Properties

Label 177.8.a.c.1.6
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
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: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
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 \(-8.14464\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.14464 q^{2} -27.0000 q^{3} -61.6648 q^{4} -54.3103 q^{5} +219.905 q^{6} -413.400 q^{7} +1544.75 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-8.14464 q^{2} -27.0000 q^{3} -61.6648 q^{4} -54.3103 q^{5} +219.905 q^{6} -413.400 q^{7} +1544.75 q^{8} +729.000 q^{9} +442.338 q^{10} +5302.90 q^{11} +1664.95 q^{12} -13494.1 q^{13} +3367.00 q^{14} +1466.38 q^{15} -4688.37 q^{16} +13725.3 q^{17} -5937.45 q^{18} -57847.2 q^{19} +3349.03 q^{20} +11161.8 q^{21} -43190.2 q^{22} -11834.5 q^{23} -41708.3 q^{24} -75175.4 q^{25} +109904. q^{26} -19683.0 q^{27} +25492.2 q^{28} -137312. q^{29} -11943.1 q^{30} +110476. q^{31} -159543. q^{32} -143178. q^{33} -111788. q^{34} +22451.9 q^{35} -44953.6 q^{36} +154147. q^{37} +471145. q^{38} +364339. q^{39} -83896.0 q^{40} +58042.4 q^{41} -90908.9 q^{42} -607469. q^{43} -327002. q^{44} -39592.2 q^{45} +96387.4 q^{46} -765556. q^{47} +126586. q^{48} -652643. q^{49} +612277. q^{50} -370584. q^{51} +832107. q^{52} +779953. q^{53} +160311. q^{54} -288002. q^{55} -638601. q^{56} +1.56188e6 q^{57} +1.11836e6 q^{58} -205379. q^{59} -90423.9 q^{60} +745383. q^{61} -899790. q^{62} -301369. q^{63} +1.89953e6 q^{64} +732866. q^{65} +1.16614e6 q^{66} -343978. q^{67} -846369. q^{68} +319530. q^{69} -182863. q^{70} +4.41562e6 q^{71} +1.12612e6 q^{72} +1.38449e6 q^{73} -1.25547e6 q^{74} +2.02974e6 q^{75} +3.56714e6 q^{76} -2.19222e6 q^{77} -2.96742e6 q^{78} -6.92496e6 q^{79} +254627. q^{80} +531441. q^{81} -472735. q^{82} +4.41025e6 q^{83} -688290. q^{84} -745427. q^{85} +4.94762e6 q^{86} +3.70742e6 q^{87} +8.19166e6 q^{88} -2.65103e6 q^{89} +322465. q^{90} +5.57844e6 q^{91} +729769. q^{92} -2.98286e6 q^{93} +6.23518e6 q^{94} +3.14170e6 q^{95} +4.30767e6 q^{96} +531245. q^{97} +5.31555e6 q^{98} +3.86581e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.14464 −0.719892 −0.359946 0.932973i \(-0.617205\pi\)
−0.359946 + 0.932973i \(0.617205\pi\)
\(3\) −27.0000 −0.577350
\(4\) −61.6648 −0.481756
\(5\) −54.3103 −0.194307 −0.0971533 0.995269i \(-0.530974\pi\)
−0.0971533 + 0.995269i \(0.530974\pi\)
\(6\) 219.905 0.415630
\(7\) −413.400 −0.455541 −0.227770 0.973715i \(-0.573144\pi\)
−0.227770 + 0.973715i \(0.573144\pi\)
\(8\) 1544.75 1.06670
\(9\) 729.000 0.333333
\(10\) 442.338 0.139880
\(11\) 5302.90 1.20126 0.600632 0.799525i \(-0.294916\pi\)
0.600632 + 0.799525i \(0.294916\pi\)
\(12\) 1664.95 0.278142
\(13\) −13494.1 −1.70349 −0.851747 0.523953i \(-0.824457\pi\)
−0.851747 + 0.523953i \(0.824457\pi\)
\(14\) 3367.00 0.327940
\(15\) 1466.38 0.112183
\(16\) −4688.37 −0.286155
\(17\) 13725.3 0.677566 0.338783 0.940865i \(-0.389985\pi\)
0.338783 + 0.940865i \(0.389985\pi\)
\(18\) −5937.45 −0.239964
\(19\) −57847.2 −1.93484 −0.967420 0.253177i \(-0.918525\pi\)
−0.967420 + 0.253177i \(0.918525\pi\)
\(20\) 3349.03 0.0936083
\(21\) 11161.8 0.263007
\(22\) −43190.2 −0.864780
\(23\) −11834.5 −0.202815 −0.101408 0.994845i \(-0.532335\pi\)
−0.101408 + 0.994845i \(0.532335\pi\)
\(24\) −41708.3 −0.615862
\(25\) −75175.4 −0.962245
\(26\) 109904. 1.22633
\(27\) −19683.0 −0.192450
\(28\) 25492.2 0.219460
\(29\) −137312. −1.04548 −0.522739 0.852493i \(-0.675090\pi\)
−0.522739 + 0.852493i \(0.675090\pi\)
\(30\) −11943.1 −0.0807596
\(31\) 110476. 0.666044 0.333022 0.942919i \(-0.391932\pi\)
0.333022 + 0.942919i \(0.391932\pi\)
\(32\) −159543. −0.860703
\(33\) −143178. −0.693550
\(34\) −111788. −0.487774
\(35\) 22451.9 0.0885146
\(36\) −44953.6 −0.160585
\(37\) 154147. 0.500297 0.250149 0.968207i \(-0.419520\pi\)
0.250149 + 0.968207i \(0.419520\pi\)
\(38\) 471145. 1.39288
\(39\) 364339. 0.983513
\(40\) −83896.0 −0.207268
\(41\) 58042.4 0.131523 0.0657615 0.997835i \(-0.479052\pi\)
0.0657615 + 0.997835i \(0.479052\pi\)
\(42\) −90908.9 −0.189336
\(43\) −607469. −1.16516 −0.582578 0.812775i \(-0.697956\pi\)
−0.582578 + 0.812775i \(0.697956\pi\)
\(44\) −327002. −0.578716
\(45\) −39592.2 −0.0647689
\(46\) 96387.4 0.146005
\(47\) −765556. −1.07556 −0.537780 0.843085i \(-0.680737\pi\)
−0.537780 + 0.843085i \(0.680737\pi\)
\(48\) 126586. 0.165212
\(49\) −652643. −0.792483
\(50\) 612277. 0.692712
\(51\) −370584. −0.391193
\(52\) 832107. 0.820668
\(53\) 779953. 0.719619 0.359810 0.933026i \(-0.382842\pi\)
0.359810 + 0.933026i \(0.382842\pi\)
\(54\) 160311. 0.138543
\(55\) −288002. −0.233414
\(56\) −638601. −0.485927
\(57\) 1.56188e6 1.11708
\(58\) 1.11836e6 0.752630
\(59\) −205379. −0.130189
\(60\) −90423.9 −0.0540448
\(61\) 745383. 0.420460 0.210230 0.977652i \(-0.432579\pi\)
0.210230 + 0.977652i \(0.432579\pi\)
\(62\) −899790. −0.479480
\(63\) −301369. −0.151847
\(64\) 1.89953e6 0.905768
\(65\) 732866. 0.331000
\(66\) 1.16614e6 0.499281
\(67\) −343978. −0.139723 −0.0698616 0.997557i \(-0.522256\pi\)
−0.0698616 + 0.997557i \(0.522256\pi\)
\(68\) −846369. −0.326421
\(69\) 319530. 0.117095
\(70\) −182863. −0.0637209
\(71\) 4.41562e6 1.46416 0.732078 0.681221i \(-0.238551\pi\)
0.732078 + 0.681221i \(0.238551\pi\)
\(72\) 1.12612e6 0.355568
\(73\) 1.38449e6 0.416544 0.208272 0.978071i \(-0.433216\pi\)
0.208272 + 0.978071i \(0.433216\pi\)
\(74\) −1.25547e6 −0.360160
\(75\) 2.02974e6 0.555552
\(76\) 3.56714e6 0.932121
\(77\) −2.19222e6 −0.547225
\(78\) −2.96742e6 −0.708023
\(79\) −6.92496e6 −1.58024 −0.790119 0.612953i \(-0.789981\pi\)
−0.790119 + 0.612953i \(0.789981\pi\)
\(80\) 254627. 0.0556019
\(81\) 531441. 0.111111
\(82\) −472735. −0.0946824
\(83\) 4.41025e6 0.846623 0.423312 0.905984i \(-0.360868\pi\)
0.423312 + 0.905984i \(0.360868\pi\)
\(84\) −688290. −0.126705
\(85\) −745427. −0.131655
\(86\) 4.94762e6 0.838787
\(87\) 3.70742e6 0.603607
\(88\) 8.19166e6 1.28139
\(89\) −2.65103e6 −0.398612 −0.199306 0.979937i \(-0.563869\pi\)
−0.199306 + 0.979937i \(0.563869\pi\)
\(90\) 322465. 0.0466266
\(91\) 5.57844e6 0.776011
\(92\) 729769. 0.0977075
\(93\) −2.98286e6 −0.384541
\(94\) 6.23518e6 0.774287
\(95\) 3.14170e6 0.375952
\(96\) 4.30767e6 0.496927
\(97\) 531245. 0.0591009 0.0295504 0.999563i \(-0.490592\pi\)
0.0295504 + 0.999563i \(0.490592\pi\)
\(98\) 5.31555e6 0.570502
\(99\) 3.86581e6 0.400422
\(100\) 4.63567e6 0.463567
\(101\) 2.90692e6 0.280742 0.140371 0.990099i \(-0.455170\pi\)
0.140371 + 0.990099i \(0.455170\pi\)
\(102\) 3.01827e6 0.281616
\(103\) 1.71445e7 1.54595 0.772974 0.634438i \(-0.218768\pi\)
0.772974 + 0.634438i \(0.218768\pi\)
\(104\) −2.08450e7 −1.81712
\(105\) −606201. −0.0511039
\(106\) −6.35244e6 −0.518048
\(107\) −7.73354e6 −0.610289 −0.305144 0.952306i \(-0.598705\pi\)
−0.305144 + 0.952306i \(0.598705\pi\)
\(108\) 1.21375e6 0.0927140
\(109\) 5.26465e6 0.389383 0.194691 0.980865i \(-0.437629\pi\)
0.194691 + 0.980865i \(0.437629\pi\)
\(110\) 2.34567e6 0.168033
\(111\) −4.16196e6 −0.288847
\(112\) 1.93817e6 0.130355
\(113\) 2.92949e7 1.90993 0.954966 0.296716i \(-0.0958915\pi\)
0.954966 + 0.296716i \(0.0958915\pi\)
\(114\) −1.27209e7 −0.804177
\(115\) 642733. 0.0394083
\(116\) 8.46729e6 0.503665
\(117\) −9.83716e6 −0.567831
\(118\) 1.67274e6 0.0937219
\(119\) −5.67405e6 −0.308659
\(120\) 2.26519e6 0.119666
\(121\) 8.63353e6 0.443037
\(122\) −6.07088e6 −0.302686
\(123\) −1.56715e6 −0.0759349
\(124\) −6.81249e6 −0.320871
\(125\) 8.32580e6 0.381277
\(126\) 2.45454e6 0.109313
\(127\) 1.91366e7 0.828995 0.414497 0.910051i \(-0.363957\pi\)
0.414497 + 0.910051i \(0.363957\pi\)
\(128\) 4.95050e6 0.208648
\(129\) 1.64017e7 0.672703
\(130\) −5.96894e6 −0.238284
\(131\) 2.05015e6 0.0796775 0.0398387 0.999206i \(-0.487316\pi\)
0.0398387 + 0.999206i \(0.487316\pi\)
\(132\) 8.82905e6 0.334122
\(133\) 2.39141e7 0.881399
\(134\) 2.80157e6 0.100586
\(135\) 1.06899e6 0.0373943
\(136\) 2.12022e7 0.722762
\(137\) −1.22263e7 −0.406232 −0.203116 0.979155i \(-0.565107\pi\)
−0.203116 + 0.979155i \(0.565107\pi\)
\(138\) −2.60246e6 −0.0842961
\(139\) 2.28834e7 0.722719 0.361359 0.932427i \(-0.382313\pi\)
0.361359 + 0.932427i \(0.382313\pi\)
\(140\) −1.38449e6 −0.0426424
\(141\) 2.06700e7 0.620975
\(142\) −3.59636e7 −1.05403
\(143\) −7.15575e7 −2.04635
\(144\) −3.41782e6 −0.0953851
\(145\) 7.45744e6 0.203143
\(146\) −1.12762e7 −0.299866
\(147\) 1.76214e7 0.457540
\(148\) −9.50542e6 −0.241021
\(149\) −1.01691e7 −0.251843 −0.125921 0.992040i \(-0.540189\pi\)
−0.125921 + 0.992040i \(0.540189\pi\)
\(150\) −1.65315e7 −0.399938
\(151\) −5.75910e7 −1.36124 −0.680620 0.732636i \(-0.738290\pi\)
−0.680620 + 0.732636i \(0.738290\pi\)
\(152\) −8.93597e7 −2.06390
\(153\) 1.00058e7 0.225855
\(154\) 1.78548e7 0.393943
\(155\) −6.00000e6 −0.129417
\(156\) −2.24669e7 −0.473813
\(157\) −1.36932e7 −0.282395 −0.141197 0.989981i \(-0.545095\pi\)
−0.141197 + 0.989981i \(0.545095\pi\)
\(158\) 5.64014e7 1.13760
\(159\) −2.10587e7 −0.415472
\(160\) 8.66484e6 0.167240
\(161\) 4.89236e6 0.0923907
\(162\) −4.32840e6 −0.0799880
\(163\) −5.01230e7 −0.906527 −0.453263 0.891377i \(-0.649740\pi\)
−0.453263 + 0.891377i \(0.649740\pi\)
\(164\) −3.57917e6 −0.0633620
\(165\) 7.77606e6 0.134761
\(166\) −3.59199e7 −0.609477
\(167\) 3.87403e6 0.0643659 0.0321829 0.999482i \(-0.489754\pi\)
0.0321829 + 0.999482i \(0.489754\pi\)
\(168\) 1.72422e7 0.280550
\(169\) 1.19341e8 1.90189
\(170\) 6.07124e6 0.0947777
\(171\) −4.21706e7 −0.644947
\(172\) 3.74594e7 0.561321
\(173\) −5.96811e7 −0.876346 −0.438173 0.898891i \(-0.644374\pi\)
−0.438173 + 0.898891i \(0.644374\pi\)
\(174\) −3.01956e7 −0.434531
\(175\) 3.10775e7 0.438342
\(176\) −2.48619e7 −0.343748
\(177\) 5.54523e6 0.0751646
\(178\) 2.15917e7 0.286957
\(179\) −1.10240e8 −1.43666 −0.718330 0.695703i \(-0.755093\pi\)
−0.718330 + 0.695703i \(0.755093\pi\)
\(180\) 2.44145e6 0.0312028
\(181\) −5.56564e7 −0.697654 −0.348827 0.937187i \(-0.613420\pi\)
−0.348827 + 0.937187i \(0.613420\pi\)
\(182\) −4.54344e7 −0.558644
\(183\) −2.01253e7 −0.242753
\(184\) −1.82813e7 −0.216344
\(185\) −8.37176e6 −0.0972111
\(186\) 2.42943e7 0.276828
\(187\) 7.27840e7 0.813936
\(188\) 4.72079e7 0.518157
\(189\) 8.13695e6 0.0876689
\(190\) −2.55881e7 −0.270645
\(191\) 1.30425e7 0.135439 0.0677194 0.997704i \(-0.478428\pi\)
0.0677194 + 0.997704i \(0.478428\pi\)
\(192\) −5.12874e7 −0.522946
\(193\) 1.67962e8 1.68175 0.840873 0.541232i \(-0.182042\pi\)
0.840873 + 0.541232i \(0.182042\pi\)
\(194\) −4.32680e6 −0.0425462
\(195\) −1.97874e7 −0.191103
\(196\) 4.02451e7 0.381783
\(197\) −3.14534e7 −0.293114 −0.146557 0.989202i \(-0.546819\pi\)
−0.146557 + 0.989202i \(0.546819\pi\)
\(198\) −3.14857e7 −0.288260
\(199\) −5.07710e6 −0.0456699 −0.0228350 0.999739i \(-0.507269\pi\)
−0.0228350 + 0.999739i \(0.507269\pi\)
\(200\) −1.16127e8 −1.02643
\(201\) 9.28739e6 0.0806692
\(202\) −2.36758e7 −0.202104
\(203\) 5.67647e7 0.476258
\(204\) 2.28520e7 0.188459
\(205\) −3.15230e6 −0.0255558
\(206\) −1.39636e8 −1.11291
\(207\) −8.62732e6 −0.0676051
\(208\) 6.32651e7 0.487464
\(209\) −3.06758e8 −2.32426
\(210\) 4.93729e6 0.0367893
\(211\) 2.48172e7 0.181871 0.0909357 0.995857i \(-0.471014\pi\)
0.0909357 + 0.995857i \(0.471014\pi\)
\(212\) −4.80956e7 −0.346681
\(213\) −1.19222e8 −0.845330
\(214\) 6.29870e7 0.439342
\(215\) 3.29918e7 0.226398
\(216\) −3.04054e7 −0.205287
\(217\) −4.56709e7 −0.303410
\(218\) −4.28787e7 −0.280313
\(219\) −3.73813e7 −0.240492
\(220\) 1.77596e7 0.112448
\(221\) −1.85210e8 −1.15423
\(222\) 3.38977e7 0.207938
\(223\) −1.11098e8 −0.670868 −0.335434 0.942064i \(-0.608883\pi\)
−0.335434 + 0.942064i \(0.608883\pi\)
\(224\) 6.59552e7 0.392085
\(225\) −5.48029e7 −0.320748
\(226\) −2.38597e8 −1.37494
\(227\) 1.42670e8 0.809549 0.404774 0.914417i \(-0.367350\pi\)
0.404774 + 0.914417i \(0.367350\pi\)
\(228\) −9.63127e7 −0.538160
\(229\) −8.79526e7 −0.483977 −0.241989 0.970279i \(-0.577800\pi\)
−0.241989 + 0.970279i \(0.577800\pi\)
\(230\) −5.23483e6 −0.0283697
\(231\) 5.91899e7 0.315941
\(232\) −2.12113e8 −1.11521
\(233\) 2.33751e8 1.21062 0.605311 0.795989i \(-0.293049\pi\)
0.605311 + 0.795989i \(0.293049\pi\)
\(234\) 8.01202e7 0.408777
\(235\) 4.15776e7 0.208988
\(236\) 1.26646e7 0.0627193
\(237\) 1.86974e8 0.912351
\(238\) 4.62131e7 0.222201
\(239\) 2.35581e8 1.11621 0.558107 0.829769i \(-0.311528\pi\)
0.558107 + 0.829769i \(0.311528\pi\)
\(240\) −6.87493e6 −0.0321017
\(241\) 1.34696e8 0.619862 0.309931 0.950759i \(-0.399694\pi\)
0.309931 + 0.950759i \(0.399694\pi\)
\(242\) −7.03171e7 −0.318938
\(243\) −1.43489e7 −0.0641500
\(244\) −4.59639e7 −0.202559
\(245\) 3.54453e7 0.153985
\(246\) 1.27638e7 0.0546649
\(247\) 7.80594e8 3.29599
\(248\) 1.70658e8 0.710472
\(249\) −1.19077e8 −0.488798
\(250\) −6.78106e7 −0.274478
\(251\) 4.35229e8 1.73724 0.868621 0.495478i \(-0.165007\pi\)
0.868621 + 0.495478i \(0.165007\pi\)
\(252\) 1.85838e7 0.0731532
\(253\) −6.27569e7 −0.243635
\(254\) −1.55861e8 −0.596786
\(255\) 2.01265e7 0.0760113
\(256\) −2.83460e8 −1.05597
\(257\) 2.77869e8 1.02112 0.510558 0.859844i \(-0.329439\pi\)
0.510558 + 0.859844i \(0.329439\pi\)
\(258\) −1.33586e8 −0.484274
\(259\) −6.37243e7 −0.227906
\(260\) −4.51920e7 −0.159461
\(261\) −1.00100e8 −0.348492
\(262\) −1.66977e7 −0.0573592
\(263\) 1.34074e8 0.454463 0.227231 0.973841i \(-0.427033\pi\)
0.227231 + 0.973841i \(0.427033\pi\)
\(264\) −2.21175e8 −0.739813
\(265\) −4.23595e7 −0.139827
\(266\) −1.94771e8 −0.634512
\(267\) 7.15779e7 0.230139
\(268\) 2.12113e7 0.0673124
\(269\) 4.06253e8 1.27252 0.636258 0.771476i \(-0.280481\pi\)
0.636258 + 0.771476i \(0.280481\pi\)
\(270\) −8.70655e6 −0.0269199
\(271\) 4.61228e8 1.40774 0.703871 0.710328i \(-0.251453\pi\)
0.703871 + 0.710328i \(0.251453\pi\)
\(272\) −6.43494e7 −0.193889
\(273\) −1.50618e8 −0.448030
\(274\) 9.95791e7 0.292443
\(275\) −3.98647e8 −1.15591
\(276\) −1.97038e7 −0.0564114
\(277\) −2.60294e8 −0.735843 −0.367922 0.929857i \(-0.619931\pi\)
−0.367922 + 0.929857i \(0.619931\pi\)
\(278\) −1.86377e8 −0.520279
\(279\) 8.05372e7 0.222015
\(280\) 3.46826e7 0.0944188
\(281\) −3.88226e8 −1.04379 −0.521894 0.853010i \(-0.674774\pi\)
−0.521894 + 0.853010i \(0.674774\pi\)
\(282\) −1.68350e8 −0.447035
\(283\) 6.85015e8 1.79658 0.898292 0.439398i \(-0.144808\pi\)
0.898292 + 0.439398i \(0.144808\pi\)
\(284\) −2.72288e8 −0.705365
\(285\) −8.48260e7 −0.217056
\(286\) 5.82811e8 1.47315
\(287\) −2.39947e7 −0.0599141
\(288\) −1.16307e8 −0.286901
\(289\) −2.21954e8 −0.540905
\(290\) −6.07382e7 −0.146241
\(291\) −1.43436e7 −0.0341219
\(292\) −8.53744e7 −0.200672
\(293\) −7.03513e7 −0.163394 −0.0816969 0.996657i \(-0.526034\pi\)
−0.0816969 + 0.996657i \(0.526034\pi\)
\(294\) −1.43520e8 −0.329379
\(295\) 1.11542e7 0.0252966
\(296\) 2.38118e8 0.533669
\(297\) −1.04377e8 −0.231183
\(298\) 8.28235e7 0.181299
\(299\) 1.59695e8 0.345495
\(300\) −1.25163e8 −0.267641
\(301\) 2.51128e8 0.530776
\(302\) 4.69058e8 0.979946
\(303\) −7.84868e7 −0.162087
\(304\) 2.71209e8 0.553665
\(305\) −4.04820e7 −0.0816982
\(306\) −8.14934e7 −0.162591
\(307\) 6.19217e8 1.22140 0.610701 0.791861i \(-0.290888\pi\)
0.610701 + 0.791861i \(0.290888\pi\)
\(308\) 1.35183e8 0.263629
\(309\) −4.62902e8 −0.892553
\(310\) 4.88679e7 0.0931661
\(311\) 6.09567e8 1.14911 0.574553 0.818468i \(-0.305176\pi\)
0.574553 + 0.818468i \(0.305176\pi\)
\(312\) 5.62814e8 1.04912
\(313\) −5.97609e8 −1.10157 −0.550785 0.834647i \(-0.685672\pi\)
−0.550785 + 0.834647i \(0.685672\pi\)
\(314\) 1.11526e8 0.203294
\(315\) 1.63674e7 0.0295049
\(316\) 4.27026e8 0.761289
\(317\) −2.89237e8 −0.509972 −0.254986 0.966945i \(-0.582071\pi\)
−0.254986 + 0.966945i \(0.582071\pi\)
\(318\) 1.71516e8 0.299095
\(319\) −7.28150e8 −1.25589
\(320\) −1.03164e8 −0.175997
\(321\) 2.08806e8 0.352350
\(322\) −3.98466e7 −0.0665113
\(323\) −7.93972e8 −1.31098
\(324\) −3.27712e7 −0.0535284
\(325\) 1.01442e9 1.63918
\(326\) 4.08234e8 0.652601
\(327\) −1.42146e8 −0.224810
\(328\) 8.96611e7 0.140296
\(329\) 3.16481e8 0.489962
\(330\) −6.33332e7 −0.0970136
\(331\) −6.01918e8 −0.912304 −0.456152 0.889902i \(-0.650773\pi\)
−0.456152 + 0.889902i \(0.650773\pi\)
\(332\) −2.71957e8 −0.407866
\(333\) 1.12373e8 0.166766
\(334\) −3.15526e7 −0.0463365
\(335\) 1.86815e7 0.0271491
\(336\) −5.23306e7 −0.0752607
\(337\) 4.06076e8 0.577966 0.288983 0.957334i \(-0.406683\pi\)
0.288983 + 0.957334i \(0.406683\pi\)
\(338\) −9.71989e8 −1.36916
\(339\) −7.90963e8 −1.10270
\(340\) 4.59666e7 0.0634258
\(341\) 5.85844e8 0.800095
\(342\) 3.43465e8 0.464292
\(343\) 6.10255e8 0.816549
\(344\) −9.38388e8 −1.24288
\(345\) −1.73538e7 −0.0227524
\(346\) 4.86082e8 0.630874
\(347\) 1.23529e8 0.158714 0.0793568 0.996846i \(-0.474713\pi\)
0.0793568 + 0.996846i \(0.474713\pi\)
\(348\) −2.28617e8 −0.290791
\(349\) −3.88573e8 −0.489309 −0.244655 0.969610i \(-0.578675\pi\)
−0.244655 + 0.969610i \(0.578675\pi\)
\(350\) −2.53115e8 −0.315559
\(351\) 2.65603e8 0.327838
\(352\) −8.46041e8 −1.03393
\(353\) 1.30943e8 0.158443 0.0792213 0.996857i \(-0.474757\pi\)
0.0792213 + 0.996857i \(0.474757\pi\)
\(354\) −4.51640e7 −0.0541104
\(355\) −2.39814e8 −0.284495
\(356\) 1.63475e8 0.192034
\(357\) 1.53199e8 0.178204
\(358\) 8.97866e8 1.03424
\(359\) 1.28579e9 1.46669 0.733346 0.679856i \(-0.237958\pi\)
0.733346 + 0.679856i \(0.237958\pi\)
\(360\) −6.11602e7 −0.0690892
\(361\) 2.45243e9 2.74361
\(362\) 4.53301e8 0.502235
\(363\) −2.33105e8 −0.255787
\(364\) −3.43993e8 −0.373848
\(365\) −7.51922e7 −0.0809372
\(366\) 1.63914e8 0.174756
\(367\) 1.23889e9 1.30828 0.654139 0.756374i \(-0.273031\pi\)
0.654139 + 0.756374i \(0.273031\pi\)
\(368\) 5.54843e7 0.0580367
\(369\) 4.23129e7 0.0438410
\(370\) 6.81850e7 0.0699814
\(371\) −3.22433e8 −0.327816
\(372\) 1.83937e8 0.185255
\(373\) −1.91108e8 −0.190676 −0.0953382 0.995445i \(-0.530393\pi\)
−0.0953382 + 0.995445i \(0.530393\pi\)
\(374\) −5.92800e8 −0.585946
\(375\) −2.24796e8 −0.220130
\(376\) −1.18259e9 −1.14730
\(377\) 1.85289e9 1.78096
\(378\) −6.62726e7 −0.0631121
\(379\) −4.82547e8 −0.455305 −0.227653 0.973742i \(-0.573105\pi\)
−0.227653 + 0.973742i \(0.573105\pi\)
\(380\) −1.93732e8 −0.181117
\(381\) −5.16688e8 −0.478620
\(382\) −1.06226e8 −0.0975012
\(383\) −4.67810e8 −0.425475 −0.212738 0.977109i \(-0.568238\pi\)
−0.212738 + 0.977109i \(0.568238\pi\)
\(384\) −1.33663e8 −0.120463
\(385\) 1.19060e8 0.106329
\(386\) −1.36799e9 −1.21068
\(387\) −4.42845e8 −0.388386
\(388\) −3.27591e7 −0.0284722
\(389\) 2.37119e8 0.204241 0.102120 0.994772i \(-0.467437\pi\)
0.102120 + 0.994772i \(0.467437\pi\)
\(390\) 1.61161e8 0.137573
\(391\) −1.62432e8 −0.137421
\(392\) −1.00817e9 −0.845344
\(393\) −5.53540e7 −0.0460018
\(394\) 2.56177e8 0.211010
\(395\) 3.76097e8 0.307051
\(396\) −2.38384e8 −0.192905
\(397\) −3.19338e8 −0.256144 −0.128072 0.991765i \(-0.540879\pi\)
−0.128072 + 0.991765i \(0.540879\pi\)
\(398\) 4.13512e7 0.0328774
\(399\) −6.45679e8 −0.508876
\(400\) 3.52450e8 0.275351
\(401\) −1.08909e9 −0.843451 −0.421725 0.906724i \(-0.638575\pi\)
−0.421725 + 0.906724i \(0.638575\pi\)
\(402\) −7.56425e7 −0.0580731
\(403\) −1.49077e9 −1.13460
\(404\) −1.79254e8 −0.135249
\(405\) −2.88627e7 −0.0215896
\(406\) −4.62328e8 −0.342854
\(407\) 8.17424e8 0.600990
\(408\) −5.72460e8 −0.417287
\(409\) 1.68562e9 1.21822 0.609112 0.793084i \(-0.291526\pi\)
0.609112 + 0.793084i \(0.291526\pi\)
\(410\) 2.56744e7 0.0183974
\(411\) 3.30111e8 0.234538
\(412\) −1.05721e9 −0.744769
\(413\) 8.49037e7 0.0593064
\(414\) 7.02664e7 0.0486684
\(415\) −2.39522e8 −0.164504
\(416\) 2.15288e9 1.46620
\(417\) −6.17853e8 −0.417262
\(418\) 2.49843e9 1.67321
\(419\) 1.13922e9 0.756588 0.378294 0.925686i \(-0.376511\pi\)
0.378294 + 0.925686i \(0.376511\pi\)
\(420\) 3.73812e7 0.0246196
\(421\) 9.61423e8 0.627953 0.313977 0.949431i \(-0.398339\pi\)
0.313977 + 0.949431i \(0.398339\pi\)
\(422\) −2.02127e8 −0.130928
\(423\) −5.58091e8 −0.358520
\(424\) 1.20483e9 0.767621
\(425\) −1.03181e9 −0.651984
\(426\) 9.71018e8 0.608546
\(427\) −3.08141e8 −0.191537
\(428\) 4.76887e8 0.294010
\(429\) 1.93205e9 1.18146
\(430\) −2.68707e8 −0.162982
\(431\) 2.62811e8 0.158115 0.0790575 0.996870i \(-0.474809\pi\)
0.0790575 + 0.996870i \(0.474809\pi\)
\(432\) 9.22812e7 0.0550706
\(433\) −2.86742e9 −1.69740 −0.848698 0.528877i \(-0.822613\pi\)
−0.848698 + 0.528877i \(0.822613\pi\)
\(434\) 3.71973e8 0.218423
\(435\) −2.01351e8 −0.117285
\(436\) −3.24643e8 −0.187587
\(437\) 6.84590e8 0.392415
\(438\) 3.04457e8 0.173128
\(439\) −2.91897e8 −0.164666 −0.0823330 0.996605i \(-0.526237\pi\)
−0.0823330 + 0.996605i \(0.526237\pi\)
\(440\) −4.44892e8 −0.248983
\(441\) −4.75777e8 −0.264161
\(442\) 1.50847e9 0.830920
\(443\) −2.33184e9 −1.27434 −0.637169 0.770724i \(-0.719895\pi\)
−0.637169 + 0.770724i \(0.719895\pi\)
\(444\) 2.56646e8 0.139154
\(445\) 1.43979e8 0.0774529
\(446\) 9.04850e8 0.482953
\(447\) 2.74565e8 0.145401
\(448\) −7.85267e8 −0.412614
\(449\) −2.29986e9 −1.19905 −0.599527 0.800355i \(-0.704645\pi\)
−0.599527 + 0.800355i \(0.704645\pi\)
\(450\) 4.46350e8 0.230904
\(451\) 3.07793e8 0.157994
\(452\) −1.80646e9 −0.920121
\(453\) 1.55496e9 0.785913
\(454\) −1.16200e9 −0.582788
\(455\) −3.02967e8 −0.150784
\(456\) 2.41271e9 1.19159
\(457\) 1.34778e9 0.660560 0.330280 0.943883i \(-0.392857\pi\)
0.330280 + 0.943883i \(0.392857\pi\)
\(458\) 7.16343e8 0.348411
\(459\) −2.70156e8 −0.130398
\(460\) −3.96340e7 −0.0189852
\(461\) 1.79615e9 0.853865 0.426932 0.904284i \(-0.359594\pi\)
0.426932 + 0.904284i \(0.359594\pi\)
\(462\) −4.82080e8 −0.227443
\(463\) −3.73066e9 −1.74684 −0.873418 0.486971i \(-0.838102\pi\)
−0.873418 + 0.486971i \(0.838102\pi\)
\(464\) 6.43768e8 0.299169
\(465\) 1.62000e8 0.0747188
\(466\) −1.90382e9 −0.871517
\(467\) −4.18677e9 −1.90226 −0.951129 0.308793i \(-0.900075\pi\)
−0.951129 + 0.308793i \(0.900075\pi\)
\(468\) 6.06606e8 0.273556
\(469\) 1.42200e8 0.0636496
\(470\) −3.38635e8 −0.150449
\(471\) 3.69717e8 0.163041
\(472\) −3.17260e8 −0.138873
\(473\) −3.22134e9 −1.39966
\(474\) −1.52284e9 −0.656794
\(475\) 4.34869e9 1.86179
\(476\) 3.49889e8 0.148698
\(477\) 5.68586e8 0.239873
\(478\) −1.91872e9 −0.803553
\(479\) 4.05633e9 1.68639 0.843197 0.537605i \(-0.180671\pi\)
0.843197 + 0.537605i \(0.180671\pi\)
\(480\) −2.33951e8 −0.0965562
\(481\) −2.08006e9 −0.852254
\(482\) −1.09705e9 −0.446233
\(483\) −1.32094e8 −0.0533418
\(484\) −5.32385e8 −0.213436
\(485\) −2.88521e7 −0.0114837
\(486\) 1.16867e8 0.0461811
\(487\) −3.44990e9 −1.35349 −0.676745 0.736218i \(-0.736610\pi\)
−0.676745 + 0.736218i \(0.736610\pi\)
\(488\) 1.15143e9 0.448507
\(489\) 1.35332e9 0.523383
\(490\) −2.88689e8 −0.110852
\(491\) −3.70056e9 −1.41086 −0.705428 0.708782i \(-0.749245\pi\)
−0.705428 + 0.708782i \(0.749245\pi\)
\(492\) 9.66376e7 0.0365821
\(493\) −1.88465e9 −0.708380
\(494\) −6.35766e9 −2.37276
\(495\) −2.09953e8 −0.0778045
\(496\) −5.17953e8 −0.190592
\(497\) −1.82542e9 −0.666982
\(498\) 9.69838e8 0.351882
\(499\) 3.49353e9 1.25867 0.629336 0.777133i \(-0.283327\pi\)
0.629336 + 0.777133i \(0.283327\pi\)
\(500\) −5.13408e8 −0.183682
\(501\) −1.04599e8 −0.0371616
\(502\) −3.54479e9 −1.25063
\(503\) 3.77577e9 1.32287 0.661436 0.750002i \(-0.269947\pi\)
0.661436 + 0.750002i \(0.269947\pi\)
\(504\) −4.65540e8 −0.161976
\(505\) −1.57876e8 −0.0545501
\(506\) 5.11132e8 0.175391
\(507\) −3.22220e9 −1.09806
\(508\) −1.18005e9 −0.399373
\(509\) −2.35210e9 −0.790575 −0.395288 0.918557i \(-0.629355\pi\)
−0.395288 + 0.918557i \(0.629355\pi\)
\(510\) −1.63923e8 −0.0547199
\(511\) −5.72349e8 −0.189753
\(512\) 1.67502e9 0.551538
\(513\) 1.13861e9 0.372360
\(514\) −2.26315e9 −0.735092
\(515\) −9.31124e8 −0.300388
\(516\) −1.01140e9 −0.324079
\(517\) −4.05967e9 −1.29203
\(518\) 5.19011e8 0.164068
\(519\) 1.61139e9 0.505959
\(520\) 1.13210e9 0.353079
\(521\) −1.51952e9 −0.470734 −0.235367 0.971907i \(-0.575629\pi\)
−0.235367 + 0.971907i \(0.575629\pi\)
\(522\) 8.15281e8 0.250877
\(523\) 5.57185e9 1.70311 0.851556 0.524264i \(-0.175659\pi\)
0.851556 + 0.524264i \(0.175659\pi\)
\(524\) −1.26422e8 −0.0383851
\(525\) −8.39093e8 −0.253077
\(526\) −1.09198e9 −0.327164
\(527\) 1.51632e9 0.451289
\(528\) 6.71272e8 0.198463
\(529\) −3.26477e9 −0.958866
\(530\) 3.45003e8 0.100660
\(531\) −1.49721e8 −0.0433963
\(532\) −1.47465e9 −0.424619
\(533\) −7.83227e8 −0.224049
\(534\) −5.82977e8 −0.165675
\(535\) 4.20011e8 0.118583
\(536\) −5.31360e8 −0.149043
\(537\) 2.97648e9 0.829456
\(538\) −3.30878e9 −0.916074
\(539\) −3.46090e9 −0.951981
\(540\) −6.59190e7 −0.0180149
\(541\) −1.01713e9 −0.276177 −0.138088 0.990420i \(-0.544096\pi\)
−0.138088 + 0.990420i \(0.544096\pi\)
\(542\) −3.75654e9 −1.01342
\(543\) 1.50272e9 0.402791
\(544\) −2.18978e9 −0.583183
\(545\) −2.85925e8 −0.0756596
\(546\) 1.22673e9 0.322533
\(547\) −3.76396e8 −0.0983308 −0.0491654 0.998791i \(-0.515656\pi\)
−0.0491654 + 0.998791i \(0.515656\pi\)
\(548\) 7.53934e8 0.195705
\(549\) 5.43384e8 0.140153
\(550\) 3.24684e9 0.832131
\(551\) 7.94310e9 2.02283
\(552\) 4.93595e8 0.124906
\(553\) 2.86278e9 0.719863
\(554\) 2.12000e9 0.529727
\(555\) 2.26038e8 0.0561248
\(556\) −1.41110e9 −0.348174
\(557\) 7.16500e9 1.75680 0.878401 0.477924i \(-0.158611\pi\)
0.878401 + 0.477924i \(0.158611\pi\)
\(558\) −6.55947e8 −0.159827
\(559\) 8.19721e9 1.98484
\(560\) −1.05263e8 −0.0253289
\(561\) −1.96517e9 −0.469926
\(562\) 3.16196e9 0.751414
\(563\) 4.65876e9 1.10025 0.550125 0.835082i \(-0.314580\pi\)
0.550125 + 0.835082i \(0.314580\pi\)
\(564\) −1.27461e9 −0.299158
\(565\) −1.59102e9 −0.371112
\(566\) −5.57921e9 −1.29335
\(567\) −2.19698e8 −0.0506157
\(568\) 6.82103e9 1.56182
\(569\) −4.71634e9 −1.07328 −0.536639 0.843812i \(-0.680306\pi\)
−0.536639 + 0.843812i \(0.680306\pi\)
\(570\) 6.90878e8 0.156257
\(571\) −8.11800e9 −1.82483 −0.912416 0.409265i \(-0.865785\pi\)
−0.912416 + 0.409265i \(0.865785\pi\)
\(572\) 4.41258e9 0.985840
\(573\) −3.52147e8 −0.0781956
\(574\) 1.95429e8 0.0431317
\(575\) 8.89660e8 0.195158
\(576\) 1.38476e9 0.301923
\(577\) −2.96457e9 −0.642460 −0.321230 0.947001i \(-0.604096\pi\)
−0.321230 + 0.947001i \(0.604096\pi\)
\(578\) 1.80774e9 0.389393
\(579\) −4.53497e9 −0.970957
\(580\) −4.59862e8 −0.0978654
\(581\) −1.82320e9 −0.385671
\(582\) 1.16824e8 0.0245641
\(583\) 4.13601e9 0.864453
\(584\) 2.13870e9 0.444329
\(585\) 5.34260e8 0.110333
\(586\) 5.72986e8 0.117626
\(587\) 4.79349e9 0.978180 0.489090 0.872233i \(-0.337329\pi\)
0.489090 + 0.872233i \(0.337329\pi\)
\(588\) −1.08662e9 −0.220423
\(589\) −6.39075e9 −1.28869
\(590\) −9.08470e7 −0.0182108
\(591\) 8.49242e8 0.169229
\(592\) −7.22697e8 −0.143163
\(593\) 6.55987e9 1.29183 0.645913 0.763411i \(-0.276477\pi\)
0.645913 + 0.763411i \(0.276477\pi\)
\(594\) 8.50113e8 0.166427
\(595\) 3.08160e8 0.0599745
\(596\) 6.27073e8 0.121327
\(597\) 1.37082e8 0.0263675
\(598\) −1.30066e9 −0.248719
\(599\) 8.15165e9 1.54971 0.774857 0.632137i \(-0.217822\pi\)
0.774857 + 0.632137i \(0.217822\pi\)
\(600\) 3.13544e9 0.592610
\(601\) −3.63784e9 −0.683570 −0.341785 0.939778i \(-0.611031\pi\)
−0.341785 + 0.939778i \(0.611031\pi\)
\(602\) −2.04534e9 −0.382102
\(603\) −2.50760e8 −0.0465744
\(604\) 3.55133e9 0.655786
\(605\) −4.68890e8 −0.0860849
\(606\) 6.39247e8 0.116685
\(607\) 4.53989e9 0.823920 0.411960 0.911202i \(-0.364844\pi\)
0.411960 + 0.911202i \(0.364844\pi\)
\(608\) 9.22913e9 1.66532
\(609\) −1.53265e9 −0.274967
\(610\) 3.29712e8 0.0588139
\(611\) 1.03305e10 1.83221
\(612\) −6.17003e8 −0.108807
\(613\) 5.71720e9 1.00247 0.501235 0.865311i \(-0.332879\pi\)
0.501235 + 0.865311i \(0.332879\pi\)
\(614\) −5.04330e9 −0.879277
\(615\) 8.51122e7 0.0147546
\(616\) −3.38643e9 −0.583727
\(617\) −2.37562e9 −0.407172 −0.203586 0.979057i \(-0.565260\pi\)
−0.203586 + 0.979057i \(0.565260\pi\)
\(618\) 3.77017e9 0.642542
\(619\) 5.60279e9 0.949483 0.474742 0.880125i \(-0.342542\pi\)
0.474742 + 0.880125i \(0.342542\pi\)
\(620\) 3.69989e8 0.0623473
\(621\) 2.32938e8 0.0390318
\(622\) −4.96470e9 −0.827231
\(623\) 1.09594e9 0.181584
\(624\) −1.70816e9 −0.281437
\(625\) 5.42090e9 0.888160
\(626\) 4.86732e9 0.793011
\(627\) 8.28246e9 1.34191
\(628\) 8.44389e8 0.136045
\(629\) 2.11571e9 0.338984
\(630\) −1.33307e8 −0.0212403
\(631\) −1.14283e10 −1.81083 −0.905415 0.424527i \(-0.860440\pi\)
−0.905415 + 0.424527i \(0.860440\pi\)
\(632\) −1.06974e10 −1.68565
\(633\) −6.70065e8 −0.105003
\(634\) 2.35573e9 0.367124
\(635\) −1.03932e9 −0.161079
\(636\) 1.29858e9 0.200156
\(637\) 8.80680e9 1.34999
\(638\) 5.93052e9 0.904108
\(639\) 3.21898e9 0.488052
\(640\) −2.68863e8 −0.0405417
\(641\) −8.14000e8 −0.122073 −0.0610367 0.998136i \(-0.519441\pi\)
−0.0610367 + 0.998136i \(0.519441\pi\)
\(642\) −1.70065e9 −0.253654
\(643\) −1.34257e10 −1.99158 −0.995792 0.0916394i \(-0.970789\pi\)
−0.995792 + 0.0916394i \(0.970789\pi\)
\(644\) −3.01686e8 −0.0445098
\(645\) −8.90779e8 −0.130711
\(646\) 6.46662e9 0.943765
\(647\) −2.58413e9 −0.375102 −0.187551 0.982255i \(-0.560055\pi\)
−0.187551 + 0.982255i \(0.560055\pi\)
\(648\) 8.20945e8 0.118523
\(649\) −1.08910e9 −0.156391
\(650\) −8.26210e9 −1.18003
\(651\) 1.23311e9 0.175174
\(652\) 3.09082e9 0.436725
\(653\) 9.14859e9 1.28575 0.642877 0.765969i \(-0.277741\pi\)
0.642877 + 0.765969i \(0.277741\pi\)
\(654\) 1.15773e9 0.161839
\(655\) −1.11344e8 −0.0154819
\(656\) −2.72124e8 −0.0376360
\(657\) 1.00929e9 0.138848
\(658\) −2.57763e9 −0.352719
\(659\) −3.05757e9 −0.416176 −0.208088 0.978110i \(-0.566724\pi\)
−0.208088 + 0.978110i \(0.566724\pi\)
\(660\) −4.79509e8 −0.0649221
\(661\) −5.81198e9 −0.782743 −0.391371 0.920233i \(-0.627999\pi\)
−0.391371 + 0.920233i \(0.627999\pi\)
\(662\) 4.90241e9 0.656760
\(663\) 5.00068e9 0.666395
\(664\) 6.81275e9 0.903096
\(665\) −1.29878e9 −0.171262
\(666\) −9.15238e8 −0.120053
\(667\) 1.62501e9 0.212039
\(668\) −2.38891e8 −0.0310086
\(669\) 2.99963e9 0.387326
\(670\) −1.52154e8 −0.0195444
\(671\) 3.95269e9 0.505084
\(672\) −1.78079e9 −0.226371
\(673\) −7.68542e9 −0.971885 −0.485943 0.873991i \(-0.661524\pi\)
−0.485943 + 0.873991i \(0.661524\pi\)
\(674\) −3.30734e9 −0.416073
\(675\) 1.47968e9 0.185184
\(676\) −7.35913e9 −0.916248
\(677\) 5.12888e9 0.635276 0.317638 0.948212i \(-0.397110\pi\)
0.317638 + 0.948212i \(0.397110\pi\)
\(678\) 6.44211e9 0.793824
\(679\) −2.19617e8 −0.0269229
\(680\) −1.15150e9 −0.140437
\(681\) −3.85210e9 −0.467393
\(682\) −4.77149e9 −0.575982
\(683\) 9.14761e9 1.09859 0.549294 0.835629i \(-0.314897\pi\)
0.549294 + 0.835629i \(0.314897\pi\)
\(684\) 2.60044e9 0.310707
\(685\) 6.64016e8 0.0789335
\(686\) −4.97031e9 −0.587827
\(687\) 2.37472e9 0.279424
\(688\) 2.84804e9 0.333416
\(689\) −1.05247e10 −1.22587
\(690\) 1.41340e8 0.0163793
\(691\) 1.46307e10 1.68691 0.843455 0.537200i \(-0.180518\pi\)
0.843455 + 0.537200i \(0.180518\pi\)
\(692\) 3.68022e9 0.422185
\(693\) −1.59813e9 −0.182408
\(694\) −1.00610e9 −0.114257
\(695\) −1.24281e9 −0.140429
\(696\) 5.72704e9 0.643869
\(697\) 7.96651e8 0.0891155
\(698\) 3.16479e9 0.352250
\(699\) −6.31129e9 −0.698953
\(700\) −1.91639e9 −0.211174
\(701\) −5.51419e9 −0.604601 −0.302300 0.953213i \(-0.597755\pi\)
−0.302300 + 0.953213i \(0.597755\pi\)
\(702\) −2.16325e9 −0.236008
\(703\) −8.91696e9 −0.967996
\(704\) 1.00730e10 1.08807
\(705\) −1.12260e9 −0.120660
\(706\) −1.06649e9 −0.114061
\(707\) −1.20172e9 −0.127890
\(708\) −3.41945e8 −0.0362110
\(709\) 4.43575e9 0.467417 0.233709 0.972307i \(-0.424914\pi\)
0.233709 + 0.972307i \(0.424914\pi\)
\(710\) 1.95320e9 0.204806
\(711\) −5.04830e9 −0.526746
\(712\) −4.09519e9 −0.425201
\(713\) −1.30743e9 −0.135084
\(714\) −1.24775e9 −0.128288
\(715\) 3.88631e9 0.397619
\(716\) 6.79793e9 0.692119
\(717\) −6.36069e9 −0.644447
\(718\) −1.04723e10 −1.05586
\(719\) −5.67887e9 −0.569785 −0.284893 0.958559i \(-0.591958\pi\)
−0.284893 + 0.958559i \(0.591958\pi\)
\(720\) 1.85623e8 0.0185340
\(721\) −7.08754e9 −0.704242
\(722\) −1.99742e10 −1.97510
\(723\) −3.63679e9 −0.357877
\(724\) 3.43204e9 0.336099
\(725\) 1.03225e10 1.00601
\(726\) 1.89856e9 0.184139
\(727\) −1.26294e10 −1.21903 −0.609514 0.792775i \(-0.708636\pi\)
−0.609514 + 0.792775i \(0.708636\pi\)
\(728\) 8.61731e9 0.827774
\(729\) 3.87420e8 0.0370370
\(730\) 6.12414e8 0.0582660
\(731\) −8.33771e9 −0.789470
\(732\) 1.24102e9 0.116948
\(733\) 8.87757e9 0.832588 0.416294 0.909230i \(-0.363329\pi\)
0.416294 + 0.909230i \(0.363329\pi\)
\(734\) −1.00903e10 −0.941819
\(735\) −9.57023e8 −0.0889030
\(736\) 1.88811e9 0.174564
\(737\) −1.82408e9 −0.167844
\(738\) −3.44624e8 −0.0315608
\(739\) −1.67534e10 −1.52703 −0.763515 0.645791i \(-0.776528\pi\)
−0.763515 + 0.645791i \(0.776528\pi\)
\(740\) 5.16243e8 0.0468320
\(741\) −2.10760e10 −1.90294
\(742\) 2.62610e9 0.235992
\(743\) 1.87784e10 1.67957 0.839784 0.542921i \(-0.182682\pi\)
0.839784 + 0.542921i \(0.182682\pi\)
\(744\) −4.60778e9 −0.410191
\(745\) 5.52286e8 0.0489347
\(746\) 1.55650e9 0.137266
\(747\) 3.21507e9 0.282208
\(748\) −4.48821e9 −0.392118
\(749\) 3.19705e9 0.278011
\(750\) 1.83089e9 0.158470
\(751\) 2.58308e9 0.222535 0.111268 0.993790i \(-0.464509\pi\)
0.111268 + 0.993790i \(0.464509\pi\)
\(752\) 3.58921e9 0.307777
\(753\) −1.17512e10 −1.00300
\(754\) −1.50911e10 −1.28210
\(755\) 3.12778e9 0.264498
\(756\) −5.01763e8 −0.0422350
\(757\) 5.60643e9 0.469733 0.234867 0.972028i \(-0.424535\pi\)
0.234867 + 0.972028i \(0.424535\pi\)
\(758\) 3.93018e9 0.327770
\(759\) 1.69444e9 0.140663
\(760\) 4.85315e9 0.401030
\(761\) 1.71960e8 0.0141443 0.00707216 0.999975i \(-0.497749\pi\)
0.00707216 + 0.999975i \(0.497749\pi\)
\(762\) 4.20824e9 0.344555
\(763\) −2.17641e9 −0.177380
\(764\) −8.04260e8 −0.0652484
\(765\) −5.43416e8 −0.0438852
\(766\) 3.81015e9 0.306296
\(767\) 2.77139e9 0.221776
\(768\) 7.65343e9 0.609666
\(769\) −1.69708e10 −1.34574 −0.672869 0.739762i \(-0.734938\pi\)
−0.672869 + 0.739762i \(0.734938\pi\)
\(770\) −9.69702e8 −0.0765457
\(771\) −7.50247e9 −0.589541
\(772\) −1.03573e10 −0.810191
\(773\) −5.87486e9 −0.457477 −0.228739 0.973488i \(-0.573460\pi\)
−0.228739 + 0.973488i \(0.573460\pi\)
\(774\) 3.60681e9 0.279596
\(775\) −8.30510e9 −0.640898
\(776\) 8.20642e8 0.0630431
\(777\) 1.72056e9 0.131582
\(778\) −1.93125e9 −0.147031
\(779\) −3.35759e9 −0.254476
\(780\) 1.22018e9 0.0920650
\(781\) 2.34156e10 1.75884
\(782\) 1.32295e9 0.0989280
\(783\) 2.70271e9 0.201202
\(784\) 3.05983e9 0.226773
\(785\) 7.43684e8 0.0548712
\(786\) 4.50839e8 0.0331163
\(787\) 1.81450e9 0.132692 0.0663461 0.997797i \(-0.478866\pi\)
0.0663461 + 0.997797i \(0.478866\pi\)
\(788\) 1.93957e9 0.141209
\(789\) −3.61999e9 −0.262384
\(790\) −3.06318e9 −0.221043
\(791\) −1.21105e10 −0.870052
\(792\) 5.97172e9 0.427131
\(793\) −1.00582e10 −0.716252
\(794\) 2.60089e9 0.184396
\(795\) 1.14371e9 0.0807290
\(796\) 3.13078e8 0.0220017
\(797\) 7.68250e9 0.537525 0.268762 0.963207i \(-0.413385\pi\)
0.268762 + 0.963207i \(0.413385\pi\)
\(798\) 5.25883e9 0.366335
\(799\) −1.05075e10 −0.728763
\(800\) 1.19937e10 0.828207
\(801\) −1.93260e9 −0.132871
\(802\) 8.87028e9 0.607193
\(803\) 7.34182e9 0.500379
\(804\) −5.72705e8 −0.0388629
\(805\) −2.65706e8 −0.0179521
\(806\) 1.21418e10 0.816791
\(807\) −1.09688e10 −0.734688
\(808\) 4.49047e9 0.299469
\(809\) −2.53283e10 −1.68185 −0.840924 0.541154i \(-0.817988\pi\)
−0.840924 + 0.541154i \(0.817988\pi\)
\(810\) 2.35077e8 0.0155422
\(811\) −1.04893e9 −0.0690513 −0.0345256 0.999404i \(-0.510992\pi\)
−0.0345256 + 0.999404i \(0.510992\pi\)
\(812\) −3.50038e9 −0.229440
\(813\) −1.24531e10 −0.812760
\(814\) −6.65763e9 −0.432647
\(815\) 2.72220e9 0.176144
\(816\) 1.73743e9 0.111942
\(817\) 3.51404e10 2.25439
\(818\) −1.37287e10 −0.876989
\(819\) 4.06668e9 0.258670
\(820\) 1.94386e8 0.0123117
\(821\) −1.22269e10 −0.771110 −0.385555 0.922685i \(-0.625990\pi\)
−0.385555 + 0.922685i \(0.625990\pi\)
\(822\) −2.68864e9 −0.168842
\(823\) 6.96726e8 0.0435675 0.0217837 0.999763i \(-0.493065\pi\)
0.0217837 + 0.999763i \(0.493065\pi\)
\(824\) 2.64840e10 1.64907
\(825\) 1.07635e10 0.667365
\(826\) −6.91510e8 −0.0426942
\(827\) −2.23328e10 −1.37301 −0.686505 0.727125i \(-0.740856\pi\)
−0.686505 + 0.727125i \(0.740856\pi\)
\(828\) 5.32001e8 0.0325692
\(829\) 1.33734e10 0.815267 0.407633 0.913146i \(-0.366354\pi\)
0.407633 + 0.913146i \(0.366354\pi\)
\(830\) 1.95082e9 0.118425
\(831\) 7.02794e9 0.424839
\(832\) −2.56324e10 −1.54297
\(833\) −8.95774e9 −0.536959
\(834\) 5.03219e9 0.300383
\(835\) −2.10400e8 −0.0125067
\(836\) 1.89162e10 1.11972
\(837\) −2.17450e9 −0.128180
\(838\) −9.27856e9 −0.544661
\(839\) −1.68025e10 −0.982217 −0.491108 0.871098i \(-0.663408\pi\)
−0.491108 + 0.871098i \(0.663408\pi\)
\(840\) −9.36430e8 −0.0545127
\(841\) 1.60463e9 0.0930226
\(842\) −7.83045e9 −0.452058
\(843\) 1.04821e10 0.602631
\(844\) −1.53035e9 −0.0876176
\(845\) −6.48145e9 −0.369550
\(846\) 4.54545e9 0.258096
\(847\) −3.56910e9 −0.201821
\(848\) −3.65671e9 −0.205923
\(849\) −1.84954e10 −1.03726
\(850\) 8.40370e9 0.469358
\(851\) −1.82424e9 −0.101468
\(852\) 7.35177e9 0.407243
\(853\) −6.95279e9 −0.383563 −0.191782 0.981438i \(-0.561427\pi\)
−0.191782 + 0.981438i \(0.561427\pi\)
\(854\) 2.50970e9 0.137886
\(855\) 2.29030e9 0.125317
\(856\) −1.19464e10 −0.650997
\(857\) −3.47269e10 −1.88466 −0.942330 0.334687i \(-0.891370\pi\)
−0.942330 + 0.334687i \(0.891370\pi\)
\(858\) −1.57359e10 −0.850523
\(859\) −1.72796e10 −0.930161 −0.465081 0.885268i \(-0.653975\pi\)
−0.465081 + 0.885268i \(0.653975\pi\)
\(860\) −2.03443e9 −0.109068
\(861\) 6.47858e8 0.0345914
\(862\) −2.14050e9 −0.113826
\(863\) −3.64236e10 −1.92906 −0.964529 0.263977i \(-0.914966\pi\)
−0.964529 + 0.263977i \(0.914966\pi\)
\(864\) 3.14029e9 0.165642
\(865\) 3.24130e9 0.170280
\(866\) 2.33541e10 1.22194
\(867\) 5.99276e9 0.312291
\(868\) 2.81628e9 0.146170
\(869\) −3.67224e10 −1.89828
\(870\) 1.63993e9 0.0844323
\(871\) 4.64165e9 0.238018
\(872\) 8.13258e9 0.415356
\(873\) 3.87278e8 0.0197003
\(874\) −5.57575e9 −0.282496
\(875\) −3.44188e9 −0.173687
\(876\) 2.30511e9 0.115858
\(877\) −2.21634e9 −0.110953 −0.0554763 0.998460i \(-0.517668\pi\)
−0.0554763 + 0.998460i \(0.517668\pi\)
\(878\) 2.37740e9 0.118542
\(879\) 1.89949e9 0.0943354
\(880\) 1.35026e9 0.0667925
\(881\) −2.32029e10 −1.14321 −0.571607 0.820528i \(-0.693680\pi\)
−0.571607 + 0.820528i \(0.693680\pi\)
\(882\) 3.87504e9 0.190167
\(883\) 1.94888e10 0.952624 0.476312 0.879276i \(-0.341973\pi\)
0.476312 + 0.879276i \(0.341973\pi\)
\(884\) 1.14209e10 0.556057
\(885\) −3.01163e8 −0.0146050
\(886\) 1.89920e10 0.917386
\(887\) 9.83405e9 0.473151 0.236575 0.971613i \(-0.423975\pi\)
0.236575 + 0.971613i \(0.423975\pi\)
\(888\) −6.42920e9 −0.308114
\(889\) −7.91107e9 −0.377641
\(890\) −1.17265e9 −0.0557577
\(891\) 2.81818e9 0.133474
\(892\) 6.85080e9 0.323195
\(893\) 4.42853e10 2.08104
\(894\) −2.23623e9 −0.104673
\(895\) 5.98717e9 0.279152
\(896\) −2.04654e9 −0.0950477
\(897\) −4.31176e9 −0.199471
\(898\) 1.87315e10 0.863189
\(899\) −1.51697e10 −0.696334
\(900\) 3.37941e9 0.154522
\(901\) 1.07051e10 0.487590
\(902\) −2.50686e9 −0.113739
\(903\) −6.78044e9 −0.306444
\(904\) 4.52534e10 2.03733
\(905\) 3.02272e9 0.135559
\(906\) −1.26646e10 −0.565772
\(907\) 2.35757e10 1.04915 0.524577 0.851363i \(-0.324224\pi\)
0.524577 + 0.851363i \(0.324224\pi\)
\(908\) −8.79773e9 −0.390005
\(909\) 2.11914e9 0.0935808
\(910\) 2.46756e9 0.108548
\(911\) −2.65692e10 −1.16430 −0.582149 0.813082i \(-0.697788\pi\)
−0.582149 + 0.813082i \(0.697788\pi\)
\(912\) −7.32265e9 −0.319658
\(913\) 2.33871e10 1.01702
\(914\) −1.09772e10 −0.475532
\(915\) 1.09301e9 0.0471685
\(916\) 5.42358e9 0.233159
\(917\) −8.47531e8 −0.0362964
\(918\) 2.20032e9 0.0938722
\(919\) 2.32295e10 0.987271 0.493636 0.869669i \(-0.335668\pi\)
0.493636 + 0.869669i \(0.335668\pi\)
\(920\) 9.92863e8 0.0420370
\(921\) −1.67189e10 −0.705177
\(922\) −1.46290e10 −0.614690
\(923\) −5.95846e10 −2.49418
\(924\) −3.64993e9 −0.152206
\(925\) −1.15880e10 −0.481409
\(926\) 3.03849e10 1.25753
\(927\) 1.24983e10 0.515316
\(928\) 2.19071e10 0.899845
\(929\) 8.90068e9 0.364224 0.182112 0.983278i \(-0.441707\pi\)
0.182112 + 0.983278i \(0.441707\pi\)
\(930\) −1.31943e9 −0.0537894
\(931\) 3.77536e10 1.53333
\(932\) −1.44142e10 −0.583224
\(933\) −1.64583e10 −0.663436
\(934\) 3.40997e10 1.36942
\(935\) −3.95292e9 −0.158153
\(936\) −1.51960e10 −0.605708
\(937\) 2.70936e10 1.07592 0.537959 0.842971i \(-0.319196\pi\)
0.537959 + 0.842971i \(0.319196\pi\)
\(938\) −1.15817e9 −0.0458208
\(939\) 1.61355e10 0.635992
\(940\) −2.56387e9 −0.100681
\(941\) −4.63143e10 −1.81197 −0.905987 0.423306i \(-0.860869\pi\)
−0.905987 + 0.423306i \(0.860869\pi\)
\(942\) −3.01121e9 −0.117372
\(943\) −6.86900e8 −0.0266749
\(944\) 9.62892e8 0.0372542
\(945\) −4.41921e8 −0.0170346
\(946\) 2.62367e10 1.00760
\(947\) 4.89914e10 1.87454 0.937271 0.348601i \(-0.113343\pi\)
0.937271 + 0.348601i \(0.113343\pi\)
\(948\) −1.15297e10 −0.439531
\(949\) −1.86824e10 −0.709580
\(950\) −3.54185e10 −1.34029
\(951\) 7.80939e9 0.294432
\(952\) −8.76500e9 −0.329248
\(953\) 3.63170e10 1.35921 0.679603 0.733580i \(-0.262152\pi\)
0.679603 + 0.733580i \(0.262152\pi\)
\(954\) −4.63093e9 −0.172683
\(955\) −7.08341e8 −0.0263166
\(956\) −1.45270e10 −0.537743
\(957\) 1.96600e10 0.725091
\(958\) −3.30374e10 −1.21402
\(959\) 5.05436e9 0.185055
\(960\) 2.78544e9 0.101612
\(961\) −1.53076e10 −0.556385
\(962\) 1.69414e10 0.613530
\(963\) −5.63775e9 −0.203430
\(964\) −8.30599e9 −0.298622
\(965\) −9.12207e9 −0.326774
\(966\) 1.07586e9 0.0384003
\(967\) −3.38245e10 −1.20293 −0.601463 0.798901i \(-0.705415\pi\)
−0.601463 + 0.798901i \(0.705415\pi\)
\(968\) 1.33367e10 0.472589
\(969\) 2.14373e10 0.756896
\(970\) 2.34990e8 0.00826701
\(971\) 4.01598e10 1.40775 0.703873 0.710326i \(-0.251452\pi\)
0.703873 + 0.710326i \(0.251452\pi\)
\(972\) 8.84822e8 0.0309047
\(973\) −9.46001e9 −0.329228
\(974\) 2.80982e10 0.974366
\(975\) −2.73894e10 −0.946380
\(976\) −3.49463e9 −0.120317
\(977\) 1.49500e10 0.512872 0.256436 0.966561i \(-0.417452\pi\)
0.256436 + 0.966561i \(0.417452\pi\)
\(978\) −1.10223e10 −0.376779
\(979\) −1.40582e10 −0.478838
\(980\) −2.18572e9 −0.0741830
\(981\) 3.83793e9 0.129794
\(982\) 3.01398e10 1.01566
\(983\) 2.98689e10 1.00295 0.501477 0.865171i \(-0.332790\pi\)
0.501477 + 0.865171i \(0.332790\pi\)
\(984\) −2.42085e9 −0.0810000
\(985\) 1.70825e9 0.0569539
\(986\) 1.53498e10 0.509957
\(987\) −8.54499e9 −0.282879
\(988\) −4.81351e10 −1.58786
\(989\) 7.18906e9 0.236312
\(990\) 1.71000e9 0.0560108
\(991\) 1.42111e10 0.463842 0.231921 0.972735i \(-0.425499\pi\)
0.231921 + 0.972735i \(0.425499\pi\)
\(992\) −1.76257e10 −0.573266
\(993\) 1.62518e10 0.526719
\(994\) 1.48674e10 0.480155
\(995\) 2.75739e8 0.00887396
\(996\) 7.34284e9 0.235481
\(997\) 2.29718e10 0.734112 0.367056 0.930199i \(-0.380366\pi\)
0.367056 + 0.930199i \(0.380366\pi\)
\(998\) −2.84536e10 −0.906108
\(999\) −3.03407e9 −0.0962823
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.c.1.6 17
3.2 odd 2 531.8.a.c.1.12 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.6 17 1.1 even 1 trivial
531.8.a.c.1.12 17 3.2 odd 2