Properties

Label 177.8.a.d.1.10
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 1.10
Root \(2.93322\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.93322 q^{2} +27.0000 q^{3} -112.530 q^{4} -305.648 q^{5} +106.197 q^{6} -1406.85 q^{7} -946.057 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+3.93322 q^{2} +27.0000 q^{3} -112.530 q^{4} -305.648 q^{5} +106.197 q^{6} -1406.85 q^{7} -946.057 q^{8} +729.000 q^{9} -1202.18 q^{10} +322.020 q^{11} -3038.30 q^{12} -6387.28 q^{13} -5533.44 q^{14} -8252.50 q^{15} +10682.8 q^{16} -14848.7 q^{17} +2867.32 q^{18} -12707.1 q^{19} +34394.5 q^{20} -37984.9 q^{21} +1266.58 q^{22} -17301.0 q^{23} -25543.5 q^{24} +15295.9 q^{25} -25122.6 q^{26} +19683.0 q^{27} +158312. q^{28} -68692.0 q^{29} -32458.9 q^{30} +153138. q^{31} +163113. q^{32} +8694.54 q^{33} -58403.3 q^{34} +430000. q^{35} -82034.2 q^{36} +198033. q^{37} -49979.8 q^{38} -172457. q^{39} +289161. q^{40} +419043. q^{41} -149403. q^{42} +104716. q^{43} -36236.8 q^{44} -222818. q^{45} -68048.6 q^{46} -738804. q^{47} +288435. q^{48} +1.15568e6 q^{49} +60162.0 q^{50} -400916. q^{51} +718760. q^{52} +248408. q^{53} +77417.6 q^{54} -98424.9 q^{55} +1.33096e6 q^{56} -343091. q^{57} -270181. q^{58} +205379. q^{59} +928652. q^{60} +915348. q^{61} +602325. q^{62} -1.02559e6 q^{63} -725835. q^{64} +1.95226e6 q^{65} +34197.5 q^{66} -828193. q^{67} +1.67092e6 q^{68} -467127. q^{69} +1.69129e6 q^{70} -302314. q^{71} -689675. q^{72} -3.36995e6 q^{73} +778909. q^{74} +412989. q^{75} +1.42993e6 q^{76} -453033. q^{77} -678310. q^{78} -608944. q^{79} -3.26517e6 q^{80} +531441. q^{81} +1.64819e6 q^{82} -3.11831e6 q^{83} +4.27443e6 q^{84} +4.53849e6 q^{85} +411872. q^{86} -1.85468e6 q^{87} -304649. q^{88} -8.70611e6 q^{89} -876391. q^{90} +8.98593e6 q^{91} +1.94688e6 q^{92} +4.13472e6 q^{93} -2.90588e6 q^{94} +3.88390e6 q^{95} +4.40405e6 q^{96} -1.80756e6 q^{97} +4.54552e6 q^{98} +234753. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.93322 0.347651 0.173825 0.984776i \(-0.444387\pi\)
0.173825 + 0.984776i \(0.444387\pi\)
\(3\) 27.0000 0.577350
\(4\) −112.530 −0.879139
\(5\) −305.648 −1.09352 −0.546760 0.837289i \(-0.684139\pi\)
−0.546760 + 0.837289i \(0.684139\pi\)
\(6\) 106.197 0.200716
\(7\) −1406.85 −1.55026 −0.775128 0.631804i \(-0.782315\pi\)
−0.775128 + 0.631804i \(0.782315\pi\)
\(8\) −946.057 −0.653284
\(9\) 729.000 0.333333
\(10\) −1202.18 −0.380163
\(11\) 322.020 0.0729472 0.0364736 0.999335i \(-0.488388\pi\)
0.0364736 + 0.999335i \(0.488388\pi\)
\(12\) −3038.30 −0.507571
\(13\) −6387.28 −0.806333 −0.403167 0.915127i \(-0.632090\pi\)
−0.403167 + 0.915127i \(0.632090\pi\)
\(14\) −5533.44 −0.538948
\(15\) −8252.50 −0.631344
\(16\) 10682.8 0.652024
\(17\) −14848.7 −0.733023 −0.366512 0.930413i \(-0.619448\pi\)
−0.366512 + 0.930413i \(0.619448\pi\)
\(18\) 2867.32 0.115884
\(19\) −12707.1 −0.425019 −0.212509 0.977159i \(-0.568164\pi\)
−0.212509 + 0.977159i \(0.568164\pi\)
\(20\) 34394.5 0.961356
\(21\) −37984.9 −0.895041
\(22\) 1266.58 0.0253601
\(23\) −17301.0 −0.296499 −0.148250 0.988950i \(-0.547364\pi\)
−0.148250 + 0.988950i \(0.547364\pi\)
\(24\) −25543.5 −0.377174
\(25\) 15295.9 0.195787
\(26\) −25122.6 −0.280322
\(27\) 19683.0 0.192450
\(28\) 158312. 1.36289
\(29\) −68692.0 −0.523013 −0.261507 0.965202i \(-0.584219\pi\)
−0.261507 + 0.965202i \(0.584219\pi\)
\(30\) −32458.9 −0.219487
\(31\) 153138. 0.923244 0.461622 0.887077i \(-0.347268\pi\)
0.461622 + 0.887077i \(0.347268\pi\)
\(32\) 163113. 0.879961
\(33\) 8694.54 0.0421161
\(34\) −58403.3 −0.254836
\(35\) 430000. 1.69524
\(36\) −82034.2 −0.293046
\(37\) 198033. 0.642736 0.321368 0.946954i \(-0.395857\pi\)
0.321368 + 0.946954i \(0.395857\pi\)
\(38\) −49979.8 −0.147758
\(39\) −172457. −0.465537
\(40\) 289161. 0.714380
\(41\) 419043. 0.949544 0.474772 0.880109i \(-0.342531\pi\)
0.474772 + 0.880109i \(0.342531\pi\)
\(42\) −149403. −0.311162
\(43\) 104716. 0.200851 0.100426 0.994945i \(-0.467980\pi\)
0.100426 + 0.994945i \(0.467980\pi\)
\(44\) −36236.8 −0.0641307
\(45\) −222818. −0.364507
\(46\) −68048.6 −0.103078
\(47\) −738804. −1.03797 −0.518987 0.854782i \(-0.673691\pi\)
−0.518987 + 0.854782i \(0.673691\pi\)
\(48\) 288435. 0.376446
\(49\) 1.15568e6 1.40330
\(50\) 60162.0 0.0680656
\(51\) −400916. −0.423211
\(52\) 718760. 0.708879
\(53\) 248408. 0.229192 0.114596 0.993412i \(-0.463443\pi\)
0.114596 + 0.993412i \(0.463443\pi\)
\(54\) 77417.6 0.0669054
\(55\) −98424.9 −0.0797692
\(56\) 1.33096e6 1.01276
\(57\) −343091. −0.245385
\(58\) −270181. −0.181826
\(59\) 205379. 0.130189
\(60\) 928652. 0.555039
\(61\) 915348. 0.516335 0.258168 0.966100i \(-0.416881\pi\)
0.258168 + 0.966100i \(0.416881\pi\)
\(62\) 602325. 0.320966
\(63\) −1.02559e6 −0.516752
\(64\) −725835. −0.346105
\(65\) 1.95226e6 0.881742
\(66\) 34197.5 0.0146417
\(67\) −828193. −0.336411 −0.168205 0.985752i \(-0.553797\pi\)
−0.168205 + 0.985752i \(0.553797\pi\)
\(68\) 1.67092e6 0.644429
\(69\) −467127. −0.171184
\(70\) 1.69129e6 0.589351
\(71\) −302314. −0.100243 −0.0501215 0.998743i \(-0.515961\pi\)
−0.0501215 + 0.998743i \(0.515961\pi\)
\(72\) −689675. −0.217761
\(73\) −3.36995e6 −1.01390 −0.506948 0.861977i \(-0.669226\pi\)
−0.506948 + 0.861977i \(0.669226\pi\)
\(74\) 778909. 0.223448
\(75\) 412989. 0.113038
\(76\) 1.42993e6 0.373651
\(77\) −453033. −0.113087
\(78\) −678310. −0.161844
\(79\) −608944. −0.138958 −0.0694789 0.997583i \(-0.522134\pi\)
−0.0694789 + 0.997583i \(0.522134\pi\)
\(80\) −3.26517e6 −0.713002
\(81\) 531441. 0.111111
\(82\) 1.64819e6 0.330110
\(83\) −3.11831e6 −0.598614 −0.299307 0.954157i \(-0.596755\pi\)
−0.299307 + 0.954157i \(0.596755\pi\)
\(84\) 4.27443e6 0.786866
\(85\) 4.53849e6 0.801576
\(86\) 411872. 0.0698260
\(87\) −1.85468e6 −0.301962
\(88\) −304649. −0.0476552
\(89\) −8.70611e6 −1.30906 −0.654529 0.756037i \(-0.727133\pi\)
−0.654529 + 0.756037i \(0.727133\pi\)
\(90\) −876391. −0.126721
\(91\) 8.98593e6 1.25002
\(92\) 1.94688e6 0.260664
\(93\) 4.13472e6 0.533035
\(94\) −2.90588e6 −0.360852
\(95\) 3.88390e6 0.464767
\(96\) 4.40405e6 0.508046
\(97\) −1.80756e6 −0.201091 −0.100545 0.994932i \(-0.532059\pi\)
−0.100545 + 0.994932i \(0.532059\pi\)
\(98\) 4.54552e6 0.487857
\(99\) 234753. 0.0243157
\(100\) −1.72124e6 −0.172124
\(101\) 8.27630e6 0.799303 0.399651 0.916667i \(-0.369131\pi\)
0.399651 + 0.916667i \(0.369131\pi\)
\(102\) −1.57689e6 −0.147130
\(103\) 1.84311e7 1.66197 0.830983 0.556298i \(-0.187779\pi\)
0.830983 + 0.556298i \(0.187779\pi\)
\(104\) 6.04273e6 0.526765
\(105\) 1.16100e7 0.978746
\(106\) 977042. 0.0796788
\(107\) 6.61305e6 0.521866 0.260933 0.965357i \(-0.415970\pi\)
0.260933 + 0.965357i \(0.415970\pi\)
\(108\) −2.21492e6 −0.169190
\(109\) 1.30251e7 0.963359 0.481679 0.876347i \(-0.340027\pi\)
0.481679 + 0.876347i \(0.340027\pi\)
\(110\) −387127. −0.0277318
\(111\) 5.34690e6 0.371084
\(112\) −1.50290e7 −1.01080
\(113\) −7.28274e6 −0.474810 −0.237405 0.971411i \(-0.576297\pi\)
−0.237405 + 0.971411i \(0.576297\pi\)
\(114\) −1.34945e6 −0.0853082
\(115\) 5.28802e6 0.324228
\(116\) 7.72989e6 0.459802
\(117\) −4.65633e6 −0.268778
\(118\) 807801. 0.0452603
\(119\) 2.08899e7 1.13637
\(120\) 7.80733e6 0.412447
\(121\) −1.93835e7 −0.994679
\(122\) 3.60026e6 0.179504
\(123\) 1.13142e7 0.548219
\(124\) −1.72326e7 −0.811660
\(125\) 1.92036e7 0.879423
\(126\) −4.03388e6 −0.179649
\(127\) 3.11648e7 1.35006 0.675028 0.737792i \(-0.264131\pi\)
0.675028 + 0.737792i \(0.264131\pi\)
\(128\) −2.37333e7 −1.00028
\(129\) 2.82734e6 0.115961
\(130\) 7.67868e6 0.306538
\(131\) −2.99857e7 −1.16537 −0.582687 0.812697i \(-0.697999\pi\)
−0.582687 + 0.812697i \(0.697999\pi\)
\(132\) −978395. −0.0370259
\(133\) 1.78769e7 0.658889
\(134\) −3.25747e6 −0.116953
\(135\) −6.01608e6 −0.210448
\(136\) 1.40477e7 0.478873
\(137\) 1.65933e7 0.551329 0.275664 0.961254i \(-0.411102\pi\)
0.275664 + 0.961254i \(0.411102\pi\)
\(138\) −1.83731e6 −0.0595122
\(139\) −5.13930e7 −1.62313 −0.811563 0.584264i \(-0.801383\pi\)
−0.811563 + 0.584264i \(0.801383\pi\)
\(140\) −4.83878e7 −1.49035
\(141\) −1.99477e7 −0.599275
\(142\) −1.18907e6 −0.0348496
\(143\) −2.05683e6 −0.0588197
\(144\) 7.78773e6 0.217341
\(145\) 2.09956e7 0.571926
\(146\) −1.32547e7 −0.352481
\(147\) 3.12032e7 0.810194
\(148\) −2.22847e7 −0.565054
\(149\) −2.55525e7 −0.632821 −0.316410 0.948622i \(-0.602478\pi\)
−0.316410 + 0.948622i \(0.602478\pi\)
\(150\) 1.62437e6 0.0392977
\(151\) −2.50959e6 −0.0593176 −0.0296588 0.999560i \(-0.509442\pi\)
−0.0296588 + 0.999560i \(0.509442\pi\)
\(152\) 1.20216e7 0.277658
\(153\) −1.08247e7 −0.244341
\(154\) −1.78188e6 −0.0393147
\(155\) −4.68063e7 −1.00959
\(156\) 1.94065e7 0.409271
\(157\) −1.51685e7 −0.312819 −0.156409 0.987692i \(-0.549992\pi\)
−0.156409 + 0.987692i \(0.549992\pi\)
\(158\) −2.39511e6 −0.0483088
\(159\) 6.70701e6 0.132324
\(160\) −4.98552e7 −0.962255
\(161\) 2.43398e7 0.459650
\(162\) 2.09027e6 0.0386279
\(163\) 1.10949e7 0.200664 0.100332 0.994954i \(-0.468010\pi\)
0.100332 + 0.994954i \(0.468010\pi\)
\(164\) −4.71548e7 −0.834781
\(165\) −2.65747e6 −0.0460548
\(166\) −1.22650e7 −0.208108
\(167\) 8.97494e7 1.49116 0.745579 0.666417i \(-0.232173\pi\)
0.745579 + 0.666417i \(0.232173\pi\)
\(168\) 3.59358e7 0.584716
\(169\) −2.19511e7 −0.349827
\(170\) 1.78509e7 0.278669
\(171\) −9.26346e6 −0.141673
\(172\) −1.17837e7 −0.176576
\(173\) 9.67840e7 1.42116 0.710578 0.703618i \(-0.248433\pi\)
0.710578 + 0.703618i \(0.248433\pi\)
\(174\) −7.29488e6 −0.104977
\(175\) −2.15189e7 −0.303520
\(176\) 3.44006e6 0.0475633
\(177\) 5.54523e6 0.0751646
\(178\) −3.42430e7 −0.455095
\(179\) −3.95244e7 −0.515086 −0.257543 0.966267i \(-0.582913\pi\)
−0.257543 + 0.966267i \(0.582913\pi\)
\(180\) 2.50736e7 0.320452
\(181\) −7.62592e6 −0.0955910 −0.0477955 0.998857i \(-0.515220\pi\)
−0.0477955 + 0.998857i \(0.515220\pi\)
\(182\) 3.53436e7 0.434572
\(183\) 2.47144e7 0.298106
\(184\) 1.63677e7 0.193698
\(185\) −6.05286e7 −0.702845
\(186\) 1.62628e7 0.185310
\(187\) −4.78159e6 −0.0534720
\(188\) 8.31374e7 0.912523
\(189\) −2.76910e7 −0.298347
\(190\) 1.52762e7 0.161577
\(191\) −7.01758e7 −0.728736 −0.364368 0.931255i \(-0.618715\pi\)
−0.364368 + 0.931255i \(0.618715\pi\)
\(192\) −1.95975e7 −0.199824
\(193\) −1.47888e8 −1.48076 −0.740378 0.672191i \(-0.765353\pi\)
−0.740378 + 0.672191i \(0.765353\pi\)
\(194\) −7.10954e6 −0.0699094
\(195\) 5.27111e7 0.509074
\(196\) −1.30048e8 −1.23369
\(197\) −2.42261e7 −0.225762 −0.112881 0.993608i \(-0.536008\pi\)
−0.112881 + 0.993608i \(0.536008\pi\)
\(198\) 923334. 0.00845338
\(199\) −1.19894e8 −1.07847 −0.539237 0.842154i \(-0.681287\pi\)
−0.539237 + 0.842154i \(0.681287\pi\)
\(200\) −1.44708e7 −0.127905
\(201\) −2.23612e7 −0.194227
\(202\) 3.25525e7 0.277878
\(203\) 9.66391e7 0.810805
\(204\) 4.51149e7 0.372062
\(205\) −1.28080e8 −1.03835
\(206\) 7.24937e7 0.577783
\(207\) −1.26124e7 −0.0988331
\(208\) −6.82338e7 −0.525749
\(209\) −4.09194e6 −0.0310039
\(210\) 4.56647e7 0.340262
\(211\) −1.87566e8 −1.37456 −0.687281 0.726391i \(-0.741196\pi\)
−0.687281 + 0.726391i \(0.741196\pi\)
\(212\) −2.79533e7 −0.201492
\(213\) −8.16249e6 −0.0578754
\(214\) 2.60106e7 0.181427
\(215\) −3.20063e7 −0.219635
\(216\) −1.86212e7 −0.125725
\(217\) −2.15441e8 −1.43127
\(218\) 5.12306e7 0.334912
\(219\) −9.09886e7 −0.585373
\(220\) 1.10757e7 0.0701282
\(221\) 9.48430e7 0.591061
\(222\) 2.10305e7 0.129008
\(223\) 5.60154e7 0.338252 0.169126 0.985594i \(-0.445905\pi\)
0.169126 + 0.985594i \(0.445905\pi\)
\(224\) −2.29475e8 −1.36417
\(225\) 1.11507e7 0.0652624
\(226\) −2.86446e7 −0.165068
\(227\) 1.96404e8 1.11445 0.557224 0.830362i \(-0.311866\pi\)
0.557224 + 0.830362i \(0.311866\pi\)
\(228\) 3.86080e7 0.215727
\(229\) 4.38337e7 0.241203 0.120602 0.992701i \(-0.461518\pi\)
0.120602 + 0.992701i \(0.461518\pi\)
\(230\) 2.07989e7 0.112718
\(231\) −1.22319e7 −0.0652907
\(232\) 6.49865e7 0.341676
\(233\) 1.57505e7 0.0815732 0.0407866 0.999168i \(-0.487014\pi\)
0.0407866 + 0.999168i \(0.487014\pi\)
\(234\) −1.83144e7 −0.0934408
\(235\) 2.25814e8 1.13505
\(236\) −2.31113e7 −0.114454
\(237\) −1.64415e7 −0.0802273
\(238\) 8.21645e7 0.395062
\(239\) −1.02176e8 −0.484126 −0.242063 0.970261i \(-0.577824\pi\)
−0.242063 + 0.970261i \(0.577824\pi\)
\(240\) −8.81595e7 −0.411652
\(241\) 1.57502e7 0.0724813 0.0362407 0.999343i \(-0.488462\pi\)
0.0362407 + 0.999343i \(0.488462\pi\)
\(242\) −7.62395e7 −0.345801
\(243\) 1.43489e7 0.0641500
\(244\) −1.03004e8 −0.453930
\(245\) −3.53230e8 −1.53453
\(246\) 4.45011e7 0.190589
\(247\) 8.11638e7 0.342707
\(248\) −1.44877e8 −0.603141
\(249\) −8.41945e7 −0.345610
\(250\) 7.55320e7 0.305732
\(251\) 1.52785e8 0.609848 0.304924 0.952377i \(-0.401369\pi\)
0.304924 + 0.952377i \(0.401369\pi\)
\(252\) 1.15410e8 0.454297
\(253\) −5.57127e6 −0.0216288
\(254\) 1.22578e8 0.469348
\(255\) 1.22539e8 0.462790
\(256\) −441486. −0.00164466
\(257\) 2.55028e8 0.937176 0.468588 0.883417i \(-0.344763\pi\)
0.468588 + 0.883417i \(0.344763\pi\)
\(258\) 1.11205e7 0.0403141
\(259\) −2.78603e8 −0.996405
\(260\) −2.19688e8 −0.775174
\(261\) −5.00764e7 −0.174338
\(262\) −1.17941e8 −0.405143
\(263\) 3.21627e8 1.09020 0.545101 0.838370i \(-0.316491\pi\)
0.545101 + 0.838370i \(0.316491\pi\)
\(264\) −8.22553e6 −0.0275138
\(265\) −7.59254e7 −0.250626
\(266\) 7.03139e7 0.229063
\(267\) −2.35065e8 −0.755785
\(268\) 9.31964e7 0.295752
\(269\) 2.32732e8 0.728992 0.364496 0.931205i \(-0.381241\pi\)
0.364496 + 0.931205i \(0.381241\pi\)
\(270\) −2.36625e7 −0.0731625
\(271\) 1.17634e8 0.359036 0.179518 0.983755i \(-0.442546\pi\)
0.179518 + 0.983755i \(0.442546\pi\)
\(272\) −1.58625e8 −0.477949
\(273\) 2.42620e8 0.721701
\(274\) 6.52651e7 0.191670
\(275\) 4.92558e6 0.0142821
\(276\) 5.25657e7 0.150494
\(277\) 2.49236e8 0.704583 0.352291 0.935890i \(-0.385403\pi\)
0.352291 + 0.935890i \(0.385403\pi\)
\(278\) −2.02140e8 −0.564281
\(279\) 1.11637e8 0.307748
\(280\) −4.06805e8 −1.10747
\(281\) 4.06099e8 1.09184 0.545921 0.837837i \(-0.316180\pi\)
0.545921 + 0.837837i \(0.316180\pi\)
\(282\) −7.84587e7 −0.208338
\(283\) −7.10346e8 −1.86302 −0.931510 0.363717i \(-0.881508\pi\)
−0.931510 + 0.363717i \(0.881508\pi\)
\(284\) 3.40194e7 0.0881276
\(285\) 1.04865e8 0.268333
\(286\) −8.08998e6 −0.0204487
\(287\) −5.89529e8 −1.47204
\(288\) 1.18909e8 0.293320
\(289\) −1.89854e8 −0.462677
\(290\) 8.25802e7 0.198831
\(291\) −4.88042e7 −0.116100
\(292\) 3.79219e8 0.891355
\(293\) 9.92751e7 0.230571 0.115285 0.993332i \(-0.463222\pi\)
0.115285 + 0.993332i \(0.463222\pi\)
\(294\) 1.22729e8 0.281664
\(295\) −6.27737e7 −0.142364
\(296\) −1.87351e8 −0.419889
\(297\) 6.33832e6 0.0140387
\(298\) −1.00503e8 −0.220001
\(299\) 1.10506e8 0.239077
\(300\) −4.64735e7 −0.0993759
\(301\) −1.47320e8 −0.311371
\(302\) −9.87078e6 −0.0206218
\(303\) 2.23460e8 0.461478
\(304\) −1.35747e8 −0.277123
\(305\) −2.79775e8 −0.564623
\(306\) −4.25760e7 −0.0849454
\(307\) 9.45615e8 1.86522 0.932610 0.360885i \(-0.117525\pi\)
0.932610 + 0.360885i \(0.117525\pi\)
\(308\) 5.09797e7 0.0994191
\(309\) 4.97641e8 0.959536
\(310\) −1.84099e8 −0.350983
\(311\) 3.51898e8 0.663370 0.331685 0.943390i \(-0.392383\pi\)
0.331685 + 0.943390i \(0.392383\pi\)
\(312\) 1.63154e8 0.304128
\(313\) 4.30491e8 0.793521 0.396761 0.917922i \(-0.370134\pi\)
0.396761 + 0.917922i \(0.370134\pi\)
\(314\) −5.96609e7 −0.108752
\(315\) 3.13470e8 0.565079
\(316\) 6.85244e7 0.122163
\(317\) 9.07736e8 1.60049 0.800243 0.599676i \(-0.204704\pi\)
0.800243 + 0.599676i \(0.204704\pi\)
\(318\) 2.63801e7 0.0460026
\(319\) −2.21202e7 −0.0381524
\(320\) 2.21850e8 0.378473
\(321\) 1.78552e8 0.301299
\(322\) 9.57340e7 0.159798
\(323\) 1.88684e8 0.311549
\(324\) −5.98029e7 −0.0976821
\(325\) −9.76991e7 −0.157870
\(326\) 4.36389e7 0.0697609
\(327\) 3.51678e8 0.556195
\(328\) −3.96438e8 −0.620322
\(329\) 1.03938e9 1.60913
\(330\) −1.04524e7 −0.0160110
\(331\) −6.30152e8 −0.955096 −0.477548 0.878606i \(-0.658474\pi\)
−0.477548 + 0.878606i \(0.658474\pi\)
\(332\) 3.50903e8 0.526264
\(333\) 1.44366e8 0.214245
\(334\) 3.53004e8 0.518402
\(335\) 2.53136e8 0.367872
\(336\) −4.05783e8 −0.583589
\(337\) 5.99949e8 0.853905 0.426952 0.904274i \(-0.359587\pi\)
0.426952 + 0.904274i \(0.359587\pi\)
\(338\) −8.63386e7 −0.121618
\(339\) −1.96634e8 −0.274132
\(340\) −5.10715e8 −0.704697
\(341\) 4.93134e7 0.0673480
\(342\) −3.64352e7 −0.0492527
\(343\) −4.67259e8 −0.625213
\(344\) −9.90674e7 −0.131213
\(345\) 1.42776e8 0.187193
\(346\) 3.80673e8 0.494066
\(347\) 9.73412e8 1.25067 0.625336 0.780355i \(-0.284962\pi\)
0.625336 + 0.780355i \(0.284962\pi\)
\(348\) 2.08707e8 0.265467
\(349\) 4.28531e7 0.0539627 0.0269813 0.999636i \(-0.491411\pi\)
0.0269813 + 0.999636i \(0.491411\pi\)
\(350\) −8.46388e7 −0.105519
\(351\) −1.25721e8 −0.155179
\(352\) 5.25256e7 0.0641907
\(353\) 1.33381e9 1.61393 0.806963 0.590603i \(-0.201110\pi\)
0.806963 + 0.590603i \(0.201110\pi\)
\(354\) 2.18106e7 0.0261310
\(355\) 9.24018e7 0.109618
\(356\) 9.79697e8 1.15084
\(357\) 5.64027e8 0.656086
\(358\) −1.55458e8 −0.179070
\(359\) 5.52883e8 0.630671 0.315336 0.948980i \(-0.397883\pi\)
0.315336 + 0.948980i \(0.397883\pi\)
\(360\) 2.10798e8 0.238127
\(361\) −7.32402e8 −0.819359
\(362\) −2.99944e7 −0.0332323
\(363\) −5.23354e8 −0.574278
\(364\) −1.01118e9 −1.09894
\(365\) 1.03002e9 1.10872
\(366\) 9.72071e7 0.103637
\(367\) −8.45526e8 −0.892885 −0.446443 0.894812i \(-0.647309\pi\)
−0.446443 + 0.894812i \(0.647309\pi\)
\(368\) −1.84822e8 −0.193325
\(369\) 3.05482e8 0.316515
\(370\) −2.38072e8 −0.244345
\(371\) −3.49471e8 −0.355306
\(372\) −4.65279e8 −0.468612
\(373\) −1.80814e9 −1.80406 −0.902031 0.431671i \(-0.857924\pi\)
−0.902031 + 0.431671i \(0.857924\pi\)
\(374\) −1.88070e7 −0.0185896
\(375\) 5.18498e8 0.507735
\(376\) 6.98950e8 0.678092
\(377\) 4.38755e8 0.421723
\(378\) −1.08915e8 −0.103721
\(379\) −1.25199e9 −1.18131 −0.590653 0.806925i \(-0.701130\pi\)
−0.590653 + 0.806925i \(0.701130\pi\)
\(380\) −4.37054e8 −0.408595
\(381\) 8.41451e8 0.779455
\(382\) −2.76017e8 −0.253346
\(383\) 2.12795e6 0.00193537 0.000967687 1.00000i \(-0.499692\pi\)
0.000967687 1.00000i \(0.499692\pi\)
\(384\) −6.40800e8 −0.577515
\(385\) 1.38469e8 0.123663
\(386\) −5.81677e8 −0.514786
\(387\) 7.63381e7 0.0669503
\(388\) 2.03405e8 0.176787
\(389\) 2.97352e8 0.256122 0.128061 0.991766i \(-0.459125\pi\)
0.128061 + 0.991766i \(0.459125\pi\)
\(390\) 2.07324e8 0.176980
\(391\) 2.56898e8 0.217341
\(392\) −1.09333e9 −0.916751
\(393\) −8.09615e8 −0.672829
\(394\) −9.52865e7 −0.0784865
\(395\) 1.86123e8 0.151953
\(396\) −2.64167e7 −0.0213769
\(397\) 7.60962e8 0.610374 0.305187 0.952292i \(-0.401281\pi\)
0.305187 + 0.952292i \(0.401281\pi\)
\(398\) −4.71568e8 −0.374933
\(399\) 4.82677e8 0.380409
\(400\) 1.63402e8 0.127658
\(401\) 4.16579e8 0.322620 0.161310 0.986904i \(-0.448428\pi\)
0.161310 + 0.986904i \(0.448428\pi\)
\(402\) −8.79516e7 −0.0675231
\(403\) −9.78134e8 −0.744442
\(404\) −9.31330e8 −0.702698
\(405\) −1.62434e8 −0.121502
\(406\) 3.80103e8 0.281877
\(407\) 6.37707e7 0.0468858
\(408\) 3.79289e8 0.276477
\(409\) −2.40984e9 −1.74163 −0.870815 0.491611i \(-0.836408\pi\)
−0.870815 + 0.491611i \(0.836408\pi\)
\(410\) −5.03766e8 −0.360982
\(411\) 4.48019e8 0.318310
\(412\) −2.07405e9 −1.46110
\(413\) −2.88937e8 −0.201826
\(414\) −4.96074e7 −0.0343594
\(415\) 9.53107e8 0.654596
\(416\) −1.04185e9 −0.709542
\(417\) −1.38761e9 −0.937113
\(418\) −1.60945e7 −0.0107785
\(419\) −2.79997e8 −0.185953 −0.0929767 0.995668i \(-0.529638\pi\)
−0.0929767 + 0.995668i \(0.529638\pi\)
\(420\) −1.30647e9 −0.860454
\(421\) 1.28732e9 0.840810 0.420405 0.907337i \(-0.361888\pi\)
0.420405 + 0.907337i \(0.361888\pi\)
\(422\) −7.37736e8 −0.477868
\(423\) −5.38588e8 −0.345991
\(424\) −2.35008e8 −0.149727
\(425\) −2.27124e8 −0.143517
\(426\) −3.21049e7 −0.0201204
\(427\) −1.28775e9 −0.800452
\(428\) −7.44165e8 −0.458792
\(429\) −5.55345e7 −0.0339596
\(430\) −1.25888e8 −0.0763562
\(431\) −1.52813e9 −0.919367 −0.459684 0.888083i \(-0.652037\pi\)
−0.459684 + 0.888083i \(0.652037\pi\)
\(432\) 2.10269e8 0.125482
\(433\) 3.05014e9 1.80556 0.902780 0.430103i \(-0.141523\pi\)
0.902780 + 0.430103i \(0.141523\pi\)
\(434\) −8.47378e8 −0.497580
\(435\) 5.66881e8 0.330202
\(436\) −1.46571e9 −0.846926
\(437\) 2.19845e8 0.126018
\(438\) −3.57878e8 −0.203505
\(439\) 1.16263e9 0.655864 0.327932 0.944701i \(-0.393648\pi\)
0.327932 + 0.944701i \(0.393648\pi\)
\(440\) 9.31155e7 0.0521120
\(441\) 8.42487e8 0.467766
\(442\) 3.73038e8 0.205483
\(443\) −1.39964e9 −0.764900 −0.382450 0.923976i \(-0.624920\pi\)
−0.382450 + 0.923976i \(0.624920\pi\)
\(444\) −6.01686e8 −0.326234
\(445\) 2.66101e9 1.43148
\(446\) 2.20321e8 0.117594
\(447\) −6.89916e8 −0.365359
\(448\) 1.02114e9 0.536552
\(449\) −9.56624e8 −0.498746 −0.249373 0.968408i \(-0.580224\pi\)
−0.249373 + 0.968408i \(0.580224\pi\)
\(450\) 4.38581e7 0.0226885
\(451\) 1.34940e8 0.0692665
\(452\) 8.19525e8 0.417424
\(453\) −6.77590e7 −0.0342471
\(454\) 7.72500e8 0.387439
\(455\) −2.74653e9 −1.36693
\(456\) 3.24584e8 0.160306
\(457\) 2.44936e9 1.20045 0.600227 0.799830i \(-0.295077\pi\)
0.600227 + 0.799830i \(0.295077\pi\)
\(458\) 1.72407e8 0.0838546
\(459\) −2.92267e8 −0.141070
\(460\) −5.95060e8 −0.285041
\(461\) 3.14456e9 1.49488 0.747441 0.664329i \(-0.231282\pi\)
0.747441 + 0.664329i \(0.231282\pi\)
\(462\) −4.81107e7 −0.0226984
\(463\) 3.16804e9 1.48340 0.741699 0.670733i \(-0.234020\pi\)
0.741699 + 0.670733i \(0.234020\pi\)
\(464\) −7.33820e8 −0.341017
\(465\) −1.26377e9 −0.582885
\(466\) 6.19500e7 0.0283590
\(467\) −1.93980e9 −0.881350 −0.440675 0.897667i \(-0.645261\pi\)
−0.440675 + 0.897667i \(0.645261\pi\)
\(468\) 5.23976e8 0.236293
\(469\) 1.16514e9 0.521523
\(470\) 8.88176e8 0.394600
\(471\) −4.09548e8 −0.180606
\(472\) −1.94300e8 −0.0850504
\(473\) 3.37207e7 0.0146515
\(474\) −6.46680e7 −0.0278911
\(475\) −1.94366e8 −0.0832133
\(476\) −2.35073e9 −0.999031
\(477\) 1.81089e8 0.0763973
\(478\) −4.01882e8 −0.168307
\(479\) −1.57859e9 −0.656287 −0.328143 0.944628i \(-0.606423\pi\)
−0.328143 + 0.944628i \(0.606423\pi\)
\(480\) −1.34609e9 −0.555558
\(481\) −1.26490e9 −0.518259
\(482\) 6.19490e7 0.0251982
\(483\) 6.57176e8 0.265379
\(484\) 2.18122e9 0.874461
\(485\) 5.52478e8 0.219897
\(486\) 5.64374e7 0.0223018
\(487\) 1.69122e9 0.663512 0.331756 0.943365i \(-0.392359\pi\)
0.331756 + 0.943365i \(0.392359\pi\)
\(488\) −8.65971e8 −0.337314
\(489\) 2.99564e8 0.115853
\(490\) −1.38933e9 −0.533482
\(491\) 3.03523e9 1.15720 0.578598 0.815613i \(-0.303600\pi\)
0.578598 + 0.815613i \(0.303600\pi\)
\(492\) −1.27318e9 −0.481961
\(493\) 1.01999e9 0.383381
\(494\) 3.19235e8 0.119142
\(495\) −7.17517e7 −0.0265897
\(496\) 1.63593e9 0.601977
\(497\) 4.25310e8 0.155403
\(498\) −3.31155e8 −0.120151
\(499\) −2.19815e9 −0.791966 −0.395983 0.918258i \(-0.629596\pi\)
−0.395983 + 0.918258i \(0.629596\pi\)
\(500\) −2.16098e9 −0.773135
\(501\) 2.42323e9 0.860921
\(502\) 6.00936e8 0.212014
\(503\) −4.49246e9 −1.57397 −0.786984 0.616973i \(-0.788359\pi\)
−0.786984 + 0.616973i \(0.788359\pi\)
\(504\) 9.70267e8 0.337586
\(505\) −2.52964e9 −0.874054
\(506\) −2.19130e7 −0.00751926
\(507\) −5.92680e8 −0.201973
\(508\) −3.50697e9 −1.18689
\(509\) −4.25095e9 −1.42881 −0.714403 0.699734i \(-0.753302\pi\)
−0.714403 + 0.699734i \(0.753302\pi\)
\(510\) 4.81973e8 0.160889
\(511\) 4.74100e9 1.57180
\(512\) 3.03613e9 0.999713
\(513\) −2.50114e8 −0.0817949
\(514\) 1.00308e9 0.325810
\(515\) −5.63345e9 −1.81739
\(516\) −3.18159e8 −0.101946
\(517\) −2.37910e8 −0.0757173
\(518\) −1.09581e9 −0.346401
\(519\) 2.61317e9 0.820505
\(520\) −1.84695e9 −0.576028
\(521\) 6.47770e8 0.200673 0.100336 0.994954i \(-0.468008\pi\)
0.100336 + 0.994954i \(0.468008\pi\)
\(522\) −1.96962e8 −0.0606087
\(523\) 2.31099e9 0.706386 0.353193 0.935551i \(-0.385096\pi\)
0.353193 + 0.935551i \(0.385096\pi\)
\(524\) 3.37429e9 1.02453
\(525\) −5.81012e8 −0.175238
\(526\) 1.26503e9 0.379010
\(527\) −2.27390e9 −0.676759
\(528\) 9.28817e7 0.0274607
\(529\) −3.10550e9 −0.912088
\(530\) −2.98631e8 −0.0871304
\(531\) 1.49721e8 0.0433963
\(532\) −2.01169e9 −0.579255
\(533\) −2.67655e9 −0.765648
\(534\) −9.24562e8 −0.262749
\(535\) −2.02127e9 −0.570671
\(536\) 7.83518e8 0.219772
\(537\) −1.06716e9 −0.297385
\(538\) 9.15386e8 0.253435
\(539\) 3.72151e8 0.102367
\(540\) 6.76988e8 0.185013
\(541\) 2.47612e8 0.0672329 0.0336165 0.999435i \(-0.489298\pi\)
0.0336165 + 0.999435i \(0.489298\pi\)
\(542\) 4.62678e8 0.124819
\(543\) −2.05900e8 −0.0551895
\(544\) −2.42202e9 −0.645032
\(545\) −3.98110e9 −1.05345
\(546\) 9.54278e8 0.250900
\(547\) −2.13776e9 −0.558474 −0.279237 0.960222i \(-0.590082\pi\)
−0.279237 + 0.960222i \(0.590082\pi\)
\(548\) −1.86724e9 −0.484694
\(549\) 6.67289e8 0.172112
\(550\) 1.93734e7 0.00496519
\(551\) 8.72874e8 0.222291
\(552\) 4.41928e8 0.111832
\(553\) 8.56691e8 0.215420
\(554\) 9.80301e8 0.244949
\(555\) −1.63427e9 −0.405788
\(556\) 5.78325e9 1.42695
\(557\) 6.39469e9 1.56793 0.783964 0.620806i \(-0.213194\pi\)
0.783964 + 0.620806i \(0.213194\pi\)
\(558\) 4.39095e8 0.106989
\(559\) −6.68852e8 −0.161953
\(560\) 4.59359e9 1.10534
\(561\) −1.29103e8 −0.0308721
\(562\) 1.59728e9 0.379580
\(563\) 7.78393e9 1.83831 0.919156 0.393893i \(-0.128872\pi\)
0.919156 + 0.393893i \(0.128872\pi\)
\(564\) 2.24471e9 0.526846
\(565\) 2.22596e9 0.519215
\(566\) −2.79395e9 −0.647680
\(567\) −7.47656e8 −0.172251
\(568\) 2.86006e8 0.0654872
\(569\) 1.45339e9 0.330742 0.165371 0.986231i \(-0.447118\pi\)
0.165371 + 0.986231i \(0.447118\pi\)
\(570\) 4.12458e8 0.0932863
\(571\) 6.80957e9 1.53071 0.765356 0.643608i \(-0.222563\pi\)
0.765356 + 0.643608i \(0.222563\pi\)
\(572\) 2.31455e8 0.0517107
\(573\) −1.89475e9 −0.420736
\(574\) −2.31875e9 −0.511755
\(575\) −2.64634e8 −0.0580507
\(576\) −5.29134e8 −0.115368
\(577\) −2.47806e9 −0.537027 −0.268513 0.963276i \(-0.586532\pi\)
−0.268513 + 0.963276i \(0.586532\pi\)
\(578\) −7.46738e8 −0.160850
\(579\) −3.99298e9 −0.854915
\(580\) −2.36263e9 −0.502802
\(581\) 4.38699e9 0.928005
\(582\) −1.91958e8 −0.0403622
\(583\) 7.99922e7 0.0167189
\(584\) 3.18816e9 0.662362
\(585\) 1.42320e9 0.293914
\(586\) 3.90471e8 0.0801580
\(587\) −1.40629e9 −0.286974 −0.143487 0.989652i \(-0.545832\pi\)
−0.143487 + 0.989652i \(0.545832\pi\)
\(588\) −3.51129e9 −0.712273
\(589\) −1.94593e9 −0.392396
\(590\) −2.46903e8 −0.0494930
\(591\) −6.54104e8 −0.130344
\(592\) 2.11554e9 0.419079
\(593\) 1.46308e8 0.0288123 0.0144062 0.999896i \(-0.495414\pi\)
0.0144062 + 0.999896i \(0.495414\pi\)
\(594\) 2.49300e7 0.00488056
\(595\) −6.38496e9 −1.24265
\(596\) 2.87541e9 0.556337
\(597\) −3.23713e9 −0.622658
\(598\) 4.34646e8 0.0831154
\(599\) −5.20381e9 −0.989299 −0.494649 0.869093i \(-0.664703\pi\)
−0.494649 + 0.869093i \(0.664703\pi\)
\(600\) −3.90710e8 −0.0738458
\(601\) 3.88435e9 0.729891 0.364945 0.931029i \(-0.381088\pi\)
0.364945 + 0.931029i \(0.381088\pi\)
\(602\) −5.79440e8 −0.108248
\(603\) −6.03753e8 −0.112137
\(604\) 2.82404e8 0.0521484
\(605\) 5.92453e9 1.08770
\(606\) 8.78917e8 0.160433
\(607\) 9.88692e9 1.79432 0.897161 0.441703i \(-0.145626\pi\)
0.897161 + 0.441703i \(0.145626\pi\)
\(608\) −2.07269e9 −0.374000
\(609\) 2.60925e9 0.468119
\(610\) −1.10041e9 −0.196292
\(611\) 4.71895e9 0.836953
\(612\) 1.21810e9 0.214810
\(613\) 3.19394e9 0.560035 0.280018 0.959995i \(-0.409660\pi\)
0.280018 + 0.959995i \(0.409660\pi\)
\(614\) 3.71931e9 0.648445
\(615\) −3.45815e9 −0.599489
\(616\) 4.28595e8 0.0738779
\(617\) −1.12180e10 −1.92272 −0.961360 0.275295i \(-0.911225\pi\)
−0.961360 + 0.275295i \(0.911225\pi\)
\(618\) 1.95733e9 0.333583
\(619\) 7.74489e9 1.31250 0.656248 0.754545i \(-0.272142\pi\)
0.656248 + 0.754545i \(0.272142\pi\)
\(620\) 5.26710e9 0.887566
\(621\) −3.40535e8 −0.0570613
\(622\) 1.38409e9 0.230621
\(623\) 1.22482e10 2.02938
\(624\) −1.84231e9 −0.303541
\(625\) −7.06454e9 −1.15745
\(626\) 1.69321e9 0.275868
\(627\) −1.10482e8 −0.0179001
\(628\) 1.70690e9 0.275011
\(629\) −2.94054e9 −0.471140
\(630\) 1.23295e9 0.196450
\(631\) 6.77645e9 1.07374 0.536871 0.843665i \(-0.319606\pi\)
0.536871 + 0.843665i \(0.319606\pi\)
\(632\) 5.76096e8 0.0907789
\(633\) −5.06427e9 −0.793604
\(634\) 3.57032e9 0.556410
\(635\) −9.52548e9 −1.47631
\(636\) −7.54738e8 −0.116331
\(637\) −7.38162e9 −1.13152
\(638\) −8.70036e7 −0.0132637
\(639\) −2.20387e8 −0.0334144
\(640\) 7.25405e9 1.09383
\(641\) 8.18828e7 0.0122797 0.00613987 0.999981i \(-0.498046\pi\)
0.00613987 + 0.999981i \(0.498046\pi\)
\(642\) 7.02286e8 0.104747
\(643\) 2.87560e9 0.426570 0.213285 0.976990i \(-0.431584\pi\)
0.213285 + 0.976990i \(0.431584\pi\)
\(644\) −2.73896e9 −0.404096
\(645\) −8.64170e8 −0.126806
\(646\) 7.42136e8 0.108310
\(647\) −4.48392e9 −0.650869 −0.325434 0.945565i \(-0.605510\pi\)
−0.325434 + 0.945565i \(0.605510\pi\)
\(648\) −5.02773e8 −0.0725871
\(649\) 6.61362e7 0.00949691
\(650\) −3.84272e8 −0.0548835
\(651\) −5.81692e9 −0.826341
\(652\) −1.24851e9 −0.176411
\(653\) −5.52310e9 −0.776224 −0.388112 0.921612i \(-0.626873\pi\)
−0.388112 + 0.921612i \(0.626873\pi\)
\(654\) 1.38322e9 0.193362
\(655\) 9.16509e9 1.27436
\(656\) 4.47654e9 0.619125
\(657\) −2.45669e9 −0.337965
\(658\) 4.08812e9 0.559414
\(659\) 5.93057e9 0.807230 0.403615 0.914929i \(-0.367753\pi\)
0.403615 + 0.914929i \(0.367753\pi\)
\(660\) 2.99045e8 0.0404886
\(661\) −1.03877e10 −1.39898 −0.699492 0.714641i \(-0.746590\pi\)
−0.699492 + 0.714641i \(0.746590\pi\)
\(662\) −2.47852e9 −0.332040
\(663\) 2.56076e9 0.341249
\(664\) 2.95010e9 0.391065
\(665\) −5.46405e9 −0.720508
\(666\) 5.67825e8 0.0744825
\(667\) 1.18844e9 0.155073
\(668\) −1.00995e10 −1.31094
\(669\) 1.51242e9 0.195290
\(670\) 9.95639e8 0.127891
\(671\) 2.94760e8 0.0376652
\(672\) −6.19582e9 −0.787601
\(673\) −1.00404e9 −0.126969 −0.0634843 0.997983i \(-0.520221\pi\)
−0.0634843 + 0.997983i \(0.520221\pi\)
\(674\) 2.35973e9 0.296861
\(675\) 3.01069e8 0.0376793
\(676\) 2.47015e9 0.307546
\(677\) 4.85982e9 0.601950 0.300975 0.953632i \(-0.402688\pi\)
0.300975 + 0.953632i \(0.402688\pi\)
\(678\) −7.73405e8 −0.0953022
\(679\) 2.54296e9 0.311742
\(680\) −4.29366e9 −0.523657
\(681\) 5.30291e9 0.643427
\(682\) 1.93961e8 0.0234136
\(683\) 1.18996e10 1.42909 0.714543 0.699592i \(-0.246635\pi\)
0.714543 + 0.699592i \(0.246635\pi\)
\(684\) 1.04242e9 0.124550
\(685\) −5.07171e9 −0.602889
\(686\) −1.83783e9 −0.217356
\(687\) 1.18351e9 0.139259
\(688\) 1.11866e9 0.130960
\(689\) −1.58665e9 −0.184805
\(690\) 5.61571e8 0.0650778
\(691\) 3.67203e9 0.423383 0.211691 0.977337i \(-0.432103\pi\)
0.211691 + 0.977337i \(0.432103\pi\)
\(692\) −1.08911e10 −1.24939
\(693\) −3.30261e8 −0.0376956
\(694\) 3.82864e9 0.434797
\(695\) 1.57082e10 1.77492
\(696\) 1.75463e9 0.197267
\(697\) −6.22225e9 −0.696038
\(698\) 1.68551e8 0.0187602
\(699\) 4.25262e8 0.0470963
\(700\) 2.42152e9 0.266837
\(701\) 1.31260e10 1.43919 0.719595 0.694394i \(-0.244328\pi\)
0.719595 + 0.694394i \(0.244328\pi\)
\(702\) −4.94488e8 −0.0539481
\(703\) −2.51643e9 −0.273175
\(704\) −2.33733e8 −0.0252474
\(705\) 6.09698e9 0.655319
\(706\) 5.24618e9 0.561082
\(707\) −1.16435e10 −1.23912
\(708\) −6.24004e8 −0.0660801
\(709\) 1.38490e10 1.45934 0.729668 0.683801i \(-0.239675\pi\)
0.729668 + 0.683801i \(0.239675\pi\)
\(710\) 3.63437e8 0.0381087
\(711\) −4.43920e8 −0.0463192
\(712\) 8.23647e9 0.855187
\(713\) −2.64944e9 −0.273741
\(714\) 2.21844e9 0.228089
\(715\) 6.28668e8 0.0643206
\(716\) 4.44767e9 0.452832
\(717\) −2.75876e9 −0.279510
\(718\) 2.17461e9 0.219253
\(719\) 5.91612e9 0.593589 0.296795 0.954941i \(-0.404082\pi\)
0.296795 + 0.954941i \(0.404082\pi\)
\(720\) −2.38031e9 −0.237667
\(721\) −2.59298e10 −2.57647
\(722\) −2.88070e9 −0.284851
\(723\) 4.25255e8 0.0418471
\(724\) 8.58143e8 0.0840378
\(725\) −1.05070e9 −0.102399
\(726\) −2.05847e9 −0.199648
\(727\) −5.62356e8 −0.0542801 −0.0271401 0.999632i \(-0.508640\pi\)
−0.0271401 + 0.999632i \(0.508640\pi\)
\(728\) −8.50120e9 −0.816621
\(729\) 3.87420e8 0.0370370
\(730\) 4.05129e9 0.385446
\(731\) −1.55490e9 −0.147229
\(732\) −2.78111e9 −0.262077
\(733\) −4.63518e9 −0.434713 −0.217356 0.976092i \(-0.569743\pi\)
−0.217356 + 0.976092i \(0.569743\pi\)
\(734\) −3.32564e9 −0.310412
\(735\) −9.53721e9 −0.885963
\(736\) −2.82201e9 −0.260908
\(737\) −2.66695e8 −0.0245402
\(738\) 1.20153e9 0.110037
\(739\) 8.68070e9 0.791224 0.395612 0.918418i \(-0.370533\pi\)
0.395612 + 0.918418i \(0.370533\pi\)
\(740\) 6.81127e9 0.617898
\(741\) 2.19142e9 0.197862
\(742\) −1.37455e9 −0.123523
\(743\) −7.23578e9 −0.647179 −0.323589 0.946198i \(-0.604890\pi\)
−0.323589 + 0.946198i \(0.604890\pi\)
\(744\) −3.91168e9 −0.348223
\(745\) 7.81006e9 0.692002
\(746\) −7.11182e9 −0.627184
\(747\) −2.27325e9 −0.199538
\(748\) 5.38071e8 0.0470093
\(749\) −9.30355e9 −0.809026
\(750\) 2.03937e9 0.176515
\(751\) −1.53619e10 −1.32344 −0.661722 0.749750i \(-0.730174\pi\)
−0.661722 + 0.749750i \(0.730174\pi\)
\(752\) −7.89246e9 −0.676784
\(753\) 4.12519e9 0.352096
\(754\) 1.72572e9 0.146612
\(755\) 7.67053e8 0.0648651
\(756\) 3.11606e9 0.262289
\(757\) −1.36460e10 −1.14332 −0.571661 0.820490i \(-0.693701\pi\)
−0.571661 + 0.820490i \(0.693701\pi\)
\(758\) −4.92434e9 −0.410682
\(759\) −1.50424e8 −0.0124874
\(760\) −3.67439e9 −0.303625
\(761\) 1.05663e10 0.869112 0.434556 0.900645i \(-0.356905\pi\)
0.434556 + 0.900645i \(0.356905\pi\)
\(762\) 3.30961e9 0.270978
\(763\) −1.83243e10 −1.49345
\(764\) 7.89686e9 0.640660
\(765\) 3.30856e9 0.267192
\(766\) 8.36968e6 0.000672834 0
\(767\) −1.31181e9 −0.104976
\(768\) −1.19201e7 −0.000949548 0
\(769\) −1.72180e10 −1.36534 −0.682668 0.730729i \(-0.739180\pi\)
−0.682668 + 0.730729i \(0.739180\pi\)
\(770\) 5.44628e8 0.0429915
\(771\) 6.88575e9 0.541079
\(772\) 1.66418e10 1.30179
\(773\) 1.20767e10 0.940419 0.470209 0.882555i \(-0.344178\pi\)
0.470209 + 0.882555i \(0.344178\pi\)
\(774\) 3.00254e8 0.0232753
\(775\) 2.34238e9 0.180759
\(776\) 1.71006e9 0.131369
\(777\) −7.52227e9 −0.575275
\(778\) 1.16955e9 0.0890410
\(779\) −5.32481e9 −0.403574
\(780\) −5.93157e9 −0.447547
\(781\) −9.73513e7 −0.00731245
\(782\) 1.01043e9 0.0755587
\(783\) −1.35206e9 −0.100654
\(784\) 1.23458e10 0.914983
\(785\) 4.63621e9 0.342074
\(786\) −3.18439e9 −0.233909
\(787\) 5.72468e9 0.418639 0.209319 0.977847i \(-0.432875\pi\)
0.209319 + 0.977847i \(0.432875\pi\)
\(788\) 2.72616e9 0.198476
\(789\) 8.68392e9 0.629428
\(790\) 7.32062e8 0.0528266
\(791\) 1.02457e10 0.736078
\(792\) −2.22089e8 −0.0158851
\(793\) −5.84659e9 −0.416338
\(794\) 2.99303e9 0.212197
\(795\) −2.04998e9 −0.144699
\(796\) 1.34916e10 0.948129
\(797\) −1.13444e10 −0.793740 −0.396870 0.917875i \(-0.629904\pi\)
−0.396870 + 0.917875i \(0.629904\pi\)
\(798\) 1.89847e9 0.132250
\(799\) 1.09703e10 0.760859
\(800\) 2.49495e9 0.172285
\(801\) −6.34675e9 −0.436353
\(802\) 1.63850e9 0.112159
\(803\) −1.08519e9 −0.0739608
\(804\) 2.51630e9 0.170752
\(805\) −7.43943e9 −0.502637
\(806\) −3.84722e9 −0.258806
\(807\) 6.28376e9 0.420884
\(808\) −7.82984e9 −0.522172
\(809\) 2.36003e10 1.56711 0.783553 0.621325i \(-0.213405\pi\)
0.783553 + 0.621325i \(0.213405\pi\)
\(810\) −6.38889e8 −0.0422404
\(811\) −5.41251e9 −0.356308 −0.178154 0.984003i \(-0.557013\pi\)
−0.178154 + 0.984003i \(0.557013\pi\)
\(812\) −1.08748e10 −0.712810
\(813\) 3.17610e9 0.207290
\(814\) 2.50824e8 0.0162999
\(815\) −3.39115e9 −0.219430
\(816\) −4.28289e9 −0.275944
\(817\) −1.33064e9 −0.0853655
\(818\) −9.47841e9 −0.605479
\(819\) 6.55074e9 0.416674
\(820\) 1.44128e10 0.912850
\(821\) 1.82304e9 0.114973 0.0574863 0.998346i \(-0.481691\pi\)
0.0574863 + 0.998346i \(0.481691\pi\)
\(822\) 1.76216e9 0.110661
\(823\) −1.51277e10 −0.945960 −0.472980 0.881073i \(-0.656822\pi\)
−0.472980 + 0.881073i \(0.656822\pi\)
\(824\) −1.74369e10 −1.08574
\(825\) 1.32991e8 0.00824579
\(826\) −1.13645e9 −0.0701651
\(827\) −2.65641e10 −1.63315 −0.816573 0.577242i \(-0.804129\pi\)
−0.816573 + 0.577242i \(0.804129\pi\)
\(828\) 1.41927e9 0.0868880
\(829\) −2.30465e10 −1.40496 −0.702479 0.711704i \(-0.747924\pi\)
−0.702479 + 0.711704i \(0.747924\pi\)
\(830\) 3.74878e9 0.227571
\(831\) 6.72938e9 0.406791
\(832\) 4.63611e9 0.279076
\(833\) −1.71603e10 −1.02865
\(834\) −5.45778e9 −0.325788
\(835\) −2.74317e10 −1.63061
\(836\) 4.60465e8 0.0272568
\(837\) 3.01421e9 0.177678
\(838\) −1.10129e9 −0.0646469
\(839\) −2.55035e10 −1.49084 −0.745422 0.666592i \(-0.767752\pi\)
−0.745422 + 0.666592i \(0.767752\pi\)
\(840\) −1.09837e10 −0.639399
\(841\) −1.25313e10 −0.726457
\(842\) 5.06330e9 0.292308
\(843\) 1.09647e10 0.630375
\(844\) 2.11067e10 1.20843
\(845\) 6.70932e9 0.382543
\(846\) −2.11838e9 −0.120284
\(847\) 2.72696e10 1.54201
\(848\) 2.65368e9 0.149439
\(849\) −1.91793e10 −1.07561
\(850\) −8.93329e8 −0.0498937
\(851\) −3.42617e9 −0.190571
\(852\) 9.18523e8 0.0508805
\(853\) −2.38277e10 −1.31450 −0.657250 0.753673i \(-0.728280\pi\)
−0.657250 + 0.753673i \(0.728280\pi\)
\(854\) −5.06502e9 −0.278278
\(855\) 2.83136e9 0.154922
\(856\) −6.25632e9 −0.340927
\(857\) 3.88776e9 0.210992 0.105496 0.994420i \(-0.466357\pi\)
0.105496 + 0.994420i \(0.466357\pi\)
\(858\) −2.18429e8 −0.0118061
\(859\) 1.22415e10 0.658961 0.329480 0.944162i \(-0.393126\pi\)
0.329480 + 0.944162i \(0.393126\pi\)
\(860\) 3.60166e9 0.193089
\(861\) −1.59173e10 −0.849881
\(862\) −6.01046e9 −0.319619
\(863\) 1.45480e10 0.770489 0.385244 0.922815i \(-0.374117\pi\)
0.385244 + 0.922815i \(0.374117\pi\)
\(864\) 3.21055e9 0.169349
\(865\) −2.95818e10 −1.55406
\(866\) 1.19969e10 0.627704
\(867\) −5.12606e9 −0.267126
\(868\) 2.42436e10 1.25828
\(869\) −1.96092e8 −0.0101366
\(870\) 2.22967e9 0.114795
\(871\) 5.28991e9 0.271259
\(872\) −1.23225e10 −0.629347
\(873\) −1.31771e9 −0.0670303
\(874\) 8.64699e8 0.0438102
\(875\) −2.70165e10 −1.36333
\(876\) 1.02389e10 0.514624
\(877\) −1.79027e10 −0.896231 −0.448115 0.893976i \(-0.647905\pi\)
−0.448115 + 0.893976i \(0.647905\pi\)
\(878\) 4.57286e9 0.228012
\(879\) 2.68043e9 0.133120
\(880\) −1.05145e9 −0.0520115
\(881\) −8.23369e9 −0.405675 −0.202838 0.979212i \(-0.565016\pi\)
−0.202838 + 0.979212i \(0.565016\pi\)
\(882\) 3.31369e9 0.162619
\(883\) −1.02587e10 −0.501452 −0.250726 0.968058i \(-0.580669\pi\)
−0.250726 + 0.968058i \(0.580669\pi\)
\(884\) −1.06727e10 −0.519625
\(885\) −1.69489e9 −0.0821940
\(886\) −5.50511e9 −0.265918
\(887\) 8.18874e9 0.393989 0.196995 0.980405i \(-0.436882\pi\)
0.196995 + 0.980405i \(0.436882\pi\)
\(888\) −5.05847e9 −0.242423
\(889\) −4.38442e10 −2.09293
\(890\) 1.04663e10 0.497656
\(891\) 1.71135e8 0.00810524
\(892\) −6.30341e9 −0.297371
\(893\) 9.38804e9 0.441159
\(894\) −2.71359e9 −0.127017
\(895\) 1.20806e10 0.563257
\(896\) 3.33891e10 1.55070
\(897\) 2.98367e9 0.138031
\(898\) −3.76261e9 −0.173389
\(899\) −1.05193e10 −0.482869
\(900\) −1.25478e9 −0.0573747
\(901\) −3.68854e9 −0.168003
\(902\) 5.30749e8 0.0240806
\(903\) −3.97763e9 −0.179770
\(904\) 6.88988e9 0.310186
\(905\) 2.33085e9 0.104531
\(906\) −2.66511e8 −0.0119060
\(907\) −1.90356e10 −0.847110 −0.423555 0.905870i \(-0.639218\pi\)
−0.423555 + 0.905870i \(0.639218\pi\)
\(908\) −2.21013e10 −0.979755
\(909\) 6.03342e9 0.266434
\(910\) −1.08027e10 −0.475213
\(911\) 1.11471e10 0.488483 0.244241 0.969714i \(-0.421461\pi\)
0.244241 + 0.969714i \(0.421461\pi\)
\(912\) −3.66516e9 −0.159997
\(913\) −1.00416e9 −0.0436672
\(914\) 9.63386e9 0.417339
\(915\) −7.55391e9 −0.325985
\(916\) −4.93259e9 −0.212051
\(917\) 4.21853e10 1.80663
\(918\) −1.14955e9 −0.0490432
\(919\) 9.09748e9 0.386649 0.193324 0.981135i \(-0.438073\pi\)
0.193324 + 0.981135i \(0.438073\pi\)
\(920\) −5.00276e9 −0.211813
\(921\) 2.55316e10 1.07689
\(922\) 1.23682e10 0.519697
\(923\) 1.93097e9 0.0808293
\(924\) 1.37645e9 0.0573996
\(925\) 3.02909e9 0.125839
\(926\) 1.24606e10 0.515705
\(927\) 1.34363e10 0.553988
\(928\) −1.12045e10 −0.460231
\(929\) −2.50717e10 −1.02596 −0.512978 0.858402i \(-0.671458\pi\)
−0.512978 + 0.858402i \(0.671458\pi\)
\(930\) −4.97069e9 −0.202640
\(931\) −1.46853e10 −0.596428
\(932\) −1.77240e9 −0.0717142
\(933\) 9.50126e9 0.382997
\(934\) −7.62967e9 −0.306402
\(935\) 1.46148e9 0.0584727
\(936\) 4.40515e9 0.175588
\(937\) 4.32100e9 0.171591 0.0857957 0.996313i \(-0.472657\pi\)
0.0857957 + 0.996313i \(0.472657\pi\)
\(938\) 4.58276e9 0.181308
\(939\) 1.16232e10 0.458140
\(940\) −2.54108e10 −0.997863
\(941\) 1.72767e10 0.675922 0.337961 0.941160i \(-0.390263\pi\)
0.337961 + 0.941160i \(0.390263\pi\)
\(942\) −1.61084e9 −0.0627878
\(943\) −7.24986e9 −0.281539
\(944\) 2.19402e9 0.0848863
\(945\) 8.46370e9 0.326249
\(946\) 1.32631e8 0.00509361
\(947\) −2.35290e10 −0.900284 −0.450142 0.892957i \(-0.648627\pi\)
−0.450142 + 0.892957i \(0.648627\pi\)
\(948\) 1.85016e9 0.0705309
\(949\) 2.15248e10 0.817537
\(950\) −7.64484e8 −0.0289292
\(951\) 2.45089e10 0.924041
\(952\) −1.97630e10 −0.742376
\(953\) 9.37890e9 0.351016 0.175508 0.984478i \(-0.443843\pi\)
0.175508 + 0.984478i \(0.443843\pi\)
\(954\) 7.12263e8 0.0265596
\(955\) 2.14491e10 0.796888
\(956\) 1.14979e10 0.425614
\(957\) −5.97245e8 −0.0220273
\(958\) −6.20892e9 −0.228159
\(959\) −2.33442e10 −0.854701
\(960\) 5.98996e9 0.218511
\(961\) −4.06144e9 −0.147621
\(962\) −4.97511e9 −0.180173
\(963\) 4.82091e9 0.173955
\(964\) −1.77237e9 −0.0637212
\(965\) 4.52018e10 1.61924
\(966\) 2.58482e9 0.0922592
\(967\) 4.13184e10 1.46944 0.734718 0.678373i \(-0.237315\pi\)
0.734718 + 0.678373i \(0.237315\pi\)
\(968\) 1.83379e10 0.649808
\(969\) 5.09447e9 0.179873
\(970\) 2.17302e9 0.0764473
\(971\) 6.75904e9 0.236929 0.118464 0.992958i \(-0.462203\pi\)
0.118464 + 0.992958i \(0.462203\pi\)
\(972\) −1.61468e9 −0.0563968
\(973\) 7.23021e10 2.51626
\(974\) 6.65194e9 0.230671
\(975\) −2.63788e9 −0.0911461
\(976\) 9.77845e9 0.336663
\(977\) 2.83700e10 0.973258 0.486629 0.873609i \(-0.338226\pi\)
0.486629 + 0.873609i \(0.338226\pi\)
\(978\) 1.17825e9 0.0402765
\(979\) −2.80354e9 −0.0954922
\(980\) 3.97489e10 1.34907
\(981\) 9.49529e9 0.321120
\(982\) 1.19382e10 0.402300
\(983\) 5.77572e10 1.93940 0.969702 0.244289i \(-0.0785546\pi\)
0.969702 + 0.244289i \(0.0785546\pi\)
\(984\) −1.07038e10 −0.358143
\(985\) 7.40466e9 0.246876
\(986\) 4.01184e9 0.133283
\(987\) 2.80634e10 0.929030
\(988\) −9.13334e9 −0.301287
\(989\) −1.81169e9 −0.0595522
\(990\) −2.82215e8 −0.00924395
\(991\) −2.44660e10 −0.798554 −0.399277 0.916830i \(-0.630739\pi\)
−0.399277 + 0.916830i \(0.630739\pi\)
\(992\) 2.49787e10 0.812418
\(993\) −1.70141e10 −0.551425
\(994\) 1.67284e9 0.0540258
\(995\) 3.66453e10 1.17933
\(996\) 9.47439e9 0.303839
\(997\) −3.98637e10 −1.27393 −0.636964 0.770894i \(-0.719810\pi\)
−0.636964 + 0.770894i \(0.719810\pi\)
\(998\) −8.64583e9 −0.275327
\(999\) 3.89789e9 0.123695
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.1.10 18
3.2 odd 2 531.8.a.e.1.9 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.10 18 1.1 even 1 trivial
531.8.a.e.1.9 18 3.2 odd 2