Properties

Label 177.8.a.d.1.6
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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 1.6
Root \(-10.5486\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.54862 q^{2} +27.0000 q^{3} -36.8240 q^{4} +248.981 q^{5} -257.813 q^{6} -1453.31 q^{7} +1573.84 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-9.54862 q^{2} +27.0000 q^{3} -36.8240 q^{4} +248.981 q^{5} -257.813 q^{6} -1453.31 q^{7} +1573.84 q^{8} +729.000 q^{9} -2377.42 q^{10} -6465.27 q^{11} -994.247 q^{12} +4142.86 q^{13} +13877.1 q^{14} +6722.48 q^{15} -10314.5 q^{16} -25815.2 q^{17} -6960.94 q^{18} +26877.5 q^{19} -9168.46 q^{20} -39239.3 q^{21} +61734.4 q^{22} +58835.3 q^{23} +42493.7 q^{24} -16133.6 q^{25} -39558.6 q^{26} +19683.0 q^{27} +53516.5 q^{28} +13734.2 q^{29} -64190.4 q^{30} +25833.8 q^{31} -102962. q^{32} -174562. q^{33} +246499. q^{34} -361845. q^{35} -26844.7 q^{36} +124281. q^{37} -256643. q^{38} +111857. q^{39} +391856. q^{40} -201398. q^{41} +374681. q^{42} +24676.8 q^{43} +238077. q^{44} +181507. q^{45} -561796. q^{46} -412000. q^{47} -278492. q^{48} +1.28856e6 q^{49} +154053. q^{50} -697010. q^{51} -152556. q^{52} -43501.3 q^{53} -187945. q^{54} -1.60973e6 q^{55} -2.28727e6 q^{56} +725694. q^{57} -131143. q^{58} +205379. q^{59} -247548. q^{60} -1.61032e6 q^{61} -246677. q^{62} -1.05946e6 q^{63} +2.30341e6 q^{64} +1.03149e6 q^{65} +1.66683e6 q^{66} +1.37101e6 q^{67} +950617. q^{68} +1.58855e6 q^{69} +3.45512e6 q^{70} +2.87276e6 q^{71} +1.14733e6 q^{72} +4.37743e6 q^{73} -1.18671e6 q^{74} -435606. q^{75} -989737. q^{76} +9.39602e6 q^{77} -1.06808e6 q^{78} -960831. q^{79} -2.56812e6 q^{80} +531441. q^{81} +1.92307e6 q^{82} +6.21351e6 q^{83} +1.44494e6 q^{84} -6.42749e6 q^{85} -235630. q^{86} +370825. q^{87} -1.01753e7 q^{88} +9.66681e6 q^{89} -1.73314e6 q^{90} -6.02084e6 q^{91} -2.16655e6 q^{92} +697514. q^{93} +3.93403e6 q^{94} +6.69199e6 q^{95} -2.77998e6 q^{96} -1.18859e7 q^{97} -1.23039e7 q^{98} -4.71318e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.54862 −0.843986 −0.421993 0.906599i \(-0.638669\pi\)
−0.421993 + 0.906599i \(0.638669\pi\)
\(3\) 27.0000 0.577350
\(4\) −36.8240 −0.287687
\(5\) 248.981 0.890781 0.445390 0.895336i \(-0.353065\pi\)
0.445390 + 0.895336i \(0.353065\pi\)
\(6\) −257.813 −0.487276
\(7\) −1453.31 −1.60145 −0.800726 0.599031i \(-0.795553\pi\)
−0.800726 + 0.599031i \(0.795553\pi\)
\(8\) 1573.84 1.08679
\(9\) 729.000 0.333333
\(10\) −2377.42 −0.751807
\(11\) −6465.27 −1.46458 −0.732288 0.680995i \(-0.761548\pi\)
−0.732288 + 0.680995i \(0.761548\pi\)
\(12\) −994.247 −0.166096
\(13\) 4142.86 0.522996 0.261498 0.965204i \(-0.415784\pi\)
0.261498 + 0.965204i \(0.415784\pi\)
\(14\) 13877.1 1.35160
\(15\) 6722.48 0.514293
\(16\) −10314.5 −0.629549
\(17\) −25815.2 −1.27440 −0.637198 0.770700i \(-0.719907\pi\)
−0.637198 + 0.770700i \(0.719907\pi\)
\(18\) −6960.94 −0.281329
\(19\) 26877.5 0.898984 0.449492 0.893284i \(-0.351605\pi\)
0.449492 + 0.893284i \(0.351605\pi\)
\(20\) −9168.46 −0.256266
\(21\) −39239.3 −0.924599
\(22\) 61734.4 1.23608
\(23\) 58835.3 1.00830 0.504151 0.863616i \(-0.331806\pi\)
0.504151 + 0.863616i \(0.331806\pi\)
\(24\) 42493.7 0.627459
\(25\) −16133.6 −0.206510
\(26\) −39558.6 −0.441401
\(27\) 19683.0 0.192450
\(28\) 53516.5 0.460717
\(29\) 13734.2 0.104571 0.0522856 0.998632i \(-0.483349\pi\)
0.0522856 + 0.998632i \(0.483349\pi\)
\(30\) −64190.4 −0.434056
\(31\) 25833.8 0.155748 0.0778741 0.996963i \(-0.475187\pi\)
0.0778741 + 0.996963i \(0.475187\pi\)
\(32\) −102962. −0.555460
\(33\) −174562. −0.845574
\(34\) 246499. 1.07557
\(35\) −361845. −1.42654
\(36\) −26844.7 −0.0958957
\(37\) 124281. 0.403365 0.201682 0.979451i \(-0.435359\pi\)
0.201682 + 0.979451i \(0.435359\pi\)
\(38\) −256643. −0.758730
\(39\) 111857. 0.301952
\(40\) 391856. 0.968092
\(41\) −201398. −0.456365 −0.228182 0.973618i \(-0.573278\pi\)
−0.228182 + 0.973618i \(0.573278\pi\)
\(42\) 374681. 0.780349
\(43\) 24676.8 0.0473314 0.0236657 0.999720i \(-0.492466\pi\)
0.0236657 + 0.999720i \(0.492466\pi\)
\(44\) 238077. 0.421340
\(45\) 181507. 0.296927
\(46\) −561796. −0.850993
\(47\) −412000. −0.578835 −0.289418 0.957203i \(-0.593462\pi\)
−0.289418 + 0.957203i \(0.593462\pi\)
\(48\) −278492. −0.363470
\(49\) 1.28856e6 1.56465
\(50\) 154053. 0.174291
\(51\) −697010. −0.735773
\(52\) −152556. −0.150459
\(53\) −43501.3 −0.0401362 −0.0200681 0.999799i \(-0.506388\pi\)
−0.0200681 + 0.999799i \(0.506388\pi\)
\(54\) −187945. −0.162425
\(55\) −1.60973e6 −1.30462
\(56\) −2.28727e6 −1.74044
\(57\) 725694. 0.519029
\(58\) −131143. −0.0882566
\(59\) 205379. 0.130189
\(60\) −247548. −0.147955
\(61\) −1.61032e6 −0.908362 −0.454181 0.890909i \(-0.650068\pi\)
−0.454181 + 0.890909i \(0.650068\pi\)
\(62\) −246677. −0.131449
\(63\) −1.05946e6 −0.533817
\(64\) 2.30341e6 1.09835
\(65\) 1.03149e6 0.465875
\(66\) 1.66683e6 0.713653
\(67\) 1.37101e6 0.556904 0.278452 0.960450i \(-0.410179\pi\)
0.278452 + 0.960450i \(0.410179\pi\)
\(68\) 950617. 0.366627
\(69\) 1.58855e6 0.582143
\(70\) 3.45512e6 1.20398
\(71\) 2.87276e6 0.952568 0.476284 0.879292i \(-0.341983\pi\)
0.476284 + 0.879292i \(0.341983\pi\)
\(72\) 1.14733e6 0.362263
\(73\) 4.37743e6 1.31701 0.658505 0.752576i \(-0.271189\pi\)
0.658505 + 0.752576i \(0.271189\pi\)
\(74\) −1.18671e6 −0.340434
\(75\) −435606. −0.119228
\(76\) −989737. −0.258626
\(77\) 9.39602e6 2.34545
\(78\) −1.06808e6 −0.254843
\(79\) −960831. −0.219256 −0.109628 0.993973i \(-0.534966\pi\)
−0.109628 + 0.993973i \(0.534966\pi\)
\(80\) −2.56812e6 −0.560790
\(81\) 531441. 0.111111
\(82\) 1.92307e6 0.385166
\(83\) 6.21351e6 1.19279 0.596395 0.802691i \(-0.296599\pi\)
0.596395 + 0.802691i \(0.296599\pi\)
\(84\) 1.44494e6 0.265995
\(85\) −6.42749e6 −1.13521
\(86\) −235630. −0.0399471
\(87\) 370825. 0.0603742
\(88\) −1.01753e7 −1.59169
\(89\) 9.66681e6 1.45351 0.726755 0.686896i \(-0.241027\pi\)
0.726755 + 0.686896i \(0.241027\pi\)
\(90\) −1.73314e6 −0.250602
\(91\) −6.02084e6 −0.837553
\(92\) −2.16655e6 −0.290075
\(93\) 697514. 0.0899213
\(94\) 3.93403e6 0.488529
\(95\) 6.69199e6 0.800798
\(96\) −2.77998e6 −0.320695
\(97\) −1.18859e7 −1.32231 −0.661153 0.750251i \(-0.729933\pi\)
−0.661153 + 0.750251i \(0.729933\pi\)
\(98\) −1.23039e7 −1.32054
\(99\) −4.71318e6 −0.488192
\(100\) 594101. 0.0594101
\(101\) 1.70394e7 1.64562 0.822808 0.568319i \(-0.192406\pi\)
0.822808 + 0.568319i \(0.192406\pi\)
\(102\) 6.65548e6 0.620982
\(103\) 1.76135e7 1.58824 0.794118 0.607764i \(-0.207933\pi\)
0.794118 + 0.607764i \(0.207933\pi\)
\(104\) 6.52020e6 0.568387
\(105\) −9.76983e6 −0.823615
\(106\) 415377. 0.0338744
\(107\) 2.01177e7 1.58758 0.793789 0.608193i \(-0.208105\pi\)
0.793789 + 0.608193i \(0.208105\pi\)
\(108\) −724806. −0.0553654
\(109\) 1.35078e7 0.999063 0.499531 0.866296i \(-0.333506\pi\)
0.499531 + 0.866296i \(0.333506\pi\)
\(110\) 1.53707e7 1.10108
\(111\) 3.35558e6 0.232883
\(112\) 1.49902e7 1.00819
\(113\) −477253. −0.0311153 −0.0155577 0.999879i \(-0.504952\pi\)
−0.0155577 + 0.999879i \(0.504952\pi\)
\(114\) −6.92937e6 −0.438053
\(115\) 1.46489e7 0.898176
\(116\) −505749. −0.0300838
\(117\) 3.02014e6 0.174332
\(118\) −1.96109e6 −0.109878
\(119\) 3.75174e7 2.04088
\(120\) 1.05801e7 0.558928
\(121\) 2.23125e7 1.14499
\(122\) 1.53764e7 0.766645
\(123\) −5.43775e6 −0.263482
\(124\) −951304. −0.0448068
\(125\) −2.34686e7 −1.07474
\(126\) 1.01164e7 0.450535
\(127\) 2.32024e7 1.00512 0.502562 0.864541i \(-0.332391\pi\)
0.502562 + 0.864541i \(0.332391\pi\)
\(128\) −8.81518e6 −0.371532
\(129\) 666274. 0.0273268
\(130\) −9.84932e6 −0.393192
\(131\) −4.19666e6 −0.163100 −0.0815501 0.996669i \(-0.525987\pi\)
−0.0815501 + 0.996669i \(0.525987\pi\)
\(132\) 6.42807e6 0.243261
\(133\) −3.90613e7 −1.43968
\(134\) −1.30913e7 −0.470019
\(135\) 4.90069e6 0.171431
\(136\) −4.06290e7 −1.38500
\(137\) −2.12221e7 −0.705126 −0.352563 0.935788i \(-0.614690\pi\)
−0.352563 + 0.935788i \(0.614690\pi\)
\(138\) −1.51685e7 −0.491321
\(139\) 5.22180e7 1.64918 0.824591 0.565729i \(-0.191405\pi\)
0.824591 + 0.565729i \(0.191405\pi\)
\(140\) 1.33246e7 0.410398
\(141\) −1.11240e7 −0.334191
\(142\) −2.74309e7 −0.803954
\(143\) −2.67847e7 −0.765968
\(144\) −7.51929e6 −0.209850
\(145\) 3.41956e6 0.0931500
\(146\) −4.17984e7 −1.11154
\(147\) 3.47910e7 0.903351
\(148\) −4.57651e6 −0.116043
\(149\) 3.35373e7 0.830569 0.415285 0.909692i \(-0.363682\pi\)
0.415285 + 0.909692i \(0.363682\pi\)
\(150\) 4.15943e6 0.100627
\(151\) −2.21646e7 −0.523890 −0.261945 0.965083i \(-0.584364\pi\)
−0.261945 + 0.965083i \(0.584364\pi\)
\(152\) 4.23010e7 0.977007
\(153\) −1.88193e7 −0.424798
\(154\) −8.97189e7 −1.97953
\(155\) 6.43213e6 0.138738
\(156\) −4.11902e6 −0.0868676
\(157\) −5.80763e7 −1.19771 −0.598853 0.800859i \(-0.704377\pi\)
−0.598853 + 0.800859i \(0.704377\pi\)
\(158\) 9.17461e6 0.185049
\(159\) −1.17453e6 −0.0231727
\(160\) −2.56356e7 −0.494793
\(161\) −8.55057e7 −1.61475
\(162\) −5.07453e6 −0.0937763
\(163\) −7.64837e7 −1.38329 −0.691643 0.722239i \(-0.743113\pi\)
−0.691643 + 0.722239i \(0.743113\pi\)
\(164\) 7.41628e6 0.131290
\(165\) −4.34627e7 −0.753221
\(166\) −5.93304e7 −1.00670
\(167\) 7.60086e7 1.26286 0.631430 0.775433i \(-0.282468\pi\)
0.631430 + 0.775433i \(0.282468\pi\)
\(168\) −6.17564e7 −1.00485
\(169\) −4.55852e7 −0.726475
\(170\) 6.13736e7 0.958099
\(171\) 1.95937e7 0.299661
\(172\) −908698. −0.0136166
\(173\) −9.21166e7 −1.35262 −0.676311 0.736616i \(-0.736422\pi\)
−0.676311 + 0.736616i \(0.736422\pi\)
\(174\) −3.54086e6 −0.0509550
\(175\) 2.34470e7 0.330715
\(176\) 6.66862e7 0.922023
\(177\) 5.54523e6 0.0751646
\(178\) −9.23047e7 −1.22674
\(179\) 5.97291e7 0.778396 0.389198 0.921154i \(-0.372752\pi\)
0.389198 + 0.921154i \(0.372752\pi\)
\(180\) −6.68381e6 −0.0854221
\(181\) −9.97102e7 −1.24987 −0.624934 0.780677i \(-0.714874\pi\)
−0.624934 + 0.780677i \(0.714874\pi\)
\(182\) 5.74907e7 0.706883
\(183\) −4.34787e7 −0.524443
\(184\) 9.25974e7 1.09581
\(185\) 3.09435e7 0.359309
\(186\) −6.66029e6 −0.0758923
\(187\) 1.66902e8 1.86645
\(188\) 1.51715e7 0.166523
\(189\) −2.86054e7 −0.308200
\(190\) −6.38993e7 −0.675862
\(191\) −2.03999e6 −0.0211841 −0.0105921 0.999944i \(-0.503372\pi\)
−0.0105921 + 0.999944i \(0.503372\pi\)
\(192\) 6.21919e7 0.634132
\(193\) 1.32752e8 1.32920 0.664602 0.747198i \(-0.268601\pi\)
0.664602 + 0.747198i \(0.268601\pi\)
\(194\) 1.13494e8 1.11601
\(195\) 2.78503e7 0.268973
\(196\) −4.74497e7 −0.450130
\(197\) −1.07186e8 −0.998867 −0.499434 0.866352i \(-0.666458\pi\)
−0.499434 + 0.866352i \(0.666458\pi\)
\(198\) 4.50043e7 0.412028
\(199\) −1.08221e8 −0.973474 −0.486737 0.873549i \(-0.661813\pi\)
−0.486737 + 0.873549i \(0.661813\pi\)
\(200\) −2.53916e7 −0.224433
\(201\) 3.70174e7 0.321528
\(202\) −1.62702e8 −1.38888
\(203\) −1.99601e7 −0.167466
\(204\) 2.56667e7 0.211672
\(205\) −5.01443e7 −0.406521
\(206\) −1.68184e8 −1.34045
\(207\) 4.28909e7 0.336101
\(208\) −4.27316e7 −0.329252
\(209\) −1.73771e8 −1.31663
\(210\) 9.32883e7 0.695120
\(211\) 1.65214e8 1.21076 0.605380 0.795937i \(-0.293021\pi\)
0.605380 + 0.795937i \(0.293021\pi\)
\(212\) 1.60189e6 0.0115467
\(213\) 7.75647e7 0.549965
\(214\) −1.92096e8 −1.33989
\(215\) 6.14406e6 0.0421619
\(216\) 3.09779e7 0.209153
\(217\) −3.75445e7 −0.249423
\(218\) −1.28981e8 −0.843195
\(219\) 1.18191e8 0.760377
\(220\) 5.92765e7 0.375321
\(221\) −1.06949e8 −0.666504
\(222\) −3.20411e7 −0.196550
\(223\) 1.30090e7 0.0785558 0.0392779 0.999228i \(-0.487494\pi\)
0.0392779 + 0.999228i \(0.487494\pi\)
\(224\) 1.49635e8 0.889542
\(225\) −1.17614e7 −0.0688365
\(226\) 4.55711e6 0.0262609
\(227\) 3.00513e8 1.70519 0.852595 0.522572i \(-0.175027\pi\)
0.852595 + 0.522572i \(0.175027\pi\)
\(228\) −2.67229e7 −0.149318
\(229\) 4.51247e7 0.248308 0.124154 0.992263i \(-0.460378\pi\)
0.124154 + 0.992263i \(0.460378\pi\)
\(230\) −1.39876e8 −0.758048
\(231\) 2.53692e8 1.35415
\(232\) 2.16155e7 0.113647
\(233\) 2.62782e8 1.36097 0.680486 0.732761i \(-0.261769\pi\)
0.680486 + 0.732761i \(0.261769\pi\)
\(234\) −2.88382e7 −0.147134
\(235\) −1.02580e8 −0.515615
\(236\) −7.56287e6 −0.0374537
\(237\) −2.59424e7 −0.126588
\(238\) −3.58239e8 −1.72248
\(239\) −3.84804e8 −1.82325 −0.911626 0.411022i \(-0.865172\pi\)
−0.911626 + 0.411022i \(0.865172\pi\)
\(240\) −6.93392e7 −0.323772
\(241\) 1.89745e8 0.873194 0.436597 0.899657i \(-0.356184\pi\)
0.436597 + 0.899657i \(0.356184\pi\)
\(242\) −2.13054e8 −0.966352
\(243\) 1.43489e7 0.0641500
\(244\) 5.92985e7 0.261324
\(245\) 3.20826e8 1.39376
\(246\) 5.19230e7 0.222375
\(247\) 1.11350e8 0.470165
\(248\) 4.06583e7 0.169266
\(249\) 1.67765e8 0.688657
\(250\) 2.24092e8 0.907062
\(251\) −1.89901e8 −0.758002 −0.379001 0.925396i \(-0.623732\pi\)
−0.379001 + 0.925396i \(0.623732\pi\)
\(252\) 3.90135e7 0.153572
\(253\) −3.80386e8 −1.47674
\(254\) −2.21550e8 −0.848310
\(255\) −1.73542e8 −0.655412
\(256\) −2.10663e8 −0.784781
\(257\) 3.23214e8 1.18775 0.593874 0.804558i \(-0.297598\pi\)
0.593874 + 0.804558i \(0.297598\pi\)
\(258\) −6.36200e6 −0.0230635
\(259\) −1.80618e8 −0.645969
\(260\) −3.79836e7 −0.134026
\(261\) 1.00123e7 0.0348571
\(262\) 4.00723e7 0.137654
\(263\) −9.96217e7 −0.337683 −0.168841 0.985643i \(-0.554003\pi\)
−0.168841 + 0.985643i \(0.554003\pi\)
\(264\) −2.74733e8 −0.918961
\(265\) −1.08310e7 −0.0357526
\(266\) 3.72981e8 1.21507
\(267\) 2.61004e8 0.839185
\(268\) −5.04861e7 −0.160214
\(269\) 5.17058e8 1.61960 0.809798 0.586709i \(-0.199577\pi\)
0.809798 + 0.586709i \(0.199577\pi\)
\(270\) −4.67948e7 −0.144685
\(271\) 9.10242e6 0.0277821 0.0138910 0.999904i \(-0.495578\pi\)
0.0138910 + 0.999904i \(0.495578\pi\)
\(272\) 2.66272e8 0.802294
\(273\) −1.62563e8 −0.483561
\(274\) 2.02642e8 0.595117
\(275\) 1.04308e8 0.302449
\(276\) −5.84968e7 −0.167475
\(277\) 1.04754e8 0.296135 0.148068 0.988977i \(-0.452695\pi\)
0.148068 + 0.988977i \(0.452695\pi\)
\(278\) −4.98610e8 −1.39189
\(279\) 1.88329e7 0.0519161
\(280\) −5.69487e8 −1.55035
\(281\) −2.20844e8 −0.593765 −0.296882 0.954914i \(-0.595947\pi\)
−0.296882 + 0.954914i \(0.595947\pi\)
\(282\) 1.06219e8 0.282052
\(283\) 4.56240e8 1.19658 0.598289 0.801280i \(-0.295847\pi\)
0.598289 + 0.801280i \(0.295847\pi\)
\(284\) −1.05787e8 −0.274041
\(285\) 1.80684e8 0.462341
\(286\) 2.55757e8 0.646466
\(287\) 2.92693e8 0.730846
\(288\) −7.50594e7 −0.185153
\(289\) 2.56086e8 0.624084
\(290\) −3.26521e7 −0.0786173
\(291\) −3.20920e8 −0.763434
\(292\) −1.61194e8 −0.378887
\(293\) 1.06837e8 0.248133 0.124067 0.992274i \(-0.460406\pi\)
0.124067 + 0.992274i \(0.460406\pi\)
\(294\) −3.32206e8 −0.762416
\(295\) 5.11354e7 0.115970
\(296\) 1.95598e8 0.438373
\(297\) −1.27256e8 −0.281858
\(298\) −3.20235e8 −0.700989
\(299\) 2.43746e8 0.527338
\(300\) 1.60407e7 0.0343005
\(301\) −3.58630e7 −0.0757991
\(302\) 2.11641e8 0.442156
\(303\) 4.60063e8 0.950097
\(304\) −2.77229e8 −0.565954
\(305\) −4.00940e8 −0.809151
\(306\) 1.79698e8 0.358524
\(307\) −4.91926e8 −0.970322 −0.485161 0.874425i \(-0.661239\pi\)
−0.485161 + 0.874425i \(0.661239\pi\)
\(308\) −3.45998e8 −0.674756
\(309\) 4.75564e8 0.916969
\(310\) −6.14179e7 −0.117093
\(311\) −34302.1 −6.46634e−5 0 −3.23317e−5 1.00000i \(-0.500010\pi\)
−3.23317e−5 1.00000i \(0.500010\pi\)
\(312\) 1.76045e8 0.328158
\(313\) −2.14984e8 −0.396279 −0.198139 0.980174i \(-0.563490\pi\)
−0.198139 + 0.980174i \(0.563490\pi\)
\(314\) 5.54548e8 1.01085
\(315\) −2.63785e8 −0.475514
\(316\) 3.53816e7 0.0630772
\(317\) −4.71181e8 −0.830769 −0.415385 0.909646i \(-0.636353\pi\)
−0.415385 + 0.909646i \(0.636353\pi\)
\(318\) 1.12152e7 0.0195574
\(319\) −8.87956e7 −0.153152
\(320\) 5.73504e8 0.978388
\(321\) 5.43178e8 0.916589
\(322\) 8.16461e8 1.36282
\(323\) −6.93849e8 −1.14566
\(324\) −1.95698e7 −0.0319652
\(325\) −6.68390e7 −0.108004
\(326\) 7.30313e8 1.16747
\(327\) 3.64711e8 0.576809
\(328\) −3.16969e8 −0.495973
\(329\) 5.98762e8 0.926977
\(330\) 4.15008e8 0.635708
\(331\) 1.20626e9 1.82829 0.914143 0.405391i \(-0.132865\pi\)
0.914143 + 0.405391i \(0.132865\pi\)
\(332\) −2.28806e8 −0.343150
\(333\) 9.06007e7 0.134455
\(334\) −7.25777e8 −1.06584
\(335\) 3.41356e8 0.496079
\(336\) 4.04735e8 0.582080
\(337\) −5.37145e8 −0.764517 −0.382258 0.924055i \(-0.624854\pi\)
−0.382258 + 0.924055i \(0.624854\pi\)
\(338\) 4.35276e8 0.613135
\(339\) −1.28858e7 −0.0179644
\(340\) 2.36686e8 0.326584
\(341\) −1.67023e8 −0.228105
\(342\) −1.87093e8 −0.252910
\(343\) −6.75807e8 −0.904259
\(344\) 3.88374e7 0.0514394
\(345\) 3.95519e8 0.518562
\(346\) 8.79586e8 1.14159
\(347\) −1.07234e9 −1.37778 −0.688888 0.724867i \(-0.741901\pi\)
−0.688888 + 0.724867i \(0.741901\pi\)
\(348\) −1.36552e7 −0.0173689
\(349\) −6.30212e8 −0.793592 −0.396796 0.917907i \(-0.629878\pi\)
−0.396796 + 0.917907i \(0.629878\pi\)
\(350\) −2.23886e8 −0.279119
\(351\) 8.15439e7 0.100651
\(352\) 6.65678e8 0.813513
\(353\) −1.30728e9 −1.58182 −0.790908 0.611936i \(-0.790391\pi\)
−0.790908 + 0.611936i \(0.790391\pi\)
\(354\) −5.29493e7 −0.0634379
\(355\) 7.15263e8 0.848529
\(356\) −3.55970e8 −0.418156
\(357\) 1.01297e9 1.17830
\(358\) −5.70330e8 −0.656956
\(359\) 1.57238e9 1.79360 0.896802 0.442432i \(-0.145884\pi\)
0.896802 + 0.442432i \(0.145884\pi\)
\(360\) 2.85663e8 0.322697
\(361\) −1.71470e8 −0.191828
\(362\) 9.52094e8 1.05487
\(363\) 6.02438e8 0.661058
\(364\) 2.21711e8 0.240953
\(365\) 1.08990e9 1.17317
\(366\) 4.15162e8 0.442623
\(367\) −1.48548e9 −1.56868 −0.784342 0.620329i \(-0.786999\pi\)
−0.784342 + 0.620329i \(0.786999\pi\)
\(368\) −6.06858e8 −0.634775
\(369\) −1.46819e8 −0.152122
\(370\) −2.95468e8 −0.303252
\(371\) 6.32207e7 0.0642763
\(372\) −2.56852e7 −0.0258692
\(373\) −9.92307e8 −0.990069 −0.495034 0.868873i \(-0.664845\pi\)
−0.495034 + 0.868873i \(0.664845\pi\)
\(374\) −1.59368e9 −1.57526
\(375\) −6.33651e8 −0.620499
\(376\) −6.48422e8 −0.629072
\(377\) 5.68990e7 0.0546903
\(378\) 2.73142e8 0.260116
\(379\) −1.37321e9 −1.29569 −0.647844 0.761773i \(-0.724329\pi\)
−0.647844 + 0.761773i \(0.724329\pi\)
\(380\) −2.46426e8 −0.230379
\(381\) 6.26464e8 0.580308
\(382\) 1.94790e7 0.0178791
\(383\) 1.58828e9 1.44455 0.722275 0.691606i \(-0.243097\pi\)
0.722275 + 0.691606i \(0.243097\pi\)
\(384\) −2.38010e8 −0.214504
\(385\) 2.33943e9 2.08928
\(386\) −1.26760e9 −1.12183
\(387\) 1.79894e7 0.0157771
\(388\) 4.37687e8 0.380410
\(389\) −1.66679e8 −0.143568 −0.0717841 0.997420i \(-0.522869\pi\)
−0.0717841 + 0.997420i \(0.522869\pi\)
\(390\) −2.65932e8 −0.227009
\(391\) −1.51884e9 −1.28497
\(392\) 2.02798e9 1.70045
\(393\) −1.13310e8 −0.0941659
\(394\) 1.02348e9 0.843030
\(395\) −2.39229e8 −0.195309
\(396\) 1.73558e8 0.140447
\(397\) −1.07153e9 −0.859482 −0.429741 0.902952i \(-0.641395\pi\)
−0.429741 + 0.902952i \(0.641395\pi\)
\(398\) 1.03336e9 0.821599
\(399\) −1.05466e9 −0.831200
\(400\) 1.66410e8 0.130008
\(401\) 1.56895e8 0.121508 0.0607538 0.998153i \(-0.480650\pi\)
0.0607538 + 0.998153i \(0.480650\pi\)
\(402\) −3.53465e8 −0.271366
\(403\) 1.07026e8 0.0814557
\(404\) −6.27457e8 −0.473423
\(405\) 1.32319e8 0.0989756
\(406\) 1.90591e8 0.141339
\(407\) −8.03508e8 −0.590758
\(408\) −1.09698e9 −0.799630
\(409\) −1.06873e9 −0.772392 −0.386196 0.922417i \(-0.626211\pi\)
−0.386196 + 0.922417i \(0.626211\pi\)
\(410\) 4.78808e8 0.343098
\(411\) −5.72997e8 −0.407105
\(412\) −6.48598e8 −0.456915
\(413\) −2.98479e8 −0.208491
\(414\) −4.09549e8 −0.283664
\(415\) 1.54705e9 1.06251
\(416\) −4.26557e8 −0.290503
\(417\) 1.40989e9 0.952156
\(418\) 1.65927e9 1.11122
\(419\) 9.24350e7 0.0613885 0.0306943 0.999529i \(-0.490228\pi\)
0.0306943 + 0.999529i \(0.490228\pi\)
\(420\) 3.59764e8 0.236943
\(421\) 8.98575e8 0.586904 0.293452 0.955974i \(-0.405196\pi\)
0.293452 + 0.955974i \(0.405196\pi\)
\(422\) −1.57756e9 −1.02186
\(423\) −3.00348e8 −0.192945
\(424\) −6.84641e7 −0.0436197
\(425\) 4.16491e8 0.263175
\(426\) −7.40635e8 −0.464163
\(427\) 2.34029e9 1.45470
\(428\) −7.40814e8 −0.456726
\(429\) −7.23187e8 −0.442232
\(430\) −5.86672e7 −0.0355841
\(431\) −7.64033e8 −0.459665 −0.229833 0.973230i \(-0.573818\pi\)
−0.229833 + 0.973230i \(0.573818\pi\)
\(432\) −2.03021e8 −0.121157
\(433\) 1.90959e8 0.113040 0.0565199 0.998401i \(-0.482000\pi\)
0.0565199 + 0.998401i \(0.482000\pi\)
\(434\) 3.58498e8 0.210510
\(435\) 9.23282e7 0.0537802
\(436\) −4.97412e8 −0.287417
\(437\) 1.58135e9 0.906447
\(438\) −1.12856e9 −0.641747
\(439\) 2.64998e9 1.49492 0.747458 0.664309i \(-0.231274\pi\)
0.747458 + 0.664309i \(0.231274\pi\)
\(440\) −2.53345e9 −1.41785
\(441\) 9.39358e8 0.521550
\(442\) 1.02121e9 0.562520
\(443\) 3.33143e9 1.82061 0.910307 0.413934i \(-0.135846\pi\)
0.910307 + 0.413934i \(0.135846\pi\)
\(444\) −1.23566e8 −0.0669973
\(445\) 2.40685e9 1.29476
\(446\) −1.24218e8 −0.0663001
\(447\) 9.05507e8 0.479529
\(448\) −3.34755e9 −1.75895
\(449\) −1.02214e8 −0.0532902 −0.0266451 0.999645i \(-0.508482\pi\)
−0.0266451 + 0.999645i \(0.508482\pi\)
\(450\) 1.12305e8 0.0580971
\(451\) 1.30209e9 0.668381
\(452\) 1.75743e7 0.00895148
\(453\) −5.98444e8 −0.302468
\(454\) −2.86948e9 −1.43916
\(455\) −1.49907e9 −0.746076
\(456\) 1.14213e9 0.564075
\(457\) −2.28925e9 −1.12198 −0.560992 0.827821i \(-0.689580\pi\)
−0.560992 + 0.827821i \(0.689580\pi\)
\(458\) −4.30879e8 −0.209568
\(459\) −5.08120e8 −0.245258
\(460\) −5.39429e8 −0.258394
\(461\) 6.39371e8 0.303948 0.151974 0.988384i \(-0.451437\pi\)
0.151974 + 0.988384i \(0.451437\pi\)
\(462\) −2.42241e9 −1.14288
\(463\) −1.36469e9 −0.638998 −0.319499 0.947587i \(-0.603515\pi\)
−0.319499 + 0.947587i \(0.603515\pi\)
\(464\) −1.41662e8 −0.0658327
\(465\) 1.73668e8 0.0801001
\(466\) −2.50920e9 −1.14864
\(467\) 2.57730e9 1.17100 0.585498 0.810674i \(-0.300899\pi\)
0.585498 + 0.810674i \(0.300899\pi\)
\(468\) −1.11214e8 −0.0501531
\(469\) −1.99250e9 −0.891855
\(470\) 9.79498e8 0.435172
\(471\) −1.56806e9 −0.691496
\(472\) 3.23234e8 0.141488
\(473\) −1.59542e8 −0.0693205
\(474\) 2.47714e8 0.106838
\(475\) −4.33630e8 −0.185649
\(476\) −1.38154e9 −0.587136
\(477\) −3.17124e7 −0.0133787
\(478\) 3.67434e9 1.53880
\(479\) 2.51029e9 1.04364 0.521818 0.853057i \(-0.325254\pi\)
0.521818 + 0.853057i \(0.325254\pi\)
\(480\) −6.92161e8 −0.285669
\(481\) 5.14877e8 0.210958
\(482\) −1.81180e9 −0.736964
\(483\) −2.30865e9 −0.932275
\(484\) −8.21635e8 −0.329398
\(485\) −2.95937e9 −1.17788
\(486\) −1.37012e8 −0.0541417
\(487\) −1.03970e9 −0.407905 −0.203952 0.978981i \(-0.565379\pi\)
−0.203952 + 0.978981i \(0.565379\pi\)
\(488\) −2.53439e9 −0.987199
\(489\) −2.06506e9 −0.798641
\(490\) −3.06344e9 −1.17631
\(491\) −4.29594e9 −1.63785 −0.818923 0.573903i \(-0.805429\pi\)
−0.818923 + 0.573903i \(0.805429\pi\)
\(492\) 2.00239e8 0.0758005
\(493\) −3.54552e8 −0.133265
\(494\) −1.06324e9 −0.396813
\(495\) −1.17349e9 −0.434872
\(496\) −2.66464e8 −0.0980511
\(497\) −4.17501e9 −1.52549
\(498\) −1.60192e9 −0.581217
\(499\) −2.36884e9 −0.853461 −0.426730 0.904379i \(-0.640335\pi\)
−0.426730 + 0.904379i \(0.640335\pi\)
\(500\) 8.64206e8 0.309188
\(501\) 2.05223e9 0.729112
\(502\) 1.81330e9 0.639743
\(503\) 3.86205e9 1.35310 0.676551 0.736396i \(-0.263474\pi\)
0.676551 + 0.736396i \(0.263474\pi\)
\(504\) −1.66742e9 −0.580148
\(505\) 4.24247e9 1.46588
\(506\) 3.63216e9 1.24634
\(507\) −1.23080e9 −0.419431
\(508\) −8.54403e8 −0.289161
\(509\) −2.82144e8 −0.0948327 −0.0474163 0.998875i \(-0.515099\pi\)
−0.0474163 + 0.998875i \(0.515099\pi\)
\(510\) 1.65709e9 0.553159
\(511\) −6.36175e9 −2.10913
\(512\) 3.13988e9 1.03388
\(513\) 5.29031e8 0.173010
\(514\) −3.08625e9 −1.00244
\(515\) 4.38542e9 1.41477
\(516\) −2.45349e7 −0.00786157
\(517\) 2.66369e9 0.847748
\(518\) 1.72465e9 0.545189
\(519\) −2.48715e9 −0.780936
\(520\) 1.62340e9 0.506308
\(521\) 1.57821e9 0.488915 0.244457 0.969660i \(-0.421390\pi\)
0.244457 + 0.969660i \(0.421390\pi\)
\(522\) −9.56033e7 −0.0294189
\(523\) −2.13247e9 −0.651818 −0.325909 0.945401i \(-0.605670\pi\)
−0.325909 + 0.945401i \(0.605670\pi\)
\(524\) 1.54538e8 0.0469218
\(525\) 6.33069e8 0.190939
\(526\) 9.51249e8 0.285000
\(527\) −6.66906e8 −0.198485
\(528\) 1.80053e9 0.532330
\(529\) 5.67661e7 0.0166722
\(530\) 1.03421e8 0.0301747
\(531\) 1.49721e8 0.0433963
\(532\) 1.43839e9 0.414177
\(533\) −8.34364e8 −0.238677
\(534\) −2.49223e9 −0.708261
\(535\) 5.00892e9 1.41418
\(536\) 2.15776e9 0.605237
\(537\) 1.61269e9 0.449407
\(538\) −4.93719e9 −1.36692
\(539\) −8.33086e9 −2.29155
\(540\) −1.80463e8 −0.0493184
\(541\) 1.40969e9 0.382766 0.191383 0.981515i \(-0.438703\pi\)
0.191383 + 0.981515i \(0.438703\pi\)
\(542\) −8.69155e7 −0.0234477
\(543\) −2.69217e9 −0.721612
\(544\) 2.65799e9 0.707875
\(545\) 3.36319e9 0.889946
\(546\) 1.55225e9 0.408119
\(547\) 2.03003e9 0.530331 0.265166 0.964203i \(-0.414573\pi\)
0.265166 + 0.964203i \(0.414573\pi\)
\(548\) 7.81482e8 0.202856
\(549\) −1.17393e9 −0.302787
\(550\) −9.95995e8 −0.255263
\(551\) 3.69143e8 0.0940078
\(552\) 2.50013e9 0.632668
\(553\) 1.39638e9 0.351129
\(554\) −1.00025e9 −0.249934
\(555\) 8.35475e8 0.207447
\(556\) −1.92287e9 −0.474448
\(557\) −1.08167e9 −0.265216 −0.132608 0.991169i \(-0.542335\pi\)
−0.132608 + 0.991169i \(0.542335\pi\)
\(558\) −1.79828e8 −0.0438165
\(559\) 1.02233e8 0.0247542
\(560\) 3.73227e9 0.898079
\(561\) 4.50636e9 1.07760
\(562\) 2.10876e9 0.501129
\(563\) 3.83004e8 0.0904533 0.0452266 0.998977i \(-0.485599\pi\)
0.0452266 + 0.998977i \(0.485599\pi\)
\(564\) 4.09630e8 0.0961423
\(565\) −1.18827e8 −0.0277169
\(566\) −4.35646e9 −1.00990
\(567\) −7.72347e8 −0.177939
\(568\) 4.52127e9 1.03524
\(569\) 2.51189e9 0.571620 0.285810 0.958286i \(-0.407737\pi\)
0.285810 + 0.958286i \(0.407737\pi\)
\(570\) −1.72528e9 −0.390209
\(571\) 3.94620e9 0.887060 0.443530 0.896259i \(-0.353726\pi\)
0.443530 + 0.896259i \(0.353726\pi\)
\(572\) 9.86318e8 0.220359
\(573\) −5.50796e7 −0.0122307
\(574\) −2.79482e9 −0.616824
\(575\) −9.49223e8 −0.208224
\(576\) 1.67918e9 0.366116
\(577\) 3.82097e9 0.828053 0.414027 0.910265i \(-0.364122\pi\)
0.414027 + 0.910265i \(0.364122\pi\)
\(578\) −2.44526e9 −0.526718
\(579\) 3.58431e9 0.767416
\(580\) −1.25922e8 −0.0267980
\(581\) −9.03014e9 −1.91020
\(582\) 3.06434e9 0.644328
\(583\) 2.81248e8 0.0587826
\(584\) 6.88938e9 1.43131
\(585\) 7.51958e8 0.155292
\(586\) −1.02014e9 −0.209421
\(587\) −5.26096e9 −1.07357 −0.536786 0.843718i \(-0.680362\pi\)
−0.536786 + 0.843718i \(0.680362\pi\)
\(588\) −1.28114e9 −0.259882
\(589\) 6.94350e8 0.140015
\(590\) −4.88273e8 −0.0978769
\(591\) −2.89403e9 −0.576696
\(592\) −1.28190e9 −0.253938
\(593\) 5.47065e9 1.07733 0.538664 0.842521i \(-0.318929\pi\)
0.538664 + 0.842521i \(0.318929\pi\)
\(594\) 1.21512e9 0.237884
\(595\) 9.34111e9 1.81798
\(596\) −1.23498e9 −0.238944
\(597\) −2.92196e9 −0.562036
\(598\) −2.32744e9 −0.445066
\(599\) −6.15736e9 −1.17058 −0.585289 0.810825i \(-0.699019\pi\)
−0.585289 + 0.810825i \(0.699019\pi\)
\(600\) −6.85574e8 −0.129576
\(601\) −7.30212e9 −1.37211 −0.686054 0.727550i \(-0.740659\pi\)
−0.686054 + 0.727550i \(0.740659\pi\)
\(602\) 3.42442e8 0.0639734
\(603\) 9.99469e8 0.185635
\(604\) 8.16187e8 0.150716
\(605\) 5.55539e9 1.01993
\(606\) −4.39296e9 −0.801869
\(607\) −6.94268e9 −1.25999 −0.629995 0.776599i \(-0.716943\pi\)
−0.629995 + 0.776599i \(0.716943\pi\)
\(608\) −2.76737e9 −0.499349
\(609\) −5.38922e8 −0.0966864
\(610\) 3.82842e9 0.682913
\(611\) −1.70686e9 −0.302728
\(612\) 6.93000e8 0.122209
\(613\) 3.42370e9 0.600323 0.300161 0.953888i \(-0.402960\pi\)
0.300161 + 0.953888i \(0.402960\pi\)
\(614\) 4.69722e9 0.818938
\(615\) −1.35390e9 −0.234705
\(616\) 1.47878e10 2.54901
\(617\) −2.18204e9 −0.373995 −0.186997 0.982360i \(-0.559876\pi\)
−0.186997 + 0.982360i \(0.559876\pi\)
\(618\) −4.54098e9 −0.773909
\(619\) 1.11884e10 1.89605 0.948025 0.318197i \(-0.103077\pi\)
0.948025 + 0.318197i \(0.103077\pi\)
\(620\) −2.36856e8 −0.0399130
\(621\) 1.15806e9 0.194048
\(622\) 327537. 5.45751e−5 0
\(623\) −1.40488e10 −2.32773
\(624\) −1.15375e9 −0.190093
\(625\) −4.58279e9 −0.750844
\(626\) 2.05280e9 0.334454
\(627\) −4.69180e9 −0.760157
\(628\) 2.13860e9 0.344564
\(629\) −3.20833e9 −0.514046
\(630\) 2.51878e9 0.401328
\(631\) 4.03660e9 0.639606 0.319803 0.947484i \(-0.396383\pi\)
0.319803 + 0.947484i \(0.396383\pi\)
\(632\) −1.51220e9 −0.238286
\(633\) 4.46077e9 0.699032
\(634\) 4.49913e9 0.701158
\(635\) 5.77694e9 0.895344
\(636\) 4.32510e7 0.00666648
\(637\) 5.33831e9 0.818305
\(638\) 8.47875e8 0.129259
\(639\) 2.09425e9 0.317523
\(640\) −2.19481e9 −0.330954
\(641\) 2.82614e9 0.423830 0.211915 0.977288i \(-0.432030\pi\)
0.211915 + 0.977288i \(0.432030\pi\)
\(642\) −5.18660e9 −0.773589
\(643\) 7.92165e9 1.17511 0.587553 0.809186i \(-0.300091\pi\)
0.587553 + 0.809186i \(0.300091\pi\)
\(644\) 3.14866e9 0.464542
\(645\) 1.65890e8 0.0243422
\(646\) 6.62530e9 0.966922
\(647\) −6.23017e9 −0.904347 −0.452173 0.891930i \(-0.649351\pi\)
−0.452173 + 0.891930i \(0.649351\pi\)
\(648\) 8.36403e8 0.120754
\(649\) −1.32783e9 −0.190672
\(650\) 6.38220e8 0.0911536
\(651\) −1.01370e9 −0.144005
\(652\) 2.81643e9 0.397954
\(653\) 8.41425e9 1.18255 0.591275 0.806470i \(-0.298625\pi\)
0.591275 + 0.806470i \(0.298625\pi\)
\(654\) −3.48249e9 −0.486819
\(655\) −1.04489e9 −0.145286
\(656\) 2.07733e9 0.287304
\(657\) 3.19115e9 0.439004
\(658\) −5.71735e9 −0.782356
\(659\) 7.96272e9 1.08383 0.541917 0.840432i \(-0.317699\pi\)
0.541917 + 0.840432i \(0.317699\pi\)
\(660\) 1.60047e9 0.216692
\(661\) 1.16395e10 1.56758 0.783790 0.621026i \(-0.213284\pi\)
0.783790 + 0.621026i \(0.213284\pi\)
\(662\) −1.15181e10 −1.54305
\(663\) −2.88761e9 −0.384806
\(664\) 9.77908e9 1.29631
\(665\) −9.72551e9 −1.28244
\(666\) −8.65111e8 −0.113478
\(667\) 8.08058e8 0.105439
\(668\) −2.79894e9 −0.363308
\(669\) 3.51244e8 0.0453542
\(670\) −3.25948e9 −0.418684
\(671\) 1.04112e10 1.33037
\(672\) 4.04016e9 0.513577
\(673\) 6.37828e8 0.0806586 0.0403293 0.999186i \(-0.487159\pi\)
0.0403293 + 0.999186i \(0.487159\pi\)
\(674\) 5.12899e9 0.645242
\(675\) −3.17557e8 −0.0397428
\(676\) 1.67863e9 0.208998
\(677\) −8.92413e9 −1.10536 −0.552682 0.833392i \(-0.686396\pi\)
−0.552682 + 0.833392i \(0.686396\pi\)
\(678\) 1.23042e8 0.0151617
\(679\) 1.72739e10 2.11761
\(680\) −1.01158e10 −1.23373
\(681\) 8.11386e9 0.984492
\(682\) 1.59484e9 0.192518
\(683\) 2.24023e9 0.269042 0.134521 0.990911i \(-0.457050\pi\)
0.134521 + 0.990911i \(0.457050\pi\)
\(684\) −7.21518e8 −0.0862087
\(685\) −5.28390e9 −0.628113
\(686\) 6.45302e9 0.763183
\(687\) 1.21837e9 0.143361
\(688\) −2.54530e8 −0.0297975
\(689\) −1.80220e8 −0.0209911
\(690\) −3.77666e9 −0.437659
\(691\) −1.59406e10 −1.83794 −0.918972 0.394322i \(-0.870979\pi\)
−0.918972 + 0.394322i \(0.870979\pi\)
\(692\) 3.39210e9 0.389132
\(693\) 6.84970e9 0.781817
\(694\) 1.02394e10 1.16282
\(695\) 1.30013e10 1.46906
\(696\) 5.83619e8 0.0656141
\(697\) 5.19913e9 0.581589
\(698\) 6.01765e9 0.669781
\(699\) 7.09511e9 0.785758
\(700\) −8.63411e8 −0.0951425
\(701\) 1.79817e10 1.97160 0.985800 0.167926i \(-0.0537071\pi\)
0.985800 + 0.167926i \(0.0537071\pi\)
\(702\) −7.78631e8 −0.0849477
\(703\) 3.34036e9 0.362618
\(704\) −1.48921e10 −1.60862
\(705\) −2.76966e9 −0.297691
\(706\) 1.24827e10 1.33503
\(707\) −2.47634e10 −2.63538
\(708\) −2.04197e8 −0.0216239
\(709\) −5.70361e9 −0.601019 −0.300510 0.953779i \(-0.597157\pi\)
−0.300510 + 0.953779i \(0.597157\pi\)
\(710\) −6.82977e9 −0.716147
\(711\) −7.00446e8 −0.0730855
\(712\) 1.52140e10 1.57966
\(713\) 1.51994e9 0.157041
\(714\) −9.67246e9 −0.994473
\(715\) −6.66887e9 −0.682309
\(716\) −2.19946e9 −0.223935
\(717\) −1.03897e10 −1.05265
\(718\) −1.50140e10 −1.51378
\(719\) 1.38876e10 1.39340 0.696702 0.717360i \(-0.254650\pi\)
0.696702 + 0.717360i \(0.254650\pi\)
\(720\) −1.87216e9 −0.186930
\(721\) −2.55978e10 −2.54348
\(722\) 1.63730e9 0.161900
\(723\) 5.12311e9 0.504139
\(724\) 3.67172e9 0.359571
\(725\) −2.21582e8 −0.0215949
\(726\) −5.75245e9 −0.557924
\(727\) −7.90006e8 −0.0762535 −0.0381268 0.999273i \(-0.512139\pi\)
−0.0381268 + 0.999273i \(0.512139\pi\)
\(728\) −9.47584e9 −0.910244
\(729\) 3.87420e8 0.0370370
\(730\) −1.04070e10 −0.990138
\(731\) −6.37037e8 −0.0603190
\(732\) 1.60106e9 0.150875
\(733\) 5.65590e8 0.0530442 0.0265221 0.999648i \(-0.491557\pi\)
0.0265221 + 0.999648i \(0.491557\pi\)
\(734\) 1.41843e10 1.32395
\(735\) 8.66230e9 0.804688
\(736\) −6.05780e9 −0.560071
\(737\) −8.86397e9 −0.815628
\(738\) 1.40192e9 0.128389
\(739\) −1.70229e10 −1.55159 −0.775796 0.630984i \(-0.782651\pi\)
−0.775796 + 0.630984i \(0.782651\pi\)
\(740\) −1.13946e9 −0.103369
\(741\) 3.00645e9 0.271450
\(742\) −6.03670e8 −0.0542483
\(743\) 1.64681e7 0.00147294 0.000736468 1.00000i \(-0.499766\pi\)
0.000736468 1.00000i \(0.499766\pi\)
\(744\) 1.09778e9 0.0977256
\(745\) 8.35014e9 0.739855
\(746\) 9.47516e9 0.835604
\(747\) 4.52965e9 0.397597
\(748\) −6.14600e9 −0.536954
\(749\) −2.92372e10 −2.54243
\(750\) 6.05049e9 0.523693
\(751\) −9.72464e9 −0.837787 −0.418893 0.908035i \(-0.637582\pi\)
−0.418893 + 0.908035i \(0.637582\pi\)
\(752\) 4.24959e9 0.364405
\(753\) −5.12734e9 −0.437633
\(754\) −5.43307e8 −0.0461578
\(755\) −5.51856e9 −0.466671
\(756\) 1.05336e9 0.0886651
\(757\) 3.21906e9 0.269708 0.134854 0.990866i \(-0.456944\pi\)
0.134854 + 0.990866i \(0.456944\pi\)
\(758\) 1.31123e10 1.09354
\(759\) −1.02704e10 −0.852594
\(760\) 1.05321e10 0.870299
\(761\) −1.87932e10 −1.54580 −0.772902 0.634526i \(-0.781195\pi\)
−0.772902 + 0.634526i \(0.781195\pi\)
\(762\) −5.98186e9 −0.489772
\(763\) −1.96310e10 −1.59995
\(764\) 7.51204e7 0.00609440
\(765\) −4.68564e9 −0.378402
\(766\) −1.51659e10 −1.21918
\(767\) 8.50856e8 0.0680883
\(768\) −5.68790e9 −0.453094
\(769\) 4.53024e9 0.359235 0.179617 0.983737i \(-0.442514\pi\)
0.179617 + 0.983737i \(0.442514\pi\)
\(770\) −2.23383e10 −1.76333
\(771\) 8.72679e9 0.685747
\(772\) −4.88846e9 −0.382395
\(773\) 2.02996e10 1.58074 0.790369 0.612631i \(-0.209889\pi\)
0.790369 + 0.612631i \(0.209889\pi\)
\(774\) −1.71774e8 −0.0133157
\(775\) −4.16792e8 −0.0321635
\(776\) −1.87066e10 −1.43707
\(777\) −4.87669e9 −0.372950
\(778\) 1.59156e9 0.121170
\(779\) −5.41309e9 −0.410265
\(780\) −1.02556e9 −0.0773800
\(781\) −1.85732e10 −1.39511
\(782\) 1.45029e10 1.08450
\(783\) 2.70331e8 0.0201247
\(784\) −1.32909e10 −0.985024
\(785\) −1.44599e10 −1.06689
\(786\) 1.08195e9 0.0794748
\(787\) 1.91261e10 1.39867 0.699333 0.714796i \(-0.253480\pi\)
0.699333 + 0.714796i \(0.253480\pi\)
\(788\) 3.94702e9 0.287361
\(789\) −2.68979e9 −0.194961
\(790\) 2.28430e9 0.164838
\(791\) 6.93595e8 0.0498297
\(792\) −7.41779e9 −0.530563
\(793\) −6.67134e9 −0.475069
\(794\) 1.02316e10 0.725391
\(795\) −2.92437e8 −0.0206418
\(796\) 3.98511e9 0.280056
\(797\) 1.30414e10 0.912470 0.456235 0.889859i \(-0.349198\pi\)
0.456235 + 0.889859i \(0.349198\pi\)
\(798\) 1.00705e10 0.701521
\(799\) 1.06359e10 0.737665
\(800\) 1.66114e9 0.114708
\(801\) 7.04711e9 0.484504
\(802\) −1.49813e9 −0.102551
\(803\) −2.83013e10 −1.92886
\(804\) −1.36313e9 −0.0924996
\(805\) −2.12893e10 −1.43839
\(806\) −1.02195e9 −0.0687475
\(807\) 1.39606e10 0.935074
\(808\) 2.68172e10 1.78844
\(809\) 5.94145e9 0.394523 0.197262 0.980351i \(-0.436795\pi\)
0.197262 + 0.980351i \(0.436795\pi\)
\(810\) −1.26346e9 −0.0835341
\(811\) 5.51763e9 0.363228 0.181614 0.983370i \(-0.441868\pi\)
0.181614 + 0.983370i \(0.441868\pi\)
\(812\) 7.35009e8 0.0481777
\(813\) 2.45765e8 0.0160400
\(814\) 7.67239e9 0.498592
\(815\) −1.90430e10 −1.23221
\(816\) 7.18933e9 0.463205
\(817\) 6.63252e8 0.0425502
\(818\) 1.02049e10 0.651889
\(819\) −4.38919e9 −0.279184
\(820\) 1.84651e9 0.116951
\(821\) 2.25214e10 1.42035 0.710173 0.704027i \(-0.248617\pi\)
0.710173 + 0.704027i \(0.248617\pi\)
\(822\) 5.47133e9 0.343591
\(823\) −1.14413e10 −0.715446 −0.357723 0.933828i \(-0.616447\pi\)
−0.357723 + 0.933828i \(0.616447\pi\)
\(824\) 2.77208e10 1.72608
\(825\) 2.81631e9 0.174619
\(826\) 2.85006e9 0.175964
\(827\) −9.39865e9 −0.577825 −0.288912 0.957356i \(-0.593294\pi\)
−0.288912 + 0.957356i \(0.593294\pi\)
\(828\) −1.57941e9 −0.0966918
\(829\) −1.97222e10 −1.20230 −0.601152 0.799135i \(-0.705291\pi\)
−0.601152 + 0.799135i \(0.705291\pi\)
\(830\) −1.47721e10 −0.896747
\(831\) 2.82835e9 0.170974
\(832\) 9.54268e9 0.574432
\(833\) −3.32643e10 −1.99398
\(834\) −1.34625e10 −0.803606
\(835\) 1.89247e10 1.12493
\(836\) 6.39892e9 0.378778
\(837\) 5.08488e8 0.0299738
\(838\) −8.82626e8 −0.0518111
\(839\) 1.88016e10 1.09908 0.549539 0.835468i \(-0.314804\pi\)
0.549539 + 0.835468i \(0.314804\pi\)
\(840\) −1.53761e10 −0.895097
\(841\) −1.70612e10 −0.989065
\(842\) −8.58014e9 −0.495339
\(843\) −5.96280e9 −0.342810
\(844\) −6.08383e9 −0.348320
\(845\) −1.13499e10 −0.647130
\(846\) 2.86791e9 0.162843
\(847\) −3.24269e10 −1.83364
\(848\) 4.48695e8 0.0252677
\(849\) 1.23185e10 0.690845
\(850\) −3.97691e9 −0.222116
\(851\) 7.31209e9 0.406713
\(852\) −2.85624e9 −0.158218
\(853\) 1.36908e10 0.755276 0.377638 0.925953i \(-0.376736\pi\)
0.377638 + 0.925953i \(0.376736\pi\)
\(854\) −2.23466e10 −1.22775
\(855\) 4.87846e9 0.266933
\(856\) 3.16621e10 1.72537
\(857\) −1.50548e10 −0.817038 −0.408519 0.912750i \(-0.633955\pi\)
−0.408519 + 0.912750i \(0.633955\pi\)
\(858\) 6.90543e9 0.373237
\(859\) −3.48451e10 −1.87571 −0.937856 0.347025i \(-0.887192\pi\)
−0.937856 + 0.347025i \(0.887192\pi\)
\(860\) −2.26248e8 −0.0121294
\(861\) 7.90272e9 0.421954
\(862\) 7.29546e9 0.387951
\(863\) 2.13694e10 1.13176 0.565879 0.824488i \(-0.308537\pi\)
0.565879 + 0.824488i \(0.308537\pi\)
\(864\) −2.02660e9 −0.106898
\(865\) −2.29353e10 −1.20489
\(866\) −1.82339e9 −0.0954041
\(867\) 6.91431e9 0.360315
\(868\) 1.38254e9 0.0717559
\(869\) 6.21203e9 0.321118
\(870\) −8.81607e8 −0.0453897
\(871\) 5.67992e9 0.291258
\(872\) 2.12592e10 1.08577
\(873\) −8.66484e9 −0.440769
\(874\) −1.50997e10 −0.765029
\(875\) 3.41070e10 1.72114
\(876\) −4.35225e9 −0.218751
\(877\) 6.17620e9 0.309188 0.154594 0.987978i \(-0.450593\pi\)
0.154594 + 0.987978i \(0.450593\pi\)
\(878\) −2.53036e10 −1.26169
\(879\) 2.88460e9 0.143260
\(880\) 1.66036e10 0.821320
\(881\) 2.32664e10 1.14634 0.573170 0.819437i \(-0.305714\pi\)
0.573170 + 0.819437i \(0.305714\pi\)
\(882\) −8.96956e9 −0.440181
\(883\) 3.59095e10 1.75528 0.877640 0.479320i \(-0.159117\pi\)
0.877640 + 0.479320i \(0.159117\pi\)
\(884\) 3.93827e9 0.191744
\(885\) 1.38066e9 0.0669552
\(886\) −3.18106e10 −1.53657
\(887\) −4.49012e9 −0.216035 −0.108018 0.994149i \(-0.534450\pi\)
−0.108018 + 0.994149i \(0.534450\pi\)
\(888\) 5.28115e9 0.253095
\(889\) −3.37201e10 −1.60966
\(890\) −2.29821e10 −1.09276
\(891\) −3.43591e9 −0.162731
\(892\) −4.79045e8 −0.0225995
\(893\) −1.10736e10 −0.520363
\(894\) −8.64633e9 −0.404716
\(895\) 1.48714e10 0.693380
\(896\) 1.28112e10 0.594991
\(897\) 6.58115e9 0.304459
\(898\) 9.76000e8 0.0449762
\(899\) 3.54808e8 0.0162868
\(900\) 4.33100e8 0.0198034
\(901\) 1.12299e9 0.0511494
\(902\) −1.24332e10 −0.564105
\(903\) −9.68301e8 −0.0437626
\(904\) −7.51120e8 −0.0338158
\(905\) −2.48259e10 −1.11336
\(906\) 5.71431e9 0.255279
\(907\) 4.38937e10 1.95333 0.976667 0.214758i \(-0.0688963\pi\)
0.976667 + 0.214758i \(0.0688963\pi\)
\(908\) −1.10661e10 −0.490561
\(909\) 1.24217e10 0.548539
\(910\) 1.43141e10 0.629678
\(911\) 1.57039e10 0.688165 0.344082 0.938939i \(-0.388190\pi\)
0.344082 + 0.938939i \(0.388190\pi\)
\(912\) −7.48519e9 −0.326754
\(913\) −4.01720e10 −1.74693
\(914\) 2.18592e10 0.946939
\(915\) −1.08254e10 −0.467164
\(916\) −1.66167e9 −0.0714350
\(917\) 6.09903e9 0.261197
\(918\) 4.85185e9 0.206994
\(919\) −2.38584e10 −1.01400 −0.506998 0.861947i \(-0.669245\pi\)
−0.506998 + 0.861947i \(0.669245\pi\)
\(920\) 2.30550e10 0.976129
\(921\) −1.32820e10 −0.560216
\(922\) −6.10511e9 −0.256528
\(923\) 1.19015e10 0.498189
\(924\) −9.34196e9 −0.389570
\(925\) −2.00509e9 −0.0832986
\(926\) 1.30309e10 0.539305
\(927\) 1.28402e10 0.529412
\(928\) −1.41411e9 −0.0580850
\(929\) 6.67984e9 0.273345 0.136672 0.990616i \(-0.456359\pi\)
0.136672 + 0.990616i \(0.456359\pi\)
\(930\) −1.65828e9 −0.0676034
\(931\) 3.46332e10 1.40659
\(932\) −9.67666e9 −0.391534
\(933\) −926156. −3.73335e−5 0
\(934\) −2.46096e10 −0.988304
\(935\) 4.15554e10 1.66260
\(936\) 4.75322e9 0.189462
\(937\) 2.07249e10 0.823009 0.411504 0.911408i \(-0.365003\pi\)
0.411504 + 0.911408i \(0.365003\pi\)
\(938\) 1.90256e10 0.752713
\(939\) −5.80456e9 −0.228792
\(940\) 3.77741e9 0.148336
\(941\) 1.64161e10 0.642255 0.321127 0.947036i \(-0.395938\pi\)
0.321127 + 0.947036i \(0.395938\pi\)
\(942\) 1.49728e10 0.583613
\(943\) −1.18493e10 −0.460153
\(944\) −2.11839e9 −0.0819603
\(945\) −7.12220e9 −0.274538
\(946\) 1.52341e9 0.0585056
\(947\) 3.58035e10 1.36994 0.684968 0.728573i \(-0.259816\pi\)
0.684968 + 0.728573i \(0.259816\pi\)
\(948\) 9.55303e8 0.0364177
\(949\) 1.81351e10 0.688791
\(950\) 4.14057e9 0.156685
\(951\) −1.27219e10 −0.479645
\(952\) 5.90464e10 2.21801
\(953\) −1.53516e10 −0.574550 −0.287275 0.957848i \(-0.592749\pi\)
−0.287275 + 0.957848i \(0.592749\pi\)
\(954\) 3.02810e8 0.0112915
\(955\) −5.07917e8 −0.0188704
\(956\) 1.41700e10 0.524526
\(957\) −2.39748e9 −0.0884226
\(958\) −2.39698e10 −0.880814
\(959\) 3.08422e10 1.12923
\(960\) 1.54846e10 0.564873
\(961\) −2.68452e10 −0.975742
\(962\) −4.91637e9 −0.178046
\(963\) 1.46658e10 0.529193
\(964\) −6.98716e9 −0.251207
\(965\) 3.30528e10 1.18403
\(966\) 2.20444e10 0.786827
\(967\) −3.79457e10 −1.34949 −0.674745 0.738051i \(-0.735747\pi\)
−0.674745 + 0.738051i \(0.735747\pi\)
\(968\) 3.51164e10 1.24436
\(969\) −1.87339e10 −0.661448
\(970\) 2.82579e10 0.994119
\(971\) −1.75229e10 −0.614242 −0.307121 0.951670i \(-0.599366\pi\)
−0.307121 + 0.951670i \(0.599366\pi\)
\(972\) −5.28383e8 −0.0184551
\(973\) −7.58888e10 −2.64109
\(974\) 9.92774e9 0.344266
\(975\) −1.80465e9 −0.0623559
\(976\) 1.66097e10 0.571858
\(977\) −2.92529e10 −1.00355 −0.501773 0.864999i \(-0.667319\pi\)
−0.501773 + 0.864999i \(0.667319\pi\)
\(978\) 1.97185e10 0.674042
\(979\) −6.24985e10 −2.12878
\(980\) −1.18141e10 −0.400967
\(981\) 9.84720e9 0.333021
\(982\) 4.10203e10 1.38232
\(983\) −1.43406e10 −0.481536 −0.240768 0.970583i \(-0.577399\pi\)
−0.240768 + 0.970583i \(0.577399\pi\)
\(984\) −8.55815e9 −0.286350
\(985\) −2.66873e10 −0.889772
\(986\) 3.38548e9 0.112474
\(987\) 1.61666e10 0.535190
\(988\) −4.10034e9 −0.135260
\(989\) 1.45187e9 0.0477244
\(990\) 1.12052e10 0.367026
\(991\) −3.77304e10 −1.23150 −0.615749 0.787942i \(-0.711146\pi\)
−0.615749 + 0.787942i \(0.711146\pi\)
\(992\) −2.65991e9 −0.0865118
\(993\) 3.25691e10 1.05556
\(994\) 3.98655e10 1.28749
\(995\) −2.69449e10 −0.867152
\(996\) −6.17776e9 −0.198118
\(997\) 4.59214e10 1.46751 0.733757 0.679412i \(-0.237765\pi\)
0.733757 + 0.679412i \(0.237765\pi\)
\(998\) 2.26191e10 0.720309
\(999\) 2.44622e9 0.0776276
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.1.6 18
3.2 odd 2 531.8.a.e.1.13 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.6 18 1.1 even 1 trivial
531.8.a.e.1.13 18 3.2 odd 2