Properties

Label 177.8.a.b.1.14
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $17$
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: \(1\)
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 1.14
Root \(15.6681\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+13.6681 q^{2} +27.0000 q^{3} +58.8158 q^{4} -207.404 q^{5} +369.038 q^{6} +882.793 q^{7} -945.614 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+13.6681 q^{2} +27.0000 q^{3} +58.8158 q^{4} -207.404 q^{5} +369.038 q^{6} +882.793 q^{7} -945.614 q^{8} +729.000 q^{9} -2834.81 q^{10} -3637.21 q^{11} +1588.03 q^{12} +208.023 q^{13} +12066.1 q^{14} -5599.90 q^{15} -20453.1 q^{16} -13120.6 q^{17} +9964.01 q^{18} -9914.89 q^{19} -12198.6 q^{20} +23835.4 q^{21} -49713.6 q^{22} -37757.5 q^{23} -25531.6 q^{24} -35108.6 q^{25} +2843.27 q^{26} +19683.0 q^{27} +51922.2 q^{28} -98092.3 q^{29} -76539.8 q^{30} +326883. q^{31} -158516. q^{32} -98204.7 q^{33} -179333. q^{34} -183095. q^{35} +42876.7 q^{36} +39434.0 q^{37} -135517. q^{38} +5616.63 q^{39} +196124. q^{40} -719442. q^{41} +325784. q^{42} -878099. q^{43} -213925. q^{44} -151197. q^{45} -516072. q^{46} -158129. q^{47} -552234. q^{48} -44219.7 q^{49} -479867. q^{50} -354257. q^{51} +12235.0 q^{52} -1.83479e6 q^{53} +269028. q^{54} +754372. q^{55} -834781. q^{56} -267702. q^{57} -1.34073e6 q^{58} -205379. q^{59} -329363. q^{60} +3.49764e6 q^{61} +4.46786e6 q^{62} +643556. q^{63} +451395. q^{64} -43144.8 q^{65} -1.34227e6 q^{66} -1.00103e6 q^{67} -771700. q^{68} -1.01945e6 q^{69} -2.50255e6 q^{70} +331392. q^{71} -689352. q^{72} -527191. q^{73} +538987. q^{74} -947933. q^{75} -583152. q^{76} -3.21090e6 q^{77} +76768.4 q^{78} -1.50960e6 q^{79} +4.24206e6 q^{80} +531441. q^{81} -9.83337e6 q^{82} +2.48775e6 q^{83} +1.40190e6 q^{84} +2.72127e6 q^{85} -1.20019e7 q^{86} -2.64849e6 q^{87} +3.43940e6 q^{88} +6.37807e6 q^{89} -2.06657e6 q^{90} +183641. q^{91} -2.22074e6 q^{92} +8.82585e6 q^{93} -2.16132e6 q^{94} +2.05639e6 q^{95} -4.27993e6 q^{96} -4.33428e6 q^{97} -604398. q^{98} -2.65153e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 13.6681 1.20810 0.604049 0.796948i \(-0.293553\pi\)
0.604049 + 0.796948i \(0.293553\pi\)
\(3\) 27.0000 0.577350
\(4\) 58.8158 0.459498
\(5\) −207.404 −0.742031 −0.371015 0.928627i \(-0.620990\pi\)
−0.371015 + 0.928627i \(0.620990\pi\)
\(6\) 369.038 0.697495
\(7\) 882.793 0.972782 0.486391 0.873741i \(-0.338313\pi\)
0.486391 + 0.873741i \(0.338313\pi\)
\(8\) −945.614 −0.652978
\(9\) 729.000 0.333333
\(10\) −2834.81 −0.896445
\(11\) −3637.21 −0.823937 −0.411969 0.911198i \(-0.635159\pi\)
−0.411969 + 0.911198i \(0.635159\pi\)
\(12\) 1588.03 0.265292
\(13\) 208.023 0.0262609 0.0131305 0.999914i \(-0.495820\pi\)
0.0131305 + 0.999914i \(0.495820\pi\)
\(14\) 12066.1 1.17522
\(15\) −5599.90 −0.428412
\(16\) −20453.1 −1.24836
\(17\) −13120.6 −0.647714 −0.323857 0.946106i \(-0.604980\pi\)
−0.323857 + 0.946106i \(0.604980\pi\)
\(18\) 9964.01 0.402699
\(19\) −9914.89 −0.331627 −0.165814 0.986157i \(-0.553025\pi\)
−0.165814 + 0.986157i \(0.553025\pi\)
\(20\) −12198.6 −0.340962
\(21\) 23835.4 0.561636
\(22\) −49713.6 −0.995396
\(23\) −37757.5 −0.647077 −0.323538 0.946215i \(-0.604872\pi\)
−0.323538 + 0.946215i \(0.604872\pi\)
\(24\) −25531.6 −0.376997
\(25\) −35108.6 −0.449391
\(26\) 2843.27 0.0317257
\(27\) 19683.0 0.192450
\(28\) 51922.2 0.446992
\(29\) −98092.3 −0.746865 −0.373432 0.927657i \(-0.621819\pi\)
−0.373432 + 0.927657i \(0.621819\pi\)
\(30\) −76539.8 −0.517563
\(31\) 326883. 1.97073 0.985364 0.170463i \(-0.0545265\pi\)
0.985364 + 0.170463i \(0.0545265\pi\)
\(32\) −158516. −0.855161
\(33\) −98204.7 −0.475700
\(34\) −179333. −0.782501
\(35\) −183095. −0.721834
\(36\) 42876.7 0.153166
\(37\) 39434.0 0.127987 0.0639934 0.997950i \(-0.479616\pi\)
0.0639934 + 0.997950i \(0.479616\pi\)
\(38\) −135517. −0.400638
\(39\) 5616.63 0.0151618
\(40\) 196124. 0.484530
\(41\) −719442. −1.63024 −0.815121 0.579291i \(-0.803330\pi\)
−0.815121 + 0.579291i \(0.803330\pi\)
\(42\) 325784. 0.678511
\(43\) −878099. −1.68424 −0.842120 0.539290i \(-0.818693\pi\)
−0.842120 + 0.539290i \(0.818693\pi\)
\(44\) −213925. −0.378598
\(45\) −151197. −0.247344
\(46\) −516072. −0.781731
\(47\) −158129. −0.222162 −0.111081 0.993811i \(-0.535431\pi\)
−0.111081 + 0.993811i \(0.535431\pi\)
\(48\) −552234. −0.720741
\(49\) −44219.7 −0.0536945
\(50\) −479867. −0.542907
\(51\) −354257. −0.373958
\(52\) 12235.0 0.0120669
\(53\) −1.83479e6 −1.69286 −0.846429 0.532502i \(-0.821252\pi\)
−0.846429 + 0.532502i \(0.821252\pi\)
\(54\) 269028. 0.232498
\(55\) 754372. 0.611387
\(56\) −834781. −0.635206
\(57\) −267702. −0.191465
\(58\) −1.34073e6 −0.902285
\(59\) −205379. −0.130189
\(60\) −329363. −0.196854
\(61\) 3.49764e6 1.97297 0.986485 0.163851i \(-0.0523917\pi\)
0.986485 + 0.163851i \(0.0523917\pi\)
\(62\) 4.46786e6 2.38083
\(63\) 643556. 0.324261
\(64\) 451395. 0.215242
\(65\) −43144.8 −0.0194864
\(66\) −1.34227e6 −0.574692
\(67\) −1.00103e6 −0.406615 −0.203308 0.979115i \(-0.565169\pi\)
−0.203308 + 0.979115i \(0.565169\pi\)
\(68\) −771700. −0.297623
\(69\) −1.01945e6 −0.373590
\(70\) −2.50255e6 −0.872046
\(71\) 331392. 0.109885 0.0549423 0.998490i \(-0.482502\pi\)
0.0549423 + 0.998490i \(0.482502\pi\)
\(72\) −689352. −0.217659
\(73\) −527191. −0.158613 −0.0793063 0.996850i \(-0.525271\pi\)
−0.0793063 + 0.996850i \(0.525271\pi\)
\(74\) 538987. 0.154620
\(75\) −947933. −0.259456
\(76\) −583152. −0.152382
\(77\) −3.21090e6 −0.801512
\(78\) 76768.4 0.0183169
\(79\) −1.50960e6 −0.344481 −0.172241 0.985055i \(-0.555101\pi\)
−0.172241 + 0.985055i \(0.555101\pi\)
\(80\) 4.24206e6 0.926321
\(81\) 531441. 0.111111
\(82\) −9.83337e6 −1.96949
\(83\) 2.48775e6 0.477566 0.238783 0.971073i \(-0.423252\pi\)
0.238783 + 0.971073i \(0.423252\pi\)
\(84\) 1.40190e6 0.258071
\(85\) 2.72127e6 0.480623
\(86\) −1.20019e7 −2.03472
\(87\) −2.64849e6 −0.431203
\(88\) 3.43940e6 0.538013
\(89\) 6.37807e6 0.959013 0.479506 0.877538i \(-0.340816\pi\)
0.479506 + 0.877538i \(0.340816\pi\)
\(90\) −2.06657e6 −0.298815
\(91\) 183641. 0.0255462
\(92\) −2.22074e6 −0.297331
\(93\) 8.82585e6 1.13780
\(94\) −2.16132e6 −0.268393
\(95\) 2.05639e6 0.246078
\(96\) −4.27993e6 −0.493728
\(97\) −4.33428e6 −0.482187 −0.241094 0.970502i \(-0.577506\pi\)
−0.241094 + 0.970502i \(0.577506\pi\)
\(98\) −604398. −0.0648682
\(99\) −2.65153e6 −0.274646
\(100\) −2.06494e6 −0.206494
\(101\) 6.87221e6 0.663700 0.331850 0.943332i \(-0.392327\pi\)
0.331850 + 0.943332i \(0.392327\pi\)
\(102\) −4.84200e6 −0.451777
\(103\) 1.71578e6 0.154715 0.0773574 0.997003i \(-0.475352\pi\)
0.0773574 + 0.997003i \(0.475352\pi\)
\(104\) −196710. −0.0171478
\(105\) −4.94356e6 −0.416751
\(106\) −2.50780e7 −2.04514
\(107\) −7.08886e6 −0.559414 −0.279707 0.960085i \(-0.590237\pi\)
−0.279707 + 0.960085i \(0.590237\pi\)
\(108\) 1.15767e6 0.0884305
\(109\) 2.29535e7 1.69768 0.848840 0.528650i \(-0.177301\pi\)
0.848840 + 0.528650i \(0.177301\pi\)
\(110\) 1.03108e7 0.738614
\(111\) 1.06472e6 0.0738932
\(112\) −1.80559e7 −1.21438
\(113\) −1.69616e7 −1.10584 −0.552919 0.833235i \(-0.686486\pi\)
−0.552919 + 0.833235i \(0.686486\pi\)
\(114\) −3.65897e6 −0.231308
\(115\) 7.83105e6 0.480151
\(116\) −5.76938e6 −0.343183
\(117\) 151649. 0.00875364
\(118\) −2.80713e6 −0.157281
\(119\) −1.15828e7 −0.630084
\(120\) 5.29535e6 0.279743
\(121\) −6.25787e6 −0.321127
\(122\) 4.78059e7 2.38354
\(123\) −1.94249e7 −0.941220
\(124\) 1.92259e7 0.905546
\(125\) 2.34851e7 1.07549
\(126\) 8.79616e6 0.391738
\(127\) −1.24664e7 −0.540044 −0.270022 0.962854i \(-0.587031\pi\)
−0.270022 + 0.962854i \(0.587031\pi\)
\(128\) 2.64597e7 1.11519
\(129\) −2.37087e7 −0.972396
\(130\) −589706. −0.0235415
\(131\) 4.68201e7 1.81963 0.909815 0.415015i \(-0.136224\pi\)
0.909815 + 0.415015i \(0.136224\pi\)
\(132\) −5.77599e6 −0.218584
\(133\) −8.75279e6 −0.322601
\(134\) −1.36821e7 −0.491231
\(135\) −4.08233e6 −0.142804
\(136\) 1.24070e7 0.422943
\(137\) −1.03397e7 −0.343545 −0.171773 0.985137i \(-0.554949\pi\)
−0.171773 + 0.985137i \(0.554949\pi\)
\(138\) −1.39339e7 −0.451333
\(139\) −3.07437e7 −0.970965 −0.485483 0.874246i \(-0.661356\pi\)
−0.485483 + 0.874246i \(0.661356\pi\)
\(140\) −1.07689e7 −0.331682
\(141\) −4.26949e6 −0.128265
\(142\) 4.52948e6 0.132751
\(143\) −756624. −0.0216374
\(144\) −1.49103e7 −0.416120
\(145\) 2.03447e7 0.554196
\(146\) −7.20567e6 −0.191619
\(147\) −1.19393e6 −0.0310005
\(148\) 2.31934e6 0.0588097
\(149\) 3.66220e6 0.0906963 0.0453482 0.998971i \(-0.485560\pi\)
0.0453482 + 0.998971i \(0.485560\pi\)
\(150\) −1.29564e7 −0.313448
\(151\) 7.28781e7 1.72257 0.861286 0.508120i \(-0.169659\pi\)
0.861286 + 0.508120i \(0.169659\pi\)
\(152\) 9.37565e6 0.216545
\(153\) −9.56493e6 −0.215905
\(154\) −4.38868e7 −0.968304
\(155\) −6.77968e7 −1.46234
\(156\) 330346. 0.00696680
\(157\) −9.09496e7 −1.87565 −0.937826 0.347106i \(-0.887164\pi\)
−0.937826 + 0.347106i \(0.887164\pi\)
\(158\) −2.06332e7 −0.416167
\(159\) −4.95393e7 −0.977372
\(160\) 3.28768e7 0.634556
\(161\) −3.33320e7 −0.629465
\(162\) 7.26377e6 0.134233
\(163\) 5.62746e7 1.01778 0.508892 0.860830i \(-0.330055\pi\)
0.508892 + 0.860830i \(0.330055\pi\)
\(164\) −4.23145e7 −0.749094
\(165\) 2.03680e7 0.352984
\(166\) 3.40027e7 0.576945
\(167\) 3.21050e7 0.533415 0.266707 0.963778i \(-0.414064\pi\)
0.266707 + 0.963778i \(0.414064\pi\)
\(168\) −2.25391e7 −0.366736
\(169\) −6.27052e7 −0.999310
\(170\) 3.71944e7 0.580640
\(171\) −7.22795e6 −0.110542
\(172\) −5.16461e7 −0.773905
\(173\) 7.85878e7 1.15397 0.576984 0.816756i \(-0.304230\pi\)
0.576984 + 0.816756i \(0.304230\pi\)
\(174\) −3.61997e7 −0.520935
\(175\) −3.09937e7 −0.437159
\(176\) 7.43923e7 1.02857
\(177\) −5.54523e6 −0.0751646
\(178\) 8.71759e7 1.15858
\(179\) −5.87666e7 −0.765853 −0.382926 0.923779i \(-0.625084\pi\)
−0.382926 + 0.923779i \(0.625084\pi\)
\(180\) −8.89280e6 −0.113654
\(181\) −4.81650e7 −0.603749 −0.301875 0.953348i \(-0.597612\pi\)
−0.301875 + 0.953348i \(0.597612\pi\)
\(182\) 2.51002e6 0.0308622
\(183\) 9.44362e7 1.13909
\(184\) 3.57040e7 0.422527
\(185\) −8.17877e6 −0.0949701
\(186\) 1.20632e8 1.37457
\(187\) 4.77225e7 0.533675
\(188\) −9.30049e6 −0.102083
\(189\) 1.73760e7 0.187212
\(190\) 2.81068e7 0.297286
\(191\) 1.65487e8 1.71849 0.859247 0.511560i \(-0.170932\pi\)
0.859247 + 0.511560i \(0.170932\pi\)
\(192\) 1.21877e7 0.124270
\(193\) −1.11360e8 −1.11501 −0.557504 0.830174i \(-0.688241\pi\)
−0.557504 + 0.830174i \(0.688241\pi\)
\(194\) −5.92412e7 −0.582529
\(195\) −1.16491e6 −0.0112505
\(196\) −2.60082e6 −0.0246725
\(197\) 3.64560e7 0.339733 0.169866 0.985467i \(-0.445666\pi\)
0.169866 + 0.985467i \(0.445666\pi\)
\(198\) −3.62412e7 −0.331799
\(199\) 7.61265e7 0.684779 0.342389 0.939558i \(-0.388764\pi\)
0.342389 + 0.939558i \(0.388764\pi\)
\(200\) 3.31992e7 0.293442
\(201\) −2.70277e7 −0.234760
\(202\) 9.39298e7 0.801814
\(203\) −8.65952e7 −0.726537
\(204\) −2.08359e7 −0.171833
\(205\) 1.49215e8 1.20969
\(206\) 2.34514e7 0.186911
\(207\) −2.75252e7 −0.215692
\(208\) −4.25472e6 −0.0327831
\(209\) 3.60625e7 0.273240
\(210\) −6.75688e7 −0.503476
\(211\) 2.35187e8 1.72355 0.861775 0.507291i \(-0.169353\pi\)
0.861775 + 0.507291i \(0.169353\pi\)
\(212\) −1.07915e8 −0.777866
\(213\) 8.94757e6 0.0634420
\(214\) −9.68910e7 −0.675826
\(215\) 1.82121e8 1.24976
\(216\) −1.86125e7 −0.125666
\(217\) 2.88570e8 1.91709
\(218\) 3.13730e8 2.05096
\(219\) −1.42342e7 −0.0915751
\(220\) 4.43690e7 0.280931
\(221\) −2.72939e6 −0.0170096
\(222\) 1.45526e7 0.0892702
\(223\) 1.57494e8 0.951036 0.475518 0.879706i \(-0.342261\pi\)
0.475518 + 0.879706i \(0.342261\pi\)
\(224\) −1.39937e8 −0.831886
\(225\) −2.55942e7 −0.149797
\(226\) −2.31832e8 −1.33596
\(227\) −4.25579e7 −0.241485 −0.120742 0.992684i \(-0.538527\pi\)
−0.120742 + 0.992684i \(0.538527\pi\)
\(228\) −1.57451e7 −0.0879779
\(229\) −5.07210e7 −0.279103 −0.139551 0.990215i \(-0.544566\pi\)
−0.139551 + 0.990215i \(0.544566\pi\)
\(230\) 1.07035e8 0.580069
\(231\) −8.66944e7 −0.462753
\(232\) 9.27574e7 0.487686
\(233\) −2.44954e8 −1.26864 −0.634321 0.773069i \(-0.718720\pi\)
−0.634321 + 0.773069i \(0.718720\pi\)
\(234\) 2.07275e6 0.0105752
\(235\) 3.27966e7 0.164851
\(236\) −1.20795e7 −0.0598216
\(237\) −4.07591e7 −0.198886
\(238\) −1.58314e8 −0.761203
\(239\) −3.38900e6 −0.0160575 −0.00802877 0.999968i \(-0.502556\pi\)
−0.00802877 + 0.999968i \(0.502556\pi\)
\(240\) 1.14536e8 0.534812
\(241\) 2.42976e8 1.11816 0.559080 0.829114i \(-0.311155\pi\)
0.559080 + 0.829114i \(0.311155\pi\)
\(242\) −8.55329e7 −0.387953
\(243\) 1.43489e7 0.0641500
\(244\) 2.05716e8 0.906577
\(245\) 9.17134e6 0.0398430
\(246\) −2.65501e8 −1.13709
\(247\) −2.06253e6 −0.00870884
\(248\) −3.09105e8 −1.28684
\(249\) 6.71692e7 0.275723
\(250\) 3.20996e8 1.29930
\(251\) −1.00242e7 −0.0400120 −0.0200060 0.999800i \(-0.506369\pi\)
−0.0200060 + 0.999800i \(0.506369\pi\)
\(252\) 3.78513e7 0.148997
\(253\) 1.37332e8 0.533150
\(254\) −1.70392e8 −0.652425
\(255\) 7.34742e7 0.277488
\(256\) 3.03875e8 1.13202
\(257\) −1.65390e8 −0.607774 −0.303887 0.952708i \(-0.598285\pi\)
−0.303887 + 0.952708i \(0.598285\pi\)
\(258\) −3.24052e8 −1.17475
\(259\) 3.48121e7 0.124503
\(260\) −2.53760e6 −0.00895397
\(261\) −7.15093e7 −0.248955
\(262\) 6.39940e8 2.19829
\(263\) −3.86525e8 −1.31019 −0.655093 0.755549i \(-0.727370\pi\)
−0.655093 + 0.755549i \(0.727370\pi\)
\(264\) 9.28637e7 0.310622
\(265\) 3.80542e8 1.25615
\(266\) −1.19634e8 −0.389734
\(267\) 1.72208e8 0.553686
\(268\) −5.88762e7 −0.186839
\(269\) 2.13282e8 0.668070 0.334035 0.942561i \(-0.391590\pi\)
0.334035 + 0.942561i \(0.391590\pi\)
\(270\) −5.57975e7 −0.172521
\(271\) −5.08751e7 −0.155279 −0.0776396 0.996981i \(-0.524738\pi\)
−0.0776396 + 0.996981i \(0.524738\pi\)
\(272\) 2.68358e8 0.808580
\(273\) 4.95832e6 0.0147491
\(274\) −1.41323e8 −0.415036
\(275\) 1.27698e8 0.370270
\(276\) −5.99599e7 −0.171664
\(277\) −4.14639e8 −1.17217 −0.586086 0.810249i \(-0.699332\pi\)
−0.586086 + 0.810249i \(0.699332\pi\)
\(278\) −4.20206e8 −1.17302
\(279\) 2.38298e8 0.656909
\(280\) 1.73137e8 0.471342
\(281\) −3.84206e8 −1.03298 −0.516491 0.856293i \(-0.672762\pi\)
−0.516491 + 0.856293i \(0.672762\pi\)
\(282\) −5.83556e7 −0.154957
\(283\) 4.66433e8 1.22331 0.611656 0.791124i \(-0.290504\pi\)
0.611656 + 0.791124i \(0.290504\pi\)
\(284\) 1.94911e7 0.0504918
\(285\) 5.55224e7 0.142073
\(286\) −1.03416e7 −0.0261400
\(287\) −6.35118e8 −1.58587
\(288\) −1.15558e8 −0.285054
\(289\) −2.38188e8 −0.580467
\(290\) 2.78073e8 0.669523
\(291\) −1.17026e8 −0.278391
\(292\) −3.10071e7 −0.0728823
\(293\) −7.57454e8 −1.75922 −0.879609 0.475697i \(-0.842196\pi\)
−0.879609 + 0.475697i \(0.842196\pi\)
\(294\) −1.63187e7 −0.0374517
\(295\) 4.25964e7 0.0966042
\(296\) −3.72894e7 −0.0835726
\(297\) −7.15912e7 −0.158567
\(298\) 5.00551e7 0.109570
\(299\) −7.85443e6 −0.0169928
\(300\) −5.57535e7 −0.119220
\(301\) −7.75180e8 −1.63840
\(302\) 9.96101e8 2.08103
\(303\) 1.85550e8 0.383187
\(304\) 2.02790e8 0.413990
\(305\) −7.25424e8 −1.46400
\(306\) −1.30734e8 −0.260834
\(307\) −5.11428e8 −1.00879 −0.504394 0.863473i \(-0.668284\pi\)
−0.504394 + 0.863473i \(0.668284\pi\)
\(308\) −1.88852e8 −0.368293
\(309\) 4.63261e7 0.0893246
\(310\) −9.26651e8 −1.76665
\(311\) 4.47095e8 0.842826 0.421413 0.906869i \(-0.361534\pi\)
0.421413 + 0.906869i \(0.361534\pi\)
\(312\) −5.31116e6 −0.00990029
\(313\) −3.05704e7 −0.0563503 −0.0281751 0.999603i \(-0.508970\pi\)
−0.0281751 + 0.999603i \(0.508970\pi\)
\(314\) −1.24310e9 −2.26597
\(315\) −1.33476e8 −0.240611
\(316\) −8.87881e7 −0.158289
\(317\) −2.57462e8 −0.453948 −0.226974 0.973901i \(-0.572883\pi\)
−0.226974 + 0.973901i \(0.572883\pi\)
\(318\) −6.77106e8 −1.18076
\(319\) 3.56782e8 0.615370
\(320\) −9.36211e7 −0.159716
\(321\) −1.91399e8 −0.322978
\(322\) −4.55584e8 −0.760454
\(323\) 1.30089e8 0.214800
\(324\) 3.12571e7 0.0510554
\(325\) −7.30341e6 −0.0118014
\(326\) 7.69164e8 1.22958
\(327\) 6.19744e8 0.980156
\(328\) 6.80314e8 1.06451
\(329\) −1.39595e8 −0.216115
\(330\) 2.78391e8 0.426439
\(331\) 6.68825e8 1.01371 0.506856 0.862031i \(-0.330808\pi\)
0.506856 + 0.862031i \(0.330808\pi\)
\(332\) 1.46319e8 0.219441
\(333\) 2.87474e7 0.0426623
\(334\) 4.38813e8 0.644417
\(335\) 2.07617e8 0.301721
\(336\) −4.87509e8 −0.701124
\(337\) 1.47419e7 0.0209821 0.0104911 0.999945i \(-0.496661\pi\)
0.0104911 + 0.999945i \(0.496661\pi\)
\(338\) −8.57059e8 −1.20726
\(339\) −4.57962e8 −0.638455
\(340\) 1.60053e8 0.220846
\(341\) −1.18894e9 −1.62376
\(342\) −9.87921e7 −0.133546
\(343\) −7.66055e8 −1.02502
\(344\) 8.30342e8 1.09977
\(345\) 2.11438e8 0.277215
\(346\) 1.07414e9 1.39410
\(347\) −6.05234e8 −0.777625 −0.388812 0.921317i \(-0.627115\pi\)
−0.388812 + 0.921317i \(0.627115\pi\)
\(348\) −1.55773e8 −0.198137
\(349\) −1.02679e9 −1.29298 −0.646491 0.762922i \(-0.723764\pi\)
−0.646491 + 0.762922i \(0.723764\pi\)
\(350\) −4.23623e8 −0.528131
\(351\) 4.09452e6 0.00505392
\(352\) 5.76556e8 0.704599
\(353\) −1.41214e9 −1.70870 −0.854352 0.519695i \(-0.826046\pi\)
−0.854352 + 0.519695i \(0.826046\pi\)
\(354\) −7.57926e7 −0.0908061
\(355\) −6.87319e7 −0.0815378
\(356\) 3.75131e8 0.440665
\(357\) −3.12735e8 −0.363779
\(358\) −8.03226e8 −0.925224
\(359\) −2.89775e8 −0.330545 −0.165272 0.986248i \(-0.552850\pi\)
−0.165272 + 0.986248i \(0.552850\pi\)
\(360\) 1.42974e8 0.161510
\(361\) −7.95567e8 −0.890023
\(362\) −6.58322e8 −0.729388
\(363\) −1.68962e8 −0.185403
\(364\) 1.08010e7 0.0117384
\(365\) 1.09341e8 0.117695
\(366\) 1.29076e9 1.37614
\(367\) −6.11428e8 −0.645676 −0.322838 0.946454i \(-0.604637\pi\)
−0.322838 + 0.946454i \(0.604637\pi\)
\(368\) 7.72259e8 0.807784
\(369\) −5.24473e8 −0.543414
\(370\) −1.11788e8 −0.114733
\(371\) −1.61974e9 −1.64678
\(372\) 5.19099e8 0.522817
\(373\) 6.09388e7 0.0608013 0.0304006 0.999538i \(-0.490322\pi\)
0.0304006 + 0.999538i \(0.490322\pi\)
\(374\) 6.52273e8 0.644732
\(375\) 6.34098e8 0.620936
\(376\) 1.49529e8 0.145067
\(377\) −2.04055e7 −0.0196134
\(378\) 2.37496e8 0.226170
\(379\) −9.94919e8 −0.938751 −0.469376 0.882999i \(-0.655521\pi\)
−0.469376 + 0.882999i \(0.655521\pi\)
\(380\) 1.20948e8 0.113072
\(381\) −3.36593e8 −0.311794
\(382\) 2.26189e9 2.07611
\(383\) −1.69498e8 −0.154159 −0.0770794 0.997025i \(-0.524560\pi\)
−0.0770794 + 0.997025i \(0.524560\pi\)
\(384\) 7.14413e8 0.643858
\(385\) 6.65954e8 0.594746
\(386\) −1.52207e9 −1.34704
\(387\) −6.40134e8 −0.561413
\(388\) −2.54924e8 −0.221564
\(389\) −1.92883e8 −0.166139 −0.0830694 0.996544i \(-0.526472\pi\)
−0.0830694 + 0.996544i \(0.526472\pi\)
\(390\) −1.59221e7 −0.0135917
\(391\) 4.95402e8 0.419120
\(392\) 4.18148e7 0.0350613
\(393\) 1.26414e9 1.05056
\(394\) 4.98283e8 0.410430
\(395\) 3.13096e8 0.255616
\(396\) −1.55952e8 −0.126199
\(397\) 1.91282e9 1.53429 0.767146 0.641472i \(-0.221676\pi\)
0.767146 + 0.641472i \(0.221676\pi\)
\(398\) 1.04050e9 0.827279
\(399\) −2.36325e8 −0.186254
\(400\) 7.18081e8 0.561001
\(401\) 2.01241e8 0.155852 0.0779258 0.996959i \(-0.475170\pi\)
0.0779258 + 0.996959i \(0.475170\pi\)
\(402\) −3.69416e8 −0.283612
\(403\) 6.79993e7 0.0517531
\(404\) 4.04195e8 0.304969
\(405\) −1.10223e8 −0.0824478
\(406\) −1.18359e9 −0.877727
\(407\) −1.43430e8 −0.105453
\(408\) 3.34990e8 0.244186
\(409\) 3.51133e8 0.253770 0.126885 0.991917i \(-0.459502\pi\)
0.126885 + 0.991917i \(0.459502\pi\)
\(410\) 2.03948e9 1.46142
\(411\) −2.79171e8 −0.198346
\(412\) 1.00915e8 0.0710912
\(413\) −1.81307e8 −0.126645
\(414\) −3.76216e8 −0.260577
\(415\) −5.15968e8 −0.354368
\(416\) −3.29750e7 −0.0224573
\(417\) −8.30079e8 −0.560587
\(418\) 4.92905e8 0.330101
\(419\) 3.67310e8 0.243940 0.121970 0.992534i \(-0.461079\pi\)
0.121970 + 0.992534i \(0.461079\pi\)
\(420\) −2.90759e8 −0.191497
\(421\) 1.20983e9 0.790202 0.395101 0.918638i \(-0.370710\pi\)
0.395101 + 0.918638i \(0.370710\pi\)
\(422\) 3.21454e9 2.08222
\(423\) −1.15276e8 −0.0740539
\(424\) 1.73500e9 1.10540
\(425\) 4.60647e8 0.291076
\(426\) 1.22296e8 0.0766440
\(427\) 3.08769e9 1.91927
\(428\) −4.16937e8 −0.257050
\(429\) −2.04289e7 −0.0124923
\(430\) 2.48924e9 1.50983
\(431\) 6.04724e8 0.363820 0.181910 0.983315i \(-0.441772\pi\)
0.181910 + 0.983315i \(0.441772\pi\)
\(432\) −4.02579e8 −0.240247
\(433\) −8.36784e8 −0.495343 −0.247671 0.968844i \(-0.579665\pi\)
−0.247671 + 0.968844i \(0.579665\pi\)
\(434\) 3.94419e9 2.31603
\(435\) 5.49308e8 0.319965
\(436\) 1.35003e9 0.780081
\(437\) 3.74361e8 0.214588
\(438\) −1.94553e8 −0.110632
\(439\) −2.70885e9 −1.52813 −0.764064 0.645140i \(-0.776799\pi\)
−0.764064 + 0.645140i \(0.776799\pi\)
\(440\) −7.13344e8 −0.399222
\(441\) −3.22362e7 −0.0178982
\(442\) −3.73055e7 −0.0205492
\(443\) −9.58898e8 −0.524034 −0.262017 0.965063i \(-0.584388\pi\)
−0.262017 + 0.965063i \(0.584388\pi\)
\(444\) 6.26223e7 0.0339538
\(445\) −1.32284e9 −0.711617
\(446\) 2.15264e9 1.14894
\(447\) 9.88793e7 0.0523635
\(448\) 3.98488e8 0.209384
\(449\) 1.53949e9 0.802630 0.401315 0.915940i \(-0.368553\pi\)
0.401315 + 0.915940i \(0.368553\pi\)
\(450\) −3.49823e8 −0.180969
\(451\) 2.61676e9 1.34322
\(452\) −9.97607e8 −0.508130
\(453\) 1.96771e9 0.994528
\(454\) −5.81683e8 −0.291737
\(455\) −3.80879e7 −0.0189560
\(456\) 2.53143e8 0.125023
\(457\) −3.33563e9 −1.63483 −0.817413 0.576052i \(-0.804592\pi\)
−0.817413 + 0.576052i \(0.804592\pi\)
\(458\) −6.93258e8 −0.337183
\(459\) −2.58253e8 −0.124653
\(460\) 4.60589e8 0.220628
\(461\) 2.62350e9 1.24718 0.623589 0.781753i \(-0.285674\pi\)
0.623589 + 0.781753i \(0.285674\pi\)
\(462\) −1.18494e9 −0.559050
\(463\) −6.85295e8 −0.320881 −0.160441 0.987045i \(-0.551292\pi\)
−0.160441 + 0.987045i \(0.551292\pi\)
\(464\) 2.00629e9 0.932356
\(465\) −1.83051e9 −0.844283
\(466\) −3.34805e9 −1.53264
\(467\) −3.13987e9 −1.42660 −0.713301 0.700858i \(-0.752801\pi\)
−0.713301 + 0.700858i \(0.752801\pi\)
\(468\) 8.91935e6 0.00402228
\(469\) −8.83699e8 −0.395548
\(470\) 4.48266e8 0.199156
\(471\) −2.45564e9 −1.08291
\(472\) 1.94209e8 0.0850105
\(473\) 3.19383e9 1.38771
\(474\) −5.57098e8 −0.240274
\(475\) 3.48098e8 0.149030
\(476\) −6.81251e8 −0.289523
\(477\) −1.33756e9 −0.564286
\(478\) −4.63211e7 −0.0193991
\(479\) 1.33909e9 0.556718 0.278359 0.960477i \(-0.410210\pi\)
0.278359 + 0.960477i \(0.410210\pi\)
\(480\) 8.87674e8 0.366361
\(481\) 8.20319e6 0.00336105
\(482\) 3.32101e9 1.35084
\(483\) −8.99965e8 −0.363422
\(484\) −3.68061e8 −0.147558
\(485\) 8.98946e8 0.357798
\(486\) 1.96122e8 0.0774995
\(487\) −3.02031e9 −1.18495 −0.592476 0.805588i \(-0.701849\pi\)
−0.592476 + 0.805588i \(0.701849\pi\)
\(488\) −3.30741e9 −1.28831
\(489\) 1.51941e9 0.587618
\(490\) 1.25354e8 0.0481342
\(491\) −2.38195e9 −0.908129 −0.454065 0.890969i \(-0.650026\pi\)
−0.454065 + 0.890969i \(0.650026\pi\)
\(492\) −1.14249e9 −0.432489
\(493\) 1.28703e9 0.483754
\(494\) −2.81907e7 −0.0105211
\(495\) 5.49937e8 0.203796
\(496\) −6.68578e9 −2.46018
\(497\) 2.92550e8 0.106894
\(498\) 9.18072e8 0.333100
\(499\) 2.51861e9 0.907421 0.453711 0.891149i \(-0.350100\pi\)
0.453711 + 0.891149i \(0.350100\pi\)
\(500\) 1.38129e9 0.494187
\(501\) 8.66835e8 0.307967
\(502\) −1.37011e8 −0.0483384
\(503\) 8.04366e8 0.281816 0.140908 0.990023i \(-0.454998\pi\)
0.140908 + 0.990023i \(0.454998\pi\)
\(504\) −6.08555e8 −0.211735
\(505\) −1.42532e9 −0.492486
\(506\) 1.87706e9 0.644098
\(507\) −1.69304e9 −0.576952
\(508\) −7.33223e8 −0.248149
\(509\) −5.19329e9 −1.74554 −0.872772 0.488128i \(-0.837680\pi\)
−0.872772 + 0.488128i \(0.837680\pi\)
\(510\) 1.00425e9 0.335232
\(511\) −4.65400e8 −0.154296
\(512\) 7.66530e8 0.252397
\(513\) −1.95155e8 −0.0638217
\(514\) −2.26055e9 −0.734250
\(515\) −3.55860e8 −0.114803
\(516\) −1.39444e9 −0.446815
\(517\) 5.75149e8 0.183047
\(518\) 4.75814e8 0.150412
\(519\) 2.12187e9 0.666243
\(520\) 4.07983e7 0.0127242
\(521\) 4.19746e9 1.30033 0.650166 0.759792i \(-0.274699\pi\)
0.650166 + 0.759792i \(0.274699\pi\)
\(522\) −9.77393e8 −0.300762
\(523\) −3.75242e9 −1.14698 −0.573490 0.819213i \(-0.694411\pi\)
−0.573490 + 0.819213i \(0.694411\pi\)
\(524\) 2.75376e9 0.836117
\(525\) −8.36829e8 −0.252394
\(526\) −5.28305e9 −1.58283
\(527\) −4.28891e9 −1.27647
\(528\) 2.00859e9 0.593845
\(529\) −1.97920e9 −0.581292
\(530\) 5.20127e9 1.51755
\(531\) −1.49721e8 −0.0433963
\(532\) −5.14803e8 −0.148235
\(533\) −1.49661e8 −0.0428117
\(534\) 2.35375e9 0.668907
\(535\) 1.47026e9 0.415102
\(536\) 9.46585e8 0.265511
\(537\) −1.58670e9 −0.442165
\(538\) 2.91516e9 0.807093
\(539\) 1.60837e8 0.0442409
\(540\) −2.40106e8 −0.0656181
\(541\) −1.64171e9 −0.445765 −0.222882 0.974845i \(-0.571547\pi\)
−0.222882 + 0.974845i \(0.571547\pi\)
\(542\) −6.95364e8 −0.187592
\(543\) −1.30046e9 −0.348575
\(544\) 2.07983e9 0.553900
\(545\) −4.76064e9 −1.25973
\(546\) 6.77706e7 0.0178183
\(547\) 6.20746e9 1.62166 0.810828 0.585285i \(-0.199017\pi\)
0.810828 + 0.585285i \(0.199017\pi\)
\(548\) −6.08135e8 −0.157858
\(549\) 2.54978e9 0.657657
\(550\) 1.74538e9 0.447322
\(551\) 9.72574e8 0.247681
\(552\) 9.64008e8 0.243946
\(553\) −1.33266e9 −0.335106
\(554\) −5.66732e9 −1.41610
\(555\) −2.20827e8 −0.0548310
\(556\) −1.80821e9 −0.446157
\(557\) 6.02362e9 1.47694 0.738472 0.674284i \(-0.235548\pi\)
0.738472 + 0.674284i \(0.235548\pi\)
\(558\) 3.25707e9 0.793610
\(559\) −1.82665e8 −0.0442297
\(560\) 3.74486e9 0.901109
\(561\) 1.28851e9 0.308118
\(562\) −5.25136e9 −1.24794
\(563\) −3.76918e9 −0.890159 −0.445079 0.895491i \(-0.646825\pi\)
−0.445079 + 0.895491i \(0.646825\pi\)
\(564\) −2.51113e8 −0.0589376
\(565\) 3.51789e9 0.820565
\(566\) 6.37524e9 1.47788
\(567\) 4.69152e8 0.108087
\(568\) −3.13368e8 −0.0717523
\(569\) −5.40701e9 −1.23045 −0.615226 0.788351i \(-0.710935\pi\)
−0.615226 + 0.788351i \(0.710935\pi\)
\(570\) 7.58884e8 0.171638
\(571\) 3.44502e7 0.00774399 0.00387200 0.999993i \(-0.498768\pi\)
0.00387200 + 0.999993i \(0.498768\pi\)
\(572\) −4.45015e7 −0.00994233
\(573\) 4.46816e9 0.992173
\(574\) −8.68083e9 −1.91589
\(575\) 1.32561e9 0.290790
\(576\) 3.29067e8 0.0717473
\(577\) 1.54181e8 0.0334131 0.0167065 0.999860i \(-0.494682\pi\)
0.0167065 + 0.999860i \(0.494682\pi\)
\(578\) −3.25557e9 −0.701261
\(579\) −3.00671e9 −0.643750
\(580\) 1.19659e9 0.254652
\(581\) 2.19617e9 0.464567
\(582\) −1.59951e9 −0.336323
\(583\) 6.67351e9 1.39481
\(584\) 4.98519e8 0.103571
\(585\) −3.14526e7 −0.00649547
\(586\) −1.03529e10 −2.12531
\(587\) 7.89426e9 1.61093 0.805467 0.592640i \(-0.201914\pi\)
0.805467 + 0.592640i \(0.201914\pi\)
\(588\) −7.02221e7 −0.0142447
\(589\) −3.24101e9 −0.653547
\(590\) 5.82210e8 0.116707
\(591\) 9.84312e8 0.196145
\(592\) −8.06549e8 −0.159774
\(593\) 1.93894e9 0.381833 0.190917 0.981606i \(-0.438854\pi\)
0.190917 + 0.981606i \(0.438854\pi\)
\(594\) −9.78513e8 −0.191564
\(595\) 2.40231e9 0.467542
\(596\) 2.15395e8 0.0416748
\(597\) 2.05542e9 0.395357
\(598\) −1.07355e8 −0.0205290
\(599\) −4.91194e9 −0.933812 −0.466906 0.884307i \(-0.654631\pi\)
−0.466906 + 0.884307i \(0.654631\pi\)
\(600\) 8.96379e8 0.169419
\(601\) −2.50771e9 −0.471211 −0.235606 0.971849i \(-0.575707\pi\)
−0.235606 + 0.971849i \(0.575707\pi\)
\(602\) −1.05952e10 −1.97934
\(603\) −7.29748e8 −0.135538
\(604\) 4.28638e9 0.791519
\(605\) 1.29791e9 0.238286
\(606\) 2.53610e9 0.462928
\(607\) −5.17646e8 −0.0939446 −0.0469723 0.998896i \(-0.514957\pi\)
−0.0469723 + 0.998896i \(0.514957\pi\)
\(608\) 1.57167e9 0.283595
\(609\) −2.33807e9 −0.419466
\(610\) −9.91513e9 −1.76866
\(611\) −3.28945e7 −0.00583417
\(612\) −5.62569e8 −0.0992078
\(613\) −2.24783e9 −0.394141 −0.197070 0.980389i \(-0.563143\pi\)
−0.197070 + 0.980389i \(0.563143\pi\)
\(614\) −6.99023e9 −1.21871
\(615\) 4.02880e9 0.698414
\(616\) 3.03627e9 0.523370
\(617\) −3.82861e9 −0.656211 −0.328106 0.944641i \(-0.606410\pi\)
−0.328106 + 0.944641i \(0.606410\pi\)
\(618\) 6.33188e8 0.107913
\(619\) 1.53060e9 0.259385 0.129693 0.991554i \(-0.458601\pi\)
0.129693 + 0.991554i \(0.458601\pi\)
\(620\) −3.98753e9 −0.671943
\(621\) −7.43181e8 −0.124530
\(622\) 6.11091e9 1.01822
\(623\) 5.63052e9 0.932911
\(624\) −1.14878e8 −0.0189273
\(625\) −2.12804e9 −0.348658
\(626\) −4.17838e8 −0.0680766
\(627\) 9.73689e8 0.157755
\(628\) −5.34928e9 −0.861859
\(629\) −5.17399e8 −0.0828988
\(630\) −1.82436e9 −0.290682
\(631\) 8.76195e9 1.38835 0.694174 0.719807i \(-0.255770\pi\)
0.694174 + 0.719807i \(0.255770\pi\)
\(632\) 1.42749e9 0.224939
\(633\) 6.35004e9 0.995092
\(634\) −3.51900e9 −0.548413
\(635\) 2.58558e9 0.400729
\(636\) −2.91369e9 −0.449101
\(637\) −9.19873e6 −0.00141007
\(638\) 4.87652e9 0.743426
\(639\) 2.41584e8 0.0366282
\(640\) −5.48785e9 −0.827508
\(641\) −8.85004e9 −1.32722 −0.663609 0.748080i \(-0.730976\pi\)
−0.663609 + 0.748080i \(0.730976\pi\)
\(642\) −2.61606e9 −0.390188
\(643\) −1.79228e9 −0.265869 −0.132935 0.991125i \(-0.542440\pi\)
−0.132935 + 0.991125i \(0.542440\pi\)
\(644\) −1.96045e9 −0.289238
\(645\) 4.91727e9 0.721548
\(646\) 1.77807e9 0.259499
\(647\) −7.01663e9 −1.01851 −0.509253 0.860617i \(-0.670078\pi\)
−0.509253 + 0.860617i \(0.670078\pi\)
\(648\) −5.02538e8 −0.0725531
\(649\) 7.47007e8 0.107267
\(650\) −9.98234e7 −0.0142572
\(651\) 7.79139e9 1.10683
\(652\) 3.30983e9 0.467670
\(653\) 5.63028e9 0.791287 0.395644 0.918404i \(-0.370522\pi\)
0.395644 + 0.918404i \(0.370522\pi\)
\(654\) 8.47070e9 1.18412
\(655\) −9.71067e9 −1.35022
\(656\) 1.47148e10 2.03513
\(657\) −3.84322e8 −0.0528709
\(658\) −1.90800e9 −0.261088
\(659\) 2.18357e9 0.297214 0.148607 0.988896i \(-0.452521\pi\)
0.148607 + 0.988896i \(0.452521\pi\)
\(660\) 1.19796e9 0.162196
\(661\) 1.27138e10 1.71227 0.856134 0.516755i \(-0.172860\pi\)
0.856134 + 0.516755i \(0.172860\pi\)
\(662\) 9.14153e9 1.22466
\(663\) −7.36936e7 −0.00982047
\(664\) −2.35245e9 −0.311840
\(665\) 1.81536e9 0.239380
\(666\) 3.92921e8 0.0515402
\(667\) 3.70372e9 0.483279
\(668\) 1.88828e9 0.245103
\(669\) 4.25234e9 0.549081
\(670\) 2.83772e9 0.364508
\(671\) −1.27216e10 −1.62560
\(672\) −3.77829e9 −0.480289
\(673\) −5.04564e9 −0.638063 −0.319031 0.947744i \(-0.603358\pi\)
−0.319031 + 0.947744i \(0.603358\pi\)
\(674\) 2.01494e8 0.0253485
\(675\) −6.91043e8 −0.0864853
\(676\) −3.68806e9 −0.459182
\(677\) −4.93561e9 −0.611337 −0.305668 0.952138i \(-0.598880\pi\)
−0.305668 + 0.952138i \(0.598880\pi\)
\(678\) −6.25945e9 −0.771316
\(679\) −3.82627e9 −0.469063
\(680\) −2.57327e9 −0.313837
\(681\) −1.14906e9 −0.139421
\(682\) −1.62505e10 −1.96165
\(683\) −8.54468e9 −1.02618 −0.513090 0.858335i \(-0.671499\pi\)
−0.513090 + 0.858335i \(0.671499\pi\)
\(684\) −4.25118e8 −0.0507941
\(685\) 2.14448e9 0.254921
\(686\) −1.04705e10 −1.23832
\(687\) −1.36947e9 −0.161140
\(688\) 1.79599e10 2.10254
\(689\) −3.81679e8 −0.0444560
\(690\) 2.88995e9 0.334903
\(691\) −2.27573e9 −0.262390 −0.131195 0.991357i \(-0.541881\pi\)
−0.131195 + 0.991357i \(0.541881\pi\)
\(692\) 4.62220e9 0.530246
\(693\) −2.34075e9 −0.267171
\(694\) −8.27237e9 −0.939446
\(695\) 6.37635e9 0.720486
\(696\) 2.50445e9 0.281566
\(697\) 9.43952e9 1.05593
\(698\) −1.40342e10 −1.56205
\(699\) −6.61377e9 −0.732451
\(700\) −1.82292e9 −0.200874
\(701\) 2.18428e8 0.0239494 0.0119747 0.999928i \(-0.496188\pi\)
0.0119747 + 0.999928i \(0.496188\pi\)
\(702\) 5.59641e7 0.00610562
\(703\) −3.90984e8 −0.0424439
\(704\) −1.64182e9 −0.177346
\(705\) 8.85508e8 0.0951767
\(706\) −1.93012e10 −2.06428
\(707\) 6.06674e9 0.645636
\(708\) −3.26147e8 −0.0345380
\(709\) 1.81988e10 1.91770 0.958849 0.283917i \(-0.0916341\pi\)
0.958849 + 0.283917i \(0.0916341\pi\)
\(710\) −9.39431e8 −0.0985056
\(711\) −1.10050e9 −0.114827
\(712\) −6.03119e9 −0.626215
\(713\) −1.23423e10 −1.27521
\(714\) −4.27448e9 −0.439481
\(715\) 1.56927e8 0.0160556
\(716\) −3.45641e9 −0.351908
\(717\) −9.15030e7 −0.00927083
\(718\) −3.96066e9 −0.399330
\(719\) −6.16960e9 −0.619022 −0.309511 0.950896i \(-0.600165\pi\)
−0.309511 + 0.950896i \(0.600165\pi\)
\(720\) 3.09246e9 0.308774
\(721\) 1.51468e9 0.150504
\(722\) −1.08739e10 −1.07523
\(723\) 6.56035e9 0.645570
\(724\) −2.83286e9 −0.277422
\(725\) 3.44389e9 0.335634
\(726\) −2.30939e9 −0.223985
\(727\) 5.20930e9 0.502816 0.251408 0.967881i \(-0.419106\pi\)
0.251408 + 0.967881i \(0.419106\pi\)
\(728\) −1.73654e8 −0.0166811
\(729\) 3.87420e8 0.0370370
\(730\) 1.49448e9 0.142188
\(731\) 1.15212e10 1.09090
\(732\) 5.55434e9 0.523412
\(733\) 9.86561e9 0.925252 0.462626 0.886554i \(-0.346907\pi\)
0.462626 + 0.886554i \(0.346907\pi\)
\(734\) −8.35704e9 −0.780039
\(735\) 2.47626e8 0.0230033
\(736\) 5.98516e9 0.553355
\(737\) 3.64095e9 0.335026
\(738\) −7.16853e9 −0.656497
\(739\) 6.68716e9 0.609517 0.304759 0.952430i \(-0.401424\pi\)
0.304759 + 0.952430i \(0.401424\pi\)
\(740\) −4.81041e8 −0.0436386
\(741\) −5.56882e7 −0.00502805
\(742\) −2.21387e10 −1.98947
\(743\) 1.17332e10 1.04944 0.524719 0.851275i \(-0.324170\pi\)
0.524719 + 0.851275i \(0.324170\pi\)
\(744\) −8.34584e9 −0.742959
\(745\) −7.59553e8 −0.0672994
\(746\) 8.32915e8 0.0734539
\(747\) 1.81357e9 0.159189
\(748\) 2.80683e9 0.245223
\(749\) −6.25800e9 −0.544188
\(750\) 8.66688e9 0.750151
\(751\) −1.63538e10 −1.40890 −0.704448 0.709756i \(-0.748805\pi\)
−0.704448 + 0.709756i \(0.748805\pi\)
\(752\) 3.23423e9 0.277338
\(753\) −2.70653e8 −0.0231010
\(754\) −2.78903e8 −0.0236948
\(755\) −1.51152e10 −1.27820
\(756\) 1.02198e9 0.0860236
\(757\) −7.22904e9 −0.605683 −0.302841 0.953041i \(-0.597935\pi\)
−0.302841 + 0.953041i \(0.597935\pi\)
\(758\) −1.35986e10 −1.13410
\(759\) 3.70796e9 0.307815
\(760\) −1.94455e9 −0.160683
\(761\) −1.16317e9 −0.0956747 −0.0478373 0.998855i \(-0.515233\pi\)
−0.0478373 + 0.998855i \(0.515233\pi\)
\(762\) −4.60058e9 −0.376678
\(763\) 2.02632e10 1.65147
\(764\) 9.73328e9 0.789646
\(765\) 1.98380e9 0.160208
\(766\) −2.31671e9 −0.186239
\(767\) −4.27236e7 −0.00341888
\(768\) 8.20461e9 0.653573
\(769\) 2.40619e10 1.90804 0.954022 0.299735i \(-0.0968984\pi\)
0.954022 + 0.299735i \(0.0968984\pi\)
\(770\) 9.10230e9 0.718511
\(771\) −4.46552e9 −0.350899
\(772\) −6.54972e9 −0.512344
\(773\) −1.37533e9 −0.107097 −0.0535486 0.998565i \(-0.517053\pi\)
−0.0535486 + 0.998565i \(0.517053\pi\)
\(774\) −8.74939e9 −0.678242
\(775\) −1.14764e10 −0.885627
\(776\) 4.09855e9 0.314858
\(777\) 9.39926e8 0.0718820
\(778\) −2.63634e9 −0.200712
\(779\) 7.13318e9 0.540633
\(780\) −6.85151e7 −0.00516958
\(781\) −1.20534e9 −0.0905381
\(782\) 6.77118e9 0.506338
\(783\) −1.93075e9 −0.143734
\(784\) 9.04432e8 0.0670301
\(785\) 1.88633e10 1.39179
\(786\) 1.72784e10 1.26918
\(787\) 1.62448e10 1.18796 0.593980 0.804480i \(-0.297556\pi\)
0.593980 + 0.804480i \(0.297556\pi\)
\(788\) 2.14419e9 0.156107
\(789\) −1.04362e10 −0.756436
\(790\) 4.27941e9 0.308809
\(791\) −1.49735e10 −1.07574
\(792\) 2.50732e9 0.179338
\(793\) 7.27590e8 0.0518120
\(794\) 2.61446e10 1.85357
\(795\) 1.02746e10 0.725240
\(796\) 4.47744e9 0.314655
\(797\) −1.10596e10 −0.773814 −0.386907 0.922119i \(-0.626457\pi\)
−0.386907 + 0.922119i \(0.626457\pi\)
\(798\) −3.23011e9 −0.225013
\(799\) 2.07475e9 0.143897
\(800\) 5.56528e9 0.384301
\(801\) 4.64961e9 0.319671
\(802\) 2.75057e9 0.188284
\(803\) 1.91750e9 0.130687
\(804\) −1.58966e9 −0.107872
\(805\) 6.91319e9 0.467082
\(806\) 9.29418e8 0.0625228
\(807\) 5.75862e9 0.385710
\(808\) −6.49846e9 −0.433382
\(809\) −2.52839e10 −1.67890 −0.839450 0.543436i \(-0.817123\pi\)
−0.839450 + 0.543436i \(0.817123\pi\)
\(810\) −1.50653e9 −0.0996050
\(811\) 7.05869e9 0.464677 0.232339 0.972635i \(-0.425362\pi\)
0.232339 + 0.972635i \(0.425362\pi\)
\(812\) −5.09317e9 −0.333843
\(813\) −1.37363e9 −0.0896504
\(814\) −1.96041e9 −0.127398
\(815\) −1.16716e10 −0.755227
\(816\) 7.24566e9 0.466834
\(817\) 8.70625e9 0.558540
\(818\) 4.79930e9 0.306578
\(819\) 1.33875e8 0.00851539
\(820\) 8.77620e9 0.555850
\(821\) −1.72937e10 −1.09066 −0.545328 0.838223i \(-0.683595\pi\)
−0.545328 + 0.838223i \(0.683595\pi\)
\(822\) −3.81572e9 −0.239621
\(823\) 1.28299e10 0.802278 0.401139 0.916017i \(-0.368614\pi\)
0.401139 + 0.916017i \(0.368614\pi\)
\(824\) −1.62247e9 −0.101025
\(825\) 3.44783e9 0.213775
\(826\) −2.47812e9 −0.153000
\(827\) 1.83922e10 1.13074 0.565372 0.824836i \(-0.308732\pi\)
0.565372 + 0.824836i \(0.308732\pi\)
\(828\) −1.61892e9 −0.0991102
\(829\) −4.09547e9 −0.249668 −0.124834 0.992178i \(-0.539840\pi\)
−0.124834 + 0.992178i \(0.539840\pi\)
\(830\) −7.05229e9 −0.428111
\(831\) −1.11953e10 −0.676754
\(832\) 9.39006e7 0.00565245
\(833\) 5.80190e8 0.0347787
\(834\) −1.13456e10 −0.677244
\(835\) −6.65870e9 −0.395810
\(836\) 2.12105e9 0.125553
\(837\) 6.43404e9 0.379267
\(838\) 5.02041e9 0.294704
\(839\) −1.88735e9 −0.110328 −0.0551641 0.998477i \(-0.517568\pi\)
−0.0551641 + 0.998477i \(0.517568\pi\)
\(840\) 4.67469e9 0.272130
\(841\) −7.62778e9 −0.442193
\(842\) 1.65361e10 0.954641
\(843\) −1.03736e10 −0.596392
\(844\) 1.38327e10 0.791969
\(845\) 1.30053e10 0.741519
\(846\) −1.57560e9 −0.0894643
\(847\) −5.52440e9 −0.312387
\(848\) 3.75272e10 2.11330
\(849\) 1.25937e10 0.706279
\(850\) 6.29615e9 0.351649
\(851\) −1.48893e9 −0.0828173
\(852\) 5.26259e8 0.0291515
\(853\) 6.81026e8 0.0375701 0.0187850 0.999824i \(-0.494020\pi\)
0.0187850 + 0.999824i \(0.494020\pi\)
\(854\) 4.22027e10 2.31866
\(855\) 1.49911e9 0.0820259
\(856\) 6.70332e9 0.365285
\(857\) 6.43948e9 0.349476 0.174738 0.984615i \(-0.444092\pi\)
0.174738 + 0.984615i \(0.444092\pi\)
\(858\) −2.79223e8 −0.0150919
\(859\) 1.88733e10 1.01595 0.507974 0.861373i \(-0.330395\pi\)
0.507974 + 0.861373i \(0.330395\pi\)
\(860\) 1.07116e10 0.574262
\(861\) −1.71482e10 −0.915603
\(862\) 8.26541e9 0.439530
\(863\) 1.84568e10 0.977505 0.488753 0.872422i \(-0.337452\pi\)
0.488753 + 0.872422i \(0.337452\pi\)
\(864\) −3.12007e9 −0.164576
\(865\) −1.62994e10 −0.856279
\(866\) −1.14372e10 −0.598422
\(867\) −6.43108e9 −0.335133
\(868\) 1.69725e10 0.880900
\(869\) 5.49072e9 0.283831
\(870\) 7.50797e9 0.386549
\(871\) −2.08237e8 −0.0106781
\(872\) −2.17051e10 −1.10855
\(873\) −3.15969e9 −0.160729
\(874\) 5.11679e9 0.259243
\(875\) 2.07325e10 1.04622
\(876\) −8.37193e8 −0.0420786
\(877\) 1.79041e10 0.896301 0.448151 0.893958i \(-0.352083\pi\)
0.448151 + 0.893958i \(0.352083\pi\)
\(878\) −3.70248e10 −1.84613
\(879\) −2.04513e10 −1.01569
\(880\) −1.54293e10 −0.763230
\(881\) −1.86101e10 −0.916922 −0.458461 0.888715i \(-0.651599\pi\)
−0.458461 + 0.888715i \(0.651599\pi\)
\(882\) −4.40606e8 −0.0216227
\(883\) −1.45294e10 −0.710207 −0.355103 0.934827i \(-0.615554\pi\)
−0.355103 + 0.934827i \(0.615554\pi\)
\(884\) −1.60531e8 −0.00781586
\(885\) 1.15010e9 0.0557744
\(886\) −1.31063e10 −0.633084
\(887\) −3.46786e9 −0.166851 −0.0834256 0.996514i \(-0.526586\pi\)
−0.0834256 + 0.996514i \(0.526586\pi\)
\(888\) −1.00681e9 −0.0482507
\(889\) −1.10053e10 −0.525345
\(890\) −1.80806e10 −0.859702
\(891\) −1.93296e9 −0.0915486
\(892\) 9.26313e9 0.436999
\(893\) 1.56783e9 0.0736749
\(894\) 1.35149e9 0.0632602
\(895\) 1.21884e10 0.568286
\(896\) 2.33585e10 1.08484
\(897\) −2.12070e8 −0.00981081
\(898\) 2.10419e10 0.969655
\(899\) −3.20647e10 −1.47187
\(900\) −1.50534e9 −0.0688314
\(901\) 2.40736e10 1.09649
\(902\) 3.57660e10 1.62274
\(903\) −2.09298e10 −0.945930
\(904\) 1.60391e10 0.722088
\(905\) 9.98961e9 0.448001
\(906\) 2.68947e10 1.20149
\(907\) −4.86854e9 −0.216657 −0.108329 0.994115i \(-0.534550\pi\)
−0.108329 + 0.994115i \(0.534550\pi\)
\(908\) −2.50308e9 −0.110962
\(909\) 5.00984e9 0.221233
\(910\) −5.20588e8 −0.0229007
\(911\) 5.41157e9 0.237142 0.118571 0.992946i \(-0.462169\pi\)
0.118571 + 0.992946i \(0.462169\pi\)
\(912\) 5.47534e9 0.239017
\(913\) −9.04846e9 −0.393484
\(914\) −4.55916e10 −1.97503
\(915\) −1.95864e10 −0.845243
\(916\) −2.98320e9 −0.128247
\(917\) 4.13325e10 1.77010
\(918\) −3.52982e9 −0.150592
\(919\) −4.36861e10 −1.85669 −0.928343 0.371724i \(-0.878767\pi\)
−0.928343 + 0.371724i \(0.878767\pi\)
\(920\) −7.40515e9 −0.313528
\(921\) −1.38086e10 −0.582424
\(922\) 3.58582e10 1.50671
\(923\) 6.89371e7 0.00288567
\(924\) −5.09900e9 −0.212634
\(925\) −1.38448e9 −0.0575161
\(926\) −9.36666e9 −0.387656
\(927\) 1.25081e9 0.0515716
\(928\) 1.55492e10 0.638690
\(929\) 1.09031e10 0.446164 0.223082 0.974800i \(-0.428388\pi\)
0.223082 + 0.974800i \(0.428388\pi\)
\(930\) −2.50196e10 −1.01998
\(931\) 4.38434e8 0.0178066
\(932\) −1.44072e10 −0.582939
\(933\) 1.20716e10 0.486606
\(934\) −4.29159e10 −1.72347
\(935\) −9.89782e9 −0.396003
\(936\) −1.43401e8 −0.00571594
\(937\) 3.25215e10 1.29146 0.645732 0.763564i \(-0.276552\pi\)
0.645732 + 0.763564i \(0.276552\pi\)
\(938\) −1.20785e10 −0.477861
\(939\) −8.25401e8 −0.0325339
\(940\) 1.92896e9 0.0757487
\(941\) −2.70964e10 −1.06010 −0.530052 0.847965i \(-0.677828\pi\)
−0.530052 + 0.847965i \(0.677828\pi\)
\(942\) −3.35638e10 −1.30826
\(943\) 2.71643e10 1.05489
\(944\) 4.20064e9 0.162523
\(945\) −3.60385e9 −0.138917
\(946\) 4.36535e10 1.67649
\(947\) 2.06829e10 0.791382 0.395691 0.918384i \(-0.370505\pi\)
0.395691 + 0.918384i \(0.370505\pi\)
\(948\) −2.39728e9 −0.0913880
\(949\) −1.09668e8 −0.00416531
\(950\) 4.75783e9 0.180043
\(951\) −6.95147e9 −0.262087
\(952\) 1.09528e10 0.411431
\(953\) −3.31031e10 −1.23892 −0.619460 0.785028i \(-0.712649\pi\)
−0.619460 + 0.785028i \(0.712649\pi\)
\(954\) −1.82819e10 −0.681712
\(955\) −3.43227e10 −1.27518
\(956\) −1.99327e8 −0.00737841
\(957\) 9.63313e9 0.355284
\(958\) 1.83027e10 0.672569
\(959\) −9.12777e9 −0.334195
\(960\) −2.52777e9 −0.0922121
\(961\) 7.93400e10 2.88377
\(962\) 1.12122e8 0.00406048
\(963\) −5.16778e9 −0.186471
\(964\) 1.42908e10 0.513792
\(965\) 2.30965e10 0.827370
\(966\) −1.23008e10 −0.439049
\(967\) −2.24560e10 −0.798618 −0.399309 0.916816i \(-0.630750\pi\)
−0.399309 + 0.916816i \(0.630750\pi\)
\(968\) 5.91752e9 0.209689
\(969\) 3.51242e9 0.124015
\(970\) 1.22868e10 0.432254
\(971\) −1.83485e10 −0.643182 −0.321591 0.946879i \(-0.604218\pi\)
−0.321591 + 0.946879i \(0.604218\pi\)
\(972\) 8.43942e8 0.0294768
\(973\) −2.71403e10 −0.944538
\(974\) −4.12818e10 −1.43154
\(975\) −1.97192e8 −0.00681355
\(976\) −7.15376e10 −2.46298
\(977\) 2.21334e10 0.759306 0.379653 0.925129i \(-0.376043\pi\)
0.379653 + 0.925129i \(0.376043\pi\)
\(978\) 2.07674e10 0.709899
\(979\) −2.31984e10 −0.790166
\(980\) 5.39420e8 0.0183078
\(981\) 1.67331e10 0.565893
\(982\) −3.25566e10 −1.09711
\(983\) 5.39105e10 1.81024 0.905121 0.425155i \(-0.139780\pi\)
0.905121 + 0.425155i \(0.139780\pi\)
\(984\) 1.83685e10 0.614597
\(985\) −7.56112e9 −0.252092
\(986\) 1.75912e10 0.584422
\(987\) −3.76907e9 −0.124774
\(988\) −1.21309e8 −0.00400170
\(989\) 3.31548e10 1.08983
\(990\) 7.51657e9 0.246205
\(991\) −1.06845e10 −0.348734 −0.174367 0.984681i \(-0.555788\pi\)
−0.174367 + 0.984681i \(0.555788\pi\)
\(992\) −5.18162e10 −1.68529
\(993\) 1.80583e10 0.585267
\(994\) 3.99859e9 0.129138
\(995\) −1.57889e10 −0.508127
\(996\) 3.95061e9 0.126694
\(997\) −3.63110e10 −1.16039 −0.580196 0.814477i \(-0.697024\pi\)
−0.580196 + 0.814477i \(0.697024\pi\)
\(998\) 3.44245e10 1.09625
\(999\) 7.76180e8 0.0246311
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.1.14 17
3.2 odd 2 531.8.a.d.1.4 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.14 17 1.1 even 1 trivial
531.8.a.d.1.4 17 3.2 odd 2