Properties

Label 177.8.a.d.1.2
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 1.2
Root \(-20.5204\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-19.5204 q^{2} +27.0000 q^{3} +253.046 q^{4} -49.6011 q^{5} -527.051 q^{6} -119.919 q^{7} -2440.94 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-19.5204 q^{2} +27.0000 q^{3} +253.046 q^{4} -49.6011 q^{5} -527.051 q^{6} -119.919 q^{7} -2440.94 q^{8} +729.000 q^{9} +968.233 q^{10} -5494.59 q^{11} +6832.23 q^{12} -12895.9 q^{13} +2340.86 q^{14} -1339.23 q^{15} +15258.3 q^{16} +25621.3 q^{17} -14230.4 q^{18} -17560.8 q^{19} -12551.3 q^{20} -3237.80 q^{21} +107256. q^{22} +63260.9 q^{23} -65905.4 q^{24} -75664.7 q^{25} +251733. q^{26} +19683.0 q^{27} -30344.9 q^{28} -29622.4 q^{29} +26142.3 q^{30} +166521. q^{31} +14593.0 q^{32} -148354. q^{33} -500138. q^{34} +5948.10 q^{35} +184470. q^{36} +107247. q^{37} +342794. q^{38} -348189. q^{39} +121073. q^{40} +520929. q^{41} +63203.2 q^{42} -943358. q^{43} -1.39038e6 q^{44} -36159.2 q^{45} -1.23488e6 q^{46} -789984. q^{47} +411973. q^{48} -809163. q^{49} +1.47701e6 q^{50} +691775. q^{51} -3.26325e6 q^{52} -930361. q^{53} -384220. q^{54} +272538. q^{55} +292714. q^{56} -474141. q^{57} +578240. q^{58} +205379. q^{59} -338886. q^{60} +2.19795e6 q^{61} -3.25055e6 q^{62} -87420.7 q^{63} -2.23792e6 q^{64} +639650. q^{65} +2.89593e6 q^{66} +1.13372e6 q^{67} +6.48336e6 q^{68} +1.70805e6 q^{69} -116109. q^{70} +2.46554e6 q^{71} -1.77945e6 q^{72} +5.09118e6 q^{73} -2.09351e6 q^{74} -2.04295e6 q^{75} -4.44368e6 q^{76} +658903. q^{77} +6.79678e6 q^{78} +4.40371e6 q^{79} -756827. q^{80} +531441. q^{81} -1.01687e7 q^{82} +172296. q^{83} -819312. q^{84} -1.27084e6 q^{85} +1.84147e7 q^{86} -799804. q^{87} +1.34120e7 q^{88} +8.28035e6 q^{89} +705842. q^{90} +1.54646e6 q^{91} +1.60079e7 q^{92} +4.49606e6 q^{93} +1.54208e7 q^{94} +871035. q^{95} +394012. q^{96} -6.60133e6 q^{97} +1.57952e7 q^{98} -4.00555e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −19.5204 −1.72538 −0.862688 0.505737i \(-0.831221\pi\)
−0.862688 + 0.505737i \(0.831221\pi\)
\(3\) 27.0000 0.577350
\(4\) 253.046 1.97692
\(5\) −49.6011 −0.177458 −0.0887292 0.996056i \(-0.528281\pi\)
−0.0887292 + 0.996056i \(0.528281\pi\)
\(6\) −527.051 −0.996146
\(7\) −119.919 −0.132143 −0.0660714 0.997815i \(-0.521047\pi\)
−0.0660714 + 0.997815i \(0.521047\pi\)
\(8\) −2440.94 −1.68555
\(9\) 729.000 0.333333
\(10\) 968.233 0.306182
\(11\) −5494.59 −1.24469 −0.622344 0.782744i \(-0.713820\pi\)
−0.622344 + 0.782744i \(0.713820\pi\)
\(12\) 6832.23 1.14137
\(13\) −12895.9 −1.62798 −0.813990 0.580879i \(-0.802709\pi\)
−0.813990 + 0.580879i \(0.802709\pi\)
\(14\) 2340.86 0.227996
\(15\) −1339.23 −0.102456
\(16\) 15258.3 0.931291
\(17\) 25621.3 1.26482 0.632411 0.774633i \(-0.282065\pi\)
0.632411 + 0.774633i \(0.282065\pi\)
\(18\) −14230.4 −0.575125
\(19\) −17560.8 −0.587363 −0.293681 0.955903i \(-0.594880\pi\)
−0.293681 + 0.955903i \(0.594880\pi\)
\(20\) −12551.3 −0.350821
\(21\) −3237.80 −0.0762927
\(22\) 107256. 2.14755
\(23\) 63260.9 1.08415 0.542074 0.840331i \(-0.317639\pi\)
0.542074 + 0.840331i \(0.317639\pi\)
\(24\) −65905.4 −0.973154
\(25\) −75664.7 −0.968509
\(26\) 251733. 2.80888
\(27\) 19683.0 0.192450
\(28\) −30344.9 −0.261236
\(29\) −29622.4 −0.225542 −0.112771 0.993621i \(-0.535973\pi\)
−0.112771 + 0.993621i \(0.535973\pi\)
\(30\) 26142.3 0.176774
\(31\) 166521. 1.00393 0.501963 0.864889i \(-0.332611\pi\)
0.501963 + 0.864889i \(0.332611\pi\)
\(32\) 14593.0 0.0787264
\(33\) −148354. −0.718621
\(34\) −500138. −2.18229
\(35\) 5948.10 0.0234498
\(36\) 184470. 0.658973
\(37\) 107247. 0.348080 0.174040 0.984739i \(-0.444318\pi\)
0.174040 + 0.984739i \(0.444318\pi\)
\(38\) 342794. 1.01342
\(39\) −348189. −0.939915
\(40\) 121073. 0.299115
\(41\) 520929. 1.18042 0.590208 0.807251i \(-0.299046\pi\)
0.590208 + 0.807251i \(0.299046\pi\)
\(42\) 63203.2 0.131633
\(43\) −943358. −1.80941 −0.904704 0.426040i \(-0.859908\pi\)
−0.904704 + 0.426040i \(0.859908\pi\)
\(44\) −1.39038e6 −2.46065
\(45\) −36159.2 −0.0591528
\(46\) −1.23488e6 −1.87056
\(47\) −789984. −1.10988 −0.554939 0.831891i \(-0.687259\pi\)
−0.554939 + 0.831891i \(0.687259\pi\)
\(48\) 411973. 0.537681
\(49\) −809163. −0.982538
\(50\) 1.47701e6 1.67104
\(51\) 691775. 0.730246
\(52\) −3.26325e6 −3.21839
\(53\) −930361. −0.858392 −0.429196 0.903211i \(-0.641203\pi\)
−0.429196 + 0.903211i \(0.641203\pi\)
\(54\) −384220. −0.332049
\(55\) 272538. 0.220880
\(56\) 292714. 0.222734
\(57\) −474141. −0.339114
\(58\) 578240. 0.389144
\(59\) 205379. 0.130189
\(60\) −338886. −0.202546
\(61\) 2.19795e6 1.23983 0.619915 0.784669i \(-0.287167\pi\)
0.619915 + 0.784669i \(0.287167\pi\)
\(62\) −3.25055e6 −1.73215
\(63\) −87420.7 −0.0440476
\(64\) −2.23792e6 −1.06712
\(65\) 639650. 0.288899
\(66\) 2.89593e6 1.23989
\(67\) 1.13372e6 0.460513 0.230257 0.973130i \(-0.426043\pi\)
0.230257 + 0.973130i \(0.426043\pi\)
\(68\) 6.48336e6 2.50045
\(69\) 1.70805e6 0.625933
\(70\) −116109. −0.0404598
\(71\) 2.46554e6 0.817537 0.408769 0.912638i \(-0.365958\pi\)
0.408769 + 0.912638i \(0.365958\pi\)
\(72\) −1.77945e6 −0.561851
\(73\) 5.09118e6 1.53175 0.765876 0.642988i \(-0.222306\pi\)
0.765876 + 0.642988i \(0.222306\pi\)
\(74\) −2.09351e6 −0.600569
\(75\) −2.04295e6 −0.559169
\(76\) −4.44368e6 −1.16117
\(77\) 658903. 0.164477
\(78\) 6.79678e6 1.62171
\(79\) 4.40371e6 1.00490 0.502452 0.864605i \(-0.332431\pi\)
0.502452 + 0.864605i \(0.332431\pi\)
\(80\) −756827. −0.165265
\(81\) 531441. 0.111111
\(82\) −1.01687e7 −2.03666
\(83\) 172296. 0.0330751 0.0165375 0.999863i \(-0.494736\pi\)
0.0165375 + 0.999863i \(0.494736\pi\)
\(84\) −819312. −0.150824
\(85\) −1.27084e6 −0.224453
\(86\) 1.84147e7 3.12191
\(87\) −799804. −0.130217
\(88\) 1.34120e7 2.09799
\(89\) 8.28035e6 1.24504 0.622521 0.782603i \(-0.286109\pi\)
0.622521 + 0.782603i \(0.286109\pi\)
\(90\) 705842. 0.102061
\(91\) 1.54646e6 0.215126
\(92\) 1.60079e7 2.14327
\(93\) 4.49606e6 0.579617
\(94\) 1.54208e7 1.91496
\(95\) 871035. 0.104232
\(96\) 394012. 0.0454527
\(97\) −6.60133e6 −0.734396 −0.367198 0.930143i \(-0.619683\pi\)
−0.367198 + 0.930143i \(0.619683\pi\)
\(98\) 1.57952e7 1.69525
\(99\) −4.00555e6 −0.414896
\(100\) −1.91466e7 −1.91466
\(101\) −1.77940e6 −0.171849 −0.0859247 0.996302i \(-0.527384\pi\)
−0.0859247 + 0.996302i \(0.527384\pi\)
\(102\) −1.35037e7 −1.25995
\(103\) −6.76389e6 −0.609911 −0.304955 0.952367i \(-0.598642\pi\)
−0.304955 + 0.952367i \(0.598642\pi\)
\(104\) 3.14781e7 2.74405
\(105\) 160599. 0.0135388
\(106\) 1.81610e7 1.48105
\(107\) 2.49224e7 1.96674 0.983371 0.181607i \(-0.0581299\pi\)
0.983371 + 0.181607i \(0.0581299\pi\)
\(108\) 4.98070e6 0.380458
\(109\) −2.15957e7 −1.59726 −0.798630 0.601823i \(-0.794441\pi\)
−0.798630 + 0.601823i \(0.794441\pi\)
\(110\) −5.32004e6 −0.381101
\(111\) 2.89567e6 0.200964
\(112\) −1.82975e6 −0.123063
\(113\) −1.84076e7 −1.20011 −0.600056 0.799958i \(-0.704855\pi\)
−0.600056 + 0.799958i \(0.704855\pi\)
\(114\) 9.25543e6 0.585099
\(115\) −3.13781e6 −0.192391
\(116\) −7.49582e6 −0.445878
\(117\) −9.40109e6 −0.542660
\(118\) −4.00908e6 −0.224625
\(119\) −3.07247e6 −0.167137
\(120\) 3.26898e6 0.172694
\(121\) 1.07033e7 0.549249
\(122\) −4.29047e7 −2.13917
\(123\) 1.40651e7 0.681513
\(124\) 4.21373e7 1.98468
\(125\) 7.62814e6 0.349328
\(126\) 1.70649e6 0.0759986
\(127\) 2.65579e7 1.15049 0.575243 0.817983i \(-0.304908\pi\)
0.575243 + 0.817983i \(0.304908\pi\)
\(128\) 4.18172e7 1.76246
\(129\) −2.54707e7 −1.04466
\(130\) −1.24862e7 −0.498459
\(131\) 3.58406e7 1.39292 0.696460 0.717595i \(-0.254757\pi\)
0.696460 + 0.717595i \(0.254757\pi\)
\(132\) −3.75403e7 −1.42066
\(133\) 2.10587e6 0.0776158
\(134\) −2.21306e7 −0.794558
\(135\) −976299. −0.0341519
\(136\) −6.25400e7 −2.13193
\(137\) 3.94491e7 1.31074 0.655368 0.755310i \(-0.272513\pi\)
0.655368 + 0.755310i \(0.272513\pi\)
\(138\) −3.33417e7 −1.07997
\(139\) −1.10784e7 −0.349886 −0.174943 0.984579i \(-0.555974\pi\)
−0.174943 + 0.984579i \(0.555974\pi\)
\(140\) 1.50514e6 0.0463584
\(141\) −2.13296e7 −0.640789
\(142\) −4.81283e7 −1.41056
\(143\) 7.08575e7 2.02633
\(144\) 1.11233e7 0.310430
\(145\) 1.46930e6 0.0400243
\(146\) −9.93819e7 −2.64285
\(147\) −2.18474e7 −0.567269
\(148\) 2.71384e7 0.688127
\(149\) 2.08170e7 0.515546 0.257773 0.966206i \(-0.417011\pi\)
0.257773 + 0.966206i \(0.417011\pi\)
\(150\) 3.98791e7 0.964776
\(151\) −2.65440e7 −0.627403 −0.313702 0.949522i \(-0.601569\pi\)
−0.313702 + 0.949522i \(0.601569\pi\)
\(152\) 4.28648e7 0.990031
\(153\) 1.86779e7 0.421608
\(154\) −1.28620e7 −0.283784
\(155\) −8.25961e6 −0.178155
\(156\) −8.81076e7 −1.85814
\(157\) 5.98029e7 1.23331 0.616656 0.787232i \(-0.288487\pi\)
0.616656 + 0.787232i \(0.288487\pi\)
\(158\) −8.59622e7 −1.73384
\(159\) −2.51197e7 −0.495593
\(160\) −723831. −0.0139707
\(161\) −7.58616e6 −0.143262
\(162\) −1.03739e7 −0.191708
\(163\) 9.55079e7 1.72736 0.863679 0.504042i \(-0.168154\pi\)
0.863679 + 0.504042i \(0.168154\pi\)
\(164\) 1.31819e8 2.33359
\(165\) 7.35852e6 0.127525
\(166\) −3.36328e6 −0.0570669
\(167\) −1.67858e7 −0.278890 −0.139445 0.990230i \(-0.544532\pi\)
−0.139445 + 0.990230i \(0.544532\pi\)
\(168\) 7.90328e6 0.128595
\(169\) 1.03555e8 1.65032
\(170\) 2.48074e7 0.387266
\(171\) −1.28018e7 −0.195788
\(172\) −2.38713e8 −3.57706
\(173\) −7.07459e7 −1.03882 −0.519410 0.854525i \(-0.673848\pi\)
−0.519410 + 0.854525i \(0.673848\pi\)
\(174\) 1.56125e7 0.224672
\(175\) 9.07361e6 0.127981
\(176\) −8.38379e7 −1.15917
\(177\) 5.54523e6 0.0751646
\(178\) −1.61636e8 −2.14816
\(179\) −9.77537e6 −0.127394 −0.0636968 0.997969i \(-0.520289\pi\)
−0.0636968 + 0.997969i \(0.520289\pi\)
\(180\) −9.14993e6 −0.116940
\(181\) 9.13710e7 1.14534 0.572668 0.819787i \(-0.305908\pi\)
0.572668 + 0.819787i \(0.305908\pi\)
\(182\) −3.01874e7 −0.371173
\(183\) 5.93445e7 0.715817
\(184\) −1.54416e8 −1.82739
\(185\) −5.31957e6 −0.0617698
\(186\) −8.77648e7 −1.00006
\(187\) −1.40778e8 −1.57431
\(188\) −1.99902e8 −2.19414
\(189\) −2.36036e6 −0.0254309
\(190\) −1.70029e7 −0.179840
\(191\) −1.44165e8 −1.49707 −0.748536 0.663094i \(-0.769243\pi\)
−0.748536 + 0.663094i \(0.769243\pi\)
\(192\) −6.04238e7 −0.616104
\(193\) 5.46350e7 0.547042 0.273521 0.961866i \(-0.411812\pi\)
0.273521 + 0.961866i \(0.411812\pi\)
\(194\) 1.28861e8 1.26711
\(195\) 1.72705e7 0.166796
\(196\) −2.04755e8 −1.94240
\(197\) 1.55456e8 1.44869 0.724347 0.689436i \(-0.242141\pi\)
0.724347 + 0.689436i \(0.242141\pi\)
\(198\) 7.81900e7 0.715852
\(199\) −2.03195e7 −0.182780 −0.0913898 0.995815i \(-0.529131\pi\)
−0.0913898 + 0.995815i \(0.529131\pi\)
\(200\) 1.84693e8 1.63247
\(201\) 3.06103e7 0.265878
\(202\) 3.47345e7 0.296505
\(203\) 3.55227e6 0.0298037
\(204\) 1.75051e8 1.44364
\(205\) −2.58386e7 −0.209475
\(206\) 1.32034e8 1.05233
\(207\) 4.61172e7 0.361382
\(208\) −1.96769e8 −1.51612
\(209\) 9.64893e7 0.731084
\(210\) −3.13495e6 −0.0233595
\(211\) −1.34211e8 −0.983557 −0.491779 0.870720i \(-0.663653\pi\)
−0.491779 + 0.870720i \(0.663653\pi\)
\(212\) −2.35424e8 −1.69697
\(213\) 6.65695e7 0.472005
\(214\) −4.86496e8 −3.39337
\(215\) 4.67916e7 0.321095
\(216\) −4.80450e7 −0.324385
\(217\) −1.99689e7 −0.132662
\(218\) 4.21557e8 2.75587
\(219\) 1.37462e8 0.884358
\(220\) 6.89645e7 0.436663
\(221\) −3.30409e8 −2.05911
\(222\) −5.65246e7 −0.346739
\(223\) 1.58205e8 0.955328 0.477664 0.878543i \(-0.341484\pi\)
0.477664 + 0.878543i \(0.341484\pi\)
\(224\) −1.74998e6 −0.0104031
\(225\) −5.51596e7 −0.322836
\(226\) 3.59323e8 2.07064
\(227\) 2.48289e8 1.40886 0.704428 0.709775i \(-0.251204\pi\)
0.704428 + 0.709775i \(0.251204\pi\)
\(228\) −1.19979e8 −0.670401
\(229\) 5.03600e6 0.0277116 0.0138558 0.999904i \(-0.495589\pi\)
0.0138558 + 0.999904i \(0.495589\pi\)
\(230\) 6.12513e7 0.331947
\(231\) 1.77904e7 0.0949606
\(232\) 7.23065e7 0.380162
\(233\) 2.76994e8 1.43458 0.717291 0.696774i \(-0.245382\pi\)
0.717291 + 0.696774i \(0.245382\pi\)
\(234\) 1.83513e8 0.936292
\(235\) 3.91841e7 0.196957
\(236\) 5.19703e7 0.257373
\(237\) 1.18900e8 0.580181
\(238\) 5.99758e7 0.288374
\(239\) 3.80453e8 1.80264 0.901319 0.433157i \(-0.142600\pi\)
0.901319 + 0.433157i \(0.142600\pi\)
\(240\) −2.04343e7 −0.0954160
\(241\) 6.73805e7 0.310080 0.155040 0.987908i \(-0.450449\pi\)
0.155040 + 0.987908i \(0.450449\pi\)
\(242\) −2.08933e8 −0.947661
\(243\) 1.43489e7 0.0641500
\(244\) 5.56181e8 2.45105
\(245\) 4.01354e7 0.174360
\(246\) −2.74556e8 −1.17587
\(247\) 2.26462e8 0.956215
\(248\) −4.06467e8 −1.69217
\(249\) 4.65198e6 0.0190959
\(250\) −1.48904e8 −0.602722
\(251\) −1.44862e8 −0.578225 −0.289113 0.957295i \(-0.593360\pi\)
−0.289113 + 0.957295i \(0.593360\pi\)
\(252\) −2.21214e7 −0.0870785
\(253\) −3.47593e8 −1.34943
\(254\) −5.18421e8 −1.98502
\(255\) −3.43128e7 −0.129588
\(256\) −5.29834e8 −1.97378
\(257\) 5.53323e7 0.203335 0.101668 0.994818i \(-0.467582\pi\)
0.101668 + 0.994818i \(0.467582\pi\)
\(258\) 4.97197e8 1.80244
\(259\) −1.28609e7 −0.0459963
\(260\) 1.61861e8 0.571129
\(261\) −2.15947e7 −0.0751806
\(262\) −6.99623e8 −2.40331
\(263\) 1.21289e8 0.411128 0.205564 0.978644i \(-0.434097\pi\)
0.205564 + 0.978644i \(0.434097\pi\)
\(264\) 3.62123e8 1.21127
\(265\) 4.61469e7 0.152329
\(266\) −4.11073e7 −0.133916
\(267\) 2.23569e8 0.718825
\(268\) 2.86882e8 0.910398
\(269\) −5.24972e8 −1.64438 −0.822191 0.569211i \(-0.807249\pi\)
−0.822191 + 0.569211i \(0.807249\pi\)
\(270\) 1.90577e7 0.0589248
\(271\) 1.03625e8 0.316280 0.158140 0.987417i \(-0.449450\pi\)
0.158140 + 0.987417i \(0.449450\pi\)
\(272\) 3.90936e8 1.17792
\(273\) 4.17543e7 0.124203
\(274\) −7.70062e8 −2.26151
\(275\) 4.15746e8 1.20549
\(276\) 4.32213e8 1.23742
\(277\) 1.95179e8 0.551765 0.275883 0.961191i \(-0.411030\pi\)
0.275883 + 0.961191i \(0.411030\pi\)
\(278\) 2.16255e8 0.603684
\(279\) 1.21394e8 0.334642
\(280\) −1.45189e7 −0.0395259
\(281\) 2.55753e8 0.687621 0.343811 0.939039i \(-0.388282\pi\)
0.343811 + 0.939039i \(0.388282\pi\)
\(282\) 4.16361e8 1.10560
\(283\) −1.42799e8 −0.374518 −0.187259 0.982311i \(-0.559960\pi\)
−0.187259 + 0.982311i \(0.559960\pi\)
\(284\) 6.23894e8 1.61621
\(285\) 2.35179e7 0.0601786
\(286\) −1.38317e9 −3.49618
\(287\) −6.24691e7 −0.155983
\(288\) 1.06383e7 0.0262421
\(289\) 2.46112e8 0.599777
\(290\) −2.86814e7 −0.0690569
\(291\) −1.78236e8 −0.424004
\(292\) 1.28830e9 3.02815
\(293\) −3.51565e8 −0.816524 −0.408262 0.912865i \(-0.633865\pi\)
−0.408262 + 0.912865i \(0.633865\pi\)
\(294\) 4.26470e8 0.978751
\(295\) −1.01870e7 −0.0231031
\(296\) −2.61784e8 −0.586708
\(297\) −1.08150e8 −0.239540
\(298\) −4.06357e8 −0.889510
\(299\) −8.15805e8 −1.76497
\(300\) −5.16959e8 −1.10543
\(301\) 1.13126e8 0.239100
\(302\) 5.18149e8 1.08251
\(303\) −4.80437e7 −0.0992172
\(304\) −2.67947e8 −0.547006
\(305\) −1.09021e8 −0.220018
\(306\) −3.64600e8 −0.727431
\(307\) −6.10924e8 −1.20504 −0.602522 0.798102i \(-0.705837\pi\)
−0.602522 + 0.798102i \(0.705837\pi\)
\(308\) 1.66733e8 0.325157
\(309\) −1.82625e8 −0.352132
\(310\) 1.61231e8 0.307384
\(311\) −5.80323e8 −1.09398 −0.546988 0.837140i \(-0.684226\pi\)
−0.546988 + 0.837140i \(0.684226\pi\)
\(312\) 8.49908e8 1.58428
\(313\) 8.00376e8 1.47533 0.737664 0.675168i \(-0.235929\pi\)
0.737664 + 0.675168i \(0.235929\pi\)
\(314\) −1.16738e9 −2.12793
\(315\) 4.33616e6 0.00781661
\(316\) 1.11434e9 1.98661
\(317\) −6.06744e8 −1.06979 −0.534895 0.844919i \(-0.679649\pi\)
−0.534895 + 0.844919i \(0.679649\pi\)
\(318\) 4.90347e8 0.855084
\(319\) 1.62763e8 0.280729
\(320\) 1.11003e8 0.189370
\(321\) 6.72906e8 1.13550
\(322\) 1.48085e8 0.247181
\(323\) −4.49930e8 −0.742910
\(324\) 1.34479e8 0.219658
\(325\) 9.75763e8 1.57671
\(326\) −1.86435e9 −2.98034
\(327\) −5.83085e8 −0.922178
\(328\) −1.27156e9 −1.98965
\(329\) 9.47337e7 0.146662
\(330\) −1.43641e8 −0.220029
\(331\) −4.95371e8 −0.750814 −0.375407 0.926860i \(-0.622497\pi\)
−0.375407 + 0.926860i \(0.622497\pi\)
\(332\) 4.35987e7 0.0653868
\(333\) 7.81831e7 0.116027
\(334\) 3.27665e8 0.481191
\(335\) −5.62335e7 −0.0817219
\(336\) −4.94033e7 −0.0710507
\(337\) −5.35635e8 −0.762367 −0.381183 0.924499i \(-0.624483\pi\)
−0.381183 + 0.924499i \(0.624483\pi\)
\(338\) −2.02144e9 −2.84742
\(339\) −4.97004e8 −0.692885
\(340\) −3.21582e8 −0.443726
\(341\) −9.14962e8 −1.24958
\(342\) 2.49896e8 0.337807
\(343\) 1.95792e8 0.261978
\(344\) 2.30268e9 3.04985
\(345\) −8.47209e7 −0.111077
\(346\) 1.38099e9 1.79235
\(347\) 1.06614e9 1.36981 0.684905 0.728633i \(-0.259844\pi\)
0.684905 + 0.728633i \(0.259844\pi\)
\(348\) −2.02387e8 −0.257428
\(349\) 9.40691e8 1.18456 0.592282 0.805731i \(-0.298227\pi\)
0.592282 + 0.805731i \(0.298227\pi\)
\(350\) −1.77120e8 −0.220816
\(351\) −2.53830e8 −0.313305
\(352\) −8.01827e7 −0.0979899
\(353\) −6.36948e8 −0.770713 −0.385357 0.922768i \(-0.625922\pi\)
−0.385357 + 0.922768i \(0.625922\pi\)
\(354\) −1.08245e8 −0.129687
\(355\) −1.22293e8 −0.145079
\(356\) 2.09531e9 2.46135
\(357\) −8.29567e7 −0.0964967
\(358\) 1.90819e8 0.219802
\(359\) 1.13966e9 1.30001 0.650003 0.759932i \(-0.274768\pi\)
0.650003 + 0.759932i \(0.274768\pi\)
\(360\) 8.82625e7 0.0997051
\(361\) −5.85490e8 −0.655005
\(362\) −1.78360e9 −1.97614
\(363\) 2.88990e8 0.317109
\(364\) 3.91324e8 0.425286
\(365\) −2.52528e8 −0.271822
\(366\) −1.15843e9 −1.23505
\(367\) −3.66527e8 −0.387057 −0.193529 0.981095i \(-0.561993\pi\)
−0.193529 + 0.981095i \(0.561993\pi\)
\(368\) 9.65252e8 1.00966
\(369\) 3.79757e8 0.393472
\(370\) 1.03840e8 0.106576
\(371\) 1.11568e8 0.113430
\(372\) 1.13771e9 1.14586
\(373\) 1.84626e9 1.84209 0.921047 0.389452i \(-0.127336\pi\)
0.921047 + 0.389452i \(0.127336\pi\)
\(374\) 2.74805e9 2.71628
\(375\) 2.05960e8 0.201685
\(376\) 1.92830e9 1.87076
\(377\) 3.82007e8 0.367178
\(378\) 4.60751e7 0.0438778
\(379\) −1.89436e8 −0.178742 −0.0893708 0.995998i \(-0.528486\pi\)
−0.0893708 + 0.995998i \(0.528486\pi\)
\(380\) 2.20412e8 0.206059
\(381\) 7.17064e8 0.664233
\(382\) 2.81415e9 2.58301
\(383\) 9.23938e8 0.840325 0.420162 0.907449i \(-0.361973\pi\)
0.420162 + 0.907449i \(0.361973\pi\)
\(384\) 1.12906e9 1.01756
\(385\) −3.26823e7 −0.0291877
\(386\) −1.06650e9 −0.943852
\(387\) −6.87708e8 −0.603136
\(388\) −1.67044e9 −1.45184
\(389\) 1.32165e8 0.113839 0.0569197 0.998379i \(-0.481872\pi\)
0.0569197 + 0.998379i \(0.481872\pi\)
\(390\) −3.37128e8 −0.287785
\(391\) 1.62083e9 1.37125
\(392\) 1.97512e9 1.65612
\(393\) 9.67697e8 0.804203
\(394\) −3.03457e9 −2.49954
\(395\) −2.18429e8 −0.178328
\(396\) −1.01359e9 −0.820216
\(397\) −3.45223e8 −0.276907 −0.138453 0.990369i \(-0.544213\pi\)
−0.138453 + 0.990369i \(0.544213\pi\)
\(398\) 3.96645e8 0.315363
\(399\) 5.68584e7 0.0448115
\(400\) −1.15451e9 −0.901963
\(401\) 1.50851e9 1.16827 0.584135 0.811656i \(-0.301434\pi\)
0.584135 + 0.811656i \(0.301434\pi\)
\(402\) −5.97525e8 −0.458738
\(403\) −2.14743e9 −1.63437
\(404\) −4.50268e8 −0.339732
\(405\) −2.63601e7 −0.0197176
\(406\) −6.93418e7 −0.0514226
\(407\) −5.89279e8 −0.433252
\(408\) −1.68858e9 −1.23087
\(409\) −8.44043e8 −0.610004 −0.305002 0.952352i \(-0.598657\pi\)
−0.305002 + 0.952352i \(0.598657\pi\)
\(410\) 5.04381e8 0.361422
\(411\) 1.06513e9 0.756754
\(412\) −1.71157e9 −1.20574
\(413\) −2.46288e7 −0.0172035
\(414\) −9.00226e8 −0.623520
\(415\) −8.54605e6 −0.00586945
\(416\) −1.88190e8 −0.128165
\(417\) −2.99118e8 −0.202007
\(418\) −1.88351e9 −1.26139
\(419\) −1.19941e9 −0.796563 −0.398281 0.917263i \(-0.630393\pi\)
−0.398281 + 0.917263i \(0.630393\pi\)
\(420\) 4.06388e7 0.0267651
\(421\) 6.99371e8 0.456794 0.228397 0.973568i \(-0.426652\pi\)
0.228397 + 0.973568i \(0.426652\pi\)
\(422\) 2.61985e9 1.69701
\(423\) −5.75898e8 −0.369960
\(424\) 2.27095e9 1.44686
\(425\) −1.93863e9 −1.22499
\(426\) −1.29946e9 −0.814386
\(427\) −2.63575e8 −0.163835
\(428\) 6.30652e9 3.88809
\(429\) 1.91315e9 1.16990
\(430\) −9.13390e8 −0.554009
\(431\) 1.57533e9 0.947768 0.473884 0.880587i \(-0.342852\pi\)
0.473884 + 0.880587i \(0.342852\pi\)
\(432\) 3.00328e8 0.179227
\(433\) 1.94780e9 1.15302 0.576511 0.817090i \(-0.304414\pi\)
0.576511 + 0.817090i \(0.304414\pi\)
\(434\) 3.89801e8 0.228891
\(435\) 3.96712e7 0.0231080
\(436\) −5.46471e9 −3.15765
\(437\) −1.11091e9 −0.636788
\(438\) −2.68331e9 −1.52585
\(439\) −1.26243e9 −0.712167 −0.356084 0.934454i \(-0.615888\pi\)
−0.356084 + 0.934454i \(0.615888\pi\)
\(440\) −6.65248e8 −0.372305
\(441\) −5.89879e8 −0.327513
\(442\) 6.44971e9 3.55273
\(443\) −2.40199e9 −1.31268 −0.656339 0.754466i \(-0.727896\pi\)
−0.656339 + 0.754466i \(0.727896\pi\)
\(444\) 7.32737e8 0.397290
\(445\) −4.10715e8 −0.220943
\(446\) −3.08822e9 −1.64830
\(447\) 5.62060e8 0.297650
\(448\) 2.68368e8 0.141013
\(449\) 7.69955e8 0.401424 0.200712 0.979650i \(-0.435675\pi\)
0.200712 + 0.979650i \(0.435675\pi\)
\(450\) 1.07674e9 0.557014
\(451\) −2.86229e9 −1.46925
\(452\) −4.65795e9 −2.37252
\(453\) −7.16687e8 −0.362231
\(454\) −4.84669e9 −2.43081
\(455\) −7.67059e7 −0.0381759
\(456\) 1.15735e9 0.571595
\(457\) 7.07638e8 0.346820 0.173410 0.984850i \(-0.444521\pi\)
0.173410 + 0.984850i \(0.444521\pi\)
\(458\) −9.83046e7 −0.0478129
\(459\) 5.04304e8 0.243415
\(460\) −7.94010e8 −0.380341
\(461\) 1.84947e8 0.0879212 0.0439606 0.999033i \(-0.486002\pi\)
0.0439606 + 0.999033i \(0.486002\pi\)
\(462\) −3.47275e8 −0.163843
\(463\) 2.34876e9 1.09978 0.549890 0.835237i \(-0.314670\pi\)
0.549890 + 0.835237i \(0.314670\pi\)
\(464\) −4.51986e8 −0.210045
\(465\) −2.23009e8 −0.102858
\(466\) −5.40704e9 −2.47519
\(467\) −2.14573e9 −0.974912 −0.487456 0.873147i \(-0.662075\pi\)
−0.487456 + 0.873147i \(0.662075\pi\)
\(468\) −2.37891e9 −1.07280
\(469\) −1.35954e8 −0.0608535
\(470\) −7.64888e8 −0.339825
\(471\) 1.61468e9 0.712053
\(472\) −5.01318e8 −0.219440
\(473\) 5.18336e9 2.25215
\(474\) −2.32098e9 −1.00103
\(475\) 1.32873e9 0.568866
\(476\) −7.77475e8 −0.330417
\(477\) −6.78233e8 −0.286131
\(478\) −7.42659e9 −3.11023
\(479\) −4.02137e8 −0.167186 −0.0835931 0.996500i \(-0.526640\pi\)
−0.0835931 + 0.996500i \(0.526640\pi\)
\(480\) −1.95434e7 −0.00806597
\(481\) −1.38305e9 −0.566668
\(482\) −1.31529e9 −0.535005
\(483\) −2.04826e8 −0.0827125
\(484\) 2.70843e9 1.08582
\(485\) 3.27433e8 0.130325
\(486\) −2.80096e8 −0.110683
\(487\) 3.29797e9 1.29389 0.646943 0.762538i \(-0.276047\pi\)
0.646943 + 0.762538i \(0.276047\pi\)
\(488\) −5.36505e9 −2.08980
\(489\) 2.57871e9 0.997291
\(490\) −7.83458e8 −0.300836
\(491\) 1.59461e9 0.607953 0.303977 0.952680i \(-0.401686\pi\)
0.303977 + 0.952680i \(0.401686\pi\)
\(492\) 3.55911e9 1.34730
\(493\) −7.58963e8 −0.285270
\(494\) −4.42062e9 −1.64983
\(495\) 1.98680e8 0.0736268
\(496\) 2.54082e9 0.934948
\(497\) −2.95664e8 −0.108032
\(498\) −9.08085e7 −0.0329476
\(499\) −4.65147e9 −1.67586 −0.837931 0.545776i \(-0.816235\pi\)
−0.837931 + 0.545776i \(0.816235\pi\)
\(500\) 1.93027e9 0.690594
\(501\) −4.53216e8 −0.161017
\(502\) 2.82777e9 0.997656
\(503\) −6.98423e8 −0.244698 −0.122349 0.992487i \(-0.539043\pi\)
−0.122349 + 0.992487i \(0.539043\pi\)
\(504\) 2.13389e8 0.0742445
\(505\) 8.82600e7 0.0304961
\(506\) 6.78515e9 2.32827
\(507\) 2.79599e9 0.952813
\(508\) 6.72037e9 2.27442
\(509\) −1.07682e9 −0.361935 −0.180967 0.983489i \(-0.557923\pi\)
−0.180967 + 0.983489i \(0.557923\pi\)
\(510\) 6.69799e8 0.223588
\(511\) −6.10527e8 −0.202410
\(512\) 4.98996e9 1.64306
\(513\) −3.45649e8 −0.113038
\(514\) −1.08011e9 −0.350830
\(515\) 3.35497e8 0.108234
\(516\) −6.44524e9 −2.06521
\(517\) 4.34063e9 1.38145
\(518\) 2.51050e8 0.0793609
\(519\) −1.91014e9 −0.599763
\(520\) −1.56135e9 −0.486954
\(521\) −3.06506e9 −0.949527 −0.474763 0.880114i \(-0.657466\pi\)
−0.474763 + 0.880114i \(0.657466\pi\)
\(522\) 4.21537e8 0.129715
\(523\) −5.83979e9 −1.78501 −0.892506 0.451036i \(-0.851055\pi\)
−0.892506 + 0.451036i \(0.851055\pi\)
\(524\) 9.06932e9 2.75369
\(525\) 2.44987e8 0.0738901
\(526\) −2.36762e9 −0.709351
\(527\) 4.26647e9 1.26979
\(528\) −2.26362e9 −0.669245
\(529\) 5.97121e8 0.175375
\(530\) −9.00806e8 −0.262824
\(531\) 1.49721e8 0.0433963
\(532\) 5.32880e8 0.153440
\(533\) −6.71783e9 −1.92169
\(534\) −4.36416e9 −1.24024
\(535\) −1.23618e9 −0.349015
\(536\) −2.76733e9 −0.776219
\(537\) −2.63935e8 −0.0735507
\(538\) 1.02477e10 2.83718
\(539\) 4.44601e9 1.22295
\(540\) −2.47048e8 −0.0675155
\(541\) −4.19074e9 −1.13789 −0.568945 0.822375i \(-0.692648\pi\)
−0.568945 + 0.822375i \(0.692648\pi\)
\(542\) −2.02280e9 −0.545701
\(543\) 2.46702e9 0.661261
\(544\) 3.73892e8 0.0995750
\(545\) 1.07117e9 0.283447
\(546\) −8.15060e8 −0.214297
\(547\) 1.83917e9 0.480470 0.240235 0.970715i \(-0.422775\pi\)
0.240235 + 0.970715i \(0.422775\pi\)
\(548\) 9.98243e9 2.59122
\(549\) 1.60230e9 0.413277
\(550\) −8.11553e9 −2.07992
\(551\) 5.20192e8 0.132475
\(552\) −4.16924e9 −1.05504
\(553\) −5.28087e8 −0.132791
\(554\) −3.80997e9 −0.952002
\(555\) −1.43629e8 −0.0356628
\(556\) −2.80335e9 −0.691696
\(557\) −3.81190e9 −0.934648 −0.467324 0.884086i \(-0.654782\pi\)
−0.467324 + 0.884086i \(0.654782\pi\)
\(558\) −2.36965e9 −0.577383
\(559\) 1.21654e10 2.94568
\(560\) 9.07576e7 0.0218386
\(561\) −3.80102e9 −0.908929
\(562\) −4.99241e9 −1.18640
\(563\) −6.18376e9 −1.46041 −0.730203 0.683231i \(-0.760574\pi\)
−0.730203 + 0.683231i \(0.760574\pi\)
\(564\) −5.39735e9 −1.26679
\(565\) 9.13035e8 0.212970
\(566\) 2.78749e9 0.646184
\(567\) −6.37297e7 −0.0146825
\(568\) −6.01823e9 −1.37800
\(569\) −5.12814e9 −1.16699 −0.583495 0.812117i \(-0.698315\pi\)
−0.583495 + 0.812117i \(0.698315\pi\)
\(570\) −4.59079e8 −0.103831
\(571\) 6.26225e8 0.140768 0.0703840 0.997520i \(-0.477578\pi\)
0.0703840 + 0.997520i \(0.477578\pi\)
\(572\) 1.79302e10 4.00589
\(573\) −3.89245e9 −0.864335
\(574\) 1.21942e9 0.269130
\(575\) −4.78662e9 −1.05001
\(576\) −1.63144e9 −0.355708
\(577\) −2.62762e9 −0.569440 −0.284720 0.958611i \(-0.591901\pi\)
−0.284720 + 0.958611i \(0.591901\pi\)
\(578\) −4.80419e9 −1.03484
\(579\) 1.47515e9 0.315835
\(580\) 3.71801e8 0.0791247
\(581\) −2.06614e7 −0.00437063
\(582\) 3.47924e9 0.731566
\(583\) 5.11195e9 1.06843
\(584\) −1.24273e10 −2.58185
\(585\) 4.66305e8 0.0962996
\(586\) 6.86269e9 1.40881
\(587\) −1.62680e9 −0.331972 −0.165986 0.986128i \(-0.553081\pi\)
−0.165986 + 0.986128i \(0.553081\pi\)
\(588\) −5.52839e9 −1.12144
\(589\) −2.92423e9 −0.589669
\(590\) 1.98855e8 0.0398615
\(591\) 4.19732e9 0.836404
\(592\) 1.63640e9 0.324164
\(593\) 2.05047e8 0.0403797 0.0201898 0.999796i \(-0.493573\pi\)
0.0201898 + 0.999796i \(0.493573\pi\)
\(594\) 2.11113e9 0.413297
\(595\) 1.52398e8 0.0296599
\(596\) 5.26766e9 1.01919
\(597\) −5.48627e8 −0.105528
\(598\) 1.59248e10 3.04524
\(599\) −2.72889e9 −0.518790 −0.259395 0.965771i \(-0.583523\pi\)
−0.259395 + 0.965771i \(0.583523\pi\)
\(600\) 4.98671e9 0.942508
\(601\) 1.75593e9 0.329949 0.164974 0.986298i \(-0.447246\pi\)
0.164974 + 0.986298i \(0.447246\pi\)
\(602\) −2.20827e9 −0.412538
\(603\) 8.26479e8 0.153504
\(604\) −6.71684e9 −1.24033
\(605\) −5.30896e8 −0.0974689
\(606\) 9.37832e8 0.171187
\(607\) 3.28692e9 0.596526 0.298263 0.954484i \(-0.403593\pi\)
0.298263 + 0.954484i \(0.403593\pi\)
\(608\) −2.56265e8 −0.0462410
\(609\) 9.59114e7 0.0172072
\(610\) 2.12812e9 0.379614
\(611\) 1.01875e10 1.80686
\(612\) 4.72637e9 0.833484
\(613\) 5.79734e9 1.01652 0.508261 0.861203i \(-0.330289\pi\)
0.508261 + 0.861203i \(0.330289\pi\)
\(614\) 1.19255e10 2.07915
\(615\) −6.97643e8 −0.120940
\(616\) −1.60834e9 −0.277234
\(617\) 1.42127e9 0.243600 0.121800 0.992555i \(-0.461133\pi\)
0.121800 + 0.992555i \(0.461133\pi\)
\(618\) 3.56491e9 0.607560
\(619\) 3.85999e9 0.654138 0.327069 0.945001i \(-0.393939\pi\)
0.327069 + 0.945001i \(0.393939\pi\)
\(620\) −2.09006e9 −0.352198
\(621\) 1.24517e9 0.208644
\(622\) 1.13281e10 1.88752
\(623\) −9.92968e8 −0.164523
\(624\) −5.31276e9 −0.875334
\(625\) 5.53294e9 0.906517
\(626\) −1.56236e10 −2.54549
\(627\) 2.60521e9 0.422091
\(628\) 1.51329e10 2.43816
\(629\) 2.74781e9 0.440260
\(630\) −8.46436e7 −0.0134866
\(631\) −5.65956e9 −0.896768 −0.448384 0.893841i \(-0.648000\pi\)
−0.448384 + 0.893841i \(0.648000\pi\)
\(632\) −1.07492e10 −1.69382
\(633\) −3.62370e9 −0.567857
\(634\) 1.18439e10 1.84579
\(635\) −1.31730e9 −0.204163
\(636\) −6.35644e9 −0.979747
\(637\) 1.04349e10 1.59955
\(638\) −3.17719e9 −0.484363
\(639\) 1.79738e9 0.272512
\(640\) −2.07418e9 −0.312763
\(641\) −2.57457e9 −0.386101 −0.193051 0.981189i \(-0.561838\pi\)
−0.193051 + 0.981189i \(0.561838\pi\)
\(642\) −1.31354e10 −1.95916
\(643\) 3.46249e9 0.513629 0.256815 0.966461i \(-0.417327\pi\)
0.256815 + 0.966461i \(0.417327\pi\)
\(644\) −1.91965e9 −0.283218
\(645\) 1.26337e9 0.185384
\(646\) 8.78281e9 1.28180
\(647\) 1.17598e9 0.170701 0.0853506 0.996351i \(-0.472799\pi\)
0.0853506 + 0.996351i \(0.472799\pi\)
\(648\) −1.29722e9 −0.187284
\(649\) −1.12847e9 −0.162045
\(650\) −1.90473e10 −2.72042
\(651\) −5.39161e8 −0.0765923
\(652\) 2.41679e10 3.41485
\(653\) 9.71067e9 1.36475 0.682375 0.731002i \(-0.260947\pi\)
0.682375 + 0.731002i \(0.260947\pi\)
\(654\) 1.13820e10 1.59110
\(655\) −1.77774e9 −0.247185
\(656\) 7.94847e9 1.09931
\(657\) 3.71147e9 0.510584
\(658\) −1.84924e9 −0.253048
\(659\) −9.69880e9 −1.32014 −0.660069 0.751205i \(-0.729473\pi\)
−0.660069 + 0.751205i \(0.729473\pi\)
\(660\) 1.86204e9 0.252107
\(661\) −1.15769e10 −1.55914 −0.779572 0.626313i \(-0.784563\pi\)
−0.779572 + 0.626313i \(0.784563\pi\)
\(662\) 9.66984e9 1.29544
\(663\) −8.92104e9 −1.18883
\(664\) −4.20563e8 −0.0557498
\(665\) −1.04453e8 −0.0137736
\(666\) −1.52617e9 −0.200190
\(667\) −1.87394e9 −0.244520
\(668\) −4.24757e9 −0.551344
\(669\) 4.27153e9 0.551559
\(670\) 1.09770e9 0.141001
\(671\) −1.20768e10 −1.54320
\(672\) −4.72494e7 −0.00600625
\(673\) −1.00199e10 −1.26710 −0.633552 0.773700i \(-0.718404\pi\)
−0.633552 + 0.773700i \(0.718404\pi\)
\(674\) 1.04558e10 1.31537
\(675\) −1.48931e9 −0.186390
\(676\) 2.62042e10 3.26255
\(677\) −2.76445e8 −0.0342411 −0.0171206 0.999853i \(-0.505450\pi\)
−0.0171206 + 0.999853i \(0.505450\pi\)
\(678\) 9.70171e9 1.19549
\(679\) 7.91622e8 0.0970452
\(680\) 3.10205e9 0.378328
\(681\) 6.70380e9 0.813403
\(682\) 1.78604e10 2.15599
\(683\) 1.25710e9 0.150973 0.0754864 0.997147i \(-0.475949\pi\)
0.0754864 + 0.997147i \(0.475949\pi\)
\(684\) −3.23944e9 −0.387056
\(685\) −1.95672e9 −0.232601
\(686\) −3.82193e9 −0.452011
\(687\) 1.35972e8 0.0159993
\(688\) −1.43940e10 −1.68509
\(689\) 1.19978e10 1.39745
\(690\) 1.65379e9 0.191649
\(691\) 1.23439e10 1.42325 0.711623 0.702562i \(-0.247960\pi\)
0.711623 + 0.702562i \(0.247960\pi\)
\(692\) −1.79020e10 −2.05366
\(693\) 4.80340e8 0.0548255
\(694\) −2.08114e10 −2.36343
\(695\) 5.49502e8 0.0620902
\(696\) 1.95227e9 0.219487
\(697\) 1.33469e10 1.49302
\(698\) −1.83627e10 −2.04382
\(699\) 7.47885e9 0.828256
\(700\) 2.29604e9 0.253009
\(701\) 4.36020e9 0.478072 0.239036 0.971011i \(-0.423169\pi\)
0.239036 + 0.971011i \(0.423169\pi\)
\(702\) 4.95485e9 0.540569
\(703\) −1.88334e9 −0.204449
\(704\) 1.22964e10 1.32824
\(705\) 1.05797e9 0.113713
\(706\) 1.24335e10 1.32977
\(707\) 2.13383e8 0.0227086
\(708\) 1.40320e9 0.148594
\(709\) −6.54244e9 −0.689411 −0.344705 0.938711i \(-0.612021\pi\)
−0.344705 + 0.938711i \(0.612021\pi\)
\(710\) 2.38722e9 0.250315
\(711\) 3.21031e9 0.334968
\(712\) −2.02118e10 −2.09858
\(713\) 1.05342e10 1.08840
\(714\) 1.61935e9 0.166493
\(715\) −3.51461e9 −0.359589
\(716\) −2.47361e9 −0.251847
\(717\) 1.02722e10 1.04075
\(718\) −2.22466e10 −2.24300
\(719\) 1.24076e9 0.124491 0.0622453 0.998061i \(-0.480174\pi\)
0.0622453 + 0.998061i \(0.480174\pi\)
\(720\) −5.51727e8 −0.0550884
\(721\) 8.11117e8 0.0805953
\(722\) 1.14290e10 1.13013
\(723\) 1.81927e9 0.179025
\(724\) 2.31210e10 2.26424
\(725\) 2.24137e9 0.218439
\(726\) −5.64119e9 −0.547133
\(727\) 8.74540e8 0.0844130 0.0422065 0.999109i \(-0.486561\pi\)
0.0422065 + 0.999109i \(0.486561\pi\)
\(728\) −3.77481e9 −0.362606
\(729\) 3.87420e8 0.0370370
\(730\) 4.92945e9 0.468995
\(731\) −2.41700e10 −2.28858
\(732\) 1.50169e10 1.41511
\(733\) −1.87673e10 −1.76010 −0.880052 0.474878i \(-0.842492\pi\)
−0.880052 + 0.474878i \(0.842492\pi\)
\(734\) 7.15476e9 0.667819
\(735\) 1.08365e9 0.100667
\(736\) 9.23169e8 0.0853510
\(737\) −6.22930e9 −0.573196
\(738\) −7.41301e9 −0.678887
\(739\) 1.50202e10 1.36905 0.684525 0.728989i \(-0.260010\pi\)
0.684525 + 0.728989i \(0.260010\pi\)
\(740\) −1.34610e9 −0.122114
\(741\) 6.11447e9 0.552071
\(742\) −2.17784e9 −0.195710
\(743\) 4.97295e9 0.444788 0.222394 0.974957i \(-0.428613\pi\)
0.222394 + 0.974957i \(0.428613\pi\)
\(744\) −1.09746e10 −0.976975
\(745\) −1.03255e9 −0.0914879
\(746\) −3.60397e10 −3.17830
\(747\) 1.25603e8 0.0110250
\(748\) −3.56234e10 −3.11228
\(749\) −2.98867e9 −0.259891
\(750\) −4.02042e9 −0.347982
\(751\) −1.94834e10 −1.67851 −0.839255 0.543737i \(-0.817009\pi\)
−0.839255 + 0.543737i \(0.817009\pi\)
\(752\) −1.20538e10 −1.03362
\(753\) −3.91128e9 −0.333839
\(754\) −7.45692e9 −0.633519
\(755\) 1.31661e9 0.111338
\(756\) −5.97278e8 −0.0502748
\(757\) 5.89170e9 0.493634 0.246817 0.969062i \(-0.420615\pi\)
0.246817 + 0.969062i \(0.420615\pi\)
\(758\) 3.69787e9 0.308396
\(759\) −9.38500e9 −0.779091
\(760\) −2.12614e9 −0.175689
\(761\) 1.46780e10 1.20731 0.603656 0.797245i \(-0.293710\pi\)
0.603656 + 0.797245i \(0.293710\pi\)
\(762\) −1.39974e10 −1.14605
\(763\) 2.58973e9 0.211066
\(764\) −3.64803e10 −2.95959
\(765\) −9.26445e8 −0.0748178
\(766\) −1.80356e10 −1.44988
\(767\) −2.64854e9 −0.211945
\(768\) −1.43055e10 −1.13956
\(769\) −2.45703e10 −1.94835 −0.974177 0.225784i \(-0.927506\pi\)
−0.974177 + 0.225784i \(0.927506\pi\)
\(770\) 6.37972e8 0.0503598
\(771\) 1.49397e9 0.117396
\(772\) 1.38252e10 1.08146
\(773\) −1.40761e10 −1.09611 −0.548057 0.836441i \(-0.684632\pi\)
−0.548057 + 0.836441i \(0.684632\pi\)
\(774\) 1.34243e10 1.04064
\(775\) −1.25997e10 −0.972312
\(776\) 1.61135e10 1.23786
\(777\) −3.47245e8 −0.0265560
\(778\) −2.57991e9 −0.196416
\(779\) −9.14792e9 −0.693332
\(780\) 4.37024e9 0.329742
\(781\) −1.35471e10 −1.01758
\(782\) −3.16392e10 −2.36593
\(783\) −5.83057e8 −0.0434055
\(784\) −1.23464e10 −0.915029
\(785\) −2.96629e9 −0.218862
\(786\) −1.88898e10 −1.38755
\(787\) 7.50211e9 0.548620 0.274310 0.961641i \(-0.411551\pi\)
0.274310 + 0.961641i \(0.411551\pi\)
\(788\) 3.93376e10 2.86395
\(789\) 3.27481e9 0.237365
\(790\) 4.26382e9 0.307683
\(791\) 2.20741e9 0.158586
\(792\) 9.77732e9 0.699329
\(793\) −2.83444e10 −2.01842
\(794\) 6.73889e9 0.477768
\(795\) 1.24597e9 0.0879471
\(796\) −5.14177e9 −0.361341
\(797\) 1.68153e10 1.17652 0.588260 0.808672i \(-0.299813\pi\)
0.588260 + 0.808672i \(0.299813\pi\)
\(798\) −1.10990e9 −0.0773166
\(799\) −2.02404e10 −1.40380
\(800\) −1.10418e9 −0.0762472
\(801\) 6.03638e9 0.415014
\(802\) −2.94467e10 −2.01570
\(803\) −2.79739e10 −1.90655
\(804\) 7.74581e9 0.525618
\(805\) 3.76282e8 0.0254231
\(806\) 4.19187e10 2.81991
\(807\) −1.41742e10 −0.949385
\(808\) 4.34340e9 0.289661
\(809\) −2.50538e10 −1.66362 −0.831810 0.555060i \(-0.812695\pi\)
−0.831810 + 0.555060i \(0.812695\pi\)
\(810\) 5.14559e8 0.0340202
\(811\) −7.70837e9 −0.507446 −0.253723 0.967277i \(-0.581655\pi\)
−0.253723 + 0.967277i \(0.581655\pi\)
\(812\) 8.98888e8 0.0589195
\(813\) 2.79787e9 0.182604
\(814\) 1.15029e10 0.747522
\(815\) −4.73730e9 −0.306534
\(816\) 1.05553e10 0.680071
\(817\) 1.65661e10 1.06278
\(818\) 1.64760e10 1.05249
\(819\) 1.12737e9 0.0717086
\(820\) −6.53836e9 −0.414114
\(821\) −1.50406e10 −0.948558 −0.474279 0.880375i \(-0.657291\pi\)
−0.474279 + 0.880375i \(0.657291\pi\)
\(822\) −2.07917e10 −1.30568
\(823\) 3.00702e10 1.88034 0.940172 0.340700i \(-0.110664\pi\)
0.940172 + 0.340700i \(0.110664\pi\)
\(824\) 1.65103e10 1.02804
\(825\) 1.12252e10 0.695991
\(826\) 4.80763e8 0.0296825
\(827\) 1.24512e10 0.765494 0.382747 0.923853i \(-0.374978\pi\)
0.382747 + 0.923853i \(0.374978\pi\)
\(828\) 1.16698e10 0.714424
\(829\) 1.67317e10 1.02000 0.509999 0.860175i \(-0.329646\pi\)
0.509999 + 0.860175i \(0.329646\pi\)
\(830\) 1.66822e8 0.0101270
\(831\) 5.26984e9 0.318562
\(832\) 2.88599e10 1.73726
\(833\) −2.07318e10 −1.24274
\(834\) 5.83889e9 0.348537
\(835\) 8.32593e8 0.0494914
\(836\) 2.44162e10 1.44529
\(837\) 3.27762e9 0.193206
\(838\) 2.34130e10 1.37437
\(839\) 1.49303e10 0.872773 0.436386 0.899759i \(-0.356258\pi\)
0.436386 + 0.899759i \(0.356258\pi\)
\(840\) −3.92012e8 −0.0228203
\(841\) −1.63724e10 −0.949131
\(842\) −1.36520e10 −0.788140
\(843\) 6.90534e9 0.396998
\(844\) −3.39615e10 −1.94441
\(845\) −5.13645e9 −0.292863
\(846\) 1.12418e10 0.638319
\(847\) −1.28353e9 −0.0725794
\(848\) −1.41957e10 −0.799413
\(849\) −3.85557e9 −0.216228
\(850\) 3.78428e10 2.11357
\(851\) 6.78455e9 0.377370
\(852\) 1.68451e10 0.933116
\(853\) −6.25227e9 −0.344918 −0.172459 0.985017i \(-0.555171\pi\)
−0.172459 + 0.985017i \(0.555171\pi\)
\(854\) 5.14508e9 0.282676
\(855\) 6.34984e8 0.0347441
\(856\) −6.08342e10 −3.31505
\(857\) 2.79248e10 1.51550 0.757751 0.652543i \(-0.226298\pi\)
0.757751 + 0.652543i \(0.226298\pi\)
\(858\) −3.73455e10 −2.01852
\(859\) 4.15604e9 0.223719 0.111860 0.993724i \(-0.464319\pi\)
0.111860 + 0.993724i \(0.464319\pi\)
\(860\) 1.18404e10 0.634778
\(861\) −1.68666e9 −0.0900570
\(862\) −3.07511e10 −1.63525
\(863\) 1.83670e10 0.972750 0.486375 0.873750i \(-0.338319\pi\)
0.486375 + 0.873750i \(0.338319\pi\)
\(864\) 2.87235e8 0.0151509
\(865\) 3.50908e9 0.184347
\(866\) −3.80219e10 −1.98939
\(867\) 6.64501e9 0.346281
\(868\) −5.05305e9 −0.262261
\(869\) −2.41966e10 −1.25079
\(870\) −7.74397e8 −0.0398700
\(871\) −1.46203e10 −0.749707
\(872\) 5.27139e10 2.69226
\(873\) −4.81237e9 −0.244799
\(874\) 2.16854e10 1.09870
\(875\) −9.14756e8 −0.0461612
\(876\) 3.47841e10 1.74830
\(877\) −2.18727e10 −1.09497 −0.547487 0.836815i \(-0.684415\pi\)
−0.547487 + 0.836815i \(0.684415\pi\)
\(878\) 2.46432e10 1.22876
\(879\) −9.49225e9 −0.471420
\(880\) 4.15845e9 0.205704
\(881\) 2.12719e10 1.04807 0.524035 0.851697i \(-0.324426\pi\)
0.524035 + 0.851697i \(0.324426\pi\)
\(882\) 1.15147e10 0.565082
\(883\) 2.46950e10 1.20711 0.603554 0.797322i \(-0.293751\pi\)
0.603554 + 0.797322i \(0.293751\pi\)
\(884\) −8.36086e10 −4.07069
\(885\) −2.75050e8 −0.0133386
\(886\) 4.68878e10 2.26486
\(887\) 2.50959e10 1.20745 0.603727 0.797191i \(-0.293682\pi\)
0.603727 + 0.797191i \(0.293682\pi\)
\(888\) −7.06816e9 −0.338736
\(889\) −3.18479e9 −0.152028
\(890\) 8.01731e9 0.381209
\(891\) −2.92005e9 −0.138299
\(892\) 4.00330e10 1.88861
\(893\) 1.38727e10 0.651902
\(894\) −1.09716e10 −0.513559
\(895\) 4.84869e8 0.0226071
\(896\) −5.01466e9 −0.232897
\(897\) −2.20267e10 −1.01901
\(898\) −1.50298e10 −0.692606
\(899\) −4.93274e9 −0.226427
\(900\) −1.39579e10 −0.638221
\(901\) −2.38370e10 −1.08571
\(902\) 5.58730e10 2.53501
\(903\) 3.05441e9 0.138045
\(904\) 4.49317e10 2.02285
\(905\) −4.53210e9 −0.203250
\(906\) 1.39900e10 0.624985
\(907\) 1.03741e10 0.461662 0.230831 0.972994i \(-0.425856\pi\)
0.230831 + 0.972994i \(0.425856\pi\)
\(908\) 6.28284e10 2.78519
\(909\) −1.29718e9 −0.0572831
\(910\) 1.49733e9 0.0658677
\(911\) −1.15863e10 −0.507728 −0.253864 0.967240i \(-0.581702\pi\)
−0.253864 + 0.967240i \(0.581702\pi\)
\(912\) −7.23458e9 −0.315814
\(913\) −9.46693e8 −0.0411682
\(914\) −1.38134e10 −0.598395
\(915\) −2.94355e9 −0.127028
\(916\) 1.27434e9 0.0547836
\(917\) −4.29796e9 −0.184064
\(918\) −9.84421e9 −0.419983
\(919\) −1.20996e10 −0.514240 −0.257120 0.966380i \(-0.582774\pi\)
−0.257120 + 0.966380i \(0.582774\pi\)
\(920\) 7.65921e9 0.324285
\(921\) −1.64949e10 −0.695732
\(922\) −3.61024e9 −0.151697
\(923\) −3.17953e10 −1.33093
\(924\) 4.50178e9 0.187729
\(925\) −8.11482e9 −0.337119
\(926\) −4.58488e10 −1.89753
\(927\) −4.93088e9 −0.203304
\(928\) −4.32280e8 −0.0177561
\(929\) −2.16304e10 −0.885134 −0.442567 0.896735i \(-0.645932\pi\)
−0.442567 + 0.896735i \(0.645932\pi\)
\(930\) 4.35323e9 0.177469
\(931\) 1.42095e10 0.577107
\(932\) 7.00922e10 2.83605
\(933\) −1.56687e10 −0.631608
\(934\) 4.18855e10 1.68209
\(935\) 6.98276e9 0.279374
\(936\) 2.29475e10 0.914682
\(937\) 1.59669e9 0.0634063 0.0317031 0.999497i \(-0.489907\pi\)
0.0317031 + 0.999497i \(0.489907\pi\)
\(938\) 2.65387e9 0.104995
\(939\) 2.16101e10 0.851781
\(940\) 9.91536e9 0.389369
\(941\) 3.39260e10 1.32730 0.663650 0.748043i \(-0.269007\pi\)
0.663650 + 0.748043i \(0.269007\pi\)
\(942\) −3.15191e10 −1.22856
\(943\) 3.29544e10 1.27974
\(944\) 3.13373e9 0.121244
\(945\) 1.17076e8 0.00451292
\(946\) −1.01181e11 −3.88580
\(947\) 2.74213e10 1.04921 0.524606 0.851345i \(-0.324213\pi\)
0.524606 + 0.851345i \(0.324213\pi\)
\(948\) 3.00872e10 1.14697
\(949\) −6.56553e10 −2.49366
\(950\) −2.59374e10 −0.981507
\(951\) −1.63821e10 −0.617643
\(952\) 7.49971e9 0.281719
\(953\) −1.02794e10 −0.384720 −0.192360 0.981324i \(-0.561614\pi\)
−0.192360 + 0.981324i \(0.561614\pi\)
\(954\) 1.32394e10 0.493683
\(955\) 7.15074e9 0.265668
\(956\) 9.62720e10 3.56367
\(957\) 4.39459e9 0.162079
\(958\) 7.84988e9 0.288459
\(959\) −4.73068e9 −0.173204
\(960\) 2.99709e9 0.109333
\(961\) 2.16493e8 0.00786887
\(962\) 2.69976e10 0.977715
\(963\) 1.81685e10 0.655581
\(964\) 1.70503e10 0.613004
\(965\) −2.70996e9 −0.0970771
\(966\) 3.99829e9 0.142710
\(967\) −2.47735e10 −0.881040 −0.440520 0.897743i \(-0.645206\pi\)
−0.440520 + 0.897743i \(0.645206\pi\)
\(968\) −2.61262e10 −0.925789
\(969\) −1.21481e10 −0.428919
\(970\) −6.39163e9 −0.224859
\(971\) 4.27396e10 1.49818 0.749088 0.662470i \(-0.230492\pi\)
0.749088 + 0.662470i \(0.230492\pi\)
\(972\) 3.63093e9 0.126819
\(973\) 1.32851e9 0.0462349
\(974\) −6.43778e10 −2.23244
\(975\) 2.63456e10 0.910316
\(976\) 3.35368e10 1.15464
\(977\) 4.18609e8 0.0143608 0.00718039 0.999974i \(-0.497714\pi\)
0.00718039 + 0.999974i \(0.497714\pi\)
\(978\) −5.03375e10 −1.72070
\(979\) −4.54971e10 −1.54969
\(980\) 1.01561e10 0.344695
\(981\) −1.57433e10 −0.532420
\(982\) −3.11275e10 −1.04895
\(983\) −8.78167e9 −0.294876 −0.147438 0.989071i \(-0.547103\pi\)
−0.147438 + 0.989071i \(0.547103\pi\)
\(984\) −3.43320e10 −1.14873
\(985\) −7.71081e9 −0.257083
\(986\) 1.48153e10 0.492198
\(987\) 2.55781e9 0.0846756
\(988\) 5.73052e10 1.89036
\(989\) −5.96777e10 −1.96167
\(990\) −3.87831e9 −0.127034
\(991\) 3.19826e10 1.04389 0.521947 0.852978i \(-0.325206\pi\)
0.521947 + 0.852978i \(0.325206\pi\)
\(992\) 2.43004e9 0.0790356
\(993\) −1.33750e10 −0.433483
\(994\) 5.77148e9 0.186395
\(995\) 1.00787e9 0.0324358
\(996\) 1.17716e9 0.0377511
\(997\) 1.39615e10 0.446170 0.223085 0.974799i \(-0.428387\pi\)
0.223085 + 0.974799i \(0.428387\pi\)
\(998\) 9.07985e10 2.89149
\(999\) 2.11094e9 0.0669881
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.1.2 18
3.2 odd 2 531.8.a.e.1.17 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.2 18 1.1 even 1 trivial
531.8.a.e.1.17 18 3.2 odd 2