(-0.849459\pi\)
−0.890233 + 0.455505i \(0.849459\pi\)
\(860\) 9.02829e12 0.562811
\(861\) 2.75682e12 0.170960
\(862\) −2.03819e13 −1.25737
\(863\) −9.12009e12 −0.559694 −0.279847 0.960045i \(-0.590284\pi\)
−0.279847 + 0.960045i \(0.590284\pi\)
\(864\) −2.56740e12 −0.156741
\(865\) 1.84588e13 1.12106
\(866\) −1.26221e13 −0.762606
\(867\) −9.59645e12 −0.576799
\(868\) −4.38491e12 −0.262194
\(869\) 1.14618e13 0.681809
\(870\) 2.37494e13 1.40545
\(871\) −1.64251e13 −0.966997
\(872\) −1.73070e13 −1.01367
\(873\) 2.23521e12 0.130243
\(874\) −1.45605e13 −0.844063
\(875\) 5.37485e12 0.309977
\(876\) −3.04026e11 −0.0174438
\(877\) 1.02362e13 0.584309 0.292154 0.956371i \(-0.405628\pi\)
0.292154 + 0.956371i \(0.405628\pi\)
\(878\) 1.18537e13 0.673177
\(879\) −1.20961e13 −0.683432
\(880\) 1.01574e13 0.570966
\(881\) 1.16506e13 0.651563 0.325781 0.945445i \(-0.394373\pi\)
0.325781 + 0.945445i \(0.394373\pi\)
\(882\) −6.30632e12 −0.350887
\(883\) 2.88623e13 1.59774 0.798872 0.601501i \(-0.205430\pi\)
0.798872 + 0.601501i \(0.205430\pi\)
\(884\) 3.46955e11 0.0191090
\(885\) −1.48990e12 −0.0816415
\(886\) −4.10620e13 −2.23866
\(887\) −2.34243e13 −1.27061 −0.635303 0.772263i \(-0.719125\pi\)
−0.635303 + 0.772263i \(0.719125\pi\)
\(888\) 2.12813e12 0.114852
\(889\) −2.81335e12 −0.151066
\(890\) −2.27220e13 −1.21392
\(891\) 8.82083e11 0.0468878
\(892\) −1.35264e13 −0.715386
\(893\) 1.83724e13 0.966796
\(894\) 5.30514e12 0.277765
\(895\) 2.63600e13 1.37323
\(896\) 7.63967e12 0.395994
\(897\) 1.75575e13 0.905518
\(898\) 1.94331e13 0.997235
\(899\) 6.35853e13 3.24667
\(900\) 5.12291e11 0.0260271
\(901\) −5.60759e11 −0.0283475
\(902\) 8.55107e12 0.430121
\(903\) −4.78797e12 −0.239639
\(904\) 9.00398e12 0.448411
\(905\) 1.02030e13 0.505603
\(906\) −2.48631e13 −1.22597
\(907\) −3.23704e13 −1.58824 −0.794118 0.607763i \(-0.792067\pi\)
−0.794118 + 0.607763i \(0.792067\pi\)
\(908\) 1.21374e13 0.592571
\(909\) −8.69593e10 −0.00422453
\(910\) −1.33304e13 −0.644401
\(911\) 4.74352e12 0.228175 0.114087 0.993471i \(-0.463606\pi\)
0.114087 + 0.993471i \(0.463606\pi\)
\(912\) 9.61286e12 0.460125
\(913\) −1.47987e13 −0.704864
\(914\) 1.11803e12 0.0529900
\(915\) −4.58098e12 −0.216054
\(916\) 4.77343e11 0.0224028
\(917\) −3.41095e12 −0.159299
\(918\) 1.53249e11 0.00712206
\(919\) −1.19100e13 −0.550800 −0.275400 0.961330i \(-0.588810\pi\)
−0.275400 + 0.961330i \(0.588810\pi\)
\(920\) 1.76132e13 0.810574
\(921\) 6.93098e12 0.317414
\(922\) 2.90206e13 1.32257
\(923\) −3.55342e13 −1.61153
\(924\) 8.15832e11 0.0368194
\(925\) −1.17536e12 −0.0527876
\(926\) −1.02006e13 −0.455906
\(927\) 1.19028e13 0.529407
\(928\) 3.44335e13 1.52411
\(929\) −1.64263e13 −0.723551 −0.361776 0.932265i \(-0.617829\pi\)
−0.361776 + 0.932265i \(0.617829\pi\)
\(930\) 2.97252e13 1.30302
\(931\) 1.28902e13 0.562325
\(932\) 1.30473e13 0.566435
\(933\) −4.06128e12 −0.175467
\(934\) −2.12551e13 −0.913907
\(935\) −3.30988e11 −0.0141631
\(936\) −7.54973e12 −0.321507
\(937\) −3.25084e13 −1.37774 −0.688869 0.724885i \(-0.741893\pi\)
−0.688869 + 0.724885i \(0.741893\pi\)
\(938\) −6.71001e12 −0.283015
\(939\) 2.63602e13 1.10651
\(940\) 1.70657e13 0.712931
\(941\) −1.89705e13 −0.788724 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(942\) 2.78932e13 1.15417
\(943\) 2.27659e13 0.937522
\(944\) −3.95693e12 −0.162175
\(945\) −1.78301e12 −0.0727294
\(946\) −1.48513e13 −0.602911
\(947\) −6.79131e12 −0.274396 −0.137198 0.990544i \(-0.543810\pi\)
−0.137198 + 0.990544i \(0.543810\pi\)
\(948\) 1.00758e13 0.405174
\(949\) −2.47455e12 −0.0990372
\(950\) −3.45793e12 −0.137740
\(951\) −1.42932e13 −0.566653
\(952\) −1.84584e11 −0.00728330
\(953\) −4.84372e12 −0.190222 −0.0951110 0.995467i \(-0.530321\pi\)
−0.0951110 + 0.995467i \(0.530321\pi\)
\(954\) −9.36978e12 −0.366237
\(955\) −1.96785e13 −0.765554
\(956\) 2.24287e11 0.00868449
\(957\) −1.18303e13 −0.455925
\(958\) 4.58682e13 1.75941
\(959\) −1.63905e12 −0.0625760
\(960\) −4.46023e12 −0.169487
\(961\) 5.31449e13 2.01005
\(962\) −1.33008e13 −0.500715
\(963\) 5.49944e12 0.206063
\(964\) 4.36396e11 0.0162755
\(965\) 4.43742e12 0.164724
\(966\) 7.17264e12 0.265022
\(967\) 3.10571e13 1.14220 0.571100 0.820880i \(-0.306517\pi\)
0.571100 + 0.820880i \(0.306517\pi\)
\(968\) 1.52106e13 0.556809
\(969\) −3.13244e11 −0.0114137
\(970\) −1.40144e13 −0.508277
\(971\) −2.31521e13 −0.835802 −0.417901 0.908493i \(-0.637234\pi\)
−0.417901 + 0.908493i \(0.637234\pi\)
\(972\) 7.75419e11 0.0278637
\(973\) −7.30673e10 −0.00261346
\(974\) 5.24292e13 1.86663
\(975\) 4.16968e12 0.147768
\(976\) −1.21664e13 −0.429177
\(977\) −3.02951e13 −1.06377 −0.531884 0.846817i \(-0.678516\pi\)
−0.531884 + 0.846817i \(0.678516\pi\)
\(978\) −3.53904e12 −0.123697
\(979\) 1.13185e13 0.393793
\(980\) 1.19734e13 0.414668
\(981\) 1.44682e13 0.498773
\(982\) 3.52690e13 1.21030
\(983\) −3.84935e13 −1.31491 −0.657456 0.753493i \(-0.728367\pi\)
−0.657456 + 0.753493i \(0.728367\pi\)
\(984\) −9.78932e12 −0.332870
\(985\) 2.38793e13 0.808275
\(986\) −2.05535e12 −0.0692531
\(987\) −9.05042e12 −0.303558
\(988\) −1.18498e13 −0.395644
\(989\) −3.95391e13 −1.31415
\(990\) −5.53051e12 −0.182981
\(991\) 2.74204e13 0.903112 0.451556 0.892243i \(-0.350869\pi\)
0.451556 + 0.892243i \(0.350869\pi\)
\(992\) 4.30976e13 1.41303
\(993\) 1.77981e13 0.580901
\(994\) −1.45165e13 −0.471654
\(995\) 2.23587e13 0.723175
\(996\) −1.30092e13 −0.418875
\(997\) −1.87482e13 −0.600942 −0.300471 0.953791i \(-0.597144\pi\)
−0.300471 + 0.953791i \(0.597144\pi\)
\(998\) 6.18350e13 1.97309
\(999\) −1.77905e12 −0.0565125
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.10.a.a.1.17 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.a.1.17 21 1.1 even 1 trivial