Properties

Label 177.8.a.b.1.6
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $17$
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: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-10.1391\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-12.1391 q^{2} +27.0000 q^{3} +19.3578 q^{4} +236.334 q^{5} -327.756 q^{6} +1426.66 q^{7} +1318.82 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-12.1391 q^{2} +27.0000 q^{3} +19.3578 q^{4} +236.334 q^{5} -327.756 q^{6} +1426.66 q^{7} +1318.82 q^{8} +729.000 q^{9} -2868.89 q^{10} -5472.58 q^{11} +522.660 q^{12} -8451.67 q^{13} -17318.4 q^{14} +6381.03 q^{15} -18487.1 q^{16} -6087.08 q^{17} -8849.40 q^{18} -12974.5 q^{19} +4574.91 q^{20} +38519.8 q^{21} +66432.2 q^{22} -53979.7 q^{23} +35608.1 q^{24} -22271.1 q^{25} +102596. q^{26} +19683.0 q^{27} +27617.0 q^{28} -14368.8 q^{29} -77459.9 q^{30} -48485.8 q^{31} +55607.6 q^{32} -147760. q^{33} +73891.7 q^{34} +337169. q^{35} +14111.8 q^{36} +82881.8 q^{37} +157499. q^{38} -228195. q^{39} +311682. q^{40} -782250. q^{41} -467596. q^{42} -369462. q^{43} -105937. q^{44} +172288. q^{45} +655265. q^{46} +368463. q^{47} -499151. q^{48} +1.21182e6 q^{49} +270351. q^{50} -164351. q^{51} -163605. q^{52} +836648. q^{53} -238934. q^{54} -1.29336e6 q^{55} +1.88151e6 q^{56} -350312. q^{57} +174424. q^{58} -205379. q^{59} +123522. q^{60} +37349.1 q^{61} +588573. q^{62} +1.04004e6 q^{63} +1.69132e6 q^{64} -1.99742e6 q^{65} +1.79367e6 q^{66} +2.64158e6 q^{67} -117832. q^{68} -1.45745e6 q^{69} -4.09293e6 q^{70} -2.03139e6 q^{71} +961419. q^{72} -3.64005e6 q^{73} -1.00611e6 q^{74} -601319. q^{75} -251158. q^{76} -7.80751e6 q^{77} +2.77008e6 q^{78} -7.63186e6 q^{79} -4.36913e6 q^{80} +531441. q^{81} +9.49581e6 q^{82} -6.69291e6 q^{83} +745658. q^{84} -1.43859e6 q^{85} +4.48494e6 q^{86} -387957. q^{87} -7.21734e6 q^{88} +717966. q^{89} -2.09142e6 q^{90} -1.20577e7 q^{91} -1.04493e6 q^{92} -1.30912e6 q^{93} -4.47281e6 q^{94} -3.06632e6 q^{95} +1.50141e6 q^{96} +1.06996e7 q^{97} -1.47104e7 q^{98} -3.98951e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −12.1391 −1.07296 −0.536478 0.843915i \(-0.680245\pi\)
−0.536478 + 0.843915i \(0.680245\pi\)
\(3\) 27.0000 0.577350
\(4\) 19.3578 0.151233
\(5\) 236.334 0.845536 0.422768 0.906238i \(-0.361059\pi\)
0.422768 + 0.906238i \(0.361059\pi\)
\(6\) −327.756 −0.619471
\(7\) 1426.66 1.57209 0.786045 0.618169i \(-0.212125\pi\)
0.786045 + 0.618169i \(0.212125\pi\)
\(8\) 1318.82 0.910689
\(9\) 729.000 0.333333
\(10\) −2868.89 −0.907222
\(11\) −5472.58 −1.23970 −0.619851 0.784719i \(-0.712807\pi\)
−0.619851 + 0.784719i \(0.712807\pi\)
\(12\) 522.660 0.0873142
\(13\) −8451.67 −1.06694 −0.533471 0.845819i \(-0.679113\pi\)
−0.533471 + 0.845819i \(0.679113\pi\)
\(14\) −17318.4 −1.68678
\(15\) 6381.03 0.488170
\(16\) −18487.1 −1.12836
\(17\) −6087.08 −0.300495 −0.150248 0.988648i \(-0.548007\pi\)
−0.150248 + 0.988648i \(0.548007\pi\)
\(18\) −8849.40 −0.357652
\(19\) −12974.5 −0.433964 −0.216982 0.976176i \(-0.569621\pi\)
−0.216982 + 0.976176i \(0.569621\pi\)
\(20\) 4574.91 0.127873
\(21\) 38519.8 0.907647
\(22\) 66432.2 1.33015
\(23\) −53979.7 −0.925087 −0.462544 0.886597i \(-0.653063\pi\)
−0.462544 + 0.886597i \(0.653063\pi\)
\(24\) 35608.1 0.525787
\(25\) −22271.1 −0.285070
\(26\) 102596. 1.14478
\(27\) 19683.0 0.192450
\(28\) 27617.0 0.237751
\(29\) −14368.8 −0.109402 −0.0547011 0.998503i \(-0.517421\pi\)
−0.0547011 + 0.998503i \(0.517421\pi\)
\(30\) −77459.9 −0.523785
\(31\) −48485.8 −0.292313 −0.146157 0.989261i \(-0.546690\pi\)
−0.146157 + 0.989261i \(0.546690\pi\)
\(32\) 55607.6 0.299992
\(33\) −147760. −0.715743
\(34\) 73891.7 0.322418
\(35\) 337169. 1.32926
\(36\) 14111.8 0.0504109
\(37\) 82881.8 0.269001 0.134500 0.990914i \(-0.457057\pi\)
0.134500 + 0.990914i \(0.457057\pi\)
\(38\) 157499. 0.465624
\(39\) −228195. −0.615999
\(40\) 311682. 0.770020
\(41\) −782250. −1.77256 −0.886282 0.463146i \(-0.846720\pi\)
−0.886282 + 0.463146i \(0.846720\pi\)
\(42\) −467596. −0.973864
\(43\) −369462. −0.708647 −0.354324 0.935123i \(-0.615289\pi\)
−0.354324 + 0.935123i \(0.615289\pi\)
\(44\) −105937. −0.187483
\(45\) 172288. 0.281845
\(46\) 655265. 0.992577
\(47\) 368463. 0.517668 0.258834 0.965922i \(-0.416662\pi\)
0.258834 + 0.965922i \(0.416662\pi\)
\(48\) −499151. −0.651460
\(49\) 1.21182e6 1.47147
\(50\) 270351. 0.305867
\(51\) −164351. −0.173491
\(52\) −163605. −0.161356
\(53\) 836648. 0.771929 0.385964 0.922514i \(-0.373869\pi\)
0.385964 + 0.922514i \(0.373869\pi\)
\(54\) −238934. −0.206490
\(55\) −1.29336e6 −1.04821
\(56\) 1.88151e6 1.43169
\(57\) −350312. −0.250549
\(58\) 174424. 0.117384
\(59\) −205379. −0.130189
\(60\) 123522. 0.0738272
\(61\) 37349.1 0.0210681 0.0105341 0.999945i \(-0.496647\pi\)
0.0105341 + 0.999945i \(0.496647\pi\)
\(62\) 588573. 0.313639
\(63\) 1.04004e6 0.524030
\(64\) 1.69132e6 0.806484
\(65\) −1.99742e6 −0.902137
\(66\) 1.79367e6 0.767960
\(67\) 2.64158e6 1.07300 0.536502 0.843899i \(-0.319745\pi\)
0.536502 + 0.843899i \(0.319745\pi\)
\(68\) −117832. −0.0454447
\(69\) −1.45745e6 −0.534099
\(70\) −4.09293e6 −1.42623
\(71\) −2.03139e6 −0.673579 −0.336790 0.941580i \(-0.609341\pi\)
−0.336790 + 0.941580i \(0.609341\pi\)
\(72\) 961419. 0.303563
\(73\) −3.64005e6 −1.09516 −0.547580 0.836753i \(-0.684451\pi\)
−0.547580 + 0.836753i \(0.684451\pi\)
\(74\) −1.00611e6 −0.288625
\(75\) −601319. −0.164585
\(76\) −251158. −0.0656295
\(77\) −7.80751e6 −1.94892
\(78\) 2.77008e6 0.660939
\(79\) −7.63186e6 −1.74155 −0.870774 0.491683i \(-0.836382\pi\)
−0.870774 + 0.491683i \(0.836382\pi\)
\(80\) −4.36913e6 −0.954070
\(81\) 531441. 0.111111
\(82\) 9.49581e6 1.90188
\(83\) −6.69291e6 −1.28482 −0.642409 0.766362i \(-0.722065\pi\)
−0.642409 + 0.766362i \(0.722065\pi\)
\(84\) 745658. 0.137266
\(85\) −1.43859e6 −0.254079
\(86\) 4.48494e6 0.760347
\(87\) −387957. −0.0631634
\(88\) −7.21734e6 −1.12898
\(89\) 717966. 0.107954 0.0539770 0.998542i \(-0.482810\pi\)
0.0539770 + 0.998542i \(0.482810\pi\)
\(90\) −2.09142e6 −0.302407
\(91\) −1.20577e7 −1.67733
\(92\) −1.04493e6 −0.139903
\(93\) −1.30912e6 −0.168767
\(94\) −4.47281e6 −0.555435
\(95\) −3.06632e6 −0.366932
\(96\) 1.50141e6 0.173200
\(97\) 1.06996e7 1.19033 0.595165 0.803604i \(-0.297087\pi\)
0.595165 + 0.803604i \(0.297087\pi\)
\(98\) −1.47104e7 −1.57882
\(99\) −3.98951e6 −0.413234
\(100\) −431118. −0.0431118
\(101\) −1.14739e7 −1.10812 −0.554058 0.832478i \(-0.686921\pi\)
−0.554058 + 0.832478i \(0.686921\pi\)
\(102\) 1.99507e6 0.186148
\(103\) 7.95531e6 0.717343 0.358671 0.933464i \(-0.383230\pi\)
0.358671 + 0.933464i \(0.383230\pi\)
\(104\) −1.11462e7 −0.971652
\(105\) 9.10356e6 0.767448
\(106\) −1.01562e7 −0.828245
\(107\) 1.10513e7 0.872107 0.436054 0.899921i \(-0.356376\pi\)
0.436054 + 0.899921i \(0.356376\pi\)
\(108\) 381019. 0.0291047
\(109\) 8.08293e6 0.597828 0.298914 0.954280i \(-0.403376\pi\)
0.298914 + 0.954280i \(0.403376\pi\)
\(110\) 1.57002e7 1.12469
\(111\) 2.23781e6 0.155308
\(112\) −2.63748e7 −1.77389
\(113\) −9.96488e6 −0.649677 −0.324839 0.945769i \(-0.605310\pi\)
−0.324839 + 0.945769i \(0.605310\pi\)
\(114\) 4.25247e6 0.268828
\(115\) −1.27572e7 −0.782194
\(116\) −278147. −0.0165452
\(117\) −6.16126e6 −0.355647
\(118\) 2.49312e6 0.139687
\(119\) −8.68420e6 −0.472406
\(120\) 8.41542e6 0.444571
\(121\) 1.04619e7 0.536863
\(122\) −453384. −0.0226051
\(123\) −2.11207e7 −1.02339
\(124\) −938576. −0.0442073
\(125\) −2.37270e7 −1.08657
\(126\) −1.26251e7 −0.562261
\(127\) 3.01471e7 1.30597 0.652984 0.757372i \(-0.273517\pi\)
0.652984 + 0.757372i \(0.273517\pi\)
\(128\) −2.76489e7 −1.16531
\(129\) −9.97547e6 −0.409138
\(130\) 2.42469e7 0.967952
\(131\) −4.85605e7 −1.88727 −0.943634 0.330990i \(-0.892617\pi\)
−0.943634 + 0.330990i \(0.892617\pi\)
\(132\) −2.86030e6 −0.108244
\(133\) −1.85102e7 −0.682231
\(134\) −3.20664e7 −1.15129
\(135\) 4.65177e6 0.162723
\(136\) −8.02775e6 −0.273658
\(137\) −1.46792e7 −0.487732 −0.243866 0.969809i \(-0.578416\pi\)
−0.243866 + 0.969809i \(0.578416\pi\)
\(138\) 1.76921e7 0.573065
\(139\) 3.93077e7 1.24144 0.620721 0.784032i \(-0.286840\pi\)
0.620721 + 0.784032i \(0.286840\pi\)
\(140\) 6.52684e6 0.201027
\(141\) 9.94850e6 0.298876
\(142\) 2.46592e7 0.722720
\(143\) 4.62524e7 1.32269
\(144\) −1.34771e7 −0.376120
\(145\) −3.39583e6 −0.0925035
\(146\) 4.41870e7 1.17506
\(147\) 3.27191e7 0.849553
\(148\) 1.60441e6 0.0406817
\(149\) −6.99960e7 −1.73349 −0.866745 0.498752i \(-0.833792\pi\)
−0.866745 + 0.498752i \(0.833792\pi\)
\(150\) 7.29947e6 0.176592
\(151\) 5.84052e7 1.38049 0.690243 0.723577i \(-0.257503\pi\)
0.690243 + 0.723577i \(0.257503\pi\)
\(152\) −1.71110e7 −0.395206
\(153\) −4.43748e6 −0.100165
\(154\) 9.47762e7 2.09111
\(155\) −1.14588e7 −0.247161
\(156\) −4.41735e6 −0.0931591
\(157\) 5.09364e7 1.05046 0.525230 0.850960i \(-0.323979\pi\)
0.525230 + 0.850960i \(0.323979\pi\)
\(158\) 9.26439e7 1.86860
\(159\) 2.25895e7 0.445673
\(160\) 1.31420e7 0.253654
\(161\) −7.70107e7 −1.45432
\(162\) −6.45122e6 −0.119217
\(163\) −2.83386e7 −0.512532 −0.256266 0.966606i \(-0.582492\pi\)
−0.256266 + 0.966606i \(0.582492\pi\)
\(164\) −1.51426e7 −0.268069
\(165\) −3.49207e7 −0.605186
\(166\) 8.12459e7 1.37855
\(167\) 9.62396e6 0.159899 0.0799496 0.996799i \(-0.474524\pi\)
0.0799496 + 0.996799i \(0.474524\pi\)
\(168\) 5.08007e7 0.826584
\(169\) 8.68213e6 0.138364
\(170\) 1.74631e7 0.272616
\(171\) −9.45842e6 −0.144655
\(172\) −7.15196e6 −0.107171
\(173\) 7.21139e7 1.05891 0.529454 0.848339i \(-0.322397\pi\)
0.529454 + 0.848339i \(0.322397\pi\)
\(174\) 4.70944e6 0.0677715
\(175\) −3.17733e7 −0.448155
\(176\) 1.01172e8 1.39883
\(177\) −5.54523e6 −0.0751646
\(178\) −8.71546e6 −0.115830
\(179\) −2.09311e7 −0.272777 −0.136388 0.990655i \(-0.543549\pi\)
−0.136388 + 0.990655i \(0.543549\pi\)
\(180\) 3.33511e6 0.0426242
\(181\) 4.58129e7 0.574266 0.287133 0.957891i \(-0.407298\pi\)
0.287133 + 0.957891i \(0.407298\pi\)
\(182\) 1.46369e8 1.79970
\(183\) 1.00843e6 0.0121637
\(184\) −7.11894e7 −0.842467
\(185\) 1.95878e7 0.227450
\(186\) 1.58915e7 0.181079
\(187\) 3.33120e7 0.372525
\(188\) 7.13262e6 0.0782883
\(189\) 2.80810e7 0.302549
\(190\) 3.72224e7 0.393701
\(191\) −1.58678e8 −1.64779 −0.823893 0.566746i \(-0.808202\pi\)
−0.823893 + 0.566746i \(0.808202\pi\)
\(192\) 4.56656e7 0.465624
\(193\) 6.81944e7 0.682807 0.341404 0.939917i \(-0.389098\pi\)
0.341404 + 0.939917i \(0.389098\pi\)
\(194\) −1.29884e8 −1.27717
\(195\) −5.39303e7 −0.520849
\(196\) 2.34581e7 0.222534
\(197\) 8.04764e7 0.749957 0.374979 0.927033i \(-0.377650\pi\)
0.374979 + 0.927033i \(0.377650\pi\)
\(198\) 4.84291e7 0.443382
\(199\) −6.86011e7 −0.617085 −0.308543 0.951211i \(-0.599841\pi\)
−0.308543 + 0.951211i \(0.599841\pi\)
\(200\) −2.93715e7 −0.259610
\(201\) 7.13226e7 0.619499
\(202\) 1.39283e8 1.18896
\(203\) −2.04993e7 −0.171990
\(204\) −3.18147e6 −0.0262375
\(205\) −1.84872e8 −1.49877
\(206\) −9.65703e7 −0.769677
\(207\) −3.93512e7 −0.308362
\(208\) 1.56247e8 1.20390
\(209\) 7.10040e7 0.537986
\(210\) −1.10509e8 −0.823437
\(211\) 2.88847e7 0.211679 0.105840 0.994383i \(-0.466247\pi\)
0.105840 + 0.994383i \(0.466247\pi\)
\(212\) 1.61956e7 0.116741
\(213\) −5.48475e7 −0.388891
\(214\) −1.34153e8 −0.935732
\(215\) −8.73166e7 −0.599187
\(216\) 2.59583e7 0.175262
\(217\) −6.91727e7 −0.459543
\(218\) −9.81195e7 −0.641442
\(219\) −9.82815e7 −0.632291
\(220\) −2.50365e7 −0.158524
\(221\) 5.14459e7 0.320611
\(222\) −2.71650e7 −0.166638
\(223\) 2.59394e8 1.56637 0.783184 0.621790i \(-0.213594\pi\)
0.783184 + 0.621790i \(0.213594\pi\)
\(224\) 7.93332e7 0.471614
\(225\) −1.62356e7 −0.0950232
\(226\) 1.20965e8 0.697074
\(227\) −2.03673e8 −1.15570 −0.577848 0.816144i \(-0.696107\pi\)
−0.577848 + 0.816144i \(0.696107\pi\)
\(228\) −6.78126e6 −0.0378912
\(229\) −3.48601e8 −1.91825 −0.959124 0.282986i \(-0.908675\pi\)
−0.959124 + 0.282986i \(0.908675\pi\)
\(230\) 1.54862e8 0.839259
\(231\) −2.10803e8 −1.12521
\(232\) −1.89498e7 −0.0996315
\(233\) 1.56687e8 0.811495 0.405748 0.913985i \(-0.367011\pi\)
0.405748 + 0.913985i \(0.367011\pi\)
\(234\) 7.47922e7 0.381593
\(235\) 8.70805e7 0.437707
\(236\) −3.97568e6 −0.0196888
\(237\) −2.06060e8 −1.00548
\(238\) 1.05418e8 0.506870
\(239\) 2.08708e8 0.988885 0.494443 0.869210i \(-0.335372\pi\)
0.494443 + 0.869210i \(0.335372\pi\)
\(240\) −1.17967e8 −0.550832
\(241\) −3.80494e8 −1.75101 −0.875505 0.483209i \(-0.839471\pi\)
−0.875505 + 0.483209i \(0.839471\pi\)
\(242\) −1.26998e8 −0.576029
\(243\) 1.43489e7 0.0641500
\(244\) 722995. 0.00318619
\(245\) 2.86394e8 1.24418
\(246\) 2.56387e8 1.09805
\(247\) 1.09656e8 0.463014
\(248\) −6.39439e7 −0.266206
\(249\) −1.80708e8 −0.741790
\(250\) 2.88025e8 1.16584
\(251\) −1.88517e8 −0.752474 −0.376237 0.926523i \(-0.622782\pi\)
−0.376237 + 0.926523i \(0.622782\pi\)
\(252\) 2.01328e7 0.0792505
\(253\) 2.95408e8 1.14683
\(254\) −3.65959e8 −1.40125
\(255\) −3.88418e7 −0.146693
\(256\) 1.19144e8 0.443844
\(257\) 3.79943e8 1.39621 0.698107 0.715993i \(-0.254026\pi\)
0.698107 + 0.715993i \(0.254026\pi\)
\(258\) 1.21093e8 0.438986
\(259\) 1.18244e8 0.422893
\(260\) −3.86656e7 −0.136433
\(261\) −1.04748e7 −0.0364674
\(262\) 5.89481e8 2.02495
\(263\) 1.63239e8 0.553322 0.276661 0.960968i \(-0.410772\pi\)
0.276661 + 0.960968i \(0.410772\pi\)
\(264\) −1.94868e8 −0.651819
\(265\) 1.97729e8 0.652693
\(266\) 2.24698e8 0.732003
\(267\) 1.93851e7 0.0623272
\(268\) 5.11350e7 0.162273
\(269\) −5.26713e8 −1.64984 −0.824918 0.565252i \(-0.808779\pi\)
−0.824918 + 0.565252i \(0.808779\pi\)
\(270\) −5.64683e7 −0.174595
\(271\) 1.21168e8 0.369824 0.184912 0.982755i \(-0.440800\pi\)
0.184912 + 0.982755i \(0.440800\pi\)
\(272\) 1.12532e8 0.339067
\(273\) −3.25557e8 −0.968406
\(274\) 1.78193e8 0.523315
\(275\) 1.21880e8 0.353402
\(276\) −2.82130e7 −0.0807732
\(277\) 3.39227e8 0.958985 0.479492 0.877546i \(-0.340821\pi\)
0.479492 + 0.877546i \(0.340821\pi\)
\(278\) −4.77161e8 −1.33201
\(279\) −3.53461e7 −0.0974377
\(280\) 4.44665e8 1.21054
\(281\) −2.48935e8 −0.669289 −0.334644 0.942344i \(-0.608616\pi\)
−0.334644 + 0.942344i \(0.608616\pi\)
\(282\) −1.20766e8 −0.320680
\(283\) −6.01535e8 −1.57764 −0.788821 0.614623i \(-0.789308\pi\)
−0.788821 + 0.614623i \(0.789308\pi\)
\(284\) −3.93232e7 −0.101867
\(285\) −8.27908e7 −0.211848
\(286\) −5.61462e8 −1.41919
\(287\) −1.11600e9 −2.78663
\(288\) 4.05379e7 0.0999972
\(289\) −3.73286e8 −0.909703
\(290\) 4.12224e7 0.0992521
\(291\) 2.88890e8 0.687237
\(292\) −7.04633e7 −0.165624
\(293\) 2.39333e8 0.555860 0.277930 0.960601i \(-0.410352\pi\)
0.277930 + 0.960601i \(0.410352\pi\)
\(294\) −3.97180e8 −0.911532
\(295\) −4.85381e7 −0.110079
\(296\) 1.09306e8 0.244976
\(297\) −1.07717e8 −0.238581
\(298\) 8.49688e8 1.85996
\(299\) 4.56218e8 0.987014
\(300\) −1.16402e7 −0.0248906
\(301\) −5.27097e8 −1.11406
\(302\) −7.08987e8 −1.48120
\(303\) −3.09795e8 −0.639771
\(304\) 2.39861e8 0.489668
\(305\) 8.82687e6 0.0178138
\(306\) 5.38670e7 0.107473
\(307\) 1.00262e8 0.197767 0.0988833 0.995099i \(-0.468473\pi\)
0.0988833 + 0.995099i \(0.468473\pi\)
\(308\) −1.51136e8 −0.294741
\(309\) 2.14793e8 0.414158
\(310\) 1.39100e8 0.265193
\(311\) 7.10291e8 1.33898 0.669491 0.742820i \(-0.266512\pi\)
0.669491 + 0.742820i \(0.266512\pi\)
\(312\) −3.00948e8 −0.560984
\(313\) −6.55081e8 −1.20751 −0.603753 0.797171i \(-0.706329\pi\)
−0.603753 + 0.797171i \(0.706329\pi\)
\(314\) −6.18322e8 −1.12710
\(315\) 2.45796e8 0.443086
\(316\) −1.47736e8 −0.263379
\(317\) −1.70031e8 −0.299793 −0.149896 0.988702i \(-0.547894\pi\)
−0.149896 + 0.988702i \(0.547894\pi\)
\(318\) −2.74216e8 −0.478187
\(319\) 7.86342e7 0.135626
\(320\) 3.99717e8 0.681911
\(321\) 2.98385e8 0.503511
\(322\) 9.34840e8 1.56042
\(323\) 7.89769e7 0.130404
\(324\) 1.02875e7 0.0168036
\(325\) 1.88228e8 0.304153
\(326\) 3.44005e8 0.549924
\(327\) 2.18239e8 0.345156
\(328\) −1.03165e9 −1.61425
\(329\) 5.25672e8 0.813821
\(330\) 4.23906e8 0.649337
\(331\) 1.29730e8 0.196626 0.0983131 0.995156i \(-0.468655\pi\)
0.0983131 + 0.995156i \(0.468655\pi\)
\(332\) −1.29560e8 −0.194306
\(333\) 6.04208e7 0.0896668
\(334\) −1.16826e8 −0.171565
\(335\) 6.24295e8 0.907263
\(336\) −7.12119e8 −1.02415
\(337\) 9.55271e7 0.135963 0.0679817 0.997687i \(-0.478344\pi\)
0.0679817 + 0.997687i \(0.478344\pi\)
\(338\) −1.05393e8 −0.148458
\(339\) −2.69052e8 −0.375091
\(340\) −2.78478e7 −0.0384251
\(341\) 2.65342e8 0.362381
\(342\) 1.14817e8 0.155208
\(343\) 5.53936e8 0.741191
\(344\) −4.87254e8 −0.645358
\(345\) −3.44446e8 −0.451600
\(346\) −8.75398e8 −1.13616
\(347\) −1.13393e8 −0.145690 −0.0728452 0.997343i \(-0.523208\pi\)
−0.0728452 + 0.997343i \(0.523208\pi\)
\(348\) −7.50998e6 −0.00955237
\(349\) −3.72221e8 −0.468719 −0.234359 0.972150i \(-0.575299\pi\)
−0.234359 + 0.972150i \(0.575299\pi\)
\(350\) 3.85699e8 0.480850
\(351\) −1.66354e8 −0.205333
\(352\) −3.04317e8 −0.371901
\(353\) 5.74672e8 0.695358 0.347679 0.937614i \(-0.386970\pi\)
0.347679 + 0.937614i \(0.386970\pi\)
\(354\) 6.73141e7 0.0806482
\(355\) −4.80087e8 −0.569535
\(356\) 1.38982e7 0.0163262
\(357\) −2.34473e8 −0.272744
\(358\) 2.54085e8 0.292677
\(359\) −5.65130e7 −0.0644640 −0.0322320 0.999480i \(-0.510262\pi\)
−0.0322320 + 0.999480i \(0.510262\pi\)
\(360\) 2.27216e8 0.256673
\(361\) −7.25534e8 −0.811675
\(362\) −5.56127e8 −0.616161
\(363\) 2.82472e8 0.309958
\(364\) −2.33409e8 −0.253667
\(365\) −8.60270e8 −0.925997
\(366\) −1.22414e7 −0.0130511
\(367\) −2.25564e8 −0.238198 −0.119099 0.992882i \(-0.538001\pi\)
−0.119099 + 0.992882i \(0.538001\pi\)
\(368\) 9.97926e8 1.04383
\(369\) −5.70260e8 −0.590855
\(370\) −2.37779e8 −0.244043
\(371\) 1.19361e9 1.21354
\(372\) −2.53416e7 −0.0255231
\(373\) 5.86159e8 0.584837 0.292418 0.956290i \(-0.405540\pi\)
0.292418 + 0.956290i \(0.405540\pi\)
\(374\) −4.04378e8 −0.399702
\(375\) −6.40630e8 −0.627333
\(376\) 4.85936e8 0.471435
\(377\) 1.21440e8 0.116726
\(378\) −3.40878e8 −0.324621
\(379\) 4.01728e8 0.379048 0.189524 0.981876i \(-0.439305\pi\)
0.189524 + 0.981876i \(0.439305\pi\)
\(380\) −5.93572e7 −0.0554921
\(381\) 8.13972e8 0.754001
\(382\) 1.92621e9 1.76800
\(383\) −9.90140e8 −0.900536 −0.450268 0.892894i \(-0.648671\pi\)
−0.450268 + 0.892894i \(0.648671\pi\)
\(384\) −7.46519e8 −0.672793
\(385\) −1.84518e9 −1.64789
\(386\) −8.27819e8 −0.732622
\(387\) −2.69338e8 −0.236216
\(388\) 2.07121e8 0.180017
\(389\) 9.31063e8 0.801965 0.400982 0.916086i \(-0.368669\pi\)
0.400982 + 0.916086i \(0.368669\pi\)
\(390\) 6.54666e8 0.558848
\(391\) 3.28578e8 0.277984
\(392\) 1.59817e9 1.34005
\(393\) −1.31113e9 −1.08962
\(394\) −9.76911e8 −0.804671
\(395\) −1.80367e9 −1.47254
\(396\) −7.72280e7 −0.0624945
\(397\) 4.18272e8 0.335500 0.167750 0.985830i \(-0.446350\pi\)
0.167750 + 0.985830i \(0.446350\pi\)
\(398\) 8.32755e8 0.662105
\(399\) −4.99776e8 −0.393886
\(400\) 4.11727e8 0.321662
\(401\) −1.89613e9 −1.46846 −0.734230 0.678900i \(-0.762457\pi\)
−0.734230 + 0.678900i \(0.762457\pi\)
\(402\) −8.65792e8 −0.664695
\(403\) 4.09785e8 0.311881
\(404\) −2.22109e8 −0.167583
\(405\) 1.25598e8 0.0939484
\(406\) 2.48844e8 0.184538
\(407\) −4.53577e8 −0.333481
\(408\) −2.16749e8 −0.157996
\(409\) −2.41716e9 −1.74693 −0.873463 0.486890i \(-0.838131\pi\)
−0.873463 + 0.486890i \(0.838131\pi\)
\(410\) 2.24419e9 1.60811
\(411\) −3.96340e8 −0.281592
\(412\) 1.53997e8 0.108486
\(413\) −2.93006e8 −0.204669
\(414\) 4.77688e8 0.330859
\(415\) −1.58176e9 −1.08636
\(416\) −4.69977e8 −0.320074
\(417\) 1.06131e9 0.716747
\(418\) −8.61925e8 −0.577235
\(419\) −1.73070e9 −1.14941 −0.574703 0.818362i \(-0.694882\pi\)
−0.574703 + 0.818362i \(0.694882\pi\)
\(420\) 1.76225e8 0.116063
\(421\) 2.38954e9 1.56073 0.780363 0.625327i \(-0.215034\pi\)
0.780363 + 0.625327i \(0.215034\pi\)
\(422\) −3.50634e8 −0.227123
\(423\) 2.68610e8 0.172556
\(424\) 1.10339e9 0.702987
\(425\) 1.35566e8 0.0856621
\(426\) 6.65799e8 0.417263
\(427\) 5.32845e7 0.0331210
\(428\) 2.13928e8 0.131891
\(429\) 1.24881e9 0.763655
\(430\) 1.05994e9 0.642900
\(431\) −1.06409e9 −0.640188 −0.320094 0.947386i \(-0.603714\pi\)
−0.320094 + 0.947386i \(0.603714\pi\)
\(432\) −3.63881e8 −0.217153
\(433\) −8.34289e8 −0.493866 −0.246933 0.969033i \(-0.579423\pi\)
−0.246933 + 0.969033i \(0.579423\pi\)
\(434\) 8.39695e8 0.493069
\(435\) −9.16875e7 −0.0534069
\(436\) 1.56467e8 0.0904110
\(437\) 7.00360e8 0.401454
\(438\) 1.19305e9 0.678420
\(439\) 1.91632e9 1.08104 0.540521 0.841331i \(-0.318227\pi\)
0.540521 + 0.841331i \(0.318227\pi\)
\(440\) −1.70571e9 −0.954596
\(441\) 8.83415e8 0.490490
\(442\) −6.24508e8 −0.344001
\(443\) 1.43118e9 0.782136 0.391068 0.920362i \(-0.372106\pi\)
0.391068 + 0.920362i \(0.372106\pi\)
\(444\) 4.33190e7 0.0234876
\(445\) 1.69680e8 0.0912789
\(446\) −3.14882e9 −1.68064
\(447\) −1.88989e9 −1.00083
\(448\) 2.41294e9 1.26787
\(449\) −1.00483e9 −0.523877 −0.261938 0.965085i \(-0.584362\pi\)
−0.261938 + 0.965085i \(0.584362\pi\)
\(450\) 1.97086e8 0.101956
\(451\) 4.28092e9 2.19745
\(452\) −1.92898e8 −0.0982524
\(453\) 1.57694e9 0.797024
\(454\) 2.47241e9 1.24001
\(455\) −2.84964e9 −1.41824
\(456\) −4.61998e8 −0.228172
\(457\) 3.31474e9 1.62459 0.812293 0.583250i \(-0.198219\pi\)
0.812293 + 0.583250i \(0.198219\pi\)
\(458\) 4.23171e9 2.05819
\(459\) −1.19812e8 −0.0578303
\(460\) −2.46952e8 −0.118293
\(461\) 2.98497e7 0.0141901 0.00709507 0.999975i \(-0.497742\pi\)
0.00709507 + 0.999975i \(0.497742\pi\)
\(462\) 2.55896e9 1.20730
\(463\) 5.11925e8 0.239702 0.119851 0.992792i \(-0.461758\pi\)
0.119851 + 0.992792i \(0.461758\pi\)
\(464\) 2.65636e8 0.123445
\(465\) −3.09389e8 −0.142699
\(466\) −1.90203e9 −0.870698
\(467\) −3.33639e9 −1.51589 −0.757946 0.652317i \(-0.773797\pi\)
−0.757946 + 0.652317i \(0.773797\pi\)
\(468\) −1.19268e8 −0.0537854
\(469\) 3.76863e9 1.68686
\(470\) −1.05708e9 −0.469640
\(471\) 1.37528e9 0.606483
\(472\) −2.70858e8 −0.118562
\(473\) 2.02191e9 0.878512
\(474\) 2.50139e9 1.07884
\(475\) 2.88956e8 0.123710
\(476\) −1.68107e8 −0.0714432
\(477\) 6.09916e8 0.257310
\(478\) −2.53352e9 −1.06103
\(479\) −2.42772e8 −0.100931 −0.0504655 0.998726i \(-0.516070\pi\)
−0.0504655 + 0.998726i \(0.516070\pi\)
\(480\) 3.54834e8 0.146447
\(481\) −7.00489e8 −0.287008
\(482\) 4.61886e9 1.87875
\(483\) −2.07929e9 −0.839653
\(484\) 2.02520e8 0.0811911
\(485\) 2.52869e9 1.00647
\(486\) −1.74183e8 −0.0688301
\(487\) −1.18746e9 −0.465872 −0.232936 0.972492i \(-0.574833\pi\)
−0.232936 + 0.972492i \(0.574833\pi\)
\(488\) 4.92567e7 0.0191865
\(489\) −7.65141e8 −0.295911
\(490\) −3.47657e9 −1.33495
\(491\) 1.93067e9 0.736077 0.368039 0.929811i \(-0.380029\pi\)
0.368039 + 0.929811i \(0.380029\pi\)
\(492\) −4.08850e8 −0.154770
\(493\) 8.74638e7 0.0328749
\(494\) −1.33113e9 −0.496793
\(495\) −9.42858e8 −0.349404
\(496\) 8.96360e8 0.329835
\(497\) −2.89810e9 −1.05893
\(498\) 2.19364e9 0.795907
\(499\) 3.11720e9 1.12308 0.561542 0.827448i \(-0.310208\pi\)
0.561542 + 0.827448i \(0.310208\pi\)
\(500\) −4.59303e8 −0.164325
\(501\) 2.59847e8 0.0923179
\(502\) 2.28842e9 0.807371
\(503\) −3.19404e8 −0.111906 −0.0559529 0.998433i \(-0.517820\pi\)
−0.0559529 + 0.998433i \(0.517820\pi\)
\(504\) 1.37162e9 0.477229
\(505\) −2.71167e9 −0.936952
\(506\) −3.58599e9 −1.23050
\(507\) 2.34418e8 0.0798845
\(508\) 5.83581e8 0.197505
\(509\) 2.35968e9 0.793124 0.396562 0.918008i \(-0.370203\pi\)
0.396562 + 0.918008i \(0.370203\pi\)
\(510\) 4.71505e8 0.157395
\(511\) −5.19312e9 −1.72169
\(512\) 2.09276e9 0.689088
\(513\) −2.55377e8 −0.0835164
\(514\) −4.61216e9 −1.49808
\(515\) 1.88011e9 0.606539
\(516\) −1.93103e8 −0.0618750
\(517\) −2.01644e9 −0.641754
\(518\) −1.43538e9 −0.453745
\(519\) 1.94708e9 0.611360
\(520\) −2.63423e9 −0.821566
\(521\) 6.10979e9 1.89276 0.946378 0.323062i \(-0.104712\pi\)
0.946378 + 0.323062i \(0.104712\pi\)
\(522\) 1.27155e8 0.0391279
\(523\) 3.79857e8 0.116109 0.0580543 0.998313i \(-0.481510\pi\)
0.0580543 + 0.998313i \(0.481510\pi\)
\(524\) −9.40023e8 −0.285417
\(525\) −8.57878e8 −0.258743
\(526\) −1.98157e9 −0.593690
\(527\) 2.95137e8 0.0878387
\(528\) 2.73164e9 0.807616
\(529\) −4.91022e8 −0.144214
\(530\) −2.40025e9 −0.700310
\(531\) −1.49721e8 −0.0433963
\(532\) −3.58317e8 −0.103176
\(533\) 6.61131e9 1.89122
\(534\) −2.35317e8 −0.0668743
\(535\) 2.61180e9 0.737398
\(536\) 3.48376e9 0.977173
\(537\) −5.65140e8 −0.157488
\(538\) 6.39382e9 1.77020
\(539\) −6.63177e9 −1.82418
\(540\) 9.00479e7 0.0246091
\(541\) −1.86189e8 −0.0505549 −0.0252774 0.999680i \(-0.508047\pi\)
−0.0252774 + 0.999680i \(0.508047\pi\)
\(542\) −1.47087e9 −0.396805
\(543\) 1.23695e9 0.331552
\(544\) −3.38488e8 −0.0901461
\(545\) 1.91027e9 0.505484
\(546\) 3.95197e9 1.03906
\(547\) 2.56203e8 0.0669311 0.0334656 0.999440i \(-0.489346\pi\)
0.0334656 + 0.999440i \(0.489346\pi\)
\(548\) −2.84157e8 −0.0737611
\(549\) 2.72275e7 0.00702270
\(550\) −1.47952e9 −0.379184
\(551\) 1.86428e8 0.0474766
\(552\) −1.92211e9 −0.486399
\(553\) −1.08881e10 −2.73787
\(554\) −4.11792e9 −1.02895
\(555\) 5.28871e8 0.131318
\(556\) 7.60911e8 0.187746
\(557\) 3.54346e9 0.868829 0.434415 0.900713i \(-0.356955\pi\)
0.434415 + 0.900713i \(0.356955\pi\)
\(558\) 4.29070e8 0.104546
\(559\) 3.12257e9 0.756085
\(560\) −6.23327e9 −1.49988
\(561\) 8.99424e8 0.215077
\(562\) 3.02184e9 0.718117
\(563\) −5.23102e9 −1.23540 −0.617699 0.786414i \(-0.711935\pi\)
−0.617699 + 0.786414i \(0.711935\pi\)
\(564\) 1.92581e8 0.0451998
\(565\) −2.35504e9 −0.549325
\(566\) 7.30209e9 1.69274
\(567\) 7.58186e8 0.174677
\(568\) −2.67903e9 −0.613421
\(569\) −2.02667e8 −0.0461200 −0.0230600 0.999734i \(-0.507341\pi\)
−0.0230600 + 0.999734i \(0.507341\pi\)
\(570\) 1.00501e9 0.227304
\(571\) 1.61917e9 0.363969 0.181985 0.983301i \(-0.441748\pi\)
0.181985 + 0.983301i \(0.441748\pi\)
\(572\) 8.95343e8 0.200034
\(573\) −4.28431e9 −0.951350
\(574\) 1.35473e10 2.98993
\(575\) 1.20218e9 0.263714
\(576\) 1.23297e9 0.268828
\(577\) −8.09171e8 −0.175358 −0.0876789 0.996149i \(-0.527945\pi\)
−0.0876789 + 0.996149i \(0.527945\pi\)
\(578\) 4.53136e9 0.976070
\(579\) 1.84125e9 0.394219
\(580\) −6.57358e7 −0.0139895
\(581\) −9.54851e9 −2.01985
\(582\) −3.50686e9 −0.737374
\(583\) −4.57862e9 −0.956962
\(584\) −4.80057e9 −0.997351
\(585\) −1.45612e9 −0.300712
\(586\) −2.90528e9 −0.596413
\(587\) −3.34084e9 −0.681746 −0.340873 0.940109i \(-0.610723\pi\)
−0.340873 + 0.940109i \(0.610723\pi\)
\(588\) 6.33369e8 0.128480
\(589\) 6.29079e8 0.126853
\(590\) 5.89209e8 0.118110
\(591\) 2.17286e9 0.432988
\(592\) −1.53224e9 −0.303530
\(593\) 4.75989e9 0.937358 0.468679 0.883369i \(-0.344730\pi\)
0.468679 + 0.883369i \(0.344730\pi\)
\(594\) 1.30758e9 0.255987
\(595\) −2.05237e9 −0.399436
\(596\) −1.35497e9 −0.262160
\(597\) −1.85223e9 −0.356274
\(598\) −5.53808e9 −1.05902
\(599\) 5.11303e9 0.972041 0.486020 0.873947i \(-0.338448\pi\)
0.486020 + 0.873947i \(0.338448\pi\)
\(600\) −7.93031e8 −0.149886
\(601\) −6.88520e9 −1.29377 −0.646883 0.762589i \(-0.723928\pi\)
−0.646883 + 0.762589i \(0.723928\pi\)
\(602\) 6.39848e9 1.19533
\(603\) 1.92571e9 0.357668
\(604\) 1.13059e9 0.208775
\(605\) 2.47251e9 0.453936
\(606\) 3.76063e9 0.686446
\(607\) −9.35321e9 −1.69746 −0.848732 0.528824i \(-0.822633\pi\)
−0.848732 + 0.528824i \(0.822633\pi\)
\(608\) −7.21482e8 −0.130186
\(609\) −5.53482e8 −0.0992986
\(610\) −1.07150e8 −0.0191134
\(611\) −3.11413e9 −0.552321
\(612\) −8.58997e7 −0.0151482
\(613\) 5.18941e9 0.909927 0.454963 0.890510i \(-0.349652\pi\)
0.454963 + 0.890510i \(0.349652\pi\)
\(614\) −1.21709e9 −0.212195
\(615\) −4.99156e9 −0.865313
\(616\) −1.02967e10 −1.77486
\(617\) 6.12264e9 1.04940 0.524700 0.851287i \(-0.324178\pi\)
0.524700 + 0.851287i \(0.324178\pi\)
\(618\) −2.60740e9 −0.444373
\(619\) 2.63490e9 0.446527 0.223263 0.974758i \(-0.428329\pi\)
0.223263 + 0.974758i \(0.428329\pi\)
\(620\) −2.21818e8 −0.0373788
\(621\) −1.06248e9 −0.178033
\(622\) −8.62230e9 −1.43667
\(623\) 1.02429e9 0.169713
\(624\) 4.21866e9 0.695069
\(625\) −3.86759e9 −0.633666
\(626\) 7.95209e9 1.29560
\(627\) 1.91711e9 0.310606
\(628\) 9.86016e8 0.158864
\(629\) −5.04508e8 −0.0808334
\(630\) −2.98374e9 −0.475412
\(631\) 2.67647e9 0.424092 0.212046 0.977260i \(-0.431987\pi\)
0.212046 + 0.977260i \(0.431987\pi\)
\(632\) −1.00650e10 −1.58601
\(633\) 7.79886e8 0.122213
\(634\) 2.06403e9 0.321664
\(635\) 7.12480e9 1.10424
\(636\) 4.37282e8 0.0674003
\(637\) −1.02419e10 −1.56997
\(638\) −9.54548e8 −0.145521
\(639\) −1.48088e9 −0.224526
\(640\) −6.53438e9 −0.985313
\(641\) −1.12657e10 −1.68949 −0.844747 0.535166i \(-0.820249\pi\)
−0.844747 + 0.535166i \(0.820249\pi\)
\(642\) −3.62212e9 −0.540245
\(643\) 5.64942e9 0.838041 0.419021 0.907977i \(-0.362373\pi\)
0.419021 + 0.907977i \(0.362373\pi\)
\(644\) −1.49075e9 −0.219941
\(645\) −2.35755e9 −0.345941
\(646\) −9.58709e8 −0.139918
\(647\) 8.48556e9 1.23173 0.615865 0.787852i \(-0.288807\pi\)
0.615865 + 0.787852i \(0.288807\pi\)
\(648\) 7.00874e8 0.101188
\(649\) 1.12395e9 0.161396
\(650\) −2.28491e9 −0.326342
\(651\) −1.86766e9 −0.265317
\(652\) −5.48572e8 −0.0775116
\(653\) −5.48317e9 −0.770611 −0.385306 0.922789i \(-0.625904\pi\)
−0.385306 + 0.922789i \(0.625904\pi\)
\(654\) −2.64923e9 −0.370337
\(655\) −1.14765e10 −1.59575
\(656\) 1.44615e10 2.00009
\(657\) −2.65360e9 −0.365054
\(658\) −6.38118e9 −0.873193
\(659\) −5.95072e9 −0.809974 −0.404987 0.914323i \(-0.632724\pi\)
−0.404987 + 0.914323i \(0.632724\pi\)
\(660\) −6.75986e8 −0.0915238
\(661\) −8.08136e9 −1.08838 −0.544188 0.838963i \(-0.683162\pi\)
−0.544188 + 0.838963i \(0.683162\pi\)
\(662\) −1.57480e9 −0.210971
\(663\) 1.38904e9 0.185105
\(664\) −8.82673e9 −1.17007
\(665\) −4.37460e9 −0.576850
\(666\) −7.33455e8 −0.0962085
\(667\) 7.75621e8 0.101207
\(668\) 1.86299e8 0.0241820
\(669\) 7.00365e9 0.904343
\(670\) −7.57838e9 −0.973453
\(671\) −2.04396e8 −0.0261182
\(672\) 2.14200e9 0.272287
\(673\) −9.07433e9 −1.14752 −0.573762 0.819022i \(-0.694517\pi\)
−0.573762 + 0.819022i \(0.694517\pi\)
\(674\) −1.15961e9 −0.145883
\(675\) −4.38361e8 −0.0548617
\(676\) 1.68067e8 0.0209251
\(677\) 1.13075e10 1.40057 0.700286 0.713862i \(-0.253056\pi\)
0.700286 + 0.713862i \(0.253056\pi\)
\(678\) 3.26605e9 0.402456
\(679\) 1.52647e10 1.87131
\(680\) −1.89723e9 −0.231387
\(681\) −5.49918e9 −0.667241
\(682\) −3.22101e9 −0.388819
\(683\) 1.42517e10 1.71157 0.855783 0.517336i \(-0.173076\pi\)
0.855783 + 0.517336i \(0.173076\pi\)
\(684\) −1.83094e8 −0.0218765
\(685\) −3.46921e9 −0.412395
\(686\) −6.72429e9 −0.795265
\(687\) −9.41224e9 −1.10750
\(688\) 6.83027e9 0.799610
\(689\) −7.07107e9 −0.823603
\(690\) 4.18126e9 0.484547
\(691\) 1.15882e10 1.33612 0.668058 0.744109i \(-0.267126\pi\)
0.668058 + 0.744109i \(0.267126\pi\)
\(692\) 1.39597e9 0.160141
\(693\) −5.69168e9 −0.649642
\(694\) 1.37648e9 0.156319
\(695\) 9.28977e9 1.04968
\(696\) −5.11644e8 −0.0575223
\(697\) 4.76161e9 0.532647
\(698\) 4.51843e9 0.502914
\(699\) 4.23054e9 0.468517
\(700\) −6.15059e8 −0.0677757
\(701\) 3.16850e9 0.347408 0.173704 0.984798i \(-0.444426\pi\)
0.173704 + 0.984798i \(0.444426\pi\)
\(702\) 2.01939e9 0.220313
\(703\) −1.07535e9 −0.116737
\(704\) −9.25587e9 −0.999800
\(705\) 2.35117e9 0.252710
\(706\) −6.97600e9 −0.746088
\(707\) −1.63693e10 −1.74206
\(708\) −1.07343e8 −0.0113673
\(709\) 7.17426e9 0.755988 0.377994 0.925808i \(-0.376614\pi\)
0.377994 + 0.925808i \(0.376614\pi\)
\(710\) 5.82782e9 0.611086
\(711\) −5.56363e9 −0.580516
\(712\) 9.46867e8 0.0983125
\(713\) 2.61724e9 0.270415
\(714\) 2.84629e9 0.292642
\(715\) 1.09310e10 1.11838
\(716\) −4.05180e8 −0.0412527
\(717\) 5.63511e9 0.570933
\(718\) 6.86017e8 0.0691670
\(719\) 3.09019e9 0.310052 0.155026 0.987910i \(-0.450454\pi\)
0.155026 + 0.987910i \(0.450454\pi\)
\(720\) −3.18510e9 −0.318023
\(721\) 1.13495e10 1.12773
\(722\) 8.80733e9 0.870891
\(723\) −1.02733e10 −1.01095
\(724\) 8.86836e8 0.0868477
\(725\) 3.20008e8 0.0311873
\(726\) −3.42896e9 −0.332571
\(727\) 1.26127e9 0.121741 0.0608707 0.998146i \(-0.480612\pi\)
0.0608707 + 0.998146i \(0.480612\pi\)
\(728\) −1.59019e10 −1.52753
\(729\) 3.87420e8 0.0370370
\(730\) 1.04429e10 0.993553
\(731\) 2.24894e9 0.212945
\(732\) 1.95209e7 0.00183954
\(733\) −1.05469e10 −0.989146 −0.494573 0.869136i \(-0.664676\pi\)
−0.494573 + 0.869136i \(0.664676\pi\)
\(734\) 2.73815e9 0.255576
\(735\) 7.73264e9 0.718327
\(736\) −3.00168e9 −0.277519
\(737\) −1.44562e10 −1.33021
\(738\) 6.92244e9 0.633960
\(739\) 7.89796e9 0.719879 0.359940 0.932976i \(-0.382797\pi\)
0.359940 + 0.932976i \(0.382797\pi\)
\(740\) 3.79177e8 0.0343978
\(741\) 2.96072e9 0.267321
\(742\) −1.44894e10 −1.30208
\(743\) −7.69335e8 −0.0688105 −0.0344053 0.999408i \(-0.510954\pi\)
−0.0344053 + 0.999408i \(0.510954\pi\)
\(744\) −1.72649e9 −0.153694
\(745\) −1.65425e10 −1.46573
\(746\) −7.11545e9 −0.627504
\(747\) −4.87913e9 −0.428272
\(748\) 6.44846e8 0.0563379
\(749\) 1.57664e10 1.37103
\(750\) 7.77667e9 0.673100
\(751\) 1.79359e10 1.54519 0.772596 0.634898i \(-0.218958\pi\)
0.772596 + 0.634898i \(0.218958\pi\)
\(752\) −6.81180e9 −0.584117
\(753\) −5.08995e9 −0.434441
\(754\) −1.47417e9 −0.125242
\(755\) 1.38032e10 1.16725
\(756\) 5.43585e8 0.0457553
\(757\) 4.00304e9 0.335394 0.167697 0.985839i \(-0.446367\pi\)
0.167697 + 0.985839i \(0.446367\pi\)
\(758\) −4.87661e9 −0.406702
\(759\) 7.97601e9 0.662124
\(760\) −4.04393e9 −0.334161
\(761\) 3.91126e9 0.321714 0.160857 0.986978i \(-0.448574\pi\)
0.160857 + 0.986978i \(0.448574\pi\)
\(762\) −9.88089e9 −0.809009
\(763\) 1.15316e10 0.939839
\(764\) −3.07166e9 −0.249199
\(765\) −1.04873e9 −0.0846932
\(766\) 1.20194e10 0.966234
\(767\) 1.73579e9 0.138904
\(768\) 3.21688e9 0.256254
\(769\) 1.77498e10 1.40751 0.703756 0.710441i \(-0.251505\pi\)
0.703756 + 0.710441i \(0.251505\pi\)
\(770\) 2.23989e10 1.76811
\(771\) 1.02585e10 0.806105
\(772\) 1.32009e9 0.103263
\(773\) 2.00177e10 1.55879 0.779393 0.626536i \(-0.215528\pi\)
0.779393 + 0.626536i \(0.215528\pi\)
\(774\) 3.26952e9 0.253449
\(775\) 1.07983e9 0.0833296
\(776\) 1.41109e10 1.08402
\(777\) 3.19259e9 0.244158
\(778\) −1.13023e10 −0.860472
\(779\) 1.01493e10 0.769229
\(780\) −1.04397e9 −0.0787693
\(781\) 1.11169e10 0.835038
\(782\) −3.98865e9 −0.298265
\(783\) −2.82820e8 −0.0210545
\(784\) −2.24030e10 −1.66035
\(785\) 1.20380e10 0.888201
\(786\) 1.59160e10 1.16911
\(787\) −1.57588e10 −1.15242 −0.576212 0.817300i \(-0.695470\pi\)
−0.576212 + 0.817300i \(0.695470\pi\)
\(788\) 1.55784e9 0.113418
\(789\) 4.40745e9 0.319461
\(790\) 2.18949e10 1.57997
\(791\) −1.42165e10 −1.02135
\(792\) −5.26144e9 −0.376328
\(793\) −3.15662e8 −0.0224784
\(794\) −5.07745e9 −0.359976
\(795\) 5.33867e9 0.376833
\(796\) −1.32796e9 −0.0933234
\(797\) −9.19325e9 −0.643228 −0.321614 0.946871i \(-0.604225\pi\)
−0.321614 + 0.946871i \(0.604225\pi\)
\(798\) 6.06684e9 0.422622
\(799\) −2.24286e9 −0.155557
\(800\) −1.23844e9 −0.0855185
\(801\) 5.23397e8 0.0359847
\(802\) 2.30173e10 1.57559
\(803\) 1.99205e10 1.35767
\(804\) 1.38065e9 0.0936885
\(805\) −1.82003e10 −1.22968
\(806\) −4.97443e9 −0.334634
\(807\) −1.42212e10 −0.952534
\(808\) −1.51320e10 −1.00915
\(809\) −1.69661e10 −1.12658 −0.563290 0.826259i \(-0.690465\pi\)
−0.563290 + 0.826259i \(0.690465\pi\)
\(810\) −1.52464e9 −0.100802
\(811\) −2.19205e10 −1.44303 −0.721517 0.692397i \(-0.756555\pi\)
−0.721517 + 0.692397i \(0.756555\pi\)
\(812\) −3.96822e8 −0.0260105
\(813\) 3.27154e9 0.213518
\(814\) 5.50602e9 0.357810
\(815\) −6.69738e9 −0.433364
\(816\) 3.03837e9 0.195761
\(817\) 4.79359e9 0.307527
\(818\) 2.93422e10 1.87437
\(819\) −8.79003e9 −0.559109
\(820\) −3.57872e9 −0.226662
\(821\) 1.97496e10 1.24554 0.622770 0.782405i \(-0.286007\pi\)
0.622770 + 0.782405i \(0.286007\pi\)
\(822\) 4.81121e9 0.302136
\(823\) 3.26341e9 0.204067 0.102033 0.994781i \(-0.467465\pi\)
0.102033 + 0.994781i \(0.467465\pi\)
\(824\) 1.04916e10 0.653276
\(825\) 3.29076e9 0.204036
\(826\) 3.55683e9 0.219600
\(827\) 7.08713e9 0.435713 0.217857 0.975981i \(-0.430093\pi\)
0.217857 + 0.975981i \(0.430093\pi\)
\(828\) −7.61751e8 −0.0466345
\(829\) 3.03587e10 1.85073 0.925365 0.379078i \(-0.123759\pi\)
0.925365 + 0.379078i \(0.123759\pi\)
\(830\) 1.92012e10 1.16561
\(831\) 9.15914e9 0.553670
\(832\) −1.42945e10 −0.860471
\(833\) −7.37643e9 −0.442169
\(834\) −1.28833e10 −0.769037
\(835\) 2.27447e9 0.135200
\(836\) 1.37448e9 0.0813611
\(837\) −9.54345e8 −0.0562557
\(838\) 2.10092e10 1.23326
\(839\) 1.35904e10 0.794448 0.397224 0.917722i \(-0.369974\pi\)
0.397224 + 0.917722i \(0.369974\pi\)
\(840\) 1.20060e10 0.698906
\(841\) −1.70434e10 −0.988031
\(842\) −2.90069e10 −1.67459
\(843\) −6.72124e9 −0.386414
\(844\) 5.59143e8 0.0320128
\(845\) 2.05189e9 0.116992
\(846\) −3.26068e9 −0.185145
\(847\) 1.49256e10 0.843997
\(848\) −1.54672e10 −0.871014
\(849\) −1.62414e10 −0.910852
\(850\) −1.64565e9 −0.0919116
\(851\) −4.47393e9 −0.248849
\(852\) −1.06173e9 −0.0588130
\(853\) 1.23490e10 0.681255 0.340628 0.940198i \(-0.389361\pi\)
0.340628 + 0.940198i \(0.389361\pi\)
\(854\) −6.46826e8 −0.0355373
\(855\) −2.23535e9 −0.122311
\(856\) 1.45747e10 0.794219
\(857\) −1.91652e10 −1.04011 −0.520057 0.854132i \(-0.674089\pi\)
−0.520057 + 0.854132i \(0.674089\pi\)
\(858\) −1.51595e10 −0.819368
\(859\) −2.69718e10 −1.45189 −0.725946 0.687751i \(-0.758598\pi\)
−0.725946 + 0.687751i \(0.758598\pi\)
\(860\) −1.69025e9 −0.0906166
\(861\) −3.01321e10 −1.60886
\(862\) 1.29171e10 0.686893
\(863\) −1.15124e9 −0.0609715 −0.0304858 0.999535i \(-0.509705\pi\)
−0.0304858 + 0.999535i \(0.509705\pi\)
\(864\) 1.09452e9 0.0577334
\(865\) 1.70430e10 0.895344
\(866\) 1.01275e10 0.529896
\(867\) −1.00787e10 −0.525217
\(868\) −1.33903e9 −0.0694978
\(869\) 4.17659e10 2.15900
\(870\) 1.11300e9 0.0573032
\(871\) −2.23257e10 −1.14483
\(872\) 1.06599e10 0.544435
\(873\) 7.80002e9 0.396776
\(874\) −8.50174e9 −0.430743
\(875\) −3.38504e10 −1.70819
\(876\) −1.90251e9 −0.0956231
\(877\) −2.50007e9 −0.125157 −0.0625784 0.998040i \(-0.519932\pi\)
−0.0625784 + 0.998040i \(0.519932\pi\)
\(878\) −2.32624e10 −1.15991
\(879\) 6.46198e9 0.320926
\(880\) 2.39104e10 1.18276
\(881\) 1.84978e10 0.911392 0.455696 0.890136i \(-0.349390\pi\)
0.455696 + 0.890136i \(0.349390\pi\)
\(882\) −1.07239e10 −0.526273
\(883\) 3.27036e10 1.59857 0.799287 0.600950i \(-0.205211\pi\)
0.799287 + 0.600950i \(0.205211\pi\)
\(884\) 9.95879e8 0.0484868
\(885\) −1.31053e9 −0.0635543
\(886\) −1.73733e10 −0.839196
\(887\) −2.66571e10 −1.28257 −0.641285 0.767303i \(-0.721598\pi\)
−0.641285 + 0.767303i \(0.721598\pi\)
\(888\) 2.95126e9 0.141437
\(889\) 4.30097e10 2.05310
\(890\) −2.05976e9 −0.0979382
\(891\) −2.90835e9 −0.137745
\(892\) 5.02130e9 0.236886
\(893\) −4.78063e9 −0.224649
\(894\) 2.29416e10 1.07385
\(895\) −4.94674e9 −0.230642
\(896\) −3.94455e10 −1.83198
\(897\) 1.23179e10 0.569853
\(898\) 1.21977e10 0.562096
\(899\) 6.96680e8 0.0319797
\(900\) −3.14285e8 −0.0143706
\(901\) −5.09274e9 −0.231961
\(902\) −5.19665e10 −2.35777
\(903\) −1.42316e10 −0.643202
\(904\) −1.31419e10 −0.591654
\(905\) 1.08272e10 0.485562
\(906\) −1.91426e10 −0.855171
\(907\) −2.98097e10 −1.32658 −0.663289 0.748364i \(-0.730840\pi\)
−0.663289 + 0.748364i \(0.730840\pi\)
\(908\) −3.94266e9 −0.174779
\(909\) −8.36446e9 −0.369372
\(910\) 3.45921e10 1.52171
\(911\) −1.60230e10 −0.702149 −0.351074 0.936348i \(-0.614184\pi\)
−0.351074 + 0.936348i \(0.614184\pi\)
\(912\) 6.47624e9 0.282710
\(913\) 3.66274e10 1.59279
\(914\) −4.02379e10 −1.74311
\(915\) 2.38326e8 0.0102848
\(916\) −6.74815e9 −0.290102
\(917\) −6.92794e10 −2.96696
\(918\) 1.45441e9 0.0620494
\(919\) 2.67157e10 1.13543 0.567717 0.823224i \(-0.307827\pi\)
0.567717 + 0.823224i \(0.307827\pi\)
\(920\) −1.68245e10 −0.712336
\(921\) 2.70708e9 0.114181
\(922\) −3.62349e8 −0.0152254
\(923\) 1.71686e10 0.718670
\(924\) −4.08067e9 −0.170169
\(925\) −1.84587e9 −0.0766839
\(926\) −6.21430e9 −0.257190
\(927\) 5.79942e9 0.239114
\(928\) −7.99012e8 −0.0328198
\(929\) −1.01936e10 −0.417130 −0.208565 0.978009i \(-0.566879\pi\)
−0.208565 + 0.978009i \(0.566879\pi\)
\(930\) 3.75570e9 0.153109
\(931\) −1.57228e10 −0.638564
\(932\) 3.03310e9 0.122725
\(933\) 1.91779e10 0.773062
\(934\) 4.05008e10 1.62648
\(935\) 7.87277e9 0.314983
\(936\) −8.12559e9 −0.323884
\(937\) −2.50793e10 −0.995923 −0.497962 0.867199i \(-0.665918\pi\)
−0.497962 + 0.867199i \(0.665918\pi\)
\(938\) −4.57478e10 −1.80992
\(939\) −1.76872e10 −0.697154
\(940\) 1.68568e9 0.0661955
\(941\) −5.38850e9 −0.210816 −0.105408 0.994429i \(-0.533615\pi\)
−0.105408 + 0.994429i \(0.533615\pi\)
\(942\) −1.66947e10 −0.650730
\(943\) 4.22256e10 1.63978
\(944\) 3.79686e9 0.146900
\(945\) 6.63650e9 0.255816
\(946\) −2.45442e10 −0.942604
\(947\) −1.51551e10 −0.579876 −0.289938 0.957045i \(-0.593635\pi\)
−0.289938 + 0.957045i \(0.593635\pi\)
\(948\) −3.98887e9 −0.152062
\(949\) 3.07645e10 1.16847
\(950\) −3.50767e9 −0.132735
\(951\) −4.59084e9 −0.173085
\(952\) −1.14529e10 −0.430215
\(953\) 4.14430e9 0.155105 0.0775526 0.996988i \(-0.475289\pi\)
0.0775526 + 0.996988i \(0.475289\pi\)
\(954\) −7.40383e9 −0.276082
\(955\) −3.75011e10 −1.39326
\(956\) 4.04012e9 0.149552
\(957\) 2.12312e9 0.0783039
\(958\) 2.94703e9 0.108294
\(959\) −2.09423e10 −0.766760
\(960\) 1.07924e10 0.393701
\(961\) −2.51617e10 −0.914553
\(962\) 8.50331e9 0.307946
\(963\) 8.05639e9 0.290702
\(964\) −7.36552e9 −0.264810
\(965\) 1.61167e10 0.577338
\(966\) 2.52407e10 0.900910
\(967\) −7.19424e9 −0.255854 −0.127927 0.991784i \(-0.540832\pi\)
−0.127927 + 0.991784i \(0.540832\pi\)
\(968\) 1.37974e10 0.488915
\(969\) 2.13238e9 0.0752888
\(970\) −3.06960e10 −1.07989
\(971\) −3.04216e10 −1.06639 −0.533193 0.845993i \(-0.679008\pi\)
−0.533193 + 0.845993i \(0.679008\pi\)
\(972\) 2.77763e8 0.00970158
\(973\) 5.60788e10 1.95166
\(974\) 1.44147e10 0.499860
\(975\) 5.08214e9 0.175603
\(976\) −6.90475e8 −0.0237724
\(977\) −1.19680e9 −0.0410572 −0.0205286 0.999789i \(-0.506535\pi\)
−0.0205286 + 0.999789i \(0.506535\pi\)
\(978\) 9.28813e9 0.317499
\(979\) −3.92912e9 −0.133831
\(980\) 5.54395e9 0.188160
\(981\) 5.89245e9 0.199276
\(982\) −2.34366e10 −0.789778
\(983\) 5.01551e10 1.68414 0.842069 0.539370i \(-0.181337\pi\)
0.842069 + 0.539370i \(0.181337\pi\)
\(984\) −2.78544e10 −0.931990
\(985\) 1.90193e10 0.634116
\(986\) −1.06173e9 −0.0352733
\(987\) 1.41931e10 0.469860
\(988\) 2.12270e9 0.0700228
\(989\) 1.99434e10 0.655561
\(990\) 1.14454e10 0.374895
\(991\) 2.70432e10 0.882674 0.441337 0.897341i \(-0.354504\pi\)
0.441337 + 0.897341i \(0.354504\pi\)
\(992\) −2.69618e9 −0.0876915
\(993\) 3.50270e9 0.113522
\(994\) 3.51803e10 1.13618
\(995\) −1.62128e10 −0.521767
\(996\) −3.49811e9 −0.112183
\(997\) 1.18645e10 0.379154 0.189577 0.981866i \(-0.439288\pi\)
0.189577 + 0.981866i \(0.439288\pi\)
\(998\) −3.78400e10 −1.20502
\(999\) 1.63136e9 0.0517692
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.b.1.6 17
3.2 odd 2 531.8.a.d.1.12 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.6 17 1.1 even 1 trivial
531.8.a.d.1.12 17 3.2 odd 2