Properties

Label 177.10.a.b.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

\(f(q)\) \(=\) \(q-36.0452 q^{2} -81.0000 q^{3} +787.255 q^{4} -242.182 q^{5} +2919.66 q^{6} +7611.77 q^{7} -9921.60 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q-36.0452 q^{2} -81.0000 q^{3} +787.255 q^{4} -242.182 q^{5} +2919.66 q^{6} +7611.77 q^{7} -9921.60 q^{8} +6561.00 q^{9} +8729.49 q^{10} +17094.0 q^{11} -63767.6 q^{12} +68268.9 q^{13} -274368. q^{14} +19616.7 q^{15} -45448.5 q^{16} +321907. q^{17} -236492. q^{18} -803167. q^{19} -190659. q^{20} -616553. q^{21} -616155. q^{22} +995598. q^{23} +803650. q^{24} -1.89447e6 q^{25} -2.46076e6 q^{26} -531441. q^{27} +5.99240e6 q^{28} -4.98426e6 q^{29} -707089. q^{30} -3.39004e6 q^{31} +6.71806e6 q^{32} -1.38461e6 q^{33} -1.16032e7 q^{34} -1.84343e6 q^{35} +5.16518e6 q^{36} +2.66106e6 q^{37} +2.89503e7 q^{38} -5.52978e6 q^{39} +2.40283e6 q^{40} +2.95240e6 q^{41} +2.22238e7 q^{42} -4.26661e7 q^{43} +1.34573e7 q^{44} -1.58896e6 q^{45} -3.58865e7 q^{46} +2.32930e7 q^{47} +3.68132e6 q^{48} +1.75854e7 q^{49} +6.82866e7 q^{50} -2.60745e7 q^{51} +5.37450e7 q^{52} +6.00847e7 q^{53} +1.91559e7 q^{54} -4.13985e6 q^{55} -7.55210e7 q^{56} +6.50565e7 q^{57} +1.79658e8 q^{58} -1.21174e7 q^{59} +1.54434e7 q^{60} -7.21065e6 q^{61} +1.22195e8 q^{62} +4.99408e7 q^{63} -2.18884e8 q^{64} -1.65335e7 q^{65} +4.99085e7 q^{66} +2.28589e8 q^{67} +2.53423e8 q^{68} -8.06434e7 q^{69} +6.64469e7 q^{70} -5.89735e7 q^{71} -6.50956e7 q^{72} +1.06671e8 q^{73} -9.59183e7 q^{74} +1.53452e8 q^{75} -6.32297e8 q^{76} +1.30115e8 q^{77} +1.99322e8 q^{78} -3.06693e7 q^{79} +1.10068e7 q^{80} +4.30467e7 q^{81} -1.06420e8 q^{82} +2.84892e8 q^{83} -4.85384e8 q^{84} -7.79601e7 q^{85} +1.53791e9 q^{86} +4.03725e8 q^{87} -1.69599e8 q^{88} +7.19999e7 q^{89} +5.72742e7 q^{90} +5.19647e8 q^{91} +7.83789e8 q^{92} +2.74593e8 q^{93} -8.39599e8 q^{94} +1.94513e8 q^{95} -5.44163e8 q^{96} -8.66357e8 q^{97} -6.33870e8 q^{98} +1.12153e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q + 20q^{2} - 1701q^{3} + 4950q^{4} + 2058q^{5} - 1620q^{6} - 17167q^{7} - 2853q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q + 20q^{2} - 1701q^{3} + 4950q^{4} + 2058q^{5} - 1620q^{6} - 17167q^{7} - 2853q^{8} + 137781q^{9} - 31559q^{10} - 38751q^{11} - 400950q^{12} - 58915q^{13} + 3453q^{14} - 166698q^{15} + 1655714q^{16} - 64233q^{17} + 131220q^{18} - 1937236q^{19} - 1065507q^{20} + 1390527q^{21} - 5386882q^{22} - 1838574q^{23} + 231093q^{24} + 4565755q^{25} - 839702q^{26} - 11160261q^{27} - 4471034q^{28} + 15658544q^{29} + 2556279q^{30} - 14282802q^{31} - 2205286q^{32} + 3138831q^{33} + 19005532q^{34} - 8633300q^{35} + 32476950q^{36} + 7531195q^{37} + 26649773q^{38} + 4772115q^{39} + 17775672q^{40} + 18338245q^{41} - 279693q^{42} - 22480305q^{43} - 80230922q^{44} + 13502538q^{45} - 83894107q^{46} - 110397260q^{47} - 134112834q^{48} + 130653638q^{49} + 65575693q^{50} + 5202873q^{51} + 177908014q^{52} + 145498338q^{53} - 10628820q^{54} + 86448944q^{55} + 354387888q^{56} + 156916116q^{57} + 115508368q^{58} - 254464581q^{59} + 86306067q^{60} + 287595506q^{61} + 819899030q^{62} - 112632687q^{63} + 822446413q^{64} + 77238206q^{65} + 436337442q^{66} - 392860610q^{67} + 167325073q^{68} + 148924494q^{69} - 424902116q^{70} - 248960491q^{71} - 18718533q^{72} - 758406074q^{73} - 923266846q^{74} - 369826155q^{75} - 2312747568q^{76} - 878126795q^{77} + 68015862q^{78} - 1925801029q^{79} - 1898919861q^{80} + 903981141q^{81} - 3249102191q^{82} - 1650336307q^{83} + 362153754q^{84} - 2342480762q^{85} - 3609864952q^{86} - 1268342064q^{87} - 5987792887q^{88} - 574997526q^{89} - 207058599q^{90} - 4481387117q^{91} - 5317166770q^{92} + 1156906962q^{93} - 5360726568q^{94} - 2789231462q^{95} + 178628166q^{96} - 4651540898q^{97} - 5566652976q^{98} - 254245311q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −36.0452 −1.59299 −0.796493 0.604647i \(-0.793314\pi\)
−0.796493 + 0.604647i \(0.793314\pi\)
\(3\) −81.0000 −0.577350
\(4\) 787.255 1.53761
\(5\) −242.182 −0.173291 −0.0866456 0.996239i \(-0.527615\pi\)
−0.0866456 + 0.996239i \(0.527615\pi\)
\(6\) 2919.66 0.919711
\(7\) 7611.77 1.19824 0.599121 0.800659i \(-0.295517\pi\)
0.599121 + 0.800659i \(0.295517\pi\)
\(8\) −9921.60 −0.856401
\(9\) 6561.00 0.333333
\(10\) 8729.49 0.276051
\(11\) 17094.0 0.352027 0.176013 0.984388i \(-0.443680\pi\)
0.176013 + 0.984388i \(0.443680\pi\)
\(12\) −63767.6 −0.887738
\(13\) 68268.9 0.662945 0.331473 0.943465i \(-0.392455\pi\)
0.331473 + 0.943465i \(0.392455\pi\)
\(14\) −274368. −1.90878
\(15\) 19616.7 0.100050
\(16\) −45448.5 −0.173372
\(17\) 321907. 0.934783 0.467391 0.884051i \(-0.345194\pi\)
0.467391 + 0.884051i \(0.345194\pi\)
\(18\) −236492. −0.530996
\(19\) −803167. −1.41389 −0.706943 0.707270i \(-0.749926\pi\)
−0.706943 + 0.707270i \(0.749926\pi\)
\(20\) −190659. −0.266454
\(21\) −616553. −0.691805
\(22\) −616155. −0.560774
\(23\) 995598. 0.741837 0.370919 0.928665i \(-0.379043\pi\)
0.370919 + 0.928665i \(0.379043\pi\)
\(24\) 803650. 0.494443
\(25\) −1.89447e6 −0.969970
\(26\) −2.46076e6 −1.05606
\(27\) −531441. −0.192450
\(28\) 5.99240e6 1.84242
\(29\) −4.98426e6 −1.30861 −0.654304 0.756232i \(-0.727038\pi\)
−0.654304 + 0.756232i \(0.727038\pi\)
\(30\) −707089. −0.159378
\(31\) −3.39004e6 −0.659291 −0.329645 0.944105i \(-0.606929\pi\)
−0.329645 + 0.944105i \(0.606929\pi\)
\(32\) 6.71806e6 1.13258
\(33\) −1.38461e6 −0.203243
\(34\) −1.16032e7 −1.48910
\(35\) −1.84343e6 −0.207645
\(36\) 5.16518e6 0.512536
\(37\) 2.66106e6 0.233424 0.116712 0.993166i \(-0.462764\pi\)
0.116712 + 0.993166i \(0.462764\pi\)
\(38\) 2.89503e7 2.25230
\(39\) −5.52978e6 −0.382752
\(40\) 2.40283e6 0.148407
\(41\) 2.95240e6 0.163173 0.0815865 0.996666i \(-0.474001\pi\)
0.0815865 + 0.996666i \(0.474001\pi\)
\(42\) 2.22238e7 1.10204
\(43\) −4.26661e7 −1.90316 −0.951580 0.307401i \(-0.900541\pi\)
−0.951580 + 0.307401i \(0.900541\pi\)
\(44\) 1.34573e7 0.541279
\(45\) −1.58896e6 −0.0577638
\(46\) −3.58865e7 −1.18174
\(47\) 2.32930e7 0.696281 0.348141 0.937442i \(-0.386813\pi\)
0.348141 + 0.937442i \(0.386813\pi\)
\(48\) 3.68132e6 0.100096
\(49\) 1.75854e7 0.435783
\(50\) 6.82866e7 1.54515
\(51\) −2.60745e7 −0.539697
\(52\) 5.37450e7 1.01935
\(53\) 6.00847e7 1.04598 0.522989 0.852340i \(-0.324817\pi\)
0.522989 + 0.852340i \(0.324817\pi\)
\(54\) 1.91559e7 0.306570
\(55\) −4.13985e6 −0.0610031
\(56\) −7.55210e7 −1.02617
\(57\) 6.50565e7 0.816308
\(58\) 1.79658e8 2.08459
\(59\) −1.21174e7 −0.130189
\(60\) 1.54434e7 0.153837
\(61\) −7.21065e6 −0.0666791 −0.0333396 0.999444i \(-0.510614\pi\)
−0.0333396 + 0.999444i \(0.510614\pi\)
\(62\) 1.22195e8 1.05024
\(63\) 4.99408e7 0.399414
\(64\) −2.18884e8 −1.63081
\(65\) −1.65335e7 −0.114883
\(66\) 4.99085e7 0.323763
\(67\) 2.28589e8 1.38586 0.692929 0.721006i \(-0.256320\pi\)
0.692929 + 0.721006i \(0.256320\pi\)
\(68\) 2.53423e8 1.43733
\(69\) −8.06434e7 −0.428300
\(70\) 6.64469e7 0.330775
\(71\) −5.89735e7 −0.275419 −0.137710 0.990473i \(-0.543974\pi\)
−0.137710 + 0.990473i \(0.543974\pi\)
\(72\) −6.50956e7 −0.285467
\(73\) 1.06671e8 0.439636 0.219818 0.975541i \(-0.429454\pi\)
0.219818 + 0.975541i \(0.429454\pi\)
\(74\) −9.59183e7 −0.371842
\(75\) 1.53452e8 0.560013
\(76\) −6.32297e8 −2.17400
\(77\) 1.30115e8 0.421813
\(78\) 1.99322e8 0.609718
\(79\) −3.06693e7 −0.0885894 −0.0442947 0.999019i \(-0.514104\pi\)
−0.0442947 + 0.999019i \(0.514104\pi\)
\(80\) 1.10068e7 0.0300439
\(81\) 4.30467e7 0.111111
\(82\) −1.06420e8 −0.259932
\(83\) 2.84892e8 0.658913 0.329457 0.944171i \(-0.393134\pi\)
0.329457 + 0.944171i \(0.393134\pi\)
\(84\) −4.85384e8 −1.06372
\(85\) −7.79601e7 −0.161990
\(86\) 1.53791e9 3.03171
\(87\) 4.03725e8 0.755525
\(88\) −1.69599e8 −0.301476
\(89\) 7.19999e7 0.121640 0.0608201 0.998149i \(-0.480628\pi\)
0.0608201 + 0.998149i \(0.480628\pi\)
\(90\) 5.72742e7 0.0920169
\(91\) 5.19647e8 0.794369
\(92\) 7.83789e8 1.14065
\(93\) 2.74593e8 0.380642
\(94\) −8.39599e8 −1.10917
\(95\) 1.94513e8 0.245014
\(96\) −5.44163e8 −0.653895
\(97\) −8.66357e8 −0.993629 −0.496814 0.867857i \(-0.665497\pi\)
−0.496814 + 0.867857i \(0.665497\pi\)
\(98\) −6.33870e8 −0.694197
\(99\) 1.12153e8 0.117342
\(100\) −1.49143e9 −1.49143
\(101\) −1.21040e9 −1.15740 −0.578698 0.815542i \(-0.696439\pi\)
−0.578698 + 0.815542i \(0.696439\pi\)
\(102\) 9.39860e8 0.859730
\(103\) −2.40395e8 −0.210454 −0.105227 0.994448i \(-0.533557\pi\)
−0.105227 + 0.994448i \(0.533557\pi\)
\(104\) −6.77337e8 −0.567747
\(105\) 1.49318e8 0.119884
\(106\) −2.16576e9 −1.66623
\(107\) −2.13921e9 −1.57771 −0.788854 0.614581i \(-0.789325\pi\)
−0.788854 + 0.614581i \(0.789325\pi\)
\(108\) −4.18379e8 −0.295913
\(109\) 1.75721e9 1.19235 0.596175 0.802855i \(-0.296687\pi\)
0.596175 + 0.802855i \(0.296687\pi\)
\(110\) 1.49222e8 0.0971772
\(111\) −2.15546e8 −0.134768
\(112\) −3.45943e8 −0.207742
\(113\) −2.26510e9 −1.30688 −0.653439 0.756979i \(-0.726674\pi\)
−0.653439 + 0.756979i \(0.726674\pi\)
\(114\) −2.34497e9 −1.30037
\(115\) −2.41116e8 −0.128554
\(116\) −3.92388e9 −2.01212
\(117\) 4.47912e8 0.220982
\(118\) 4.36772e8 0.207389
\(119\) 2.45028e9 1.12010
\(120\) −1.94629e8 −0.0856827
\(121\) −2.06574e9 −0.876077
\(122\) 2.59909e8 0.106219
\(123\) −2.39145e8 −0.0942080
\(124\) −2.66882e9 −1.01373
\(125\) 9.31819e8 0.341379
\(126\) −1.80013e9 −0.636261
\(127\) −5.19361e9 −1.77155 −0.885774 0.464117i \(-0.846372\pi\)
−0.885774 + 0.464117i \(0.846372\pi\)
\(128\) 4.45007e9 1.46528
\(129\) 3.45596e9 1.09879
\(130\) 5.95952e8 0.183006
\(131\) −4.53077e9 −1.34416 −0.672081 0.740478i \(-0.734599\pi\)
−0.672081 + 0.740478i \(0.734599\pi\)
\(132\) −1.09004e9 −0.312507
\(133\) −6.11352e9 −1.69418
\(134\) −8.23953e9 −2.20765
\(135\) 1.28705e8 0.0333499
\(136\) −3.19384e9 −0.800548
\(137\) 5.15763e9 1.25086 0.625428 0.780282i \(-0.284924\pi\)
0.625428 + 0.780282i \(0.284924\pi\)
\(138\) 2.90681e9 0.682276
\(139\) 1.20180e9 0.273064 0.136532 0.990636i \(-0.456404\pi\)
0.136532 + 0.990636i \(0.456404\pi\)
\(140\) −1.45125e9 −0.319276
\(141\) −1.88673e9 −0.401998
\(142\) 2.12571e9 0.438739
\(143\) 1.16699e9 0.233374
\(144\) −2.98187e8 −0.0577907
\(145\) 1.20710e9 0.226770
\(146\) −3.84497e9 −0.700334
\(147\) −1.42442e9 −0.251600
\(148\) 2.09493e9 0.358915
\(149\) 7.95137e9 1.32161 0.660806 0.750557i \(-0.270215\pi\)
0.660806 + 0.750557i \(0.270215\pi\)
\(150\) −5.53122e9 −0.892093
\(151\) −4.03039e9 −0.630886 −0.315443 0.948945i \(-0.602153\pi\)
−0.315443 + 0.948945i \(0.602153\pi\)
\(152\) 7.96870e9 1.21085
\(153\) 2.11203e9 0.311594
\(154\) −4.69003e9 −0.671943
\(155\) 8.21006e8 0.114249
\(156\) −4.35334e9 −0.588522
\(157\) 4.48151e9 0.588675 0.294338 0.955702i \(-0.404901\pi\)
0.294338 + 0.955702i \(0.404901\pi\)
\(158\) 1.10548e9 0.141122
\(159\) −4.86686e9 −0.603895
\(160\) −1.62699e9 −0.196266
\(161\) 7.57826e9 0.888901
\(162\) −1.55163e9 −0.176999
\(163\) 1.22081e10 1.35458 0.677288 0.735718i \(-0.263155\pi\)
0.677288 + 0.735718i \(0.263155\pi\)
\(164\) 2.32429e9 0.250896
\(165\) 3.35328e8 0.0352202
\(166\) −1.02690e10 −1.04964
\(167\) 1.46186e9 0.145439 0.0727197 0.997352i \(-0.476832\pi\)
0.0727197 + 0.997352i \(0.476832\pi\)
\(168\) 6.11720e9 0.592462
\(169\) −5.94386e9 −0.560504
\(170\) 2.81009e9 0.258047
\(171\) −5.26958e9 −0.471296
\(172\) −3.35891e10 −2.92631
\(173\) 1.06740e10 0.905982 0.452991 0.891515i \(-0.350357\pi\)
0.452991 + 0.891515i \(0.350357\pi\)
\(174\) −1.45523e10 −1.20354
\(175\) −1.44203e10 −1.16226
\(176\) −7.76894e8 −0.0610316
\(177\) 9.81506e8 0.0751646
\(178\) −2.59525e9 −0.193771
\(179\) −1.20853e9 −0.0879872 −0.0439936 0.999032i \(-0.514008\pi\)
−0.0439936 + 0.999032i \(0.514008\pi\)
\(180\) −1.25091e9 −0.0888179
\(181\) −1.40693e9 −0.0974358 −0.0487179 0.998813i \(-0.515514\pi\)
−0.0487179 + 0.998813i \(0.515514\pi\)
\(182\) −1.87308e10 −1.26542
\(183\) 5.84062e8 0.0384972
\(184\) −9.87793e9 −0.635310
\(185\) −6.44460e8 −0.0404504
\(186\) −9.89776e9 −0.606357
\(187\) 5.50267e9 0.329068
\(188\) 1.83375e10 1.07061
\(189\) −4.04521e9 −0.230602
\(190\) −7.01124e9 −0.390304
\(191\) −1.01040e10 −0.549343 −0.274672 0.961538i \(-0.588569\pi\)
−0.274672 + 0.961538i \(0.588569\pi\)
\(192\) 1.77296e10 0.941550
\(193\) −3.13082e10 −1.62424 −0.812120 0.583490i \(-0.801687\pi\)
−0.812120 + 0.583490i \(0.801687\pi\)
\(194\) 3.12280e10 1.58284
\(195\) 1.33921e9 0.0663275
\(196\) 1.38442e10 0.670063
\(197\) −4.34991e9 −0.205770 −0.102885 0.994693i \(-0.532807\pi\)
−0.102885 + 0.994693i \(0.532807\pi\)
\(198\) −4.04259e9 −0.186925
\(199\) −2.18718e10 −0.988657 −0.494328 0.869275i \(-0.664586\pi\)
−0.494328 + 0.869275i \(0.664586\pi\)
\(200\) 1.87962e10 0.830683
\(201\) −1.85157e10 −0.800125
\(202\) 4.36290e10 1.84372
\(203\) −3.79390e10 −1.56803
\(204\) −2.05273e10 −0.829842
\(205\) −7.15019e8 −0.0282765
\(206\) 8.66508e9 0.335251
\(207\) 6.53212e9 0.247279
\(208\) −3.10272e9 −0.114936
\(209\) −1.37293e10 −0.497726
\(210\) −5.38220e9 −0.190973
\(211\) 4.42064e10 1.53537 0.767686 0.640826i \(-0.221408\pi\)
0.767686 + 0.640826i \(0.221408\pi\)
\(212\) 4.73019e10 1.60830
\(213\) 4.77685e9 0.159013
\(214\) 7.71082e10 2.51327
\(215\) 1.03330e10 0.329801
\(216\) 5.27275e9 0.164814
\(217\) −2.58042e10 −0.789990
\(218\) −6.33388e10 −1.89940
\(219\) −8.64034e9 −0.253824
\(220\) −3.25911e9 −0.0937989
\(221\) 2.19762e10 0.619710
\(222\) 7.76938e9 0.214683
\(223\) 5.04420e10 1.36590 0.682952 0.730463i \(-0.260696\pi\)
0.682952 + 0.730463i \(0.260696\pi\)
\(224\) 5.11363e10 1.35710
\(225\) −1.24296e10 −0.323323
\(226\) 8.16461e10 2.08184
\(227\) −1.49660e10 −0.374101 −0.187050 0.982350i \(-0.559893\pi\)
−0.187050 + 0.982350i \(0.559893\pi\)
\(228\) 5.12161e10 1.25516
\(229\) 1.63674e10 0.393297 0.196648 0.980474i \(-0.436994\pi\)
0.196648 + 0.980474i \(0.436994\pi\)
\(230\) 8.69106e9 0.204785
\(231\) −1.05393e10 −0.243534
\(232\) 4.94518e10 1.12069
\(233\) 2.72820e10 0.606421 0.303211 0.952924i \(-0.401941\pi\)
0.303211 + 0.952924i \(0.401941\pi\)
\(234\) −1.61451e10 −0.352021
\(235\) −5.64114e9 −0.120659
\(236\) −9.53945e9 −0.200179
\(237\) 2.48421e9 0.0511471
\(238\) −8.83209e10 −1.78430
\(239\) 3.93459e10 0.780026 0.390013 0.920809i \(-0.372471\pi\)
0.390013 + 0.920809i \(0.372471\pi\)
\(240\) −8.91550e8 −0.0173458
\(241\) −8.35801e10 −1.59598 −0.797988 0.602673i \(-0.794102\pi\)
−0.797988 + 0.602673i \(0.794102\pi\)
\(242\) 7.44601e10 1.39558
\(243\) −3.48678e9 −0.0641500
\(244\) −5.67662e9 −0.102526
\(245\) −4.25887e9 −0.0755174
\(246\) 8.62001e9 0.150072
\(247\) −5.48313e10 −0.937330
\(248\) 3.36346e10 0.564617
\(249\) −2.30762e10 −0.380424
\(250\) −3.35876e10 −0.543812
\(251\) 8.76618e10 1.39405 0.697025 0.717046i \(-0.254506\pi\)
0.697025 + 0.717046i \(0.254506\pi\)
\(252\) 3.93161e10 0.614142
\(253\) 1.70187e10 0.261147
\(254\) 1.87205e11 2.82205
\(255\) 6.31477e9 0.0935248
\(256\) −4.83348e10 −0.703364
\(257\) −6.46345e10 −0.924198 −0.462099 0.886828i \(-0.652904\pi\)
−0.462099 + 0.886828i \(0.652904\pi\)
\(258\) −1.24571e11 −1.75036
\(259\) 2.02554e10 0.279699
\(260\) −1.30161e10 −0.176644
\(261\) −3.27017e10 −0.436203
\(262\) 1.63313e11 2.14123
\(263\) 8.56627e10 1.10406 0.552028 0.833826i \(-0.313854\pi\)
0.552028 + 0.833826i \(0.313854\pi\)
\(264\) 1.37376e10 0.174057
\(265\) −1.45514e10 −0.181259
\(266\) 2.20363e11 2.69880
\(267\) −5.83199e9 −0.0702290
\(268\) 1.79958e11 2.13090
\(269\) 1.25389e11 1.46007 0.730034 0.683410i \(-0.239504\pi\)
0.730034 + 0.683410i \(0.239504\pi\)
\(270\) −4.63921e9 −0.0531260
\(271\) −2.38301e10 −0.268389 −0.134194 0.990955i \(-0.542845\pi\)
−0.134194 + 0.990955i \(0.542845\pi\)
\(272\) −1.46302e10 −0.162065
\(273\) −4.20914e10 −0.458629
\(274\) −1.85908e11 −1.99260
\(275\) −3.23840e10 −0.341455
\(276\) −6.34869e10 −0.658557
\(277\) −1.00357e11 −1.02421 −0.512103 0.858924i \(-0.671133\pi\)
−0.512103 + 0.858924i \(0.671133\pi\)
\(278\) −4.33189e10 −0.434987
\(279\) −2.22420e10 −0.219764
\(280\) 1.82898e10 0.177827
\(281\) −1.79252e11 −1.71508 −0.857542 0.514414i \(-0.828010\pi\)
−0.857542 + 0.514414i \(0.828010\pi\)
\(282\) 6.80075e10 0.640378
\(283\) −2.07641e11 −1.92430 −0.962151 0.272517i \(-0.912144\pi\)
−0.962151 + 0.272517i \(0.912144\pi\)
\(284\) −4.64272e10 −0.423486
\(285\) −1.57555e10 −0.141459
\(286\) −4.20642e10 −0.371762
\(287\) 2.24730e10 0.195521
\(288\) 4.40772e10 0.377527
\(289\) −1.49636e10 −0.126181
\(290\) −4.35100e10 −0.361242
\(291\) 7.01749e10 0.573672
\(292\) 8.39771e10 0.675987
\(293\) −1.22646e11 −0.972182 −0.486091 0.873908i \(-0.661578\pi\)
−0.486091 + 0.873908i \(0.661578\pi\)
\(294\) 5.13435e10 0.400795
\(295\) 2.93461e9 0.0225606
\(296\) −2.64020e10 −0.199905
\(297\) −9.08443e9 −0.0677476
\(298\) −2.86608e11 −2.10531
\(299\) 6.79684e10 0.491798
\(300\) 1.20806e11 0.861079
\(301\) −3.24765e11 −2.28045
\(302\) 1.45276e11 1.00499
\(303\) 9.80422e10 0.668223
\(304\) 3.65027e10 0.245129
\(305\) 1.74629e9 0.0115549
\(306\) −7.61286e10 −0.496365
\(307\) 1.93309e11 1.24202 0.621012 0.783801i \(-0.286722\pi\)
0.621012 + 0.783801i \(0.286722\pi\)
\(308\) 1.02434e11 0.648583
\(309\) 1.94720e10 0.121506
\(310\) −2.95933e10 −0.181998
\(311\) −1.41995e9 −0.00860702 −0.00430351 0.999991i \(-0.501370\pi\)
−0.00430351 + 0.999991i \(0.501370\pi\)
\(312\) 5.48643e10 0.327789
\(313\) −9.72842e9 −0.0572918 −0.0286459 0.999590i \(-0.509120\pi\)
−0.0286459 + 0.999590i \(0.509120\pi\)
\(314\) −1.61537e11 −0.937752
\(315\) −1.20948e10 −0.0692149
\(316\) −2.41446e10 −0.136216
\(317\) −2.46412e10 −0.137055 −0.0685276 0.997649i \(-0.521830\pi\)
−0.0685276 + 0.997649i \(0.521830\pi\)
\(318\) 1.75427e11 0.961997
\(319\) −8.52007e10 −0.460665
\(320\) 5.30097e10 0.282606
\(321\) 1.73276e11 0.910890
\(322\) −2.73160e11 −1.41601
\(323\) −2.58545e11 −1.32168
\(324\) 3.38887e10 0.170845
\(325\) −1.29334e11 −0.643037
\(326\) −4.40043e11 −2.15782
\(327\) −1.42334e11 −0.688403
\(328\) −2.92926e10 −0.139741
\(329\) 1.77301e11 0.834313
\(330\) −1.20869e10 −0.0561053
\(331\) −9.23716e10 −0.422973 −0.211486 0.977381i \(-0.567830\pi\)
−0.211486 + 0.977381i \(0.567830\pi\)
\(332\) 2.24282e11 1.01315
\(333\) 1.74592e10 0.0778082
\(334\) −5.26931e10 −0.231683
\(335\) −5.53601e10 −0.240157
\(336\) 2.80214e10 0.119940
\(337\) −3.30308e11 −1.39503 −0.697516 0.716569i \(-0.745711\pi\)
−0.697516 + 0.716569i \(0.745711\pi\)
\(338\) 2.14247e11 0.892875
\(339\) 1.83473e11 0.754526
\(340\) −6.13745e10 −0.249076
\(341\) −5.79492e10 −0.232088
\(342\) 1.89943e11 0.750768
\(343\) −1.73306e11 −0.676068
\(344\) 4.23317e11 1.62987
\(345\) 1.95304e10 0.0742207
\(346\) −3.84746e11 −1.44322
\(347\) −1.83796e10 −0.0680538 −0.0340269 0.999421i \(-0.510833\pi\)
−0.0340269 + 0.999421i \(0.510833\pi\)
\(348\) 3.17834e11 1.16170
\(349\) 3.63360e11 1.31106 0.655530 0.755169i \(-0.272445\pi\)
0.655530 + 0.755169i \(0.272445\pi\)
\(350\) 5.19782e11 1.85146
\(351\) −3.62809e10 −0.127584
\(352\) 1.14838e11 0.398698
\(353\) −1.44721e11 −0.496073 −0.248036 0.968751i \(-0.579785\pi\)
−0.248036 + 0.968751i \(0.579785\pi\)
\(354\) −3.53786e10 −0.119736
\(355\) 1.42823e10 0.0477277
\(356\) 5.66823e10 0.187035
\(357\) −1.98473e11 −0.646688
\(358\) 4.35617e10 0.140162
\(359\) −4.51872e11 −1.43579 −0.717894 0.696152i \(-0.754894\pi\)
−0.717894 + 0.696152i \(0.754894\pi\)
\(360\) 1.57650e10 0.0494689
\(361\) 3.22390e11 0.999076
\(362\) 5.07130e10 0.155214
\(363\) 1.67325e11 0.505803
\(364\) 4.09095e11 1.22143
\(365\) −2.58337e10 −0.0761850
\(366\) −2.10526e10 −0.0613256
\(367\) −1.99239e11 −0.573294 −0.286647 0.958036i \(-0.592541\pi\)
−0.286647 + 0.958036i \(0.592541\pi\)
\(368\) −4.52484e10 −0.128614
\(369\) 1.93707e10 0.0543910
\(370\) 2.32297e10 0.0644370
\(371\) 4.57351e11 1.25333
\(372\) 2.16175e11 0.585277
\(373\) 4.88966e11 1.30794 0.653972 0.756519i \(-0.273101\pi\)
0.653972 + 0.756519i \(0.273101\pi\)
\(374\) −1.98345e11 −0.524202
\(375\) −7.54773e10 −0.197095
\(376\) −2.31104e11 −0.596295
\(377\) −3.40270e11 −0.867535
\(378\) 1.45810e11 0.367346
\(379\) 2.52680e11 0.629063 0.314532 0.949247i \(-0.398153\pi\)
0.314532 + 0.949247i \(0.398153\pi\)
\(380\) 1.53131e11 0.376736
\(381\) 4.20683e11 1.02280
\(382\) 3.64201e11 0.875096
\(383\) −7.44311e11 −1.76750 −0.883752 0.467956i \(-0.844990\pi\)
−0.883752 + 0.467956i \(0.844990\pi\)
\(384\) −3.60455e11 −0.845982
\(385\) −3.15116e10 −0.0730965
\(386\) 1.12851e12 2.58739
\(387\) −2.79933e11 −0.634387
\(388\) −6.82044e11 −1.52781
\(389\) −6.06698e11 −1.34338 −0.671691 0.740831i \(-0.734432\pi\)
−0.671691 + 0.740831i \(0.734432\pi\)
\(390\) −4.82721e10 −0.105659
\(391\) 3.20490e11 0.693457
\(392\) −1.74476e11 −0.373205
\(393\) 3.66993e11 0.776052
\(394\) 1.56793e11 0.327789
\(395\) 7.42755e9 0.0153518
\(396\) 8.82933e10 0.180426
\(397\) −1.25702e11 −0.253971 −0.126985 0.991905i \(-0.540530\pi\)
−0.126985 + 0.991905i \(0.540530\pi\)
\(398\) 7.88373e11 1.57492
\(399\) 4.95195e11 0.978134
\(400\) 8.61009e10 0.168166
\(401\) 4.96752e11 0.959378 0.479689 0.877439i \(-0.340750\pi\)
0.479689 + 0.877439i \(0.340750\pi\)
\(402\) 6.67402e11 1.27459
\(403\) −2.31434e11 −0.437074
\(404\) −9.52891e11 −1.77962
\(405\) −1.04251e10 −0.0192546
\(406\) 1.36752e12 2.49785
\(407\) 4.54880e10 0.0821716
\(408\) 2.58701e11 0.462197
\(409\) −8.77033e11 −1.54975 −0.774874 0.632116i \(-0.782187\pi\)
−0.774874 + 0.632116i \(0.782187\pi\)
\(410\) 2.57730e10 0.0450440
\(411\) −4.17768e11 −0.722182
\(412\) −1.89252e11 −0.323596
\(413\) −9.22346e10 −0.155998
\(414\) −2.35451e11 −0.393912
\(415\) −6.89956e10 −0.114184
\(416\) 4.58634e11 0.750839
\(417\) −9.73455e10 −0.157653
\(418\) 4.94875e11 0.792871
\(419\) −1.30070e11 −0.206165 −0.103083 0.994673i \(-0.532871\pi\)
−0.103083 + 0.994673i \(0.532871\pi\)
\(420\) 1.17551e11 0.184334
\(421\) −9.55911e11 −1.48302 −0.741512 0.670940i \(-0.765891\pi\)
−0.741512 + 0.670940i \(0.765891\pi\)
\(422\) −1.59343e12 −2.44583
\(423\) 1.52825e11 0.232094
\(424\) −5.96136e11 −0.895775
\(425\) −6.09845e11 −0.906711
\(426\) −1.72182e11 −0.253306
\(427\) −5.48858e10 −0.0798977
\(428\) −1.68410e12 −2.42589
\(429\) −9.45258e10 −0.134739
\(430\) −3.72454e11 −0.525369
\(431\) 5.09635e11 0.711396 0.355698 0.934601i \(-0.384243\pi\)
0.355698 + 0.934601i \(0.384243\pi\)
\(432\) 2.41532e10 0.0333655
\(433\) −8.04193e10 −0.109942 −0.0549712 0.998488i \(-0.517507\pi\)
−0.0549712 + 0.998488i \(0.517507\pi\)
\(434\) 9.30117e11 1.25844
\(435\) −9.77749e10 −0.130926
\(436\) 1.38337e12 1.83336
\(437\) −7.99631e11 −1.04887
\(438\) 3.11443e11 0.404338
\(439\) 1.22947e11 0.157989 0.0789945 0.996875i \(-0.474829\pi\)
0.0789945 + 0.996875i \(0.474829\pi\)
\(440\) 4.10739e10 0.0522431
\(441\) 1.15378e11 0.145261
\(442\) −7.92138e11 −0.987189
\(443\) −9.69820e11 −1.19639 −0.598197 0.801349i \(-0.704116\pi\)
−0.598197 + 0.801349i \(0.704116\pi\)
\(444\) −1.69689e11 −0.207220
\(445\) −1.74371e10 −0.0210792
\(446\) −1.81819e12 −2.17587
\(447\) −6.44061e11 −0.763033
\(448\) −1.66609e12 −1.95411
\(449\) −8.84280e11 −1.02679 −0.513394 0.858153i \(-0.671612\pi\)
−0.513394 + 0.858153i \(0.671612\pi\)
\(450\) 4.48028e11 0.515050
\(451\) 5.04683e10 0.0574413
\(452\) −1.78321e12 −2.00946
\(453\) 3.26461e11 0.364242
\(454\) 5.39451e11 0.595938
\(455\) −1.25849e11 −0.137657
\(456\) −6.45465e11 −0.699087
\(457\) 4.81205e11 0.516068 0.258034 0.966136i \(-0.416925\pi\)
0.258034 + 0.966136i \(0.416925\pi\)
\(458\) −5.89966e11 −0.626517
\(459\) −1.71075e11 −0.179899
\(460\) −1.89820e11 −0.197665
\(461\) 1.38965e12 1.43301 0.716507 0.697580i \(-0.245740\pi\)
0.716507 + 0.697580i \(0.245740\pi\)
\(462\) 3.79892e11 0.387946
\(463\) −4.35217e11 −0.440141 −0.220070 0.975484i \(-0.570629\pi\)
−0.220070 + 0.975484i \(0.570629\pi\)
\(464\) 2.26527e11 0.226876
\(465\) −6.65015e10 −0.0659619
\(466\) −9.83384e11 −0.966021
\(467\) 9.99032e10 0.0971971 0.0485985 0.998818i \(-0.484525\pi\)
0.0485985 + 0.998818i \(0.484525\pi\)
\(468\) 3.52621e11 0.339783
\(469\) 1.73997e12 1.66059
\(470\) 2.03336e11 0.192209
\(471\) −3.63002e11 −0.339872
\(472\) 1.20224e11 0.111494
\(473\) −7.29333e11 −0.669963
\(474\) −8.95439e10 −0.0814767
\(475\) 1.52158e12 1.37143
\(476\) 1.92900e12 1.72227
\(477\) 3.94215e11 0.348659
\(478\) −1.41823e12 −1.24257
\(479\) −1.44103e12 −1.25073 −0.625363 0.780334i \(-0.715049\pi\)
−0.625363 + 0.780334i \(0.715049\pi\)
\(480\) 1.31786e11 0.113314
\(481\) 1.81667e11 0.154748
\(482\) 3.01266e12 2.54237
\(483\) −6.13839e11 −0.513207
\(484\) −1.62627e12 −1.34706
\(485\) 2.09816e11 0.172187
\(486\) 1.25682e11 0.102190
\(487\) 1.91897e12 1.54593 0.772963 0.634452i \(-0.218774\pi\)
0.772963 + 0.634452i \(0.218774\pi\)
\(488\) 7.15412e10 0.0571041
\(489\) −9.88855e11 −0.782065
\(490\) 1.53512e11 0.120298
\(491\) 2.21466e11 0.171965 0.0859827 0.996297i \(-0.472597\pi\)
0.0859827 + 0.996297i \(0.472597\pi\)
\(492\) −1.88268e11 −0.144855
\(493\) −1.60447e12 −1.22326
\(494\) 1.97640e12 1.49315
\(495\) −2.71615e10 −0.0203344
\(496\) 1.54072e11 0.114303
\(497\) −4.48893e11 −0.330019
\(498\) 8.31787e11 0.606010
\(499\) −7.58060e11 −0.547332 −0.273666 0.961825i \(-0.588236\pi\)
−0.273666 + 0.961825i \(0.588236\pi\)
\(500\) 7.33579e11 0.524906
\(501\) −1.18411e11 −0.0839695
\(502\) −3.15978e12 −2.22070
\(503\) 4.35553e10 0.0303378 0.0151689 0.999885i \(-0.495171\pi\)
0.0151689 + 0.999885i \(0.495171\pi\)
\(504\) −4.95493e11 −0.342058
\(505\) 2.93136e11 0.200567
\(506\) −6.13442e11 −0.416003
\(507\) 4.81453e11 0.323607
\(508\) −4.08870e12 −2.72394
\(509\) 2.46695e12 1.62903 0.814516 0.580141i \(-0.197003\pi\)
0.814516 + 0.580141i \(0.197003\pi\)
\(510\) −2.27617e11 −0.148984
\(511\) 8.11954e11 0.526790
\(512\) −5.36197e11 −0.344834
\(513\) 4.26836e11 0.272103
\(514\) 2.32976e12 1.47224
\(515\) 5.82193e10 0.0364699
\(516\) 2.72072e12 1.68951
\(517\) 3.98169e11 0.245109
\(518\) −7.30108e11 −0.445557
\(519\) −8.64593e11 −0.523069
\(520\) 1.64039e11 0.0983855
\(521\) −1.47965e12 −0.879809 −0.439905 0.898045i \(-0.644988\pi\)
−0.439905 + 0.898045i \(0.644988\pi\)
\(522\) 1.17874e12 0.694865
\(523\) 4.59042e9 0.00268284 0.00134142 0.999999i \(-0.499573\pi\)
0.00134142 + 0.999999i \(0.499573\pi\)
\(524\) −3.56687e12 −2.06679
\(525\) 1.16804e12 0.671030
\(526\) −3.08773e12 −1.75875
\(527\) −1.09128e12 −0.616294
\(528\) 6.29284e10 0.0352366
\(529\) −8.09937e11 −0.449677
\(530\) 5.24508e11 0.288743
\(531\) −7.95020e10 −0.0433963
\(532\) −4.81290e12 −2.60498
\(533\) 2.01557e11 0.108175
\(534\) 2.10215e11 0.111874
\(535\) 5.18078e11 0.273403
\(536\) −2.26797e12 −1.18685
\(537\) 9.78911e10 0.0507994
\(538\) −4.51966e12 −2.32587
\(539\) 3.00605e11 0.153407
\(540\) 1.01324e11 0.0512791
\(541\) −1.00252e12 −0.503157 −0.251579 0.967837i \(-0.580950\pi\)
−0.251579 + 0.967837i \(0.580950\pi\)
\(542\) 8.58960e11 0.427539
\(543\) 1.13961e11 0.0562546
\(544\) 2.16259e12 1.05872
\(545\) −4.25564e11 −0.206624
\(546\) 1.51719e12 0.730590
\(547\) 2.37893e12 1.13616 0.568079 0.822974i \(-0.307687\pi\)
0.568079 + 0.822974i \(0.307687\pi\)
\(548\) 4.06036e12 1.92332
\(549\) −4.73091e10 −0.0222264
\(550\) 1.16729e12 0.543934
\(551\) 4.00319e12 1.85022
\(552\) 8.00112e11 0.366796
\(553\) −2.33448e11 −0.106152
\(554\) 3.61737e12 1.63155
\(555\) 5.22012e10 0.0233541
\(556\) 9.46119e11 0.419865
\(557\) −2.81309e11 −0.123833 −0.0619164 0.998081i \(-0.519721\pi\)
−0.0619164 + 0.998081i \(0.519721\pi\)
\(558\) 8.01718e11 0.350081
\(559\) −2.91277e12 −1.26169
\(560\) 8.37812e10 0.0359998
\(561\) −4.45716e11 −0.189988
\(562\) 6.46117e12 2.73211
\(563\) 1.32901e12 0.557495 0.278748 0.960364i \(-0.410081\pi\)
0.278748 + 0.960364i \(0.410081\pi\)
\(564\) −1.48534e12 −0.618115
\(565\) 5.48567e11 0.226471
\(566\) 7.48444e12 3.06539
\(567\) 3.27662e11 0.133138
\(568\) 5.85112e11 0.235869
\(569\) −2.03849e12 −0.815273 −0.407636 0.913144i \(-0.633647\pi\)
−0.407636 + 0.913144i \(0.633647\pi\)
\(570\) 5.67910e11 0.225342
\(571\) −2.71625e12 −1.06932 −0.534659 0.845068i \(-0.679560\pi\)
−0.534659 + 0.845068i \(0.679560\pi\)
\(572\) 9.18715e11 0.358838
\(573\) 8.18425e11 0.317163
\(574\) −8.10044e11 −0.311462
\(575\) −1.88613e12 −0.719560
\(576\) −1.43610e12 −0.543604
\(577\) 1.61395e12 0.606175 0.303087 0.952963i \(-0.401983\pi\)
0.303087 + 0.952963i \(0.401983\pi\)
\(578\) 5.39365e11 0.201005
\(579\) 2.53597e12 0.937756
\(580\) 9.50293e11 0.348684
\(581\) 2.16853e12 0.789538
\(582\) −2.52947e12 −0.913852
\(583\) 1.02708e12 0.368212
\(584\) −1.05835e12 −0.376504
\(585\) −1.08476e11 −0.0382942
\(586\) 4.42078e12 1.54867
\(587\) −8.60925e11 −0.299291 −0.149646 0.988740i \(-0.547813\pi\)
−0.149646 + 0.988740i \(0.547813\pi\)
\(588\) −1.12138e12 −0.386861
\(589\) 2.72277e12 0.932163
\(590\) −1.05778e11 −0.0359387
\(591\) 3.52343e11 0.118802
\(592\) −1.20941e11 −0.0404693
\(593\) 2.08900e11 0.0693732 0.0346866 0.999398i \(-0.488957\pi\)
0.0346866 + 0.999398i \(0.488957\pi\)
\(594\) 3.27450e11 0.107921
\(595\) −5.93414e11 −0.194103
\(596\) 6.25975e12 2.03212
\(597\) 1.77162e12 0.570801
\(598\) −2.44993e12 −0.783427
\(599\) −7.49045e11 −0.237732 −0.118866 0.992910i \(-0.537926\pi\)
−0.118866 + 0.992910i \(0.537926\pi\)
\(600\) −1.52249e12 −0.479595
\(601\) −2.92254e12 −0.913745 −0.456872 0.889532i \(-0.651030\pi\)
−0.456872 + 0.889532i \(0.651030\pi\)
\(602\) 1.17062e13 3.63272
\(603\) 1.49977e12 0.461952
\(604\) −3.17294e12 −0.970054
\(605\) 5.00286e11 0.151817
\(606\) −3.53395e12 −1.06447
\(607\) −2.01355e12 −0.602024 −0.301012 0.953620i \(-0.597324\pi\)
−0.301012 + 0.953620i \(0.597324\pi\)
\(608\) −5.39572e12 −1.60134
\(609\) 3.07306e12 0.905302
\(610\) −6.29453e10 −0.0184068
\(611\) 1.59019e12 0.461596
\(612\) 1.66271e12 0.479109
\(613\) 3.83557e12 1.09713 0.548564 0.836109i \(-0.315175\pi\)
0.548564 + 0.836109i \(0.315175\pi\)
\(614\) −6.96786e12 −1.97853
\(615\) 5.79165e10 0.0163254
\(616\) −1.29095e12 −0.361241
\(617\) 4.53937e12 1.26099 0.630496 0.776192i \(-0.282851\pi\)
0.630496 + 0.776192i \(0.282851\pi\)
\(618\) −7.01871e11 −0.193557
\(619\) 2.84541e11 0.0779000 0.0389500 0.999241i \(-0.487599\pi\)
0.0389500 + 0.999241i \(0.487599\pi\)
\(620\) 6.46341e11 0.175671
\(621\) −5.29102e11 −0.142767
\(622\) 5.11825e10 0.0137109
\(623\) 5.48047e11 0.145754
\(624\) 2.51320e11 0.0663585
\(625\) 3.47447e12 0.910812
\(626\) 3.50663e11 0.0912651
\(627\) 1.11207e12 0.287362
\(628\) 3.52809e12 0.905151
\(629\) 8.56614e11 0.218201
\(630\) 4.35958e11 0.110258
\(631\) −1.03135e12 −0.258985 −0.129492 0.991580i \(-0.541335\pi\)
−0.129492 + 0.991580i \(0.541335\pi\)
\(632\) 3.04289e11 0.0758680
\(633\) −3.58072e12 −0.886448
\(634\) 8.88198e11 0.218327
\(635\) 1.25780e12 0.306994
\(636\) −3.83146e12 −0.928553
\(637\) 1.20054e12 0.288900
\(638\) 3.07107e12 0.733833
\(639\) −3.86925e11 −0.0918064
\(640\) −1.07773e12 −0.253921
\(641\) −8.00081e11 −0.187186 −0.0935929 0.995611i \(-0.529835\pi\)
−0.0935929 + 0.995611i \(0.529835\pi\)
\(642\) −6.24576e12 −1.45104
\(643\) −4.85967e12 −1.12113 −0.560567 0.828109i \(-0.689417\pi\)
−0.560567 + 0.828109i \(0.689417\pi\)
\(644\) 5.96602e12 1.36678
\(645\) −8.36970e11 −0.190411
\(646\) 9.31931e12 2.10541
\(647\) −8.72956e11 −0.195850 −0.0979249 0.995194i \(-0.531220\pi\)
−0.0979249 + 0.995194i \(0.531220\pi\)
\(648\) −4.27093e11 −0.0951556
\(649\) −2.07134e11 −0.0458300
\(650\) 4.66185e12 1.02435
\(651\) 2.09014e12 0.456101
\(652\) 9.61087e12 2.08280
\(653\) −4.17937e12 −0.899500 −0.449750 0.893155i \(-0.648487\pi\)
−0.449750 + 0.893155i \(0.648487\pi\)
\(654\) 5.13044e12 1.09662
\(655\) 1.09727e12 0.232932
\(656\) −1.34182e11 −0.0282896
\(657\) 6.99867e11 0.146545
\(658\) −6.39084e12 −1.32905
\(659\) −4.28518e12 −0.885084 −0.442542 0.896748i \(-0.645923\pi\)
−0.442542 + 0.896748i \(0.645923\pi\)
\(660\) 2.63988e11 0.0541548
\(661\) 5.09397e12 1.03789 0.518943 0.854809i \(-0.326326\pi\)
0.518943 + 0.854809i \(0.326326\pi\)
\(662\) 3.32955e12 0.673790
\(663\) −1.78008e12 −0.357790
\(664\) −2.82658e12 −0.564294
\(665\) 1.48058e12 0.293586
\(666\) −6.29320e11 −0.123947
\(667\) −4.96232e12 −0.970774
\(668\) 1.15086e12 0.223629
\(669\) −4.08580e12 −0.788605
\(670\) 1.99546e12 0.382567
\(671\) −1.23259e11 −0.0234728
\(672\) −4.14204e12 −0.783525
\(673\) −1.75158e12 −0.329125 −0.164563 0.986367i \(-0.552621\pi\)
−0.164563 + 0.986367i \(0.552621\pi\)
\(674\) 1.19060e13 2.22227
\(675\) 1.00680e12 0.186671
\(676\) −4.67933e12 −0.861834
\(677\) 1.69864e12 0.310780 0.155390 0.987853i \(-0.450337\pi\)
0.155390 + 0.987853i \(0.450337\pi\)
\(678\) −6.61333e12 −1.20195
\(679\) −6.59451e12 −1.19061
\(680\) 7.73489e11 0.138728
\(681\) 1.21224e12 0.215987
\(682\) 2.08879e12 0.369713
\(683\) −7.96541e12 −1.40060 −0.700302 0.713847i \(-0.746951\pi\)
−0.700302 + 0.713847i \(0.746951\pi\)
\(684\) −4.14850e12 −0.724667
\(685\) −1.24908e12 −0.216762
\(686\) 6.24685e12 1.07697
\(687\) −1.32576e12 −0.227070
\(688\) 1.93911e12 0.329955
\(689\) 4.10191e12 0.693426
\(690\) −7.03976e11 −0.118233
\(691\) −9.26701e12 −1.54628 −0.773140 0.634235i \(-0.781315\pi\)
−0.773140 + 0.634235i \(0.781315\pi\)
\(692\) 8.40315e12 1.39304
\(693\) 8.53686e11 0.140604
\(694\) 6.62494e11 0.108409
\(695\) −2.91053e11 −0.0473196
\(696\) −4.00560e12 −0.647032
\(697\) 9.50400e11 0.152531
\(698\) −1.30974e13 −2.08850
\(699\) −2.20984e12 −0.350118
\(700\) −1.13524e13 −1.78710
\(701\) −3.48399e12 −0.544937 −0.272469 0.962165i \(-0.587840\pi\)
−0.272469 + 0.962165i \(0.587840\pi\)
\(702\) 1.30775e12 0.203239
\(703\) −2.13727e12 −0.330036
\(704\) −3.74159e12 −0.574090
\(705\) 4.56932e11 0.0696627
\(706\) 5.21649e12 0.790237
\(707\) −9.21327e12 −1.38684
\(708\) 7.72695e11 0.115574
\(709\) 3.47197e12 0.516022 0.258011 0.966142i \(-0.416933\pi\)
0.258011 + 0.966142i \(0.416933\pi\)
\(710\) −5.14808e11 −0.0760297
\(711\) −2.01221e11 −0.0295298
\(712\) −7.14355e11 −0.104173
\(713\) −3.37512e12 −0.489087
\(714\) 7.15399e12 1.03016
\(715\) −2.82623e11 −0.0404417
\(716\) −9.51422e11 −0.135290
\(717\) −3.18702e12 −0.450348
\(718\) 1.62878e13 2.28719
\(719\) −8.56963e12 −1.19587 −0.597933 0.801546i \(-0.704011\pi\)
−0.597933 + 0.801546i \(0.704011\pi\)
\(720\) 7.22156e10 0.0100146
\(721\) −1.82983e12 −0.252175
\(722\) −1.16206e13 −1.59151
\(723\) 6.76999e12 0.921437
\(724\) −1.10761e12 −0.149818
\(725\) 9.44254e12 1.26931
\(726\) −6.03127e12 −0.805738
\(727\) 1.77144e12 0.235191 0.117596 0.993062i \(-0.462481\pi\)
0.117596 + 0.993062i \(0.462481\pi\)
\(728\) −5.15573e12 −0.680298
\(729\) 2.82430e11 0.0370370
\(730\) 9.31182e11 0.121362
\(731\) −1.37345e13 −1.77904
\(732\) 4.59806e11 0.0591936
\(733\) 7.38789e12 0.945262 0.472631 0.881260i \(-0.343304\pi\)
0.472631 + 0.881260i \(0.343304\pi\)
\(734\) 7.18161e12 0.913250
\(735\) 3.44969e11 0.0436000
\(736\) 6.68849e12 0.840190
\(737\) 3.90749e12 0.487859
\(738\) −6.98221e11 −0.0866442
\(739\) −5.26699e12 −0.649624 −0.324812 0.945779i \(-0.605301\pi\)
−0.324812 + 0.945779i \(0.605301\pi\)
\(740\) −5.07354e11 −0.0621968
\(741\) 4.44134e12 0.541168
\(742\) −1.64853e13 −1.99654
\(743\) 8.11827e12 0.977269 0.488634 0.872489i \(-0.337495\pi\)
0.488634 + 0.872489i \(0.337495\pi\)
\(744\) −2.72440e12 −0.325982
\(745\) −1.92568e12 −0.229024
\(746\) −1.76249e13 −2.08354
\(747\) 1.86917e12 0.219638
\(748\) 4.33200e12 0.505978
\(749\) −1.62832e13 −1.89047
\(750\) 2.72059e12 0.313970
\(751\) −6.42270e12 −0.736780 −0.368390 0.929671i \(-0.620091\pi\)
−0.368390 + 0.929671i \(0.620091\pi\)
\(752\) −1.05863e12 −0.120716
\(753\) −7.10061e12 −0.804856
\(754\) 1.22651e13 1.38197
\(755\) 9.76087e11 0.109327
\(756\) −3.18461e12 −0.354575
\(757\) −1.29515e13 −1.43347 −0.716736 0.697344i \(-0.754365\pi\)
−0.716736 + 0.697344i \(0.754365\pi\)
\(758\) −9.10789e12 −1.00209
\(759\) −1.37852e12 −0.150773
\(760\) −1.92988e12 −0.209830
\(761\) 1.45543e13 1.57311 0.786556 0.617519i \(-0.211862\pi\)
0.786556 + 0.617519i \(0.211862\pi\)
\(762\) −1.51636e13 −1.62931
\(763\) 1.33754e13 1.42872
\(764\) −7.95443e12 −0.844674
\(765\) −5.11496e11 −0.0539966
\(766\) 2.68288e13 2.81561
\(767\) −8.27239e11 −0.0863081
\(768\) 3.91512e12 0.406087
\(769\) −1.48958e13 −1.53601 −0.768006 0.640443i \(-0.778751\pi\)
−0.768006 + 0.640443i \(0.778751\pi\)
\(770\) 1.13584e12 0.116442
\(771\) 5.23539e12 0.533586
\(772\) −2.46475e13 −2.49744
\(773\) −1.43291e13 −1.44349 −0.721743 0.692162i \(-0.756659\pi\)
−0.721743 + 0.692162i \(0.756659\pi\)
\(774\) 1.00902e13 1.01057
\(775\) 6.42234e12 0.639492
\(776\) 8.59565e12 0.850944
\(777\) −1.64068e12 −0.161484
\(778\) 2.18685e13 2.13999
\(779\) −2.37127e12 −0.230708
\(780\) 1.05430e12 0.101986
\(781\) −1.00809e12 −0.0969549
\(782\) −1.15521e13 −1.10467
\(783\) 2.64884e12 0.251842
\(784\) −7.99230e11 −0.0755527
\(785\) −1.08534e12 −0.102012
\(786\) −1.32283e13 −1.23624
\(787\) 1.13054e12 0.105051 0.0525255 0.998620i \(-0.483273\pi\)
0.0525255 + 0.998620i \(0.483273\pi\)
\(788\) −3.42449e12 −0.316394
\(789\) −6.93868e12 −0.637427
\(790\) −2.67727e11 −0.0244552
\(791\) −1.72414e13 −1.56596
\(792\) −1.11274e12 −0.100492
\(793\) −4.92263e11 −0.0442046
\(794\) 4.53094e12 0.404572
\(795\) 1.17866e12 0.104650
\(796\) −1.72187e13 −1.52017
\(797\) −2.22807e13 −1.95599 −0.977996 0.208622i \(-0.933102\pi\)
−0.977996 + 0.208622i \(0.933102\pi\)
\(798\) −1.78494e13 −1.55815
\(799\) 7.49818e12 0.650871
\(800\) −1.27272e13 −1.09857
\(801\) 4.72391e11 0.0405467
\(802\) −1.79055e13 −1.52828
\(803\) 1.82343e12 0.154763
\(804\) −1.45766e13 −1.23028
\(805\) −1.83532e12 −0.154039
\(806\) 8.34209e12 0.696253
\(807\) −1.01565e13 −0.842971
\(808\) 1.20091e13 0.991195
\(809\) −7.94931e12 −0.652470 −0.326235 0.945289i \(-0.605780\pi\)
−0.326235 + 0.945289i \(0.605780\pi\)
\(810\) 3.75776e11 0.0306723
\(811\) −5.75368e12 −0.467038 −0.233519 0.972352i \(-0.575024\pi\)
−0.233519 + 0.972352i \(0.575024\pi\)
\(812\) −2.98677e13 −2.41101
\(813\) 1.93024e12 0.154954
\(814\) −1.63962e12 −0.130898
\(815\) −2.95658e12 −0.234736
\(816\) 1.18505e12 0.0935684
\(817\) 3.42680e13 2.69085
\(818\) 3.16128e13 2.46873
\(819\) 3.40940e12 0.264790
\(820\) −5.62902e11 −0.0434781
\(821\) 3.18644e12 0.244772 0.122386 0.992483i \(-0.460945\pi\)
0.122386 + 0.992483i \(0.460945\pi\)
\(822\) 1.50585e13 1.15043
\(823\) −9.54724e12 −0.725401 −0.362701 0.931906i \(-0.618145\pi\)
−0.362701 + 0.931906i \(0.618145\pi\)
\(824\) 2.38510e12 0.180233
\(825\) 2.62311e12 0.197139
\(826\) 3.32461e12 0.248502
\(827\) −6.39294e12 −0.475254 −0.237627 0.971356i \(-0.576370\pi\)
−0.237627 + 0.971356i \(0.576370\pi\)
\(828\) 5.14244e12 0.380218
\(829\) 2.35682e13 1.73313 0.866563 0.499068i \(-0.166324\pi\)
0.866563 + 0.499068i \(0.166324\pi\)
\(830\) 2.48696e12 0.181893
\(831\) 8.12889e12 0.591325
\(832\) −1.49430e13 −1.08114
\(833\) 5.66088e12 0.407363
\(834\) 3.50883e12 0.251140
\(835\) −3.54037e11 −0.0252034
\(836\) −1.08085e13 −0.765307
\(837\) 1.80161e12 0.126881
\(838\) 4.68841e12 0.328418
\(839\) −6.86890e12 −0.478584 −0.239292 0.970948i \(-0.576915\pi\)
−0.239292 + 0.970948i \(0.576915\pi\)
\(840\) −1.48147e12 −0.102669
\(841\) 1.03357e13 0.712454
\(842\) 3.44560e13 2.36244
\(843\) 1.45194e13 0.990204
\(844\) 3.48017e13 2.36080
\(845\) 1.43950e12 0.0971304
\(846\) −5.50861e12 −0.369722
\(847\) −1.57240e13 −1.04975
\(848\) −2.73075e12 −0.181343
\(849\) 1.68189e13 1.11100
\(850\) 2.19820e13 1.44438
\(851\) 2.64934e12 0.173163
\(852\) 3.76060e12 0.244500
\(853\) 1.02548e13 0.663221 0.331610 0.943416i \(-0.392408\pi\)
0.331610 + 0.943416i \(0.392408\pi\)
\(854\) 1.97837e12 0.127276
\(855\) 1.27620e12 0.0816714
\(856\) 2.12244e13 1.35115
\(857\) 1.05349e13 0.667139 0.333570 0.942725i \(-0.391747\pi\)
0.333570 + 0.942725i \(0.391747\pi\)
\(858\) 3.40720e12 0.214637
\(859\) 9.86201e12 0.618011 0.309006 0.951060i \(-0.400004\pi\)
0.309006 + 0.951060i \(0.400004\pi\)
\(860\) 8.13468e12 0.507104
\(861\) −1.82031e12 −0.112884
\(862\) −1.83699e13 −1.13324
\(863\) 1.13582e13 0.697044 0.348522 0.937301i \(-0.386684\pi\)
0.348522 + 0.937301i \(0.386684\pi\)
\(864\) −3.57025e12 −0.217965
\(865\) −2.58505e12 −0.156999
\(866\) 2.89873e12 0.175137
\(867\) 1.21205e12 0.0728508
\(868\) −2.03145e13 −1.21469
\(869\) −5.24260e11 −0.0311858
\(870\) 3.52431e12 0.208563
\(871\) 1.56055e13 0.918747
\(872\) −1.74343e13 −1.02113
\(873\) −5.68417e12 −0.331210
\(874\) 2.88229e13 1.67084
\(875\) 7.09279e12 0.409054
\(876\) −6.80215e12 −0.390281
\(877\) −2.38147e12 −0.135940 −0.0679699 0.997687i \(-0.521652\pi\)
−0.0679699 + 0.997687i \(0.521652\pi\)
\(878\) −4.43164e12 −0.251674
\(879\) 9.93429e12 0.561289
\(880\) 1.88150e11 0.0105762
\(881\) −1.42926e13 −0.799321 −0.399660 0.916663i \(-0.630872\pi\)
−0.399660 + 0.916663i \(0.630872\pi\)
\(882\) −4.15882e12 −0.231399
\(883\) −2.73680e13 −1.51502 −0.757512 0.652822i \(-0.773585\pi\)
−0.757512 + 0.652822i \(0.773585\pi\)
\(884\) 1.73009e13 0.952870
\(885\) −2.37703e11 −0.0130254
\(886\) 3.49573e13 1.90584
\(887\) −2.10999e13 −1.14452 −0.572260 0.820072i \(-0.693933\pi\)
−0.572260 + 0.820072i \(0.693933\pi\)
\(888\) 2.13856e12 0.115415
\(889\) −3.95326e13 −2.12274
\(890\) 6.28522e11 0.0335788
\(891\) 7.35839e11 0.0391141
\(892\) 3.97107e13 2.10022
\(893\) −1.87081e13 −0.984463
\(894\) 2.32153e13 1.21550
\(895\) 2.92685e11 0.0152474
\(896\) 3.38729e13 1.75576
\(897\) −5.50544e12 −0.283939
\(898\) 3.18740e13 1.63566
\(899\) 1.68968e13 0.862753
\(900\) −9.78529e12 −0.497144
\(901\) 1.93417e13 0.977761
\(902\) −1.81914e12 −0.0915032
\(903\) 2.63060e13 1.31662
\(904\) 2.24735e13 1.11921
\(905\) 3.40733e11 0.0168848
\(906\) −1.17674e13 −0.580233
\(907\) −1.82296e13 −0.894428 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(908\) −1.17820e13 −0.575220
\(909\) −7.94142e12 −0.385799
\(910\) 4.53625e12 0.219286
\(911\) −2.40842e13 −1.15851 −0.579256 0.815146i \(-0.696657\pi\)
−0.579256 + 0.815146i \(0.696657\pi\)
\(912\) −2.95672e12 −0.141525
\(913\) 4.86993e12 0.231955
\(914\) −1.73451e13 −0.822089
\(915\) −1.41449e11 −0.00667123
\(916\) 1.28853e13 0.604736
\(917\) −3.44872e13 −1.61063
\(918\) 6.16642e12 0.286577
\(919\) 2.76663e13 1.27947 0.639737 0.768594i \(-0.279043\pi\)
0.639737 + 0.768594i \(0.279043\pi\)
\(920\) 2.39226e12 0.110094
\(921\) −1.56580e13 −0.717083
\(922\) −5.00901e13 −2.28277
\(923\) −4.02605e12 −0.182588
\(924\) −8.29714e12 −0.374459
\(925\) −5.04130e12 −0.226415
\(926\) 1.56875e13 0.701139
\(927\) −1.57723e12 −0.0701514
\(928\) −3.34845e13 −1.48210
\(929\) 2.01602e13 0.888024 0.444012 0.896021i \(-0.353555\pi\)
0.444012 + 0.896021i \(0.353555\pi\)
\(930\) 2.39706e12 0.105076
\(931\) −1.41240e13 −0.616148
\(932\) 2.14779e13 0.932438
\(933\) 1.15016e11 0.00496926
\(934\) −3.60103e12 −0.154834
\(935\) −1.33265e12 −0.0570247
\(936\) −4.44401e12 −0.189249
\(937\) 1.74977e13 0.741569 0.370785 0.928719i \(-0.379089\pi\)
0.370785 + 0.928719i \(0.379089\pi\)
\(938\) −6.27174e13 −2.64530
\(939\) 7.88002e11 0.0330775
\(940\) −4.44101e12 −0.185527
\(941\) 6.82182e12 0.283627 0.141813 0.989893i \(-0.454707\pi\)
0.141813 + 0.989893i \(0.454707\pi\)
\(942\) 1.30845e13 0.541411
\(943\) 2.93941e12 0.121048
\(944\) 5.50715e11 0.0225711
\(945\) 9.79676e11 0.0399613
\(946\) 2.62890e13 1.06724
\(947\) −1.82410e13 −0.737011 −0.368506 0.929625i \(-0.620131\pi\)
−0.368506 + 0.929625i \(0.620131\pi\)
\(948\) 1.95571e12 0.0786442
\(949\) 7.28230e12 0.291454
\(950\) −5.48456e13 −2.18467
\(951\) 1.99594e12 0.0791289
\(952\) −2.43107e13 −0.959250
\(953\) 3.07532e12 0.120774 0.0603868 0.998175i \(-0.480767\pi\)
0.0603868 + 0.998175i \(0.480767\pi\)
\(954\) −1.42096e13 −0.555409
\(955\) 2.44701e12 0.0951964
\(956\) 3.09752e13 1.19937
\(957\) 6.90126e12 0.265965
\(958\) 5.19421e13 1.99239
\(959\) 3.92587e13 1.49883
\(960\) −4.29379e12 −0.163162
\(961\) −1.49473e13 −0.565335
\(962\) −6.54823e12 −0.246511
\(963\) −1.40354e13 −0.525902
\(964\) −6.57989e13 −2.45398
\(965\) 7.58228e12 0.281467
\(966\) 2.21259e13 0.817532
\(967\) −3.97010e12 −0.146010 −0.0730049 0.997332i \(-0.523259\pi\)
−0.0730049 + 0.997332i \(0.523259\pi\)
\(968\) 2.04955e13 0.750273
\(969\) 2.09422e13 0.763071
\(970\) −7.56285e12 −0.274292
\(971\) 4.95697e12 0.178949 0.0894746 0.995989i \(-0.471481\pi\)
0.0894746 + 0.995989i \(0.471481\pi\)
\(972\) −2.74499e12 −0.0986375
\(973\) 9.14779e12 0.327196
\(974\) −6.91697e13 −2.46264
\(975\) 1.04760e13 0.371258
\(976\) 3.27713e11 0.0115603
\(977\) 2.85651e13 1.00302 0.501510 0.865152i \(-0.332778\pi\)
0.501510 + 0.865152i \(0.332778\pi\)
\(978\) 3.56434e13 1.24582
\(979\) 1.23076e12 0.0428206
\(980\) −3.35282e12 −0.116116
\(981\) 1.15290e13 0.397450
\(982\) −7.98279e12 −0.273938
\(983\) 7.23756e12 0.247230 0.123615 0.992330i \(-0.460551\pi\)
0.123615 + 0.992330i \(0.460551\pi\)
\(984\) 2.37270e12 0.0806798
\(985\) 1.05347e12 0.0356582
\(986\) 5.78334e13 1.94864
\(987\) −1.43614e13 −0.481691
\(988\) −4.31662e13 −1.44124
\(989\) −4.24783e13 −1.41184
\(990\) 9.79042e11 0.0323924
\(991\) −2.56375e13 −0.844393 −0.422196 0.906504i \(-0.638741\pi\)
−0.422196 + 0.906504i \(0.638741\pi\)
\(992\) −2.27745e13 −0.746700
\(993\) 7.48210e12 0.244204
\(994\) 1.61804e13 0.525716
\(995\) 5.29695e12 0.171326
\(996\) −1.81669e13 −0.584942
\(997\) 9.24324e12 0.296276 0.148138 0.988967i \(-0.452672\pi\)
0.148138 + 0.988967i \(0.452672\pi\)
\(998\) 2.73244e13 0.871893
\(999\) −1.41419e12 −0.0449226
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.10.a.b.1.3 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.b.1.3 21 1.1 even 1 trivial