Properties

Label 177.10.a.a.1.3
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 1.3
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-35.8479 q^{2} +81.0000 q^{3} +773.074 q^{4} +1881.24 q^{5} -2903.68 q^{6} -9737.47 q^{7} -9358.98 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q-35.8479 q^{2} +81.0000 q^{3} +773.074 q^{4} +1881.24 q^{5} -2903.68 q^{6} -9737.47 q^{7} -9358.98 q^{8} +6561.00 q^{9} -67438.4 q^{10} -61627.4 q^{11} +62619.0 q^{12} +169936. q^{13} +349068. q^{14} +152380. q^{15} -60314.1 q^{16} +560769. q^{17} -235198. q^{18} -659805. q^{19} +1.45434e6 q^{20} -788735. q^{21} +2.20922e6 q^{22} -535600. q^{23} -758077. q^{24} +1.58592e6 q^{25} -6.09185e6 q^{26} +531441. q^{27} -7.52779e6 q^{28} +1.76519e6 q^{29} -5.46251e6 q^{30} -7.35837e6 q^{31} +6.95393e6 q^{32} -4.99182e6 q^{33} -2.01024e7 q^{34} -1.83185e7 q^{35} +5.07214e6 q^{36} -1.11914e7 q^{37} +2.36527e7 q^{38} +1.37648e7 q^{39} -1.76064e7 q^{40} +9.17749e6 q^{41} +2.82745e7 q^{42} +1.27647e7 q^{43} -4.76426e7 q^{44} +1.23428e7 q^{45} +1.92002e7 q^{46} +7.13479e6 q^{47} -4.88544e6 q^{48} +5.44648e7 q^{49} -5.68520e7 q^{50} +4.54223e7 q^{51} +1.31373e8 q^{52} -1.07603e8 q^{53} -1.90511e7 q^{54} -1.15936e8 q^{55} +9.11328e7 q^{56} -5.34442e7 q^{57} -6.32783e7 q^{58} +1.21174e7 q^{59} +1.17801e8 q^{60} -2.18590e7 q^{61} +2.63782e8 q^{62} -6.38876e7 q^{63} -2.18403e8 q^{64} +3.19690e8 q^{65} +1.78947e8 q^{66} +3.04061e8 q^{67} +4.33516e8 q^{68} -4.33836e7 q^{69} +6.56680e8 q^{70} -2.46812e8 q^{71} -6.14042e7 q^{72} -9.89167e7 q^{73} +4.01190e8 q^{74} +1.28460e8 q^{75} -5.10079e8 q^{76} +6.00096e8 q^{77} -4.93440e8 q^{78} -1.89951e7 q^{79} -1.13465e8 q^{80} +4.30467e7 q^{81} -3.28994e8 q^{82} -3.82819e8 q^{83} -6.09751e8 q^{84} +1.05494e9 q^{85} -4.57587e8 q^{86} +1.42980e8 q^{87} +5.76770e8 q^{88} +8.54743e8 q^{89} -4.42463e8 q^{90} -1.65475e9 q^{91} -4.14059e8 q^{92} -5.96028e8 q^{93} -2.55767e8 q^{94} -1.24125e9 q^{95} +5.63268e8 q^{96} +1.05008e9 q^{97} -1.95245e9 q^{98} -4.04338e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} - 54663q^{10} - 151769q^{11} + 421686q^{12} - 153611q^{13} - 286771q^{14} - 240084q^{15} + 805530q^{16} - 723621q^{17} - 433026q^{18} - 549388q^{19} - 527311q^{20} - 2492775q^{21} + 2973158q^{22} + 169962q^{23} - 1994301q^{24} + 8035779q^{25} - 2337392q^{26} + 11160261q^{27} - 22659054q^{28} - 16845442q^{29} - 4427703q^{30} - 19307976q^{31} - 44923568q^{32} - 12293289q^{33} - 35547496q^{34} - 34882596q^{35} + 34156566q^{36} - 41561129q^{37} - 52335371q^{38} - 12442491q^{39} - 125735038q^{40} - 68169291q^{41} - 23228451q^{42} - 25719587q^{43} - 126277032q^{44} - 19446804q^{45} - 292814271q^{46} - 174095332q^{47} + 65247930q^{48} + 7479350q^{49} - 227877439q^{50} - 58613301q^{51} - 232397708q^{52} - 228390500q^{53} - 35075106q^{54} - 29426208q^{55} + 326778474q^{56} - 44500428q^{57} + 480343762q^{58} + 254464581q^{59} - 42712191q^{60} - 183928964q^{61} - 21753862q^{62} - 201914775q^{63} + 310571245q^{64} + 5308466q^{65} + 240825798q^{66} - 82724114q^{67} - 138336205q^{68} + 13766922q^{69} + 1030274876q^{70} - 404721965q^{71} - 161538381q^{72} + 154162574q^{73} + 36352054q^{74} + 650898099q^{75} + 1068940636q^{76} - 448535481q^{77} - 189328752q^{78} + 272529635q^{79} - 345587859q^{80} + 903981141q^{81} - 38412637q^{82} + 432518643q^{83} - 1835383374q^{84} - 126211490q^{85} - 3699273072q^{86} - 1364480802q^{87} + 170111045q^{88} - 1255621070q^{89} - 358643943q^{90} + 1448885849q^{91} + 1568933320q^{92} - 1563946056q^{93} - 1908445164q^{94} - 2896546490q^{95} - 3638809008q^{96} + 1007235486q^{97} - 9506868248q^{98} - 995756409q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −35.8479 −1.58427 −0.792135 0.610346i \(-0.791030\pi\)
−0.792135 + 0.610346i \(0.791030\pi\)
\(3\) 81.0000 0.577350
\(4\) 773.074 1.50991
\(5\) 1881.24 1.34610 0.673051 0.739596i \(-0.264983\pi\)
0.673051 + 0.739596i \(0.264983\pi\)
\(6\) −2903.68 −0.914679
\(7\) −9737.47 −1.53287 −0.766435 0.642322i \(-0.777971\pi\)
−0.766435 + 0.642322i \(0.777971\pi\)
\(8\) −9358.98 −0.807836
\(9\) 6561.00 0.333333
\(10\) −67438.4 −2.13259
\(11\) −61627.4 −1.26913 −0.634566 0.772868i \(-0.718821\pi\)
−0.634566 + 0.772868i \(0.718821\pi\)
\(12\) 62619.0 0.871747
\(13\) 169936. 1.65021 0.825107 0.564977i \(-0.191115\pi\)
0.825107 + 0.564977i \(0.191115\pi\)
\(14\) 349068. 2.42848
\(15\) 152380. 0.777173
\(16\) −60314.1 −0.230080
\(17\) 560769. 1.62841 0.814205 0.580578i \(-0.197173\pi\)
0.814205 + 0.580578i \(0.197173\pi\)
\(18\) −235198. −0.528090
\(19\) −659805. −1.16151 −0.580757 0.814077i \(-0.697243\pi\)
−0.580757 + 0.814077i \(0.697243\pi\)
\(20\) 1.45434e6 2.03250
\(21\) −788735. −0.885002
\(22\) 2.20922e6 2.01065
\(23\) −535600. −0.399085 −0.199543 0.979889i \(-0.563946\pi\)
−0.199543 + 0.979889i \(0.563946\pi\)
\(24\) −758077. −0.466405
\(25\) 1.58592e6 0.811992
\(26\) −6.09185e6 −2.61438
\(27\) 531441. 0.192450
\(28\) −7.52779e6 −2.31450
\(29\) 1.76519e6 0.463446 0.231723 0.972782i \(-0.425564\pi\)
0.231723 + 0.972782i \(0.425564\pi\)
\(30\) −5.46251e6 −1.23125
\(31\) −7.35837e6 −1.43105 −0.715523 0.698589i \(-0.753812\pi\)
−0.715523 + 0.698589i \(0.753812\pi\)
\(32\) 6.95393e6 1.17235
\(33\) −4.99182e6 −0.732734
\(34\) −2.01024e7 −2.57984
\(35\) −1.83185e7 −2.06340
\(36\) 5.07214e6 0.503304
\(37\) −1.11914e7 −0.981699 −0.490849 0.871245i \(-0.663313\pi\)
−0.490849 + 0.871245i \(0.663313\pi\)
\(38\) 2.36527e7 1.84015
\(39\) 1.37648e7 0.952751
\(40\) −1.76064e7 −1.08743
\(41\) 9.17749e6 0.507220 0.253610 0.967307i \(-0.418382\pi\)
0.253610 + 0.967307i \(0.418382\pi\)
\(42\) 2.82745e7 1.40208
\(43\) 1.27647e7 0.569379 0.284689 0.958620i \(-0.408110\pi\)
0.284689 + 0.958620i \(0.408110\pi\)
\(44\) −4.76426e7 −1.91628
\(45\) 1.23428e7 0.448701
\(46\) 1.92002e7 0.632258
\(47\) 7.13479e6 0.213275 0.106638 0.994298i \(-0.465991\pi\)
0.106638 + 0.994298i \(0.465991\pi\)
\(48\) −4.88544e6 −0.132837
\(49\) 5.44648e7 1.34969
\(50\) −5.68520e7 −1.28641
\(51\) 4.54223e7 0.940163
\(52\) 1.31373e8 2.49168
\(53\) −1.07603e8 −1.87320 −0.936602 0.350396i \(-0.886047\pi\)
−0.936602 + 0.350396i \(0.886047\pi\)
\(54\) −1.90511e7 −0.304893
\(55\) −1.15936e8 −1.70838
\(56\) 9.11328e7 1.23831
\(57\) −5.34442e7 −0.670601
\(58\) −6.32783e7 −0.734224
\(59\) 1.21174e7 0.130189
\(60\) 1.17801e8 1.17346
\(61\) −2.18590e7 −0.202138 −0.101069 0.994879i \(-0.532226\pi\)
−0.101069 + 0.994879i \(0.532226\pi\)
\(62\) 2.63782e8 2.26716
\(63\) −6.38876e7 −0.510956
\(64\) −2.18403e8 −1.62723
\(65\) 3.19690e8 2.22136
\(66\) 1.78947e8 1.16085
\(67\) 3.04061e8 1.84342 0.921709 0.387883i \(-0.126794\pi\)
0.921709 + 0.387883i \(0.126794\pi\)
\(68\) 4.33516e8 2.45875
\(69\) −4.33836e7 −0.230412
\(70\) 6.56680e8 3.26898
\(71\) −2.46812e8 −1.15267 −0.576334 0.817214i \(-0.695517\pi\)
−0.576334 + 0.817214i \(0.695517\pi\)
\(72\) −6.14042e7 −0.269279
\(73\) −9.89167e7 −0.407678 −0.203839 0.979004i \(-0.565342\pi\)
−0.203839 + 0.979004i \(0.565342\pi\)
\(74\) 4.01190e8 1.55528
\(75\) 1.28460e8 0.468804
\(76\) −5.10079e8 −1.75378
\(77\) 6.00096e8 1.94541
\(78\) −4.93440e8 −1.50941
\(79\) −1.89951e7 −0.0548679 −0.0274340 0.999624i \(-0.508734\pi\)
−0.0274340 + 0.999624i \(0.508734\pi\)
\(80\) −1.13465e8 −0.309711
\(81\) 4.30467e7 0.111111
\(82\) −3.28994e8 −0.803574
\(83\) −3.82819e8 −0.885405 −0.442703 0.896668i \(-0.645980\pi\)
−0.442703 + 0.896668i \(0.645980\pi\)
\(84\) −6.09751e8 −1.33627
\(85\) 1.05494e9 2.19201
\(86\) −4.57587e8 −0.902049
\(87\) 1.42980e8 0.267571
\(88\) 5.76770e8 1.02525
\(89\) 8.54743e8 1.44405 0.722023 0.691869i \(-0.243213\pi\)
0.722023 + 0.691869i \(0.243213\pi\)
\(90\) −4.42463e8 −0.710863
\(91\) −1.65475e9 −2.52956
\(92\) −4.14059e8 −0.602583
\(93\) −5.96028e8 −0.826215
\(94\) −2.55767e8 −0.337886
\(95\) −1.24125e9 −1.56352
\(96\) 5.63268e8 0.676854
\(97\) 1.05008e9 1.20434 0.602169 0.798369i \(-0.294303\pi\)
0.602169 + 0.798369i \(0.294303\pi\)
\(98\) −1.95245e9 −2.13827
\(99\) −4.04338e8 −0.423044
\(100\) 1.22604e9 1.22604
\(101\) −1.45452e9 −1.39083 −0.695415 0.718608i \(-0.744779\pi\)
−0.695415 + 0.718608i \(0.744779\pi\)
\(102\) −1.62829e9 −1.48947
\(103\) 1.64504e9 1.44015 0.720076 0.693895i \(-0.244107\pi\)
0.720076 + 0.693895i \(0.244107\pi\)
\(104\) −1.59043e9 −1.33310
\(105\) −1.48380e9 −1.19130
\(106\) 3.85736e9 2.96766
\(107\) 4.75094e8 0.350391 0.175195 0.984534i \(-0.443944\pi\)
0.175195 + 0.984534i \(0.443944\pi\)
\(108\) 4.10843e8 0.290582
\(109\) −2.37857e9 −1.61398 −0.806988 0.590568i \(-0.798904\pi\)
−0.806988 + 0.590568i \(0.798904\pi\)
\(110\) 4.15606e9 2.70654
\(111\) −9.06506e8 −0.566784
\(112\) 5.87307e8 0.352682
\(113\) −1.39288e9 −0.803636 −0.401818 0.915720i \(-0.631622\pi\)
−0.401818 + 0.915720i \(0.631622\pi\)
\(114\) 1.91587e9 1.06241
\(115\) −1.00759e9 −0.537209
\(116\) 1.36462e9 0.699763
\(117\) 1.11495e9 0.550071
\(118\) −4.34382e8 −0.206254
\(119\) −5.46047e9 −2.49614
\(120\) −1.42612e9 −0.627828
\(121\) 1.43999e9 0.610698
\(122\) 7.83602e8 0.320240
\(123\) 7.43377e8 0.292844
\(124\) −5.68856e9 −2.16075
\(125\) −6.90795e8 −0.253078
\(126\) 2.29024e9 0.809493
\(127\) 4.26841e8 0.145596 0.0727980 0.997347i \(-0.476807\pi\)
0.0727980 + 0.997347i \(0.476807\pi\)
\(128\) 4.26889e9 1.40563
\(129\) 1.03394e9 0.328731
\(130\) −1.14602e10 −3.51923
\(131\) −2.55369e9 −0.757614 −0.378807 0.925476i \(-0.623666\pi\)
−0.378807 + 0.925476i \(0.623666\pi\)
\(132\) −3.85905e9 −1.10636
\(133\) 6.42484e9 1.78045
\(134\) −1.08999e10 −2.92047
\(135\) 9.99766e8 0.259058
\(136\) −5.24822e9 −1.31549
\(137\) −1.89733e9 −0.460150 −0.230075 0.973173i \(-0.573897\pi\)
−0.230075 + 0.973173i \(0.573897\pi\)
\(138\) 1.55521e9 0.365035
\(139\) 1.42180e9 0.323052 0.161526 0.986868i \(-0.448358\pi\)
0.161526 + 0.986868i \(0.448358\pi\)
\(140\) −1.41615e10 −3.11555
\(141\) 5.77918e8 0.123135
\(142\) 8.84771e9 1.82614
\(143\) −1.04727e10 −2.09434
\(144\) −3.95721e8 −0.0766933
\(145\) 3.32073e9 0.623846
\(146\) 3.54596e9 0.645871
\(147\) 4.41165e9 0.779243
\(148\) −8.65181e9 −1.48228
\(149\) −9.87928e9 −1.64205 −0.821027 0.570890i \(-0.806598\pi\)
−0.821027 + 0.570890i \(0.806598\pi\)
\(150\) −4.60502e9 −0.742712
\(151\) −6.67305e9 −1.04455 −0.522274 0.852778i \(-0.674916\pi\)
−0.522274 + 0.852778i \(0.674916\pi\)
\(152\) 6.17510e9 0.938314
\(153\) 3.67920e9 0.542803
\(154\) −2.15122e10 −3.08206
\(155\) −1.38428e10 −1.92634
\(156\) 1.06412e10 1.43857
\(157\) 6.15305e9 0.808243 0.404122 0.914705i \(-0.367577\pi\)
0.404122 + 0.914705i \(0.367577\pi\)
\(158\) 6.80934e8 0.0869256
\(159\) −8.71588e9 −1.08149
\(160\) 1.30820e10 1.57810
\(161\) 5.21539e9 0.611745
\(162\) −1.54314e9 −0.176030
\(163\) 1.59815e10 1.77326 0.886630 0.462480i \(-0.153040\pi\)
0.886630 + 0.462480i \(0.153040\pi\)
\(164\) 7.09488e9 0.765857
\(165\) −9.39080e9 −0.986335
\(166\) 1.37233e10 1.40272
\(167\) −3.33847e9 −0.332141 −0.166071 0.986114i \(-0.553108\pi\)
−0.166071 + 0.986114i \(0.553108\pi\)
\(168\) 7.38176e9 0.714937
\(169\) 1.82737e10 1.72320
\(170\) −3.78174e10 −3.47273
\(171\) −4.32898e9 −0.387171
\(172\) 9.86803e9 0.859711
\(173\) −6.91460e9 −0.586894 −0.293447 0.955975i \(-0.594802\pi\)
−0.293447 + 0.955975i \(0.594802\pi\)
\(174\) −5.12554e9 −0.423904
\(175\) −1.54429e10 −1.24468
\(176\) 3.71700e9 0.292002
\(177\) 9.81506e8 0.0751646
\(178\) −3.06408e10 −2.28776
\(179\) −3.30267e9 −0.240451 −0.120226 0.992747i \(-0.538362\pi\)
−0.120226 + 0.992747i \(0.538362\pi\)
\(180\) 9.54189e9 0.677498
\(181\) −2.48966e10 −1.72419 −0.862096 0.506744i \(-0.830849\pi\)
−0.862096 + 0.506744i \(0.830849\pi\)
\(182\) 5.93192e10 4.00751
\(183\) −1.77058e9 −0.116704
\(184\) 5.01267e9 0.322395
\(185\) −2.10537e10 −1.32147
\(186\) 2.13664e10 1.30895
\(187\) −3.45587e10 −2.06667
\(188\) 5.51572e9 0.322027
\(189\) −5.17489e9 −0.295001
\(190\) 4.44962e10 2.47703
\(191\) −1.74235e9 −0.0947295 −0.0473648 0.998878i \(-0.515082\pi\)
−0.0473648 + 0.998878i \(0.515082\pi\)
\(192\) −1.76907e10 −0.939482
\(193\) −1.25460e10 −0.650875 −0.325438 0.945563i \(-0.605512\pi\)
−0.325438 + 0.945563i \(0.605512\pi\)
\(194\) −3.76431e10 −1.90800
\(195\) 2.58949e10 1.28250
\(196\) 4.21053e10 2.03791
\(197\) 5.37321e9 0.254177 0.127088 0.991891i \(-0.459437\pi\)
0.127088 + 0.991891i \(0.459437\pi\)
\(198\) 1.44947e10 0.670216
\(199\) −2.42720e10 −1.09715 −0.548576 0.836101i \(-0.684830\pi\)
−0.548576 + 0.836101i \(0.684830\pi\)
\(200\) −1.48426e10 −0.655957
\(201\) 2.46289e10 1.06430
\(202\) 5.21416e10 2.20345
\(203\) −1.71884e10 −0.710403
\(204\) 3.51148e10 1.41956
\(205\) 1.72650e10 0.682770
\(206\) −5.89712e10 −2.28159
\(207\) −3.51407e9 −0.133028
\(208\) −1.02495e10 −0.379681
\(209\) 4.06621e10 1.47412
\(210\) 5.31911e10 1.88735
\(211\) 8.97002e9 0.311546 0.155773 0.987793i \(-0.450213\pi\)
0.155773 + 0.987793i \(0.450213\pi\)
\(212\) −8.31855e10 −2.82837
\(213\) −1.99918e10 −0.665493
\(214\) −1.70311e10 −0.555113
\(215\) 2.40133e10 0.766442
\(216\) −4.97374e9 −0.155468
\(217\) 7.16519e10 2.19361
\(218\) 8.52669e10 2.55697
\(219\) −8.01226e9 −0.235373
\(220\) −8.96270e10 −2.57951
\(221\) 9.52948e10 2.68722
\(222\) 3.24964e10 0.897939
\(223\) −5.14153e10 −1.39226 −0.696130 0.717916i \(-0.745096\pi\)
−0.696130 + 0.717916i \(0.745096\pi\)
\(224\) −6.77137e10 −1.79705
\(225\) 1.04052e10 0.270664
\(226\) 4.99317e10 1.27318
\(227\) −2.13816e10 −0.534470 −0.267235 0.963631i \(-0.586110\pi\)
−0.267235 + 0.963631i \(0.586110\pi\)
\(228\) −4.13164e10 −1.01255
\(229\) 4.05363e10 0.974057 0.487028 0.873386i \(-0.338081\pi\)
0.487028 + 0.873386i \(0.338081\pi\)
\(230\) 3.61200e10 0.851085
\(231\) 4.86077e10 1.12319
\(232\) −1.65203e10 −0.374389
\(233\) 3.97623e10 0.883833 0.441916 0.897056i \(-0.354299\pi\)
0.441916 + 0.897056i \(0.354299\pi\)
\(234\) −3.99686e10 −0.871461
\(235\) 1.34222e10 0.287091
\(236\) 9.36762e9 0.196574
\(237\) −1.53860e9 −0.0316780
\(238\) 1.95747e11 3.95456
\(239\) −3.15480e10 −0.625434 −0.312717 0.949846i \(-0.601239\pi\)
−0.312717 + 0.949846i \(0.601239\pi\)
\(240\) −9.19067e9 −0.178812
\(241\) −8.14750e10 −1.55578 −0.777889 0.628402i \(-0.783709\pi\)
−0.777889 + 0.628402i \(0.783709\pi\)
\(242\) −5.16208e10 −0.967511
\(243\) 3.48678e9 0.0641500
\(244\) −1.68987e10 −0.305210
\(245\) 1.02461e11 1.81682
\(246\) −2.66485e10 −0.463943
\(247\) −1.12125e11 −1.91675
\(248\) 6.88668e10 1.15605
\(249\) −3.10083e10 −0.511189
\(250\) 2.47636e10 0.400943
\(251\) −3.36364e9 −0.0534907 −0.0267453 0.999642i \(-0.508514\pi\)
−0.0267453 + 0.999642i \(0.508514\pi\)
\(252\) −4.93898e10 −0.771499
\(253\) 3.30077e10 0.506492
\(254\) −1.53014e10 −0.230663
\(255\) 8.54500e10 1.26556
\(256\) −4.12085e10 −0.599663
\(257\) −6.90974e10 −0.988013 −0.494007 0.869458i \(-0.664468\pi\)
−0.494007 + 0.869458i \(0.664468\pi\)
\(258\) −3.70645e10 −0.520798
\(259\) 1.08976e11 1.50482
\(260\) 2.47144e11 3.35405
\(261\) 1.15814e10 0.154482
\(262\) 9.15447e10 1.20027
\(263\) −8.29232e10 −1.06875 −0.534374 0.845248i \(-0.679453\pi\)
−0.534374 + 0.845248i \(0.679453\pi\)
\(264\) 4.67184e10 0.591929
\(265\) −2.02427e11 −2.52152
\(266\) −2.30317e11 −2.82071
\(267\) 6.92342e10 0.833720
\(268\) 2.35062e11 2.78340
\(269\) 4.87849e10 0.568068 0.284034 0.958814i \(-0.408327\pi\)
0.284034 + 0.958814i \(0.408327\pi\)
\(270\) −3.58395e10 −0.410417
\(271\) −1.15620e11 −1.30218 −0.651088 0.759002i \(-0.725687\pi\)
−0.651088 + 0.759002i \(0.725687\pi\)
\(272\) −3.38223e10 −0.374664
\(273\) −1.34034e11 −1.46044
\(274\) 6.80152e10 0.729002
\(275\) −9.77363e10 −1.03053
\(276\) −3.35388e10 −0.347901
\(277\) −7.34803e10 −0.749914 −0.374957 0.927042i \(-0.622343\pi\)
−0.374957 + 0.927042i \(0.622343\pi\)
\(278\) −5.09686e10 −0.511801
\(279\) −4.82782e10 −0.477016
\(280\) 1.71442e11 1.66689
\(281\) −4.09307e10 −0.391626 −0.195813 0.980641i \(-0.562735\pi\)
−0.195813 + 0.980641i \(0.562735\pi\)
\(282\) −2.07172e10 −0.195078
\(283\) −1.88250e11 −1.74460 −0.872300 0.488970i \(-0.837373\pi\)
−0.872300 + 0.488970i \(0.837373\pi\)
\(284\) −1.90804e11 −1.74043
\(285\) −1.00541e11 −0.902697
\(286\) 3.75425e11 3.31800
\(287\) −8.93655e10 −0.777502
\(288\) 4.56247e10 0.390782
\(289\) 1.95874e11 1.65172
\(290\) −1.19041e11 −0.988341
\(291\) 8.50563e10 0.695325
\(292\) −7.64700e10 −0.615557
\(293\) 9.87036e10 0.782400 0.391200 0.920306i \(-0.372060\pi\)
0.391200 + 0.920306i \(0.372060\pi\)
\(294\) −1.58148e11 −1.23453
\(295\) 2.27956e10 0.175248
\(296\) 1.04740e11 0.793052
\(297\) −3.27514e10 −0.244245
\(298\) 3.54152e11 2.60146
\(299\) −9.10177e10 −0.658575
\(300\) 9.93089e10 0.707852
\(301\) −1.24295e11 −0.872783
\(302\) 2.39215e11 1.65484
\(303\) −1.17816e11 −0.802996
\(304\) 3.97956e10 0.267241
\(305\) −4.11220e10 −0.272098
\(306\) −1.31892e11 −0.859947
\(307\) −1.14378e11 −0.734885 −0.367442 0.930046i \(-0.619766\pi\)
−0.367442 + 0.930046i \(0.619766\pi\)
\(308\) 4.63919e11 2.93740
\(309\) 1.33248e11 0.831472
\(310\) 4.96236e11 3.05184
\(311\) −2.11753e10 −0.128354 −0.0641768 0.997939i \(-0.520442\pi\)
−0.0641768 + 0.997939i \(0.520442\pi\)
\(312\) −1.28825e11 −0.769667
\(313\) −9.81324e10 −0.577914 −0.288957 0.957342i \(-0.593308\pi\)
−0.288957 + 0.957342i \(0.593308\pi\)
\(314\) −2.20574e11 −1.28048
\(315\) −1.20188e11 −0.687800
\(316\) −1.46846e10 −0.0828457
\(317\) −4.64728e10 −0.258483 −0.129242 0.991613i \(-0.541254\pi\)
−0.129242 + 0.991613i \(0.541254\pi\)
\(318\) 3.12446e11 1.71338
\(319\) −1.08784e11 −0.588175
\(320\) −4.10868e11 −2.19042
\(321\) 3.84826e10 0.202298
\(322\) −1.86961e11 −0.969169
\(323\) −3.69998e11 −1.89142
\(324\) 3.32783e10 0.167768
\(325\) 2.69505e11 1.33996
\(326\) −5.72902e11 −2.80932
\(327\) −1.92664e11 −0.931829
\(328\) −8.58919e10 −0.409751
\(329\) −6.94748e10 −0.326923
\(330\) 3.36641e11 1.56262
\(331\) 5.07093e10 0.232200 0.116100 0.993238i \(-0.462961\pi\)
0.116100 + 0.993238i \(0.462961\pi\)
\(332\) −2.95948e11 −1.33688
\(333\) −7.34270e10 −0.327233
\(334\) 1.19677e11 0.526201
\(335\) 5.72010e11 2.48143
\(336\) 4.75718e10 0.203621
\(337\) −3.44025e11 −1.45297 −0.726484 0.687184i \(-0.758847\pi\)
−0.726484 + 0.687184i \(0.758847\pi\)
\(338\) −6.55075e11 −2.73002
\(339\) −1.12823e11 −0.463980
\(340\) 8.15546e11 3.30973
\(341\) 4.53477e11 1.81619
\(342\) 1.55185e11 0.613384
\(343\) −1.37407e11 −0.536026
\(344\) −1.19464e11 −0.459965
\(345\) −8.16148e10 −0.310158
\(346\) 2.47874e11 0.929799
\(347\) −1.06614e11 −0.394758 −0.197379 0.980327i \(-0.563243\pi\)
−0.197379 + 0.980327i \(0.563243\pi\)
\(348\) 1.10534e11 0.404008
\(349\) −3.09166e11 −1.11552 −0.557761 0.830002i \(-0.688339\pi\)
−0.557761 + 0.830002i \(0.688339\pi\)
\(350\) 5.53595e11 1.97191
\(351\) 9.03109e10 0.317584
\(352\) −4.28553e11 −1.48786
\(353\) −3.60113e11 −1.23439 −0.617195 0.786810i \(-0.711731\pi\)
−0.617195 + 0.786810i \(0.711731\pi\)
\(354\) −3.51850e10 −0.119081
\(355\) −4.64312e11 −1.55161
\(356\) 6.60780e11 2.18038
\(357\) −4.42298e11 −1.44115
\(358\) 1.18394e11 0.380940
\(359\) 7.09609e10 0.225473 0.112736 0.993625i \(-0.464038\pi\)
0.112736 + 0.993625i \(0.464038\pi\)
\(360\) −1.15516e11 −0.362477
\(361\) 1.12655e11 0.349116
\(362\) 8.92490e11 2.73159
\(363\) 1.16640e11 0.352587
\(364\) −1.27924e12 −3.81941
\(365\) −1.86086e11 −0.548776
\(366\) 6.34717e10 0.184891
\(367\) 3.13142e10 0.0901041 0.0450520 0.998985i \(-0.485655\pi\)
0.0450520 + 0.998985i \(0.485655\pi\)
\(368\) 3.23042e10 0.0918215
\(369\) 6.02135e10 0.169073
\(370\) 7.54733e11 2.09356
\(371\) 1.04779e12 2.87138
\(372\) −4.60774e11 −1.24751
\(373\) 6.03351e11 1.61391 0.806957 0.590610i \(-0.201113\pi\)
0.806957 + 0.590610i \(0.201113\pi\)
\(374\) 1.23886e12 3.27416
\(375\) −5.59544e10 −0.146114
\(376\) −6.67743e10 −0.172292
\(377\) 2.99968e11 0.764785
\(378\) 1.85509e11 0.467361
\(379\) −1.57900e11 −0.393101 −0.196551 0.980494i \(-0.562974\pi\)
−0.196551 + 0.980494i \(0.562974\pi\)
\(380\) −9.59578e11 −2.36077
\(381\) 3.45741e10 0.0840599
\(382\) 6.24597e10 0.150077
\(383\) −7.59563e11 −1.80372 −0.901861 0.432027i \(-0.857799\pi\)
−0.901861 + 0.432027i \(0.857799\pi\)
\(384\) 3.45780e11 0.811540
\(385\) 1.12892e12 2.61873
\(386\) 4.49749e11 1.03116
\(387\) 8.37489e10 0.189793
\(388\) 8.11788e11 1.81844
\(389\) −1.69262e11 −0.374788 −0.187394 0.982285i \(-0.560004\pi\)
−0.187394 + 0.982285i \(0.560004\pi\)
\(390\) −9.28277e11 −2.03183
\(391\) −3.00348e11 −0.649874
\(392\) −5.09735e11 −1.09033
\(393\) −2.06849e11 −0.437409
\(394\) −1.92619e11 −0.402685
\(395\) −3.57342e10 −0.0738579
\(396\) −3.12583e11 −0.638759
\(397\) 2.11292e10 0.0426899 0.0213450 0.999772i \(-0.493205\pi\)
0.0213450 + 0.999772i \(0.493205\pi\)
\(398\) 8.70101e11 1.73818
\(399\) 5.20412e11 1.02794
\(400\) −9.56535e10 −0.186823
\(401\) −1.24151e11 −0.239774 −0.119887 0.992788i \(-0.538253\pi\)
−0.119887 + 0.992788i \(0.538253\pi\)
\(402\) −8.82896e11 −1.68613
\(403\) −1.25045e12 −2.36153
\(404\) −1.12445e12 −2.10003
\(405\) 8.09810e10 0.149567
\(406\) 6.16170e11 1.12547
\(407\) 6.89700e11 1.24591
\(408\) −4.25106e11 −0.759498
\(409\) 5.91126e11 1.04454 0.522270 0.852780i \(-0.325085\pi\)
0.522270 + 0.852780i \(0.325085\pi\)
\(410\) −6.18915e11 −1.08169
\(411\) −1.53683e11 −0.265668
\(412\) 1.27174e12 2.17450
\(413\) −1.17992e11 −0.199563
\(414\) 1.25972e11 0.210753
\(415\) −7.20173e11 −1.19185
\(416\) 1.18172e12 1.93462
\(417\) 1.15166e11 0.186514
\(418\) −1.45765e12 −2.33540
\(419\) −5.16849e11 −0.819221 −0.409610 0.912261i \(-0.634335\pi\)
−0.409610 + 0.912261i \(0.634335\pi\)
\(420\) −1.14709e12 −1.79876
\(421\) 4.07079e11 0.631552 0.315776 0.948834i \(-0.397735\pi\)
0.315776 + 0.948834i \(0.397735\pi\)
\(422\) −3.21557e11 −0.493573
\(423\) 4.68114e10 0.0710918
\(424\) 1.00706e12 1.51324
\(425\) 8.89336e11 1.32226
\(426\) 7.16665e11 1.05432
\(427\) 2.12852e11 0.309850
\(428\) 3.67283e11 0.529059
\(429\) −8.48290e11 −1.20917
\(430\) −8.60828e11 −1.21425
\(431\) −8.85168e11 −1.23560 −0.617800 0.786335i \(-0.711976\pi\)
−0.617800 + 0.786335i \(0.711976\pi\)
\(432\) −3.20534e10 −0.0442789
\(433\) 6.74952e11 0.922735 0.461368 0.887209i \(-0.347359\pi\)
0.461368 + 0.887209i \(0.347359\pi\)
\(434\) −2.56857e12 −3.47527
\(435\) 2.68979e11 0.360178
\(436\) −1.83881e12 −2.43696
\(437\) 3.53392e11 0.463543
\(438\) 2.87223e11 0.372894
\(439\) 1.28318e12 1.64892 0.824458 0.565922i \(-0.191480\pi\)
0.824458 + 0.565922i \(0.191480\pi\)
\(440\) 1.08504e12 1.38009
\(441\) 3.57343e11 0.449896
\(442\) −3.41612e12 −4.25729
\(443\) 1.44560e12 1.78332 0.891662 0.452701i \(-0.149539\pi\)
0.891662 + 0.452701i \(0.149539\pi\)
\(444\) −7.00797e11 −0.855793
\(445\) 1.60797e12 1.94383
\(446\) 1.84313e12 2.20572
\(447\) −8.00222e11 −0.948040
\(448\) 2.12670e12 2.49433
\(449\) −7.76770e11 −0.901953 −0.450977 0.892536i \(-0.648924\pi\)
−0.450977 + 0.892536i \(0.648924\pi\)
\(450\) −3.73006e11 −0.428805
\(451\) −5.65585e11 −0.643730
\(452\) −1.07680e12 −1.21342
\(453\) −5.40517e11 −0.603070
\(454\) 7.66486e11 0.846745
\(455\) −3.11297e12 −3.40505
\(456\) 5.00183e11 0.541736
\(457\) 7.01907e11 0.752760 0.376380 0.926465i \(-0.377169\pi\)
0.376380 + 0.926465i \(0.377169\pi\)
\(458\) −1.45314e12 −1.54317
\(459\) 2.98016e11 0.313388
\(460\) −7.78942e11 −0.811138
\(461\) −4.87181e11 −0.502385 −0.251192 0.967937i \(-0.580823\pi\)
−0.251192 + 0.967937i \(0.580823\pi\)
\(462\) −1.74249e12 −1.77943
\(463\) 1.22256e12 1.23639 0.618194 0.786025i \(-0.287864\pi\)
0.618194 + 0.786025i \(0.287864\pi\)
\(464\) −1.06466e11 −0.106630
\(465\) −1.12127e12 −1.11217
\(466\) −1.42540e12 −1.40023
\(467\) 1.39487e12 1.35709 0.678543 0.734561i \(-0.262612\pi\)
0.678543 + 0.734561i \(0.262612\pi\)
\(468\) 8.61939e11 0.830558
\(469\) −2.96078e12 −2.82572
\(470\) −4.81159e11 −0.454829
\(471\) 4.98397e11 0.466639
\(472\) −1.13406e11 −0.105171
\(473\) −7.86653e11 −0.722617
\(474\) 5.51556e10 0.0501865
\(475\) −1.04640e12 −0.943141
\(476\) −4.22135e12 −3.76895
\(477\) −7.05986e11 −0.624401
\(478\) 1.13093e12 0.990856
\(479\) 7.97510e11 0.692192 0.346096 0.938199i \(-0.387507\pi\)
0.346096 + 0.938199i \(0.387507\pi\)
\(480\) 1.05964e12 0.911115
\(481\) −1.90183e12 −1.62001
\(482\) 2.92071e12 2.46477
\(483\) 4.22447e11 0.353191
\(484\) 1.11322e12 0.922100
\(485\) 1.97544e12 1.62116
\(486\) −1.24994e11 −0.101631
\(487\) −1.76341e12 −1.42060 −0.710302 0.703897i \(-0.751442\pi\)
−0.710302 + 0.703897i \(0.751442\pi\)
\(488\) 2.04578e11 0.163294
\(489\) 1.29450e12 1.02379
\(490\) −3.67302e12 −2.87833
\(491\) 9.35179e11 0.726153 0.363077 0.931759i \(-0.381726\pi\)
0.363077 + 0.931759i \(0.381726\pi\)
\(492\) 5.74685e11 0.442168
\(493\) 9.89861e11 0.754680
\(494\) 4.01944e12 3.03664
\(495\) −7.60654e11 −0.569461
\(496\) 4.43813e11 0.329255
\(497\) 2.40333e12 1.76689
\(498\) 1.11158e12 0.809861
\(499\) 2.32592e12 1.67935 0.839676 0.543088i \(-0.182745\pi\)
0.839676 + 0.543088i \(0.182745\pi\)
\(500\) −5.34036e11 −0.382125
\(501\) −2.70416e11 −0.191762
\(502\) 1.20580e11 0.0847436
\(503\) −2.02638e12 −1.41145 −0.705723 0.708488i \(-0.749378\pi\)
−0.705723 + 0.708488i \(0.749378\pi\)
\(504\) 5.97922e11 0.412769
\(505\) −2.73630e12 −1.87220
\(506\) −1.18326e12 −0.802420
\(507\) 1.48017e12 0.994893
\(508\) 3.29980e11 0.219837
\(509\) 2.28341e12 1.50784 0.753919 0.656968i \(-0.228161\pi\)
0.753919 + 0.656968i \(0.228161\pi\)
\(510\) −3.06321e12 −2.00498
\(511\) 9.63199e11 0.624917
\(512\) −7.08433e11 −0.455600
\(513\) −3.50648e11 −0.223534
\(514\) 2.47700e12 1.56528
\(515\) 3.09470e12 1.93859
\(516\) 7.99310e11 0.496354
\(517\) −4.39699e11 −0.270675
\(518\) −3.90658e12 −2.38403
\(519\) −5.60083e11 −0.338844
\(520\) −2.99197e12 −1.79449
\(521\) 1.64372e12 0.977367 0.488683 0.872461i \(-0.337477\pi\)
0.488683 + 0.872461i \(0.337477\pi\)
\(522\) −4.15169e11 −0.244741
\(523\) −3.11181e11 −0.181868 −0.0909339 0.995857i \(-0.528985\pi\)
−0.0909339 + 0.995857i \(0.528985\pi\)
\(524\) −1.97420e12 −1.14393
\(525\) −1.25087e12 −0.718615
\(526\) 2.97263e12 1.69319
\(527\) −4.12634e12 −2.33033
\(528\) 3.01077e11 0.168587
\(529\) −1.51429e12 −0.840731
\(530\) 7.25661e12 3.99477
\(531\) 7.95020e10 0.0433963
\(532\) 4.96688e12 2.68832
\(533\) 1.55958e12 0.837021
\(534\) −2.48190e12 −1.32084
\(535\) 8.93764e11 0.471662
\(536\) −2.84570e12 −1.48918
\(537\) −2.67517e11 −0.138825
\(538\) −1.74884e12 −0.899973
\(539\) −3.35652e12 −1.71293
\(540\) 7.72893e11 0.391154
\(541\) −8.12242e11 −0.407659 −0.203830 0.979006i \(-0.565339\pi\)
−0.203830 + 0.979006i \(0.565339\pi\)
\(542\) 4.14473e12 2.06300
\(543\) −2.01662e12 −0.995463
\(544\) 3.89955e12 1.90906
\(545\) −4.47465e12 −2.17258
\(546\) 4.80486e12 2.31374
\(547\) −5.36570e11 −0.256261 −0.128131 0.991757i \(-0.540898\pi\)
−0.128131 + 0.991757i \(0.540898\pi\)
\(548\) −1.46677e12 −0.694786
\(549\) −1.43417e11 −0.0673792
\(550\) 3.50365e12 1.63263
\(551\) −1.16468e12 −0.538299
\(552\) 4.06026e11 0.186135
\(553\) 1.84964e11 0.0841054
\(554\) 2.63412e12 1.18807
\(555\) −1.70535e12 −0.762949
\(556\) 1.09916e12 0.487779
\(557\) 6.35994e11 0.279965 0.139983 0.990154i \(-0.455295\pi\)
0.139983 + 0.990154i \(0.455295\pi\)
\(558\) 1.73068e12 0.755721
\(559\) 2.16917e12 0.939596
\(560\) 1.10486e12 0.474747
\(561\) −2.79926e12 −1.19319
\(562\) 1.46728e12 0.620441
\(563\) 5.11920e11 0.214741 0.107370 0.994219i \(-0.465757\pi\)
0.107370 + 0.994219i \(0.465757\pi\)
\(564\) 4.46774e11 0.185922
\(565\) −2.62033e12 −1.08178
\(566\) 6.74837e12 2.76392
\(567\) −4.19166e11 −0.170319
\(568\) 2.30991e12 0.931167
\(569\) 4.59360e11 0.183716 0.0918581 0.995772i \(-0.470719\pi\)
0.0918581 + 0.995772i \(0.470719\pi\)
\(570\) 3.60419e12 1.43012
\(571\) 3.99628e12 1.57323 0.786616 0.617442i \(-0.211831\pi\)
0.786616 + 0.617442i \(0.211831\pi\)
\(572\) −8.09619e12 −3.16227
\(573\) −1.41130e11 −0.0546921
\(574\) 3.20357e12 1.23177
\(575\) −8.49420e11 −0.324054
\(576\) −1.43294e12 −0.542410
\(577\) −1.20235e12 −0.451584 −0.225792 0.974176i \(-0.572497\pi\)
−0.225792 + 0.974176i \(0.572497\pi\)
\(578\) −7.02167e12 −2.61677
\(579\) −1.01623e12 −0.375783
\(580\) 2.56717e12 0.941952
\(581\) 3.72769e12 1.35721
\(582\) −3.04909e12 −1.10158
\(583\) 6.63133e12 2.37734
\(584\) 9.25760e11 0.329337
\(585\) 2.09748e12 0.740452
\(586\) −3.53832e12 −1.23953
\(587\) −4.83604e12 −1.68120 −0.840598 0.541660i \(-0.817796\pi\)
−0.840598 + 0.541660i \(0.817796\pi\)
\(588\) 3.41053e12 1.17659
\(589\) 4.85509e12 1.66218
\(590\) −8.17176e11 −0.277640
\(591\) 4.35230e11 0.146749
\(592\) 6.75001e11 0.225869
\(593\) −3.57818e12 −1.18827 −0.594136 0.804365i \(-0.702506\pi\)
−0.594136 + 0.804365i \(0.702506\pi\)
\(594\) 1.17407e12 0.386950
\(595\) −1.02724e13 −3.36006
\(596\) −7.63742e12 −2.47935
\(597\) −1.96603e12 −0.633441
\(598\) 3.26280e12 1.04336
\(599\) 5.28387e12 1.67699 0.838496 0.544907i \(-0.183435\pi\)
0.838496 + 0.544907i \(0.183435\pi\)
\(600\) −1.20225e12 −0.378717
\(601\) 2.67502e12 0.836359 0.418179 0.908365i \(-0.362668\pi\)
0.418179 + 0.908365i \(0.362668\pi\)
\(602\) 4.45574e12 1.38272
\(603\) 1.99494e12 0.614472
\(604\) −5.15876e12 −1.57717
\(605\) 2.70897e12 0.822062
\(606\) 4.22347e12 1.27216
\(607\) 3.70644e12 1.10818 0.554088 0.832458i \(-0.313067\pi\)
0.554088 + 0.832458i \(0.313067\pi\)
\(608\) −4.58824e12 −1.36170
\(609\) −1.39226e12 −0.410151
\(610\) 1.47414e12 0.431076
\(611\) 1.21246e12 0.351950
\(612\) 2.84430e12 0.819584
\(613\) 6.45333e12 1.84592 0.922958 0.384900i \(-0.125764\pi\)
0.922958 + 0.384900i \(0.125764\pi\)
\(614\) 4.10021e12 1.16426
\(615\) 1.39847e12 0.394198
\(616\) −5.61628e12 −1.57158
\(617\) −6.71687e12 −1.86588 −0.932941 0.360031i \(-0.882766\pi\)
−0.932941 + 0.360031i \(0.882766\pi\)
\(618\) −4.77667e12 −1.31728
\(619\) −4.48065e12 −1.22669 −0.613343 0.789817i \(-0.710176\pi\)
−0.613343 + 0.789817i \(0.710176\pi\)
\(620\) −1.07015e13 −2.90859
\(621\) −2.84640e11 −0.0768039
\(622\) 7.59092e11 0.203347
\(623\) −8.32304e12 −2.21353
\(624\) −8.30212e11 −0.219209
\(625\) −4.39705e12 −1.15266
\(626\) 3.51784e12 0.915571
\(627\) 3.29363e12 0.851081
\(628\) 4.75676e12 1.22037
\(629\) −6.27581e12 −1.59861
\(630\) 4.30848e12 1.08966
\(631\) 3.31200e12 0.831683 0.415842 0.909437i \(-0.363487\pi\)
0.415842 + 0.909437i \(0.363487\pi\)
\(632\) 1.77774e11 0.0443243
\(633\) 7.26571e11 0.179871
\(634\) 1.66595e12 0.409507
\(635\) 8.02988e11 0.195987
\(636\) −6.73803e12 −1.63296
\(637\) 9.25552e12 2.22727
\(638\) 3.89968e12 0.931828
\(639\) −1.61934e12 −0.384223
\(640\) 8.03079e12 1.89212
\(641\) 3.34924e12 0.783584 0.391792 0.920054i \(-0.371855\pi\)
0.391792 + 0.920054i \(0.371855\pi\)
\(642\) −1.37952e12 −0.320495
\(643\) −3.43017e12 −0.791345 −0.395673 0.918392i \(-0.629489\pi\)
−0.395673 + 0.918392i \(0.629489\pi\)
\(644\) 4.03189e12 0.923681
\(645\) 1.94508e12 0.442505
\(646\) 1.32637e13 2.99652
\(647\) 5.54972e12 1.24509 0.622546 0.782583i \(-0.286098\pi\)
0.622546 + 0.782583i \(0.286098\pi\)
\(648\) −4.02873e11 −0.0897596
\(649\) −7.46762e11 −0.165227
\(650\) −9.66120e12 −2.12286
\(651\) 5.80380e12 1.26648
\(652\) 1.23549e13 2.67746
\(653\) −2.85530e11 −0.0614529 −0.0307265 0.999528i \(-0.509782\pi\)
−0.0307265 + 0.999528i \(0.509782\pi\)
\(654\) 6.90662e12 1.47627
\(655\) −4.80410e12 −1.01983
\(656\) −5.53532e11 −0.116701
\(657\) −6.48993e11 −0.135893
\(658\) 2.49053e12 0.517935
\(659\) 7.57432e11 0.156444 0.0782221 0.996936i \(-0.475076\pi\)
0.0782221 + 0.996936i \(0.475076\pi\)
\(660\) −7.25978e12 −1.48928
\(661\) 5.28983e12 1.07779 0.538896 0.842372i \(-0.318841\pi\)
0.538896 + 0.842372i \(0.318841\pi\)
\(662\) −1.81782e12 −0.367867
\(663\) 7.71888e12 1.55147
\(664\) 3.58279e12 0.715263
\(665\) 1.20866e13 2.39667
\(666\) 2.63221e12 0.518425
\(667\) −9.45434e11 −0.184954
\(668\) −2.58088e12 −0.501504
\(669\) −4.16464e12 −0.803822
\(670\) −2.05054e13 −3.93125
\(671\) 1.34712e12 0.256539
\(672\) −5.48481e12 −1.03753
\(673\) 3.50601e12 0.658788 0.329394 0.944193i \(-0.393156\pi\)
0.329394 + 0.944193i \(0.393156\pi\)
\(674\) 1.23326e13 2.30189
\(675\) 8.42824e11 0.156268
\(676\) 1.41269e13 2.60189
\(677\) −3.61379e12 −0.661171 −0.330585 0.943776i \(-0.607246\pi\)
−0.330585 + 0.943776i \(0.607246\pi\)
\(678\) 4.04447e12 0.735069
\(679\) −1.02251e13 −1.84609
\(680\) −9.87314e12 −1.77078
\(681\) −1.73191e12 −0.308577
\(682\) −1.62562e13 −2.87733
\(683\) −1.51664e12 −0.266680 −0.133340 0.991070i \(-0.542570\pi\)
−0.133340 + 0.991070i \(0.542570\pi\)
\(684\) −3.34663e12 −0.584594
\(685\) −3.56932e12 −0.619409
\(686\) 4.92576e12 0.849209
\(687\) 3.28344e12 0.562372
\(688\) −7.69889e11 −0.131003
\(689\) −1.82857e13 −3.09119
\(690\) 2.92572e12 0.491374
\(691\) 7.69784e12 1.28445 0.642226 0.766515i \(-0.278011\pi\)
0.642226 + 0.766515i \(0.278011\pi\)
\(692\) −5.34550e12 −0.886158
\(693\) 3.93723e12 0.648472
\(694\) 3.82189e12 0.625404
\(695\) 2.67474e12 0.434861
\(696\) −1.33815e12 −0.216153
\(697\) 5.14645e12 0.825962
\(698\) 1.10830e13 1.76729
\(699\) 3.22075e12 0.510281
\(700\) −1.19385e13 −1.87935
\(701\) −6.31846e11 −0.0988280 −0.0494140 0.998778i \(-0.515735\pi\)
−0.0494140 + 0.998778i \(0.515735\pi\)
\(702\) −3.23746e12 −0.503138
\(703\) 7.38417e12 1.14026
\(704\) 1.34596e13 2.06517
\(705\) 1.08720e12 0.165752
\(706\) 1.29093e13 1.95561
\(707\) 1.41634e13 2.13196
\(708\) 7.58777e11 0.113492
\(709\) −7.86914e12 −1.16955 −0.584775 0.811195i \(-0.698817\pi\)
−0.584775 + 0.811195i \(0.698817\pi\)
\(710\) 1.66446e13 2.45817
\(711\) −1.24627e11 −0.0182893
\(712\) −7.99952e12 −1.16655
\(713\) 3.94114e12 0.571109
\(714\) 1.58555e13 2.28316
\(715\) −1.97016e13 −2.81920
\(716\) −2.55321e12 −0.363060
\(717\) −2.55539e12 −0.361094
\(718\) −2.54380e12 −0.357210
\(719\) 1.18994e13 1.66052 0.830259 0.557378i \(-0.188193\pi\)
0.830259 + 0.557378i \(0.188193\pi\)
\(720\) −7.44444e11 −0.103237
\(721\) −1.60185e13 −2.20756
\(722\) −4.03846e12 −0.553093
\(723\) −6.59947e12 −0.898229
\(724\) −1.92469e13 −2.60338
\(725\) 2.79945e12 0.376315
\(726\) −4.18129e12 −0.558593
\(727\) −9.95970e12 −1.32233 −0.661167 0.750238i \(-0.729939\pi\)
−0.661167 + 0.750238i \(0.729939\pi\)
\(728\) 1.54867e13 2.04347
\(729\) 2.82430e11 0.0370370
\(730\) 6.67079e12 0.869409
\(731\) 7.15802e12 0.927181
\(732\) −1.36879e12 −0.176213
\(733\) −2.10318e12 −0.269097 −0.134549 0.990907i \(-0.542958\pi\)
−0.134549 + 0.990907i \(0.542958\pi\)
\(734\) −1.12255e12 −0.142749
\(735\) 8.29935e12 1.04894
\(736\) −3.72453e12 −0.467865
\(737\) −1.87385e13 −2.33954
\(738\) −2.15853e12 −0.267858
\(739\) −1.19388e13 −1.47251 −0.736256 0.676703i \(-0.763408\pi\)
−0.736256 + 0.676703i \(0.763408\pi\)
\(740\) −1.62761e13 −1.99530
\(741\) −9.08209e12 −1.10663
\(742\) −3.75610e13 −4.54903
\(743\) −9.79773e12 −1.17944 −0.589720 0.807608i \(-0.700762\pi\)
−0.589720 + 0.807608i \(0.700762\pi\)
\(744\) 5.57821e12 0.667447
\(745\) −1.85853e13 −2.21037
\(746\) −2.16289e13 −2.55687
\(747\) −2.51168e12 −0.295135
\(748\) −2.67165e13 −3.12048
\(749\) −4.62621e12 −0.537103
\(750\) 2.00585e12 0.231485
\(751\) −1.06936e13 −1.22672 −0.613359 0.789804i \(-0.710182\pi\)
−0.613359 + 0.789804i \(0.710182\pi\)
\(752\) −4.30328e11 −0.0490704
\(753\) −2.72455e11 −0.0308828
\(754\) −1.07532e13 −1.21163
\(755\) −1.25536e13 −1.40607
\(756\) −4.00058e12 −0.445425
\(757\) −5.82516e12 −0.644728 −0.322364 0.946616i \(-0.604477\pi\)
−0.322364 + 0.946616i \(0.604477\pi\)
\(758\) 5.66037e12 0.622778
\(759\) 2.67362e12 0.292423
\(760\) 1.16168e13 1.26307
\(761\) 6.06747e12 0.655808 0.327904 0.944711i \(-0.393658\pi\)
0.327904 + 0.944711i \(0.393658\pi\)
\(762\) −1.23941e12 −0.133174
\(763\) 2.31613e13 2.47401
\(764\) −1.34697e12 −0.143033
\(765\) 6.92145e12 0.730669
\(766\) 2.72288e13 2.85758
\(767\) 2.05918e12 0.214839
\(768\) −3.33789e12 −0.346216
\(769\) −8.69120e12 −0.896213 −0.448107 0.893980i \(-0.647902\pi\)
−0.448107 + 0.893980i \(0.647902\pi\)
\(770\) −4.04695e13 −4.14877
\(771\) −5.59689e12 −0.570430
\(772\) −9.69900e12 −0.982764
\(773\) −6.91962e12 −0.697067 −0.348534 0.937296i \(-0.613320\pi\)
−0.348534 + 0.937296i \(0.613320\pi\)
\(774\) −3.00223e12 −0.300683
\(775\) −1.16698e13 −1.16200
\(776\) −9.82765e12 −0.972908
\(777\) 8.82708e12 0.868806
\(778\) 6.06769e12 0.593766
\(779\) −6.05535e12 −0.589143
\(780\) 2.00186e13 1.93646
\(781\) 1.52104e13 1.46289
\(782\) 1.07668e13 1.02958
\(783\) 9.38092e11 0.0891903
\(784\) −3.28499e12 −0.310536
\(785\) 1.15753e13 1.08798
\(786\) 7.41512e12 0.692973
\(787\) −1.52860e13 −1.42039 −0.710195 0.704005i \(-0.751393\pi\)
−0.710195 + 0.704005i \(0.751393\pi\)
\(788\) 4.15389e12 0.383784
\(789\) −6.71678e12 −0.617042
\(790\) 1.28100e12 0.117011
\(791\) 1.35631e13 1.23187
\(792\) 3.78419e12 0.341751
\(793\) −3.71464e12 −0.333570
\(794\) −7.57438e11 −0.0676324
\(795\) −1.63966e13 −1.45580
\(796\) −1.87641e13 −1.65660
\(797\) −1.37425e13 −1.20644 −0.603219 0.797576i \(-0.706115\pi\)
−0.603219 + 0.797576i \(0.706115\pi\)
\(798\) −1.86557e13 −1.62854
\(799\) 4.00097e12 0.347300
\(800\) 1.10284e13 0.951935
\(801\) 5.60797e12 0.481348
\(802\) 4.45056e12 0.379866
\(803\) 6.09599e12 0.517397
\(804\) 1.90400e13 1.60699
\(805\) 9.81138e12 0.823472
\(806\) 4.48261e13 3.74130
\(807\) 3.95158e12 0.327974
\(808\) 1.36128e13 1.12356
\(809\) −3.57292e12 −0.293262 −0.146631 0.989191i \(-0.546843\pi\)
−0.146631 + 0.989191i \(0.546843\pi\)
\(810\) −2.90300e12 −0.236954
\(811\) 1.71499e13 1.39209 0.696046 0.717998i \(-0.254941\pi\)
0.696046 + 0.717998i \(0.254941\pi\)
\(812\) −1.32879e13 −1.07264
\(813\) −9.36519e12 −0.751812
\(814\) −2.47243e13 −1.97385
\(815\) 3.00649e13 2.38699
\(816\) −2.73960e12 −0.216313
\(817\) −8.42219e12 −0.661341
\(818\) −2.11906e13 −1.65483
\(819\) −1.08568e13 −0.843187
\(820\) 1.33471e13 1.03092
\(821\) −6.68653e12 −0.513637 −0.256819 0.966460i \(-0.582674\pi\)
−0.256819 + 0.966460i \(0.582674\pi\)
\(822\) 5.50924e12 0.420890
\(823\) 1.36851e13 1.03979 0.519897 0.854229i \(-0.325970\pi\)
0.519897 + 0.854229i \(0.325970\pi\)
\(824\) −1.53959e13 −1.16341
\(825\) −7.91664e12 −0.594974
\(826\) 4.22979e12 0.316161
\(827\) 1.79247e13 1.33253 0.666265 0.745715i \(-0.267892\pi\)
0.666265 + 0.745715i \(0.267892\pi\)
\(828\) −2.71664e12 −0.200861
\(829\) 6.90632e12 0.507869 0.253934 0.967221i \(-0.418275\pi\)
0.253934 + 0.967221i \(0.418275\pi\)
\(830\) 2.58167e13 1.88821
\(831\) −5.95190e12 −0.432963
\(832\) −3.71146e13 −2.68528
\(833\) 3.05421e13 2.19784
\(834\) −4.12846e12 −0.295488
\(835\) −6.28044e12 −0.447096
\(836\) 3.14348e13 2.22578
\(837\) −3.91054e12 −0.275405
\(838\) 1.85280e13 1.29787
\(839\) −2.60104e13 −1.81225 −0.906124 0.423011i \(-0.860973\pi\)
−0.906124 + 0.423011i \(0.860973\pi\)
\(840\) 1.38868e13 0.962379
\(841\) −1.13913e13 −0.785218
\(842\) −1.45929e13 −1.00055
\(843\) −3.31539e12 −0.226105
\(844\) 6.93449e12 0.470407
\(845\) 3.43772e13 2.31961
\(846\) −1.67809e12 −0.112629
\(847\) −1.40219e13 −0.936120
\(848\) 6.49001e12 0.430987
\(849\) −1.52483e13 −1.00725
\(850\) −3.18808e13 −2.09481
\(851\) 5.99414e12 0.391781
\(852\) −1.54551e13 −1.00484
\(853\) 1.41771e13 0.916888 0.458444 0.888723i \(-0.348407\pi\)
0.458444 + 0.888723i \(0.348407\pi\)
\(854\) −7.63030e12 −0.490887
\(855\) −8.14384e12 −0.521172
\(856\) −4.44639e12 −0.283058
\(857\) 1.12350e13 0.711476 0.355738 0.934586i \(-0.384230\pi\)
0.355738 + 0.934586i \(0.384230\pi\)
\(858\) 3.04094e13 1.91565
\(859\) 2.44792e13 1.53401 0.767003 0.641643i \(-0.221747\pi\)
0.767003 + 0.641643i \(0.221747\pi\)
\(860\) 1.85641e13 1.15726
\(861\) −7.23861e12 −0.448891
\(862\) 3.17315e13 1.95752
\(863\) 2.35658e13 1.44622 0.723110 0.690733i \(-0.242712\pi\)
0.723110 + 0.690733i \(0.242712\pi\)
\(864\) 3.69560e12 0.225618
\(865\) −1.30080e13 −0.790020
\(866\) −2.41956e13 −1.46186
\(867\) 1.58658e13 0.953620
\(868\) 5.53922e13 3.31215
\(869\) 1.17062e12 0.0696347
\(870\) −9.64235e12 −0.570619
\(871\) 5.16708e13 3.04203
\(872\) 2.22610e13 1.30383
\(873\) 6.88956e12 0.401446
\(874\) −1.26684e13 −0.734377
\(875\) 6.72659e12 0.387935
\(876\) −6.19407e12 −0.355392
\(877\) −2.44619e12 −0.139634 −0.0698170 0.997560i \(-0.522242\pi\)
−0.0698170 + 0.997560i \(0.522242\pi\)
\(878\) −4.59995e13 −2.61233
\(879\) 7.99499e12 0.451719
\(880\) 6.99256e12 0.393065
\(881\) −9.21734e12 −0.515483 −0.257741 0.966214i \(-0.582978\pi\)
−0.257741 + 0.966214i \(0.582978\pi\)
\(882\) −1.28100e13 −0.712757
\(883\) 2.92106e12 0.161703 0.0808515 0.996726i \(-0.474236\pi\)
0.0808515 + 0.996726i \(0.474236\pi\)
\(884\) 7.36699e13 4.05747
\(885\) 1.84644e12 0.101179
\(886\) −5.18217e13 −2.82527
\(887\) 1.35029e13 0.732436 0.366218 0.930529i \(-0.380652\pi\)
0.366218 + 0.930529i \(0.380652\pi\)
\(888\) 8.48397e12 0.457869
\(889\) −4.15635e12 −0.223180
\(890\) −5.76425e13 −3.07956
\(891\) −2.65286e12 −0.141015
\(892\) −3.97478e13 −2.10219
\(893\) −4.70757e12 −0.247722
\(894\) 2.86863e13 1.50195
\(895\) −6.21311e12 −0.323672
\(896\) −4.15682e13 −2.15464
\(897\) −7.37243e12 −0.380229
\(898\) 2.78456e13 1.42894
\(899\) −1.29889e13 −0.663213
\(900\) 8.04402e12 0.408679
\(901\) −6.03407e13 −3.05034
\(902\) 2.02751e13 1.01984
\(903\) −1.00679e13 −0.503901
\(904\) 1.30359e13 0.649207
\(905\) −4.68363e13 −2.32094
\(906\) 1.93764e13 0.955425
\(907\) −8.48404e12 −0.416265 −0.208133 0.978101i \(-0.566739\pi\)
−0.208133 + 0.978101i \(0.566739\pi\)
\(908\) −1.65296e13 −0.807002
\(909\) −9.54312e12 −0.463610
\(910\) 1.11593e14 5.39452
\(911\) 2.66713e13 1.28295 0.641477 0.767142i \(-0.278322\pi\)
0.641477 + 0.767142i \(0.278322\pi\)
\(912\) 3.22344e12 0.154292
\(913\) 2.35922e13 1.12370
\(914\) −2.51619e13 −1.19257
\(915\) −3.33088e12 −0.157096
\(916\) 3.13376e13 1.47074
\(917\) 2.48665e13 1.16132
\(918\) −1.06832e13 −0.496490
\(919\) 2.08506e13 0.964270 0.482135 0.876097i \(-0.339861\pi\)
0.482135 + 0.876097i \(0.339861\pi\)
\(920\) 9.43001e12 0.433977
\(921\) −9.26461e12 −0.424286
\(922\) 1.74644e13 0.795913
\(923\) −4.19423e13 −1.90215
\(924\) 3.75774e13 1.69591
\(925\) −1.77488e13 −0.797132
\(926\) −4.38262e13 −1.95877
\(927\) 1.07931e13 0.480051
\(928\) 1.22750e13 0.543319
\(929\) 1.22073e13 0.537712 0.268856 0.963180i \(-0.413355\pi\)
0.268856 + 0.963180i \(0.413355\pi\)
\(930\) 4.01952e13 1.76198
\(931\) −3.59361e13 −1.56768
\(932\) 3.07392e13 1.33451
\(933\) −1.71520e12 −0.0741050
\(934\) −5.00031e13 −2.14999
\(935\) −6.50131e13 −2.78195
\(936\) −1.04348e13 −0.444368
\(937\) 3.42377e13 1.45103 0.725514 0.688207i \(-0.241602\pi\)
0.725514 + 0.688207i \(0.241602\pi\)
\(938\) 1.06138e14 4.47670
\(939\) −7.94873e12 −0.333659
\(940\) 1.03764e13 0.433481
\(941\) −7.47857e12 −0.310932 −0.155466 0.987841i \(-0.549688\pi\)
−0.155466 + 0.987841i \(0.549688\pi\)
\(942\) −1.78665e13 −0.739283
\(943\) −4.91546e12 −0.202424
\(944\) −7.30848e11 −0.0299539
\(945\) −9.73519e12 −0.397101
\(946\) 2.81999e13 1.14482
\(947\) −2.42514e13 −0.979857 −0.489928 0.871763i \(-0.662977\pi\)
−0.489928 + 0.871763i \(0.662977\pi\)
\(948\) −1.18945e12 −0.0478310
\(949\) −1.68095e13 −0.672755
\(950\) 3.75113e13 1.49419
\(951\) −3.76430e12 −0.149235
\(952\) 5.11044e13 2.01647
\(953\) 2.59499e13 1.01910 0.509551 0.860440i \(-0.329811\pi\)
0.509551 + 0.860440i \(0.329811\pi\)
\(954\) 2.53082e13 0.989220
\(955\) −3.27777e12 −0.127516
\(956\) −2.43890e13 −0.944350
\(957\) −8.81149e12 −0.339583
\(958\) −2.85891e13 −1.09662
\(959\) 1.84752e13 0.705350
\(960\) −3.32803e13 −1.26464
\(961\) 2.77059e13 1.04789
\(962\) 6.81766e13 2.56654
\(963\) 3.11709e12 0.116797
\(964\) −6.29862e13 −2.34909
\(965\) −2.36020e13 −0.876145
\(966\) −1.51438e13 −0.559550
\(967\) 1.84647e13 0.679085 0.339542 0.940591i \(-0.389728\pi\)
0.339542 + 0.940591i \(0.389728\pi\)
\(968\) −1.34769e13 −0.493344
\(969\) −2.99699e13 −1.09201
\(970\) −7.08155e13 −2.56836
\(971\) 5.23071e13 1.88831 0.944157 0.329495i \(-0.106878\pi\)
0.944157 + 0.329495i \(0.106878\pi\)
\(972\) 2.69554e12 0.0968608
\(973\) −1.38447e13 −0.495196
\(974\) 6.32146e13 2.25062
\(975\) 2.18299e13 0.773627
\(976\) 1.31841e12 0.0465078
\(977\) 2.96462e13 1.04098 0.520491 0.853867i \(-0.325749\pi\)
0.520491 + 0.853867i \(0.325749\pi\)
\(978\) −4.64051e13 −1.62196
\(979\) −5.26757e13 −1.83269
\(980\) 7.92100e13 2.74323
\(981\) −1.56058e13 −0.537992
\(982\) −3.35242e13 −1.15042
\(983\) −1.15962e13 −0.396117 −0.198059 0.980190i \(-0.563464\pi\)
−0.198059 + 0.980190i \(0.563464\pi\)
\(984\) −6.95724e12 −0.236570
\(985\) 1.01083e13 0.342148
\(986\) −3.54845e13 −1.19562
\(987\) −5.62746e12 −0.188749
\(988\) −8.66807e13 −2.89412
\(989\) −6.83675e12 −0.227230
\(990\) 2.72679e13 0.902180
\(991\) 2.04116e13 0.672273 0.336137 0.941813i \(-0.390880\pi\)
0.336137 + 0.941813i \(0.390880\pi\)
\(992\) −5.11696e13 −1.67768
\(993\) 4.10745e12 0.134060
\(994\) −8.61543e13 −2.79923
\(995\) −4.56614e13 −1.47688
\(996\) −2.39718e13 −0.771850
\(997\) −5.11008e13 −1.63794 −0.818972 0.573833i \(-0.805456\pi\)
−0.818972 + 0.573833i \(0.805456\pi\)
\(998\) −8.33793e13 −2.66055
\(999\) −5.94759e12 −0.188928
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.1.3 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.a.1.3 21 1.1 even 1 trivial