Properties

Label 177.8.a.d.1.3
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + 3375594921 x^{11} + 115310342333 x^{10} - 774932111214 x^{9} - 14600047830166 x^{8} + 102185027148481 x^{7} + 988557475638619 x^{6} - 7379206238519716 x^{5} - 30152342836849520 x^{4} + 260578770749067175 x^{3} + 182609347488069978 x^{2} - 3481290425710753600 x + 5164646074739714048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-17.7235\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-16.7235 q^{2} +27.0000 q^{3} +151.676 q^{4} -298.819 q^{5} -451.535 q^{6} +200.095 q^{7} -395.941 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-16.7235 q^{2} +27.0000 q^{3} +151.676 q^{4} -298.819 q^{5} -451.535 q^{6} +200.095 q^{7} -395.941 q^{8} +729.000 q^{9} +4997.30 q^{10} +6819.80 q^{11} +4095.24 q^{12} -7653.89 q^{13} -3346.30 q^{14} -8068.11 q^{15} -12793.0 q^{16} -23868.2 q^{17} -12191.4 q^{18} +41453.1 q^{19} -45323.6 q^{20} +5402.58 q^{21} -114051. q^{22} +87045.3 q^{23} -10690.4 q^{24} +11167.7 q^{25} +128000. q^{26} +19683.0 q^{27} +30349.6 q^{28} -28849.4 q^{29} +134927. q^{30} -136082. q^{31} +264624. q^{32} +184135. q^{33} +399161. q^{34} -59792.3 q^{35} +110572. q^{36} -238332. q^{37} -693241. q^{38} -206655. q^{39} +118315. q^{40} -387202. q^{41} -90350.0 q^{42} -560156. q^{43} +1.03440e6 q^{44} -217839. q^{45} -1.45570e6 q^{46} +343766. q^{47} -345410. q^{48} -783505. q^{49} -186763. q^{50} -644443. q^{51} -1.16091e6 q^{52} +1.90828e6 q^{53} -329169. q^{54} -2.03788e6 q^{55} -79226.0 q^{56} +1.11923e6 q^{57} +482464. q^{58} +205379. q^{59} -1.22374e6 q^{60} +2.52219e6 q^{61} +2.27577e6 q^{62} +145870. q^{63} -2.78794e6 q^{64} +2.28713e6 q^{65} -3.07938e6 q^{66} +1.23640e6 q^{67} -3.62023e6 q^{68} +2.35022e6 q^{69} +999937. q^{70} -5.34185e6 q^{71} -288641. q^{72} -4.51521e6 q^{73} +3.98574e6 q^{74} +301528. q^{75} +6.28743e6 q^{76} +1.36461e6 q^{77} +3.45600e6 q^{78} -1.40570e6 q^{79} +3.82278e6 q^{80} +531441. q^{81} +6.47538e6 q^{82} +8.95946e6 q^{83} +819440. q^{84} +7.13228e6 q^{85} +9.36777e6 q^{86} -778935. q^{87} -2.70024e6 q^{88} -1.30962e6 q^{89} +3.64303e6 q^{90} -1.53151e6 q^{91} +1.32027e7 q^{92} -3.67422e6 q^{93} -5.74897e6 q^{94} -1.23870e7 q^{95} +7.14484e6 q^{96} +1.27503e7 q^{97} +1.31029e7 q^{98} +4.97163e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.7235 −1.47816 −0.739082 0.673616i \(-0.764740\pi\)
−0.739082 + 0.673616i \(0.764740\pi\)
\(3\) 27.0000 0.577350
\(4\) 151.676 1.18497
\(5\) −298.819 −1.06909 −0.534543 0.845141i \(-0.679516\pi\)
−0.534543 + 0.845141i \(0.679516\pi\)
\(6\) −451.535 −0.853418
\(7\) 200.095 0.220493 0.110246 0.993904i \(-0.464836\pi\)
0.110246 + 0.993904i \(0.464836\pi\)
\(8\) −395.941 −0.273411
\(9\) 729.000 0.333333
\(10\) 4997.30 1.58028
\(11\) 6819.80 1.54489 0.772444 0.635082i \(-0.219034\pi\)
0.772444 + 0.635082i \(0.219034\pi\)
\(12\) 4095.24 0.684141
\(13\) −7653.89 −0.966230 −0.483115 0.875557i \(-0.660495\pi\)
−0.483115 + 0.875557i \(0.660495\pi\)
\(14\) −3346.30 −0.325924
\(15\) −8068.11 −0.617238
\(16\) −12793.0 −0.780821
\(17\) −23868.2 −1.17828 −0.589141 0.808030i \(-0.700534\pi\)
−0.589141 + 0.808030i \(0.700534\pi\)
\(18\) −12191.4 −0.492721
\(19\) 41453.1 1.38650 0.693249 0.720698i \(-0.256179\pi\)
0.693249 + 0.720698i \(0.256179\pi\)
\(20\) −45323.6 −1.26683
\(21\) 5402.58 0.127301
\(22\) −114051. −2.28360
\(23\) 87045.3 1.49176 0.745878 0.666082i \(-0.232030\pi\)
0.745878 + 0.666082i \(0.232030\pi\)
\(24\) −10690.4 −0.157854
\(25\) 11167.7 0.142947
\(26\) 128000. 1.42825
\(27\) 19683.0 0.192450
\(28\) 30349.6 0.261276
\(29\) −28849.4 −0.219657 −0.109828 0.993951i \(-0.535030\pi\)
−0.109828 + 0.993951i \(0.535030\pi\)
\(30\) 134927. 0.912378
\(31\) −136082. −0.820419 −0.410209 0.911991i \(-0.634544\pi\)
−0.410209 + 0.911991i \(0.634544\pi\)
\(32\) 264624. 1.42759
\(33\) 184135. 0.891942
\(34\) 399161. 1.74169
\(35\) −59792.3 −0.235726
\(36\) 110572. 0.394989
\(37\) −238332. −0.773527 −0.386764 0.922179i \(-0.626407\pi\)
−0.386764 + 0.922179i \(0.626407\pi\)
\(38\) −693241. −2.04947
\(39\) −206655. −0.557853
\(40\) 118315. 0.292300
\(41\) −387202. −0.877394 −0.438697 0.898635i \(-0.644560\pi\)
−0.438697 + 0.898635i \(0.644560\pi\)
\(42\) −90350.0 −0.188172
\(43\) −560156. −1.07441 −0.537204 0.843452i \(-0.680519\pi\)
−0.537204 + 0.843452i \(0.680519\pi\)
\(44\) 1.03440e6 1.83064
\(45\) −217839. −0.356362
\(46\) −1.45570e6 −2.20506
\(47\) 343766. 0.482970 0.241485 0.970405i \(-0.422365\pi\)
0.241485 + 0.970405i \(0.422365\pi\)
\(48\) −345410. −0.450807
\(49\) −783505. −0.951383
\(50\) −186763. −0.211298
\(51\) −644443. −0.680281
\(52\) −1.16091e6 −1.14495
\(53\) 1.90828e6 1.76066 0.880332 0.474358i \(-0.157320\pi\)
0.880332 + 0.474358i \(0.157320\pi\)
\(54\) −329169. −0.284473
\(55\) −2.03788e6 −1.65162
\(56\) −79226.0 −0.0602851
\(57\) 1.11923e6 0.800495
\(58\) 482464. 0.324689
\(59\) 205379. 0.130189
\(60\) −1.22374e6 −0.731406
\(61\) 2.52219e6 1.42273 0.711366 0.702822i \(-0.248077\pi\)
0.711366 + 0.702822i \(0.248077\pi\)
\(62\) 2.27577e6 1.21271
\(63\) 145870. 0.0734975
\(64\) −2.78794e6 −1.32939
\(65\) 2.28713e6 1.03298
\(66\) −3.07938e6 −1.31844
\(67\) 1.23640e6 0.502224 0.251112 0.967958i \(-0.419204\pi\)
0.251112 + 0.967958i \(0.419204\pi\)
\(68\) −3.62023e6 −1.39622
\(69\) 2.35022e6 0.861266
\(70\) 999937. 0.348441
\(71\) −5.34185e6 −1.77128 −0.885640 0.464373i \(-0.846280\pi\)
−0.885640 + 0.464373i \(0.846280\pi\)
\(72\) −288641. −0.0911369
\(73\) −4.51521e6 −1.35846 −0.679232 0.733923i \(-0.737687\pi\)
−0.679232 + 0.733923i \(0.737687\pi\)
\(74\) 3.98574e6 1.14340
\(75\) 301528. 0.0825303
\(76\) 6.28743e6 1.64295
\(77\) 1.36461e6 0.340637
\(78\) 3.45600e6 0.824598
\(79\) −1.40570e6 −0.320772 −0.160386 0.987054i \(-0.551274\pi\)
−0.160386 + 0.987054i \(0.551274\pi\)
\(80\) 3.82278e6 0.834765
\(81\) 531441. 0.111111
\(82\) 6.47538e6 1.29693
\(83\) 8.95946e6 1.71992 0.859961 0.510361i \(-0.170488\pi\)
0.859961 + 0.510361i \(0.170488\pi\)
\(84\) 819440. 0.150848
\(85\) 7.13228e6 1.25969
\(86\) 9.36777e6 1.58815
\(87\) −778935. −0.126819
\(88\) −2.70024e6 −0.422389
\(89\) −1.30962e6 −0.196916 −0.0984581 0.995141i \(-0.531391\pi\)
−0.0984581 + 0.995141i \(0.531391\pi\)
\(90\) 3.64303e6 0.526762
\(91\) −1.53151e6 −0.213047
\(92\) 1.32027e7 1.76768
\(93\) −3.67422e6 −0.473669
\(94\) −5.74897e6 −0.713909
\(95\) −1.23870e7 −1.48229
\(96\) 7.14484e6 0.824220
\(97\) 1.27503e7 1.41847 0.709236 0.704971i \(-0.249040\pi\)
0.709236 + 0.704971i \(0.249040\pi\)
\(98\) 1.31029e7 1.40630
\(99\) 4.97163e6 0.514963
\(100\) 1.69387e6 0.169387
\(101\) 1.88135e7 1.81696 0.908478 0.417933i \(-0.137245\pi\)
0.908478 + 0.417933i \(0.137245\pi\)
\(102\) 1.07773e7 1.00557
\(103\) 1.54829e7 1.39611 0.698057 0.716042i \(-0.254048\pi\)
0.698057 + 0.716042i \(0.254048\pi\)
\(104\) 3.03049e6 0.264178
\(105\) −1.61439e6 −0.136096
\(106\) −3.19131e7 −2.60255
\(107\) 2.71022e6 0.213876 0.106938 0.994266i \(-0.465895\pi\)
0.106938 + 0.994266i \(0.465895\pi\)
\(108\) 2.98543e6 0.228047
\(109\) 4.25363e6 0.314606 0.157303 0.987550i \(-0.449720\pi\)
0.157303 + 0.987550i \(0.449720\pi\)
\(110\) 3.40806e7 2.44136
\(111\) −6.43495e6 −0.446596
\(112\) −2.55981e6 −0.172165
\(113\) 1.36740e7 0.891501 0.445750 0.895157i \(-0.352937\pi\)
0.445750 + 0.895157i \(0.352937\pi\)
\(114\) −1.87175e7 −1.18326
\(115\) −2.60108e7 −1.59482
\(116\) −4.37576e6 −0.260286
\(117\) −5.57969e6 −0.322077
\(118\) −3.43466e6 −0.192440
\(119\) −4.77593e6 −0.259802
\(120\) 3.19450e6 0.168759
\(121\) 2.70225e7 1.38668
\(122\) −4.21798e7 −2.10303
\(123\) −1.04545e7 −0.506563
\(124\) −2.06404e7 −0.972169
\(125\) 2.00081e7 0.916264
\(126\) −2.43945e6 −0.108641
\(127\) 3.13310e7 1.35726 0.678628 0.734482i \(-0.262575\pi\)
0.678628 + 0.734482i \(0.262575\pi\)
\(128\) 1.27523e7 0.537468
\(129\) −1.51242e7 −0.620309
\(130\) −3.82488e7 −1.52692
\(131\) −2.66364e6 −0.103520 −0.0517602 0.998660i \(-0.516483\pi\)
−0.0517602 + 0.998660i \(0.516483\pi\)
\(132\) 2.79287e7 1.05692
\(133\) 8.29457e6 0.305713
\(134\) −2.06770e7 −0.742369
\(135\) −5.88165e6 −0.205746
\(136\) 9.45042e6 0.322155
\(137\) 2.83551e6 0.0942126 0.0471063 0.998890i \(-0.485000\pi\)
0.0471063 + 0.998890i \(0.485000\pi\)
\(138\) −3.93040e7 −1.27309
\(139\) 3.75740e7 1.18669 0.593343 0.804950i \(-0.297808\pi\)
0.593343 + 0.804950i \(0.297808\pi\)
\(140\) −9.06904e6 −0.279327
\(141\) 9.28168e6 0.278843
\(142\) 8.93344e7 2.61824
\(143\) −5.21980e7 −1.49272
\(144\) −9.32607e6 −0.260274
\(145\) 8.62076e6 0.234832
\(146\) 7.55102e7 2.00803
\(147\) −2.11546e7 −0.549281
\(148\) −3.61491e7 −0.916604
\(149\) 2.26234e7 0.560280 0.280140 0.959959i \(-0.409619\pi\)
0.280140 + 0.959959i \(0.409619\pi\)
\(150\) −5.04261e6 −0.121993
\(151\) −2.07436e7 −0.490303 −0.245152 0.969485i \(-0.578838\pi\)
−0.245152 + 0.969485i \(0.578838\pi\)
\(152\) −1.64130e7 −0.379084
\(153\) −1.74000e7 −0.392761
\(154\) −2.28211e7 −0.503516
\(155\) 4.06639e7 0.877099
\(156\) −3.13446e7 −0.661037
\(157\) −5.37534e7 −1.10855 −0.554277 0.832332i \(-0.687005\pi\)
−0.554277 + 0.832332i \(0.687005\pi\)
\(158\) 2.35082e7 0.474154
\(159\) 5.15236e7 1.01652
\(160\) −7.90746e7 −1.52622
\(161\) 1.74174e7 0.328921
\(162\) −8.88756e6 −0.164240
\(163\) 2.39837e7 0.433769 0.216885 0.976197i \(-0.430410\pi\)
0.216885 + 0.976197i \(0.430410\pi\)
\(164\) −5.87292e7 −1.03968
\(165\) −5.50229e7 −0.953563
\(166\) −1.49834e8 −2.54232
\(167\) 7.25545e7 1.20547 0.602736 0.797941i \(-0.294077\pi\)
0.602736 + 0.797941i \(0.294077\pi\)
\(168\) −2.13910e6 −0.0348056
\(169\) −4.16645e6 −0.0663992
\(170\) −1.19277e8 −1.86202
\(171\) 3.02193e7 0.462166
\(172\) −8.49620e7 −1.27314
\(173\) 8.99082e7 1.32020 0.660098 0.751180i \(-0.270515\pi\)
0.660098 + 0.751180i \(0.270515\pi\)
\(174\) 1.30265e7 0.187459
\(175\) 2.23461e6 0.0315187
\(176\) −8.72455e7 −1.20628
\(177\) 5.54523e6 0.0751646
\(178\) 2.19015e7 0.291074
\(179\) 2.66685e7 0.347547 0.173774 0.984786i \(-0.444404\pi\)
0.173774 + 0.984786i \(0.444404\pi\)
\(180\) −3.30409e7 −0.422277
\(181\) −1.27888e8 −1.60308 −0.801539 0.597943i \(-0.795985\pi\)
−0.801539 + 0.597943i \(0.795985\pi\)
\(182\) 2.56122e7 0.314918
\(183\) 6.80991e7 0.821414
\(184\) −3.44648e7 −0.407862
\(185\) 7.12180e7 0.826968
\(186\) 6.14459e7 0.700160
\(187\) −1.62777e8 −1.82031
\(188\) 5.21410e7 0.572304
\(189\) 3.93848e6 0.0424338
\(190\) 2.07153e8 2.19106
\(191\) −3.33882e7 −0.346718 −0.173359 0.984859i \(-0.555462\pi\)
−0.173359 + 0.984859i \(0.555462\pi\)
\(192\) −7.52743e7 −0.767525
\(193\) 6.61151e7 0.661988 0.330994 0.943633i \(-0.392616\pi\)
0.330994 + 0.943633i \(0.392616\pi\)
\(194\) −2.13230e8 −2.09673
\(195\) 6.17524e7 0.596394
\(196\) −1.18839e8 −1.12736
\(197\) −7.50492e6 −0.0699381 −0.0349691 0.999388i \(-0.511133\pi\)
−0.0349691 + 0.999388i \(0.511133\pi\)
\(198\) −8.31432e7 −0.761199
\(199\) 2.88579e7 0.259584 0.129792 0.991541i \(-0.458569\pi\)
0.129792 + 0.991541i \(0.458569\pi\)
\(200\) −4.42175e6 −0.0390832
\(201\) 3.33828e7 0.289959
\(202\) −3.14627e8 −2.68576
\(203\) −5.77264e6 −0.0484327
\(204\) −9.77463e7 −0.806111
\(205\) 1.15703e8 0.938010
\(206\) −2.58928e8 −2.06368
\(207\) 6.34560e7 0.497252
\(208\) 9.79160e7 0.754453
\(209\) 2.82702e8 2.14199
\(210\) 2.69983e7 0.201173
\(211\) −1.61521e8 −1.18370 −0.591849 0.806049i \(-0.701602\pi\)
−0.591849 + 0.806049i \(0.701602\pi\)
\(212\) 2.89440e8 2.08633
\(213\) −1.44230e8 −1.02265
\(214\) −4.53245e7 −0.316144
\(215\) 1.67385e8 1.14863
\(216\) −7.79331e6 −0.0526179
\(217\) −2.72294e7 −0.180896
\(218\) −7.11356e7 −0.465039
\(219\) −1.21911e8 −0.784310
\(220\) −3.09098e8 −1.95711
\(221\) 1.82685e8 1.13849
\(222\) 1.07615e8 0.660142
\(223\) 1.37272e8 0.828924 0.414462 0.910067i \(-0.363970\pi\)
0.414462 + 0.910067i \(0.363970\pi\)
\(224\) 5.29500e7 0.314773
\(225\) 8.14126e6 0.0476489
\(226\) −2.28678e8 −1.31778
\(227\) −5.03590e7 −0.285750 −0.142875 0.989741i \(-0.545635\pi\)
−0.142875 + 0.989741i \(0.545635\pi\)
\(228\) 1.69761e8 0.948560
\(229\) 3.29634e8 1.81388 0.906938 0.421264i \(-0.138413\pi\)
0.906938 + 0.421264i \(0.138413\pi\)
\(230\) 4.34991e8 2.35740
\(231\) 3.68445e7 0.196667
\(232\) 1.14227e7 0.0600565
\(233\) 4.35459e7 0.225529 0.112764 0.993622i \(-0.464029\pi\)
0.112764 + 0.993622i \(0.464029\pi\)
\(234\) 9.33119e7 0.476082
\(235\) −1.02724e8 −0.516337
\(236\) 3.11510e7 0.154270
\(237\) −3.79538e7 −0.185198
\(238\) 7.98703e7 0.384030
\(239\) −1.16895e8 −0.553865 −0.276933 0.960889i \(-0.589318\pi\)
−0.276933 + 0.960889i \(0.589318\pi\)
\(240\) 1.03215e8 0.481952
\(241\) −3.76459e8 −1.73244 −0.866219 0.499664i \(-0.833457\pi\)
−0.866219 + 0.499664i \(0.833457\pi\)
\(242\) −4.51911e8 −2.04974
\(243\) 1.43489e7 0.0641500
\(244\) 3.82555e8 1.68589
\(245\) 2.34126e8 1.01711
\(246\) 1.74835e8 0.748783
\(247\) −3.17277e8 −1.33968
\(248\) 5.38806e7 0.224311
\(249\) 2.41905e8 0.992997
\(250\) −3.34606e8 −1.35439
\(251\) 3.72328e8 1.48617 0.743084 0.669198i \(-0.233362\pi\)
0.743084 + 0.669198i \(0.233362\pi\)
\(252\) 2.21249e7 0.0870921
\(253\) 5.93631e8 2.30460
\(254\) −5.23965e8 −2.00625
\(255\) 1.92572e8 0.727280
\(256\) 1.43594e8 0.534928
\(257\) −4.73110e8 −1.73859 −0.869294 0.494296i \(-0.835426\pi\)
−0.869294 + 0.494296i \(0.835426\pi\)
\(258\) 2.52930e8 0.916919
\(259\) −4.76891e7 −0.170557
\(260\) 3.46902e8 1.22405
\(261\) −2.10312e7 −0.0732189
\(262\) 4.45454e7 0.153020
\(263\) −4.08150e8 −1.38349 −0.691743 0.722144i \(-0.743157\pi\)
−0.691743 + 0.722144i \(0.743157\pi\)
\(264\) −7.29065e7 −0.243867
\(265\) −5.70230e8 −1.88230
\(266\) −1.38714e8 −0.451893
\(267\) −3.53598e7 −0.113690
\(268\) 1.87532e8 0.595119
\(269\) 5.16986e8 1.61937 0.809684 0.586865i \(-0.199638\pi\)
0.809684 + 0.586865i \(0.199638\pi\)
\(270\) 9.83618e7 0.304126
\(271\) −5.66936e7 −0.173038 −0.0865189 0.996250i \(-0.527574\pi\)
−0.0865189 + 0.996250i \(0.527574\pi\)
\(272\) 3.05346e8 0.920027
\(273\) −4.13507e7 −0.123003
\(274\) −4.74196e7 −0.139262
\(275\) 7.61615e7 0.220837
\(276\) 3.56472e8 1.02057
\(277\) 2.31128e8 0.653392 0.326696 0.945129i \(-0.394065\pi\)
0.326696 + 0.945129i \(0.394065\pi\)
\(278\) −6.28370e8 −1.75412
\(279\) −9.92040e7 −0.273473
\(280\) 2.36742e7 0.0644500
\(281\) 3.68378e8 0.990426 0.495213 0.868772i \(-0.335090\pi\)
0.495213 + 0.868772i \(0.335090\pi\)
\(282\) −1.55222e8 −0.412176
\(283\) 5.19507e8 1.36251 0.681254 0.732047i \(-0.261435\pi\)
0.681254 + 0.732047i \(0.261435\pi\)
\(284\) −8.10228e8 −2.09891
\(285\) −3.34448e8 −0.855799
\(286\) 8.72934e8 2.20648
\(287\) −7.74774e7 −0.193459
\(288\) 1.92911e8 0.475864
\(289\) 1.59354e8 0.388349
\(290\) −1.44169e8 −0.347120
\(291\) 3.44259e8 0.818955
\(292\) −6.84848e8 −1.60974
\(293\) −8.25048e8 −1.91621 −0.958104 0.286421i \(-0.907534\pi\)
−0.958104 + 0.286421i \(0.907534\pi\)
\(294\) 3.53780e8 0.811927
\(295\) −6.13711e7 −0.139183
\(296\) 9.43653e7 0.211491
\(297\) 1.34234e8 0.297314
\(298\) −3.78342e8 −0.828186
\(299\) −6.66235e8 −1.44138
\(300\) 4.57345e7 0.0977956
\(301\) −1.12085e8 −0.236899
\(302\) 3.46906e8 0.724748
\(303\) 5.07964e8 1.04902
\(304\) −5.30308e8 −1.08261
\(305\) −7.53677e8 −1.52102
\(306\) 2.90988e8 0.580564
\(307\) −2.01480e8 −0.397418 −0.198709 0.980059i \(-0.563675\pi\)
−0.198709 + 0.980059i \(0.563675\pi\)
\(308\) 2.06978e8 0.403643
\(309\) 4.18037e8 0.806047
\(310\) −6.80044e8 −1.29650
\(311\) 1.01602e8 0.191533 0.0957663 0.995404i \(-0.469470\pi\)
0.0957663 + 0.995404i \(0.469470\pi\)
\(312\) 8.18233e7 0.152523
\(313\) −5.09494e8 −0.939148 −0.469574 0.882893i \(-0.655592\pi\)
−0.469574 + 0.882893i \(0.655592\pi\)
\(314\) 8.98946e8 1.63863
\(315\) −4.35886e7 −0.0785752
\(316\) −2.13210e8 −0.380104
\(317\) 5.25634e8 0.926779 0.463389 0.886155i \(-0.346633\pi\)
0.463389 + 0.886155i \(0.346633\pi\)
\(318\) −8.61655e8 −1.50258
\(319\) −1.96747e8 −0.339345
\(320\) 8.33088e8 1.42124
\(321\) 7.31761e7 0.123481
\(322\) −2.91279e8 −0.486199
\(323\) −9.89412e8 −1.63369
\(324\) 8.06067e7 0.131663
\(325\) −8.54764e7 −0.138119
\(326\) −4.01091e8 −0.641182
\(327\) 1.14848e8 0.181638
\(328\) 1.53309e8 0.239889
\(329\) 6.87860e7 0.106491
\(330\) 9.20176e8 1.40952
\(331\) 2.47062e8 0.374463 0.187231 0.982316i \(-0.440049\pi\)
0.187231 + 0.982316i \(0.440049\pi\)
\(332\) 1.35893e9 2.03805
\(333\) −1.73744e8 −0.257842
\(334\) −1.21337e9 −1.78188
\(335\) −3.69460e8 −0.536921
\(336\) −6.91150e7 −0.0993996
\(337\) 1.28783e9 1.83296 0.916482 0.400076i \(-0.131016\pi\)
0.916482 + 0.400076i \(0.131016\pi\)
\(338\) 6.96777e7 0.0981488
\(339\) 3.69199e8 0.514708
\(340\) 1.08179e9 1.49269
\(341\) −9.28054e8 −1.26746
\(342\) −5.05373e8 −0.683157
\(343\) −3.21563e8 −0.430266
\(344\) 2.21789e8 0.293755
\(345\) −7.02291e8 −0.920768
\(346\) −1.50358e9 −1.95146
\(347\) −1.16047e9 −1.49101 −0.745504 0.666502i \(-0.767791\pi\)
−0.745504 + 0.666502i \(0.767791\pi\)
\(348\) −1.18146e8 −0.150276
\(349\) 1.00669e9 1.26767 0.633836 0.773467i \(-0.281479\pi\)
0.633836 + 0.773467i \(0.281479\pi\)
\(350\) −3.73705e7 −0.0465897
\(351\) −1.50652e8 −0.185951
\(352\) 1.80468e9 2.20547
\(353\) 2.36378e8 0.286019 0.143010 0.989721i \(-0.454322\pi\)
0.143010 + 0.989721i \(0.454322\pi\)
\(354\) −9.27357e7 −0.111106
\(355\) 1.59624e9 1.89365
\(356\) −1.98638e8 −0.233339
\(357\) −1.28950e8 −0.149997
\(358\) −4.45992e8 −0.513731
\(359\) −4.28699e8 −0.489015 −0.244507 0.969647i \(-0.578626\pi\)
−0.244507 + 0.969647i \(0.578626\pi\)
\(360\) 8.62514e7 0.0974333
\(361\) 8.24486e8 0.922377
\(362\) 2.13873e9 2.36961
\(363\) 7.29608e8 0.800601
\(364\) −2.32293e8 −0.252453
\(365\) 1.34923e9 1.45232
\(366\) −1.13886e9 −1.21418
\(367\) 1.21528e9 1.28335 0.641677 0.766975i \(-0.278239\pi\)
0.641677 + 0.766975i \(0.278239\pi\)
\(368\) −1.11357e9 −1.16479
\(369\) −2.82270e8 −0.292465
\(370\) −1.19101e9 −1.22239
\(371\) 3.81838e8 0.388213
\(372\) −5.57290e8 −0.561282
\(373\) 9.41218e8 0.939094 0.469547 0.882907i \(-0.344417\pi\)
0.469547 + 0.882907i \(0.344417\pi\)
\(374\) 2.72220e9 2.69072
\(375\) 5.40219e8 0.529006
\(376\) −1.36111e8 −0.132049
\(377\) 2.20811e8 0.212239
\(378\) −6.58652e7 −0.0627241
\(379\) 2.27326e8 0.214492 0.107246 0.994233i \(-0.465797\pi\)
0.107246 + 0.994233i \(0.465797\pi\)
\(380\) −1.87880e9 −1.75646
\(381\) 8.45938e8 0.783612
\(382\) 5.58368e8 0.512506
\(383\) 1.40852e9 1.28105 0.640525 0.767938i \(-0.278717\pi\)
0.640525 + 0.767938i \(0.278717\pi\)
\(384\) 3.44311e8 0.310307
\(385\) −4.07771e8 −0.364170
\(386\) −1.10568e9 −0.978527
\(387\) −4.08353e8 −0.358136
\(388\) 1.93392e9 1.68084
\(389\) 1.22433e9 1.05457 0.527285 0.849689i \(-0.323210\pi\)
0.527285 + 0.849689i \(0.323210\pi\)
\(390\) −1.03272e9 −0.881567
\(391\) −2.07762e9 −1.75771
\(392\) 3.10222e8 0.260118
\(393\) −7.19183e7 −0.0597676
\(394\) 1.25509e8 0.103380
\(395\) 4.20049e8 0.342933
\(396\) 7.54076e8 0.610214
\(397\) −3.38844e8 −0.271790 −0.135895 0.990723i \(-0.543391\pi\)
−0.135895 + 0.990723i \(0.543391\pi\)
\(398\) −4.82605e8 −0.383708
\(399\) 2.23953e8 0.176503
\(400\) −1.42868e8 −0.111616
\(401\) 1.32434e8 0.102564 0.0512820 0.998684i \(-0.483669\pi\)
0.0512820 + 0.998684i \(0.483669\pi\)
\(402\) −5.58278e8 −0.428607
\(403\) 1.04156e9 0.792713
\(404\) 2.85355e9 2.15303
\(405\) −1.58805e8 −0.118787
\(406\) 9.65388e7 0.0715914
\(407\) −1.62537e9 −1.19501
\(408\) 2.55161e8 0.185996
\(409\) 1.81103e9 1.30886 0.654431 0.756122i \(-0.272908\pi\)
0.654431 + 0.756122i \(0.272908\pi\)
\(410\) −1.93497e9 −1.38653
\(411\) 7.65587e7 0.0543936
\(412\) 2.34837e9 1.65435
\(413\) 4.10954e7 0.0287057
\(414\) −1.06121e9 −0.735020
\(415\) −2.67726e9 −1.83874
\(416\) −2.02540e9 −1.37938
\(417\) 1.01450e9 0.685134
\(418\) −4.72776e9 −3.16620
\(419\) −2.75900e9 −1.83233 −0.916164 0.400805i \(-0.868731\pi\)
−0.916164 + 0.400805i \(0.868731\pi\)
\(420\) −2.44864e8 −0.161270
\(421\) −6.24044e8 −0.407594 −0.203797 0.979013i \(-0.565328\pi\)
−0.203797 + 0.979013i \(0.565328\pi\)
\(422\) 2.70120e9 1.74970
\(423\) 2.50605e8 0.160990
\(424\) −7.55566e8 −0.481385
\(425\) −2.66554e8 −0.168431
\(426\) 2.41203e9 1.51164
\(427\) 5.04678e8 0.313702
\(428\) 4.11075e8 0.253436
\(429\) −1.40935e9 −0.861821
\(430\) −2.79927e9 −1.69787
\(431\) 2.11891e9 1.27480 0.637401 0.770532i \(-0.280010\pi\)
0.637401 + 0.770532i \(0.280010\pi\)
\(432\) −2.51804e8 −0.150269
\(433\) −1.12211e9 −0.664244 −0.332122 0.943236i \(-0.607765\pi\)
−0.332122 + 0.943236i \(0.607765\pi\)
\(434\) 4.55372e8 0.267394
\(435\) 2.32760e8 0.135580
\(436\) 6.45172e8 0.372797
\(437\) 3.60830e9 2.06832
\(438\) 2.03878e9 1.15934
\(439\) −1.19818e9 −0.675922 −0.337961 0.941160i \(-0.609737\pi\)
−0.337961 + 0.941160i \(0.609737\pi\)
\(440\) 8.06882e8 0.451571
\(441\) −5.71175e8 −0.317128
\(442\) −3.05513e9 −1.68288
\(443\) −5.64409e8 −0.308447 −0.154224 0.988036i \(-0.549288\pi\)
−0.154224 + 0.988036i \(0.549288\pi\)
\(444\) −9.76026e8 −0.529201
\(445\) 3.91340e8 0.210521
\(446\) −2.29567e9 −1.22529
\(447\) 6.10831e8 0.323478
\(448\) −5.57854e8 −0.293121
\(449\) −1.48373e9 −0.773558 −0.386779 0.922172i \(-0.626412\pi\)
−0.386779 + 0.922172i \(0.626412\pi\)
\(450\) −1.36150e8 −0.0704328
\(451\) −2.64064e9 −1.35548
\(452\) 2.07402e9 1.05640
\(453\) −5.60077e8 −0.283077
\(454\) 8.42179e8 0.422386
\(455\) 4.57644e8 0.227765
\(456\) −4.43150e8 −0.218864
\(457\) −1.51906e9 −0.744505 −0.372253 0.928131i \(-0.621415\pi\)
−0.372253 + 0.928131i \(0.621415\pi\)
\(458\) −5.51264e9 −2.68121
\(459\) −4.69799e8 −0.226760
\(460\) −3.94520e9 −1.88980
\(461\) 8.02602e8 0.381546 0.190773 0.981634i \(-0.438901\pi\)
0.190773 + 0.981634i \(0.438901\pi\)
\(462\) −6.16169e8 −0.290705
\(463\) 9.47165e8 0.443499 0.221749 0.975104i \(-0.428823\pi\)
0.221749 + 0.975104i \(0.428823\pi\)
\(464\) 3.69070e8 0.171513
\(465\) 1.09793e9 0.506393
\(466\) −7.28241e8 −0.333368
\(467\) 1.58449e9 0.719914 0.359957 0.932969i \(-0.382791\pi\)
0.359957 + 0.932969i \(0.382791\pi\)
\(468\) −8.46303e8 −0.381650
\(469\) 2.47398e8 0.110737
\(470\) 1.71790e9 0.763231
\(471\) −1.45134e9 −0.640025
\(472\) −8.13180e7 −0.0355951
\(473\) −3.82015e9 −1.65984
\(474\) 6.34721e8 0.273753
\(475\) 4.62936e8 0.198195
\(476\) −7.24392e8 −0.307857
\(477\) 1.39114e9 0.586888
\(478\) 1.95490e9 0.818703
\(479\) −2.85017e6 −0.00118494 −0.000592471 1.00000i \(-0.500189\pi\)
−0.000592471 1.00000i \(0.500189\pi\)
\(480\) −2.13501e9 −0.881163
\(481\) 1.82416e9 0.747405
\(482\) 6.29571e9 2.56083
\(483\) 4.70269e8 0.189903
\(484\) 4.09866e9 1.64317
\(485\) −3.81004e9 −1.51647
\(486\) −2.39964e8 −0.0948242
\(487\) 1.28087e9 0.502521 0.251260 0.967920i \(-0.419155\pi\)
0.251260 + 0.967920i \(0.419155\pi\)
\(488\) −9.98638e8 −0.388990
\(489\) 6.47559e8 0.250437
\(490\) −3.91541e9 −1.50346
\(491\) −1.37484e8 −0.0524164 −0.0262082 0.999657i \(-0.508343\pi\)
−0.0262082 + 0.999657i \(0.508343\pi\)
\(492\) −1.58569e9 −0.600261
\(493\) 6.88586e8 0.258818
\(494\) 5.30599e9 1.98026
\(495\) −1.48562e9 −0.550540
\(496\) 1.74090e9 0.640600
\(497\) −1.06888e9 −0.390554
\(498\) −4.04551e9 −1.46781
\(499\) 1.97156e8 0.0710327 0.0355163 0.999369i \(-0.488692\pi\)
0.0355163 + 0.999369i \(0.488692\pi\)
\(500\) 3.03474e9 1.08574
\(501\) 1.95897e9 0.695979
\(502\) −6.22663e9 −2.19680
\(503\) −2.00636e9 −0.702946 −0.351473 0.936198i \(-0.614319\pi\)
−0.351473 + 0.936198i \(0.614319\pi\)
\(504\) −5.77558e7 −0.0200950
\(505\) −5.62182e9 −1.94248
\(506\) −9.92760e9 −3.40657
\(507\) −1.12494e8 −0.0383356
\(508\) 4.75216e9 1.60830
\(509\) 4.88057e9 1.64043 0.820216 0.572053i \(-0.193853\pi\)
0.820216 + 0.572053i \(0.193853\pi\)
\(510\) −3.22047e9 −1.07504
\(511\) −9.03474e8 −0.299531
\(512\) −4.03368e9 −1.32818
\(513\) 8.15921e8 0.266832
\(514\) 7.91206e9 2.56992
\(515\) −4.62657e9 −1.49257
\(516\) −2.29397e9 −0.735046
\(517\) 2.34442e9 0.746135
\(518\) 7.97528e8 0.252111
\(519\) 2.42752e9 0.762215
\(520\) −9.05568e8 −0.282429
\(521\) 3.48111e9 1.07842 0.539208 0.842173i \(-0.318724\pi\)
0.539208 + 0.842173i \(0.318724\pi\)
\(522\) 3.51716e8 0.108230
\(523\) −8.79645e8 −0.268876 −0.134438 0.990922i \(-0.542923\pi\)
−0.134438 + 0.990922i \(0.542923\pi\)
\(524\) −4.04010e8 −0.122668
\(525\) 6.03344e7 0.0181973
\(526\) 6.82570e9 2.04502
\(527\) 3.24804e9 0.966685
\(528\) −2.35563e9 −0.696447
\(529\) 4.17206e9 1.22534
\(530\) 9.53625e9 2.78235
\(531\) 1.49721e8 0.0433963
\(532\) 1.25809e9 0.362259
\(533\) 2.96360e9 0.847764
\(534\) 5.91341e8 0.168052
\(535\) −8.09866e8 −0.228652
\(536\) −4.89542e8 −0.137314
\(537\) 7.20051e8 0.200656
\(538\) −8.64582e9 −2.39369
\(539\) −5.34335e9 −1.46978
\(540\) −8.92104e8 −0.243802
\(541\) −5.31588e9 −1.44339 −0.721697 0.692209i \(-0.756638\pi\)
−0.721697 + 0.692209i \(0.756638\pi\)
\(542\) 9.48115e8 0.255778
\(543\) −3.45297e9 −0.925537
\(544\) −6.31611e9 −1.68211
\(545\) −1.27106e9 −0.336341
\(546\) 6.91529e8 0.181818
\(547\) 4.68079e9 1.22282 0.611412 0.791313i \(-0.290602\pi\)
0.611412 + 0.791313i \(0.290602\pi\)
\(548\) 4.30078e8 0.111639
\(549\) 1.83867e9 0.474244
\(550\) −1.27369e9 −0.326433
\(551\) −1.19590e9 −0.304554
\(552\) −9.30550e8 −0.235479
\(553\) −2.81274e8 −0.0707279
\(554\) −3.86527e9 −0.965820
\(555\) 1.92288e9 0.477450
\(556\) 5.69907e9 1.40618
\(557\) 9.19162e8 0.225371 0.112686 0.993631i \(-0.464055\pi\)
0.112686 + 0.993631i \(0.464055\pi\)
\(558\) 1.65904e9 0.404238
\(559\) 4.28737e9 1.03813
\(560\) 7.64921e8 0.184060
\(561\) −4.39497e9 −1.05096
\(562\) −6.16058e9 −1.46401
\(563\) 2.14197e9 0.505865 0.252933 0.967484i \(-0.418605\pi\)
0.252933 + 0.967484i \(0.418605\pi\)
\(564\) 1.40781e9 0.330420
\(565\) −4.08606e9 −0.953092
\(566\) −8.68798e9 −2.01401
\(567\) 1.06339e8 0.0244992
\(568\) 2.11506e9 0.484287
\(569\) −6.07580e9 −1.38265 −0.691323 0.722546i \(-0.742972\pi\)
−0.691323 + 0.722546i \(0.742972\pi\)
\(570\) 5.59314e9 1.26501
\(571\) −6.78352e9 −1.52486 −0.762428 0.647073i \(-0.775993\pi\)
−0.762428 + 0.647073i \(0.775993\pi\)
\(572\) −7.91717e9 −1.76882
\(573\) −9.01481e8 −0.200178
\(574\) 1.29569e9 0.285964
\(575\) 9.72096e8 0.213242
\(576\) −2.03241e9 −0.443131
\(577\) −2.20797e9 −0.478495 −0.239248 0.970959i \(-0.576901\pi\)
−0.239248 + 0.970959i \(0.576901\pi\)
\(578\) −2.66497e9 −0.574043
\(579\) 1.78511e9 0.382199
\(580\) 1.30756e9 0.278268
\(581\) 1.79275e9 0.379230
\(582\) −5.75722e9 −1.21055
\(583\) 1.30141e10 2.72003
\(584\) 1.78776e9 0.371419
\(585\) 1.66732e9 0.344328
\(586\) 1.37977e10 2.83247
\(587\) −3.43972e9 −0.701924 −0.350962 0.936390i \(-0.614145\pi\)
−0.350962 + 0.936390i \(0.614145\pi\)
\(588\) −3.20864e9 −0.650880
\(589\) −5.64103e9 −1.13751
\(590\) 1.02634e9 0.205736
\(591\) −2.02633e8 −0.0403788
\(592\) 3.04897e9 0.603986
\(593\) −3.62804e9 −0.714464 −0.357232 0.934016i \(-0.616279\pi\)
−0.357232 + 0.934016i \(0.616279\pi\)
\(594\) −2.24487e9 −0.439479
\(595\) 1.42714e9 0.277751
\(596\) 3.43142e9 0.663914
\(597\) 7.79162e8 0.149871
\(598\) 1.11418e10 2.13059
\(599\) 3.45165e9 0.656194 0.328097 0.944644i \(-0.393593\pi\)
0.328097 + 0.944644i \(0.393593\pi\)
\(600\) −1.19387e8 −0.0225647
\(601\) 7.18642e9 1.35037 0.675184 0.737649i \(-0.264064\pi\)
0.675184 + 0.737649i \(0.264064\pi\)
\(602\) 1.87445e9 0.350175
\(603\) 9.01337e8 0.167408
\(604\) −3.14630e9 −0.580993
\(605\) −8.07483e9 −1.48248
\(606\) −8.49494e9 −1.55062
\(607\) 7.76353e9 1.40896 0.704480 0.709724i \(-0.251180\pi\)
0.704480 + 0.709724i \(0.251180\pi\)
\(608\) 1.09695e10 1.97935
\(609\) −1.55861e8 −0.0279626
\(610\) 1.26041e10 2.24832
\(611\) −2.63115e9 −0.466660
\(612\) −2.63915e9 −0.465408
\(613\) 2.87864e9 0.504750 0.252375 0.967630i \(-0.418788\pi\)
0.252375 + 0.967630i \(0.418788\pi\)
\(614\) 3.36945e9 0.587449
\(615\) 3.12399e9 0.541560
\(616\) −5.40306e8 −0.0931337
\(617\) 9.59531e9 1.64460 0.822301 0.569053i \(-0.192690\pi\)
0.822301 + 0.569053i \(0.192690\pi\)
\(618\) −6.99105e9 −1.19147
\(619\) 6.93284e8 0.117488 0.0587441 0.998273i \(-0.481290\pi\)
0.0587441 + 0.998273i \(0.481290\pi\)
\(620\) 6.16773e9 1.03933
\(621\) 1.71331e9 0.287089
\(622\) −1.69915e9 −0.283116
\(623\) −2.62050e8 −0.0434186
\(624\) 2.64373e9 0.435583
\(625\) −6.85127e9 −1.12251
\(626\) 8.52053e9 1.38821
\(627\) 7.63295e9 1.23668
\(628\) −8.15309e9 −1.31360
\(629\) 5.68856e9 0.911433
\(630\) 7.28954e8 0.116147
\(631\) 4.90432e9 0.777099 0.388549 0.921428i \(-0.372976\pi\)
0.388549 + 0.921428i \(0.372976\pi\)
\(632\) 5.56573e8 0.0877026
\(633\) −4.36107e9 −0.683408
\(634\) −8.79044e9 −1.36993
\(635\) −9.36231e9 −1.45102
\(636\) 7.81487e9 1.20454
\(637\) 5.99686e9 0.919255
\(638\) 3.29031e9 0.501608
\(639\) −3.89421e9 −0.590427
\(640\) −3.81062e9 −0.574599
\(641\) −9.16838e9 −1.37496 −0.687479 0.726204i \(-0.741283\pi\)
−0.687479 + 0.726204i \(0.741283\pi\)
\(642\) −1.22376e9 −0.182526
\(643\) 4.70285e9 0.697626 0.348813 0.937192i \(-0.386585\pi\)
0.348813 + 0.937192i \(0.386585\pi\)
\(644\) 2.64179e9 0.389761
\(645\) 4.51940e9 0.663165
\(646\) 1.65464e10 2.41485
\(647\) 5.90195e9 0.856703 0.428352 0.903612i \(-0.359094\pi\)
0.428352 + 0.903612i \(0.359094\pi\)
\(648\) −2.10419e8 −0.0303790
\(649\) 1.40064e9 0.201127
\(650\) 1.42947e9 0.204163
\(651\) −7.35195e8 −0.104441
\(652\) 3.63774e9 0.514002
\(653\) 7.53884e9 1.05952 0.529759 0.848148i \(-0.322282\pi\)
0.529759 + 0.848148i \(0.322282\pi\)
\(654\) −1.92066e9 −0.268490
\(655\) 7.95947e8 0.110672
\(656\) 4.95347e9 0.685087
\(657\) −3.29159e9 −0.452822
\(658\) −1.15034e9 −0.157412
\(659\) −8.90395e9 −1.21195 −0.605974 0.795484i \(-0.707216\pi\)
−0.605974 + 0.795484i \(0.707216\pi\)
\(660\) −8.34564e9 −1.12994
\(661\) −2.51361e9 −0.338527 −0.169264 0.985571i \(-0.554139\pi\)
−0.169264 + 0.985571i \(0.554139\pi\)
\(662\) −4.13175e9 −0.553517
\(663\) 4.93249e9 0.657309
\(664\) −3.54742e9 −0.470245
\(665\) −2.47857e9 −0.326833
\(666\) 2.90560e9 0.381133
\(667\) −2.51121e9 −0.327674
\(668\) 1.10048e10 1.42844
\(669\) 3.70634e9 0.478580
\(670\) 6.17867e9 0.793657
\(671\) 1.72008e10 2.19796
\(672\) 1.42965e9 0.181734
\(673\) 1.80021e9 0.227651 0.113826 0.993501i \(-0.463690\pi\)
0.113826 + 0.993501i \(0.463690\pi\)
\(674\) −2.15370e10 −2.70942
\(675\) 2.19814e8 0.0275101
\(676\) −6.31949e8 −0.0786808
\(677\) 7.89862e9 0.978343 0.489171 0.872188i \(-0.337299\pi\)
0.489171 + 0.872188i \(0.337299\pi\)
\(678\) −6.17430e9 −0.760823
\(679\) 2.55128e9 0.312762
\(680\) −2.82396e9 −0.344412
\(681\) −1.35969e9 −0.164978
\(682\) 1.55203e10 1.87351
\(683\) 3.31446e9 0.398053 0.199026 0.979994i \(-0.436222\pi\)
0.199026 + 0.979994i \(0.436222\pi\)
\(684\) 4.58353e9 0.547651
\(685\) −8.47303e8 −0.100721
\(686\) 5.37766e9 0.636003
\(687\) 8.90012e9 1.04724
\(688\) 7.16605e9 0.838920
\(689\) −1.46058e10 −1.70121
\(690\) 1.17448e10 1.36105
\(691\) −5.78938e9 −0.667512 −0.333756 0.942659i \(-0.608316\pi\)
−0.333756 + 0.942659i \(0.608316\pi\)
\(692\) 1.36369e10 1.56439
\(693\) 9.94801e8 0.113546
\(694\) 1.94071e10 2.20395
\(695\) −1.12278e10 −1.26867
\(696\) 3.08412e8 0.0346737
\(697\) 9.24184e9 1.03382
\(698\) −1.68354e10 −1.87383
\(699\) 1.17574e9 0.130209
\(700\) 3.38936e8 0.0373486
\(701\) −1.14194e10 −1.25207 −0.626036 0.779794i \(-0.715324\pi\)
−0.626036 + 0.779794i \(0.715324\pi\)
\(702\) 2.51942e9 0.274866
\(703\) −9.87958e9 −1.07249
\(704\) −1.90132e10 −2.05376
\(705\) −2.77354e9 −0.298107
\(706\) −3.95307e9 −0.422783
\(707\) 3.76449e9 0.400625
\(708\) 8.41077e8 0.0890675
\(709\) −1.23857e10 −1.30515 −0.652575 0.757725i \(-0.726311\pi\)
−0.652575 + 0.757725i \(0.726311\pi\)
\(710\) −2.66948e10 −2.79913
\(711\) −1.02475e9 −0.106924
\(712\) 5.18534e8 0.0538390
\(713\) −1.18453e10 −1.22386
\(714\) 2.15650e9 0.221720
\(715\) 1.55978e10 1.59585
\(716\) 4.04497e9 0.411832
\(717\) −3.15617e9 −0.319774
\(718\) 7.16935e9 0.722844
\(719\) −1.16266e10 −1.16655 −0.583274 0.812275i \(-0.698229\pi\)
−0.583274 + 0.812275i \(0.698229\pi\)
\(720\) 2.78681e9 0.278255
\(721\) 3.09805e9 0.307833
\(722\) −1.37883e10 −1.36342
\(723\) −1.01644e10 −1.00022
\(724\) −1.93975e10 −1.89959
\(725\) −3.22182e8 −0.0313992
\(726\) −1.22016e10 −1.18342
\(727\) 8.72736e9 0.842389 0.421194 0.906970i \(-0.361611\pi\)
0.421194 + 0.906970i \(0.361611\pi\)
\(728\) 6.06387e8 0.0582492
\(729\) 3.87420e8 0.0370370
\(730\) −2.25639e10 −2.14676
\(731\) 1.33699e10 1.26596
\(732\) 1.03290e10 0.973349
\(733\) 2.02816e10 1.90212 0.951062 0.309000i \(-0.0999943\pi\)
0.951062 + 0.309000i \(0.0999943\pi\)
\(734\) −2.03238e10 −1.89701
\(735\) 6.32140e9 0.587229
\(736\) 2.30343e10 2.12962
\(737\) 8.43201e9 0.775881
\(738\) 4.72055e9 0.432310
\(739\) −3.02731e9 −0.275932 −0.137966 0.990437i \(-0.544056\pi\)
−0.137966 + 0.990437i \(0.544056\pi\)
\(740\) 1.08020e10 0.979929
\(741\) −8.56649e9 −0.773462
\(742\) −6.38567e9 −0.573843
\(743\) 1.35696e10 1.21369 0.606843 0.794822i \(-0.292436\pi\)
0.606843 + 0.794822i \(0.292436\pi\)
\(744\) 1.45478e9 0.129506
\(745\) −6.76029e9 −0.598988
\(746\) −1.57405e10 −1.38813
\(747\) 6.53145e9 0.573307
\(748\) −2.46893e10 −2.15701
\(749\) 5.42304e8 0.0471581
\(750\) −9.03435e9 −0.781957
\(751\) −1.38238e10 −1.19094 −0.595469 0.803378i \(-0.703034\pi\)
−0.595469 + 0.803378i \(0.703034\pi\)
\(752\) −4.39779e9 −0.377113
\(753\) 1.00529e10 0.858039
\(754\) −3.69273e9 −0.313724
\(755\) 6.19858e9 0.524177
\(756\) 5.97372e8 0.0502827
\(757\) −1.53342e9 −0.128477 −0.0642386 0.997935i \(-0.520462\pi\)
−0.0642386 + 0.997935i \(0.520462\pi\)
\(758\) −3.80168e9 −0.317054
\(759\) 1.60280e10 1.33056
\(760\) 4.90451e9 0.405273
\(761\) −8.31351e9 −0.683814 −0.341907 0.939734i \(-0.611073\pi\)
−0.341907 + 0.939734i \(0.611073\pi\)
\(762\) −1.41471e10 −1.15831
\(763\) 8.51131e8 0.0693682
\(764\) −5.06418e9 −0.410849
\(765\) 5.19943e9 0.419895
\(766\) −2.35553e10 −1.89360
\(767\) −1.57195e9 −0.125792
\(768\) 3.87703e9 0.308841
\(769\) 1.21279e10 0.961710 0.480855 0.876800i \(-0.340326\pi\)
0.480855 + 0.876800i \(0.340326\pi\)
\(770\) 6.81937e9 0.538303
\(771\) −1.27740e10 −1.00377
\(772\) 1.00281e10 0.784434
\(773\) −8.47150e9 −0.659678 −0.329839 0.944037i \(-0.606994\pi\)
−0.329839 + 0.944037i \(0.606994\pi\)
\(774\) 6.82910e9 0.529383
\(775\) −1.51973e9 −0.117276
\(776\) −5.04838e9 −0.387825
\(777\) −1.28760e9 −0.0984711
\(778\) −2.04751e10 −1.55883
\(779\) −1.60507e10 −1.21650
\(780\) 9.36635e9 0.706706
\(781\) −3.64303e10 −2.73643
\(782\) 3.47451e10 2.59818
\(783\) −5.67844e8 −0.0422730
\(784\) 1.00234e10 0.742860
\(785\) 1.60625e10 1.18514
\(786\) 1.20273e9 0.0883463
\(787\) −1.35820e10 −0.993233 −0.496617 0.867970i \(-0.665424\pi\)
−0.496617 + 0.867970i \(0.665424\pi\)
\(788\) −1.13831e9 −0.0828744
\(789\) −1.10200e10 −0.798756
\(790\) −7.02469e9 −0.506912
\(791\) 2.73611e9 0.196569
\(792\) −1.96847e9 −0.140796
\(793\) −1.93045e10 −1.37469
\(794\) 5.66666e9 0.401750
\(795\) −1.53962e10 −1.08675
\(796\) 4.37704e9 0.307599
\(797\) −2.11682e10 −1.48108 −0.740542 0.672010i \(-0.765431\pi\)
−0.740542 + 0.672010i \(0.765431\pi\)
\(798\) −3.74529e9 −0.260901
\(799\) −8.20509e9 −0.569075
\(800\) 2.95524e9 0.204069
\(801\) −9.54716e8 −0.0656388
\(802\) −2.21477e9 −0.151606
\(803\) −3.07929e10 −2.09868
\(804\) 5.06337e9 0.343592
\(805\) −5.20464e9 −0.351645
\(806\) −1.74185e10 −1.17176
\(807\) 1.39586e10 0.934943
\(808\) −7.44903e9 −0.496775
\(809\) −5.28973e9 −0.351248 −0.175624 0.984457i \(-0.556194\pi\)
−0.175624 + 0.984457i \(0.556194\pi\)
\(810\) 2.65577e9 0.175587
\(811\) 2.57240e10 1.69342 0.846710 0.532054i \(-0.178580\pi\)
0.846710 + 0.532054i \(0.178580\pi\)
\(812\) −8.75570e8 −0.0573911
\(813\) −1.53073e9 −0.0999035
\(814\) 2.71819e10 1.76642
\(815\) −7.16677e9 −0.463737
\(816\) 8.24433e9 0.531178
\(817\) −2.32202e10 −1.48966
\(818\) −3.02868e10 −1.93471
\(819\) −1.11647e9 −0.0710155
\(820\) 1.75494e10 1.11151
\(821\) −3.97061e8 −0.0250413 −0.0125206 0.999922i \(-0.503986\pi\)
−0.0125206 + 0.999922i \(0.503986\pi\)
\(822\) −1.28033e9 −0.0804027
\(823\) −2.17465e10 −1.35985 −0.679923 0.733283i \(-0.737987\pi\)
−0.679923 + 0.733283i \(0.737987\pi\)
\(824\) −6.13030e9 −0.381713
\(825\) 2.05636e9 0.127500
\(826\) −6.87259e8 −0.0424317
\(827\) 1.11227e10 0.683821 0.341910 0.939733i \(-0.388926\pi\)
0.341910 + 0.939733i \(0.388926\pi\)
\(828\) 9.62474e9 0.589227
\(829\) −9.06605e9 −0.552684 −0.276342 0.961059i \(-0.589122\pi\)
−0.276342 + 0.961059i \(0.589122\pi\)
\(830\) 4.47731e10 2.71797
\(831\) 6.24046e9 0.377236
\(832\) 2.13386e10 1.28450
\(833\) 1.87009e10 1.12100
\(834\) −1.69660e10 −1.01274
\(835\) −2.16807e10 −1.28875
\(836\) 4.28790e10 2.53818
\(837\) −2.67851e9 −0.157890
\(838\) 4.61402e10 2.70848
\(839\) 5.43132e9 0.317496 0.158748 0.987319i \(-0.449254\pi\)
0.158748 + 0.987319i \(0.449254\pi\)
\(840\) 6.39204e8 0.0372102
\(841\) −1.64176e10 −0.951751
\(842\) 1.04362e10 0.602491
\(843\) 9.94622e9 0.571823
\(844\) −2.44988e10 −1.40264
\(845\) 1.24501e9 0.0709865
\(846\) −4.19100e9 −0.237970
\(847\) 5.40708e9 0.305753
\(848\) −2.44126e10 −1.37476
\(849\) 1.40267e10 0.786644
\(850\) 4.45771e9 0.248969
\(851\) −2.07456e10 −1.15391
\(852\) −2.18762e10 −1.21180
\(853\) −6.82993e9 −0.376786 −0.188393 0.982094i \(-0.560328\pi\)
−0.188393 + 0.982094i \(0.560328\pi\)
\(854\) −8.43999e9 −0.463702
\(855\) −9.03010e9 −0.494096
\(856\) −1.07309e9 −0.0584760
\(857\) 3.60369e10 1.95576 0.977878 0.209178i \(-0.0670788\pi\)
0.977878 + 0.209178i \(0.0670788\pi\)
\(858\) 2.35692e10 1.27391
\(859\) 1.90837e10 1.02727 0.513637 0.858008i \(-0.328298\pi\)
0.513637 + 0.858008i \(0.328298\pi\)
\(860\) 2.53882e10 1.36109
\(861\) −2.09189e9 −0.111693
\(862\) −3.54357e10 −1.88437
\(863\) −3.33278e10 −1.76510 −0.882550 0.470218i \(-0.844175\pi\)
−0.882550 + 0.470218i \(0.844175\pi\)
\(864\) 5.20859e9 0.274740
\(865\) −2.68663e10 −1.41140
\(866\) 1.87656e10 0.981861
\(867\) 4.30257e9 0.224213
\(868\) −4.13004e9 −0.214356
\(869\) −9.58657e9 −0.495558
\(870\) −3.89257e9 −0.200410
\(871\) −9.46328e9 −0.485264
\(872\) −1.68419e9 −0.0860166
\(873\) 9.29500e9 0.472824
\(874\) −6.03434e10 −3.05731
\(875\) 4.00353e9 0.202030
\(876\) −1.84909e10 −0.929381
\(877\) 2.92873e10 1.46616 0.733079 0.680143i \(-0.238082\pi\)
0.733079 + 0.680143i \(0.238082\pi\)
\(878\) 2.00378e10 0.999124
\(879\) −2.22763e10 −1.10632
\(880\) 2.60706e10 1.28962
\(881\) −7.06390e9 −0.348040 −0.174020 0.984742i \(-0.555676\pi\)
−0.174020 + 0.984742i \(0.555676\pi\)
\(882\) 9.55205e9 0.468766
\(883\) −2.11700e10 −1.03480 −0.517401 0.855743i \(-0.673100\pi\)
−0.517401 + 0.855743i \(0.673100\pi\)
\(884\) 2.77089e10 1.34907
\(885\) −1.65702e9 −0.0803575
\(886\) 9.43890e9 0.455935
\(887\) 3.92843e9 0.189011 0.0945053 0.995524i \(-0.469873\pi\)
0.0945053 + 0.995524i \(0.469873\pi\)
\(888\) 2.54786e9 0.122104
\(889\) 6.26920e9 0.299265
\(890\) −6.54458e9 −0.311184
\(891\) 3.62432e9 0.171654
\(892\) 2.08208e10 0.982248
\(893\) 1.42502e10 0.669637
\(894\) −1.02152e10 −0.478153
\(895\) −7.96906e9 −0.371558
\(896\) 2.55167e9 0.118508
\(897\) −1.79884e10 −0.832181
\(898\) 2.48132e10 1.14345
\(899\) 3.92590e9 0.180211
\(900\) 1.23483e9 0.0564623
\(901\) −4.55473e10 −2.07456
\(902\) 4.41608e10 2.00361
\(903\) −3.02628e9 −0.136774
\(904\) −5.41411e9 −0.243746
\(905\) 3.82153e10 1.71383
\(906\) 9.36645e9 0.418434
\(907\) 1.73810e10 0.773482 0.386741 0.922188i \(-0.373601\pi\)
0.386741 + 0.922188i \(0.373601\pi\)
\(908\) −7.63824e9 −0.338605
\(909\) 1.37150e10 0.605652
\(910\) −7.65341e9 −0.336674
\(911\) 3.94515e9 0.172882 0.0864409 0.996257i \(-0.472451\pi\)
0.0864409 + 0.996257i \(0.472451\pi\)
\(912\) −1.43183e10 −0.625043
\(913\) 6.11017e10 2.65709
\(914\) 2.54040e10 1.10050
\(915\) −2.03493e10 −0.878163
\(916\) 4.99975e10 2.14938
\(917\) −5.32983e8 −0.0228255
\(918\) 7.85668e9 0.335189
\(919\) −1.68687e10 −0.716930 −0.358465 0.933543i \(-0.616700\pi\)
−0.358465 + 0.933543i \(0.616700\pi\)
\(920\) 1.02987e10 0.436040
\(921\) −5.43996e9 −0.229449
\(922\) −1.34223e10 −0.563987
\(923\) 4.08859e10 1.71146
\(924\) 5.58841e9 0.233043
\(925\) −2.66162e9 −0.110573
\(926\) −1.58399e10 −0.655564
\(927\) 1.12870e10 0.465371
\(928\) −7.63425e9 −0.313580
\(929\) 5.86634e9 0.240056 0.120028 0.992771i \(-0.461702\pi\)
0.120028 + 0.992771i \(0.461702\pi\)
\(930\) −1.83612e10 −0.748532
\(931\) −3.24787e10 −1.31909
\(932\) 6.60486e9 0.267244
\(933\) 2.74326e9 0.110581
\(934\) −2.64983e10 −1.06415
\(935\) 4.86407e10 1.94607
\(936\) 2.20923e9 0.0880593
\(937\) 2.42022e10 0.961093 0.480546 0.876969i \(-0.340438\pi\)
0.480546 + 0.876969i \(0.340438\pi\)
\(938\) −4.13737e9 −0.163687
\(939\) −1.37563e10 −0.542217
\(940\) −1.55807e10 −0.611842
\(941\) 2.22320e10 0.869792 0.434896 0.900481i \(-0.356785\pi\)
0.434896 + 0.900481i \(0.356785\pi\)
\(942\) 2.42715e10 0.946061
\(943\) −3.37041e10 −1.30886
\(944\) −2.62741e9 −0.101654
\(945\) −1.17689e9 −0.0453654
\(946\) 6.38863e10 2.45351
\(947\) 2.37539e9 0.0908889 0.0454444 0.998967i \(-0.485530\pi\)
0.0454444 + 0.998967i \(0.485530\pi\)
\(948\) −5.75667e9 −0.219453
\(949\) 3.45590e10 1.31259
\(950\) −7.74191e9 −0.292965
\(951\) 1.41921e10 0.535076
\(952\) 1.89099e9 0.0710328
\(953\) −2.11402e10 −0.791197 −0.395599 0.918423i \(-0.629463\pi\)
−0.395599 + 0.918423i \(0.629463\pi\)
\(954\) −2.32647e10 −0.867516
\(955\) 9.97702e9 0.370671
\(956\) −1.77302e10 −0.656312
\(957\) −5.31218e9 −0.195921
\(958\) 4.76649e7 0.00175154
\(959\) 5.67372e8 0.0207732
\(960\) 2.24934e10 0.820551
\(961\) −8.99423e9 −0.326913
\(962\) −3.05064e10 −1.10479
\(963\) 1.97575e9 0.0712920
\(964\) −5.70996e10 −2.05288
\(965\) −1.97564e10 −0.707723
\(966\) −7.86454e9 −0.280707
\(967\) −9.04540e9 −0.321688 −0.160844 0.986980i \(-0.551422\pi\)
−0.160844 + 0.986980i \(0.551422\pi\)
\(968\) −1.06993e10 −0.379134
\(969\) −2.67141e10 −0.943209
\(970\) 6.37173e10 2.24159
\(971\) −4.75592e9 −0.166712 −0.0833560 0.996520i \(-0.526564\pi\)
−0.0833560 + 0.996520i \(0.526564\pi\)
\(972\) 2.17638e9 0.0760156
\(973\) 7.51839e9 0.261656
\(974\) −2.14207e10 −0.742808
\(975\) −2.30786e9 −0.0797432
\(976\) −3.22663e10 −1.11090
\(977\) 1.97475e9 0.0677456 0.0338728 0.999426i \(-0.489216\pi\)
0.0338728 + 0.999426i \(0.489216\pi\)
\(978\) −1.08295e10 −0.370187
\(979\) −8.93137e9 −0.304214
\(980\) 3.55112e10 1.20524
\(981\) 3.10089e9 0.104869
\(982\) 2.29922e9 0.0774801
\(983\) 1.84818e10 0.620592 0.310296 0.950640i \(-0.399572\pi\)
0.310296 + 0.950640i \(0.399572\pi\)
\(984\) 4.13935e9 0.138500
\(985\) 2.24261e9 0.0747699
\(986\) −1.15156e10 −0.382575
\(987\) 1.85722e9 0.0614828
\(988\) −4.81233e10 −1.58747
\(989\) −4.87589e10 −1.60275
\(990\) 2.48447e10 0.813788
\(991\) 1.09246e9 0.0356573 0.0178286 0.999841i \(-0.494325\pi\)
0.0178286 + 0.999841i \(0.494325\pi\)
\(992\) −3.60106e10 −1.17122
\(993\) 6.67069e9 0.216196
\(994\) 1.78754e10 0.577303
\(995\) −8.62327e9 −0.277518
\(996\) 3.66912e10 1.17667
\(997\) −7.48711e9 −0.239266 −0.119633 0.992818i \(-0.538172\pi\)
−0.119633 + 0.992818i \(0.538172\pi\)
\(998\) −3.29714e9 −0.104998
\(999\) −4.69108e9 −0.148865
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 177.8.a.d.1.3 18
3.2 odd 2 531.8.a.e.1.16 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.3 18 1.1 even 1 trivial
531.8.a.e.1.16 18 3.2 odd 2