Properties

Label 177.8.a.c.1.11
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 1.11
Root \(7.49302\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+7.49302 q^{2} -27.0000 q^{3} -71.8546 q^{4} +400.807 q^{5} -202.312 q^{6} -272.200 q^{7} -1497.52 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+7.49302 q^{2} -27.0000 q^{3} -71.8546 q^{4} +400.807 q^{5} -202.312 q^{6} -272.200 q^{7} -1497.52 q^{8} +729.000 q^{9} +3003.25 q^{10} +1287.50 q^{11} +1940.08 q^{12} +9535.11 q^{13} -2039.60 q^{14} -10821.8 q^{15} -2023.52 q^{16} -35497.8 q^{17} +5462.41 q^{18} -14418.3 q^{19} -28799.8 q^{20} +7349.41 q^{21} +9647.27 q^{22} +51239.1 q^{23} +40432.9 q^{24} +82520.9 q^{25} +71446.8 q^{26} -19683.0 q^{27} +19558.9 q^{28} -79916.8 q^{29} -81087.8 q^{30} +228626. q^{31} +176520. q^{32} -34762.5 q^{33} -265986. q^{34} -109100. q^{35} -52382.0 q^{36} -139882. q^{37} -108037. q^{38} -257448. q^{39} -600214. q^{40} +497679. q^{41} +55069.3 q^{42} +299840. q^{43} -92512.8 q^{44} +292188. q^{45} +383936. q^{46} +280623. q^{47} +54635.0 q^{48} -749450. q^{49} +618331. q^{50} +958441. q^{51} -685142. q^{52} +1.78864e6 q^{53} -147485. q^{54} +516038. q^{55} +407624. q^{56} +389294. q^{57} -598818. q^{58} -205379. q^{59} +777595. q^{60} +3.36147e6 q^{61} +1.71310e6 q^{62} -198434. q^{63} +1.58168e6 q^{64} +3.82174e6 q^{65} -260476. q^{66} -585455. q^{67} +2.55068e6 q^{68} -1.38346e6 q^{69} -817487. q^{70} +2.83547e6 q^{71} -1.09169e6 q^{72} -758408. q^{73} -1.04814e6 q^{74} -2.22806e6 q^{75} +1.03602e6 q^{76} -350458. q^{77} -1.92906e6 q^{78} +3.27559e6 q^{79} -811040. q^{80} +531441. q^{81} +3.72912e6 q^{82} +9.92122e6 q^{83} -528089. q^{84} -1.42278e7 q^{85} +2.24671e6 q^{86} +2.15775e6 q^{87} -1.92805e6 q^{88} -634907. q^{89} +2.18937e6 q^{90} -2.59546e6 q^{91} -3.68177e6 q^{92} -6.17290e6 q^{93} +2.10271e6 q^{94} -5.77895e6 q^{95} -4.76603e6 q^{96} +8.12879e6 q^{97} -5.61564e6 q^{98} +938588. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.49302 0.662296 0.331148 0.943579i \(-0.392564\pi\)
0.331148 + 0.943579i \(0.392564\pi\)
\(3\) −27.0000 −0.577350
\(4\) −71.8546 −0.561364
\(5\) 400.807 1.43397 0.716985 0.697089i \(-0.245522\pi\)
0.716985 + 0.697089i \(0.245522\pi\)
\(6\) −202.312 −0.382377
\(7\) −272.200 −0.299948 −0.149974 0.988690i \(-0.547919\pi\)
−0.149974 + 0.988690i \(0.547919\pi\)
\(8\) −1497.52 −1.03408
\(9\) 729.000 0.333333
\(10\) 3003.25 0.949712
\(11\) 1287.50 0.291657 0.145829 0.989310i \(-0.453415\pi\)
0.145829 + 0.989310i \(0.453415\pi\)
\(12\) 1940.08 0.324104
\(13\) 9535.11 1.20372 0.601858 0.798603i \(-0.294427\pi\)
0.601858 + 0.798603i \(0.294427\pi\)
\(14\) −2039.60 −0.198654
\(15\) −10821.8 −0.827902
\(16\) −2023.52 −0.123506
\(17\) −35497.8 −1.75239 −0.876195 0.481957i \(-0.839926\pi\)
−0.876195 + 0.481957i \(0.839926\pi\)
\(18\) 5462.41 0.220765
\(19\) −14418.3 −0.482255 −0.241128 0.970493i \(-0.577517\pi\)
−0.241128 + 0.970493i \(0.577517\pi\)
\(20\) −28799.8 −0.804979
\(21\) 7349.41 0.173175
\(22\) 9647.27 0.193163
\(23\) 51239.1 0.878120 0.439060 0.898458i \(-0.355312\pi\)
0.439060 + 0.898458i \(0.355312\pi\)
\(24\) 40432.9 0.597029
\(25\) 82520.9 1.05627
\(26\) 71446.8 0.797216
\(27\) −19683.0 −0.192450
\(28\) 19558.9 0.168380
\(29\) −79916.8 −0.608478 −0.304239 0.952596i \(-0.598402\pi\)
−0.304239 + 0.952596i \(0.598402\pi\)
\(30\) −81087.8 −0.548316
\(31\) 228626. 1.37835 0.689175 0.724595i \(-0.257973\pi\)
0.689175 + 0.724595i \(0.257973\pi\)
\(32\) 176520. 0.952288
\(33\) −34762.5 −0.168388
\(34\) −265986. −1.16060
\(35\) −109100. −0.430116
\(36\) −52382.0 −0.187121
\(37\) −139882. −0.454001 −0.227000 0.973895i \(-0.572892\pi\)
−0.227000 + 0.973895i \(0.572892\pi\)
\(38\) −108037. −0.319395
\(39\) −257448. −0.694966
\(40\) −600214. −1.48285
\(41\) 497679. 1.12773 0.563866 0.825866i \(-0.309314\pi\)
0.563866 + 0.825866i \(0.309314\pi\)
\(42\) 55069.3 0.114693
\(43\) 299840. 0.575109 0.287554 0.957764i \(-0.407158\pi\)
0.287554 + 0.957764i \(0.407158\pi\)
\(44\) −92512.8 −0.163726
\(45\) 292188. 0.477990
\(46\) 383936. 0.581575
\(47\) 280623. 0.394258 0.197129 0.980378i \(-0.436838\pi\)
0.197129 + 0.980378i \(0.436838\pi\)
\(48\) 54635.0 0.0713061
\(49\) −749450. −0.910031
\(50\) 618331. 0.699561
\(51\) 958441. 1.01174
\(52\) −685142. −0.675723
\(53\) 1.78864e6 1.65028 0.825140 0.564928i \(-0.191096\pi\)
0.825140 + 0.564928i \(0.191096\pi\)
\(54\) −147485. −0.127459
\(55\) 516038. 0.418228
\(56\) 407624. 0.310172
\(57\) 389294. 0.278430
\(58\) −598818. −0.402992
\(59\) −205379. −0.130189
\(60\) 777595. 0.464755
\(61\) 3.36147e6 1.89616 0.948080 0.318033i \(-0.103022\pi\)
0.948080 + 0.318033i \(0.103022\pi\)
\(62\) 1.71310e6 0.912875
\(63\) −198434. −0.0999826
\(64\) 1.58168e6 0.754202
\(65\) 3.82174e6 1.72609
\(66\) −260476. −0.111523
\(67\) −585455. −0.237811 −0.118905 0.992906i \(-0.537939\pi\)
−0.118905 + 0.992906i \(0.537939\pi\)
\(68\) 2.55068e6 0.983729
\(69\) −1.38346e6 −0.506983
\(70\) −817487. −0.284864
\(71\) 2.83547e6 0.940202 0.470101 0.882612i \(-0.344217\pi\)
0.470101 + 0.882612i \(0.344217\pi\)
\(72\) −1.09169e6 −0.344695
\(73\) −758408. −0.228178 −0.114089 0.993471i \(-0.536395\pi\)
−0.114089 + 0.993471i \(0.536395\pi\)
\(74\) −1.04814e6 −0.300683
\(75\) −2.22806e6 −0.609836
\(76\) 1.03602e6 0.270721
\(77\) −350458. −0.0874820
\(78\) −1.92906e6 −0.460273
\(79\) 3.27559e6 0.747471 0.373736 0.927535i \(-0.378077\pi\)
0.373736 + 0.927535i \(0.378077\pi\)
\(80\) −811040. −0.177104
\(81\) 531441. 0.111111
\(82\) 3.72912e6 0.746892
\(83\) 9.92122e6 1.90455 0.952273 0.305246i \(-0.0987388\pi\)
0.952273 + 0.305246i \(0.0987388\pi\)
\(84\) −528089. −0.0972142
\(85\) −1.42278e7 −2.51287
\(86\) 2.24671e6 0.380892
\(87\) 2.15775e6 0.351305
\(88\) −1.92805e6 −0.301598
\(89\) −634907. −0.0954652 −0.0477326 0.998860i \(-0.515200\pi\)
−0.0477326 + 0.998860i \(0.515200\pi\)
\(90\) 2.18937e6 0.316571
\(91\) −2.59546e6 −0.361052
\(92\) −3.68177e6 −0.492945
\(93\) −6.17290e6 −0.795791
\(94\) 2.10271e6 0.261116
\(95\) −5.77895e6 −0.691539
\(96\) −4.76603e6 −0.549804
\(97\) 8.12879e6 0.904325 0.452163 0.891936i \(-0.350653\pi\)
0.452163 + 0.891936i \(0.350653\pi\)
\(98\) −5.61564e6 −0.602710
\(99\) 938588. 0.0972191
\(100\) −5.92951e6 −0.592951
\(101\) 4.69794e6 0.453715 0.226857 0.973928i \(-0.427155\pi\)
0.226857 + 0.973928i \(0.427155\pi\)
\(102\) 7.18162e6 0.670073
\(103\) 6.17390e6 0.556710 0.278355 0.960478i \(-0.410211\pi\)
0.278355 + 0.960478i \(0.410211\pi\)
\(104\) −1.42790e7 −1.24474
\(105\) 2.94569e6 0.248328
\(106\) 1.34023e7 1.09297
\(107\) −1.35523e6 −0.106948 −0.0534738 0.998569i \(-0.517029\pi\)
−0.0534738 + 0.998569i \(0.517029\pi\)
\(108\) 1.41431e6 0.108035
\(109\) 1.04846e7 0.775457 0.387729 0.921774i \(-0.373260\pi\)
0.387729 + 0.921774i \(0.373260\pi\)
\(110\) 3.86669e6 0.276990
\(111\) 3.77682e6 0.262118
\(112\) 550803. 0.0370453
\(113\) −7.14003e6 −0.465506 −0.232753 0.972536i \(-0.574773\pi\)
−0.232753 + 0.972536i \(0.574773\pi\)
\(114\) 2.91699e6 0.184403
\(115\) 2.05370e7 1.25920
\(116\) 5.74239e6 0.341578
\(117\) 6.95110e6 0.401239
\(118\) −1.53891e6 −0.0862236
\(119\) 9.66253e6 0.525625
\(120\) 1.62058e7 0.856121
\(121\) −1.78295e7 −0.914936
\(122\) 2.51876e7 1.25582
\(123\) −1.34373e7 −0.651096
\(124\) −1.64278e7 −0.773756
\(125\) 1.76189e6 0.0806854
\(126\) −1.48687e6 −0.0662181
\(127\) −7.91805e6 −0.343009 −0.171504 0.985183i \(-0.554863\pi\)
−0.171504 + 0.985183i \(0.554863\pi\)
\(128\) −1.07430e7 −0.452783
\(129\) −8.09568e6 −0.332039
\(130\) 2.86363e7 1.14318
\(131\) −2.93961e7 −1.14246 −0.571230 0.820790i \(-0.693534\pi\)
−0.571230 + 0.820790i \(0.693534\pi\)
\(132\) 2.49785e6 0.0945273
\(133\) 3.92467e6 0.144651
\(134\) −4.38683e6 −0.157501
\(135\) −7.88907e6 −0.275967
\(136\) 5.31585e7 1.81212
\(137\) 9.93494e6 0.330098 0.165049 0.986285i \(-0.447222\pi\)
0.165049 + 0.986285i \(0.447222\pi\)
\(138\) −1.03663e7 −0.335773
\(139\) 2.35079e7 0.742440 0.371220 0.928545i \(-0.378939\pi\)
0.371220 + 0.928545i \(0.378939\pi\)
\(140\) 7.83932e6 0.241452
\(141\) −7.57682e6 −0.227625
\(142\) 2.12463e7 0.622692
\(143\) 1.22765e7 0.351073
\(144\) −1.47515e6 −0.0411686
\(145\) −3.20312e7 −0.872539
\(146\) −5.68277e6 −0.151121
\(147\) 2.02351e7 0.525407
\(148\) 1.00512e7 0.254860
\(149\) −2.98820e7 −0.740044 −0.370022 0.929023i \(-0.620650\pi\)
−0.370022 + 0.929023i \(0.620650\pi\)
\(150\) −1.66949e7 −0.403892
\(151\) −3.58463e7 −0.847277 −0.423639 0.905831i \(-0.639247\pi\)
−0.423639 + 0.905831i \(0.639247\pi\)
\(152\) 2.15916e7 0.498693
\(153\) −2.58779e7 −0.584130
\(154\) −2.62599e6 −0.0579389
\(155\) 9.16347e7 1.97651
\(156\) 1.84988e7 0.390129
\(157\) −6.40578e7 −1.32106 −0.660531 0.750799i \(-0.729669\pi\)
−0.660531 + 0.750799i \(0.729669\pi\)
\(158\) 2.45440e7 0.495047
\(159\) −4.82933e7 −0.952790
\(160\) 7.07502e7 1.36555
\(161\) −1.39473e7 −0.263390
\(162\) 3.98210e6 0.0735884
\(163\) −4.23392e7 −0.765749 −0.382874 0.923800i \(-0.625066\pi\)
−0.382874 + 0.923800i \(0.625066\pi\)
\(164\) −3.57605e7 −0.633068
\(165\) −1.39330e7 −0.241464
\(166\) 7.43399e7 1.26137
\(167\) 6.08959e7 1.01177 0.505883 0.862602i \(-0.331167\pi\)
0.505883 + 0.862602i \(0.331167\pi\)
\(168\) −1.10059e7 −0.179078
\(169\) 2.81699e7 0.448933
\(170\) −1.06609e8 −1.66426
\(171\) −1.05109e7 −0.160752
\(172\) −2.15449e7 −0.322845
\(173\) −1.00814e8 −1.48033 −0.740165 0.672425i \(-0.765253\pi\)
−0.740165 + 0.672425i \(0.765253\pi\)
\(174\) 1.61681e7 0.232668
\(175\) −2.24622e7 −0.316825
\(176\) −2.60528e6 −0.0360214
\(177\) 5.54523e6 0.0751646
\(178\) −4.75737e6 −0.0632262
\(179\) 1.28687e8 1.67707 0.838533 0.544851i \(-0.183414\pi\)
0.838533 + 0.544851i \(0.183414\pi\)
\(180\) −2.09951e7 −0.268326
\(181\) −7.12899e7 −0.893620 −0.446810 0.894629i \(-0.647440\pi\)
−0.446810 + 0.894629i \(0.647440\pi\)
\(182\) −1.94479e7 −0.239123
\(183\) −9.07597e7 −1.09475
\(184\) −7.67313e7 −0.908051
\(185\) −5.60657e7 −0.651023
\(186\) −4.62537e7 −0.527049
\(187\) −4.57035e7 −0.511097
\(188\) −2.01641e7 −0.221322
\(189\) 5.35772e6 0.0577250
\(190\) −4.33018e7 −0.458003
\(191\) 7.68302e7 0.797839 0.398920 0.916986i \(-0.369385\pi\)
0.398920 + 0.916986i \(0.369385\pi\)
\(192\) −4.27053e7 −0.435439
\(193\) 1.34702e8 1.34873 0.674364 0.738399i \(-0.264418\pi\)
0.674364 + 0.738399i \(0.264418\pi\)
\(194\) 6.09092e7 0.598931
\(195\) −1.03187e8 −0.996560
\(196\) 5.38514e7 0.510859
\(197\) 1.09326e8 1.01881 0.509405 0.860527i \(-0.329866\pi\)
0.509405 + 0.860527i \(0.329866\pi\)
\(198\) 7.03286e6 0.0643878
\(199\) −5.09514e7 −0.458322 −0.229161 0.973389i \(-0.573598\pi\)
−0.229161 + 0.973389i \(0.573598\pi\)
\(200\) −1.23576e8 −1.09227
\(201\) 1.58073e7 0.137300
\(202\) 3.52018e7 0.300493
\(203\) 2.17534e7 0.182512
\(204\) −6.88685e7 −0.567956
\(205\) 1.99473e8 1.61713
\(206\) 4.62612e7 0.368707
\(207\) 3.73533e7 0.292707
\(208\) −1.92945e7 −0.148666
\(209\) −1.85636e7 −0.140653
\(210\) 2.20721e7 0.164466
\(211\) −1.44274e8 −1.05730 −0.528651 0.848839i \(-0.677302\pi\)
−0.528651 + 0.848839i \(0.677302\pi\)
\(212\) −1.28522e8 −0.926408
\(213\) −7.65578e7 −0.542826
\(214\) −1.01548e7 −0.0708310
\(215\) 1.20178e8 0.824688
\(216\) 2.94756e7 0.199010
\(217\) −6.22321e7 −0.413433
\(218\) 7.85611e7 0.513582
\(219\) 2.04770e7 0.131738
\(220\) −3.70798e7 −0.234778
\(221\) −3.38476e8 −2.10938
\(222\) 2.82998e7 0.173599
\(223\) 1.97785e8 1.19433 0.597167 0.802117i \(-0.296293\pi\)
0.597167 + 0.802117i \(0.296293\pi\)
\(224\) −4.80487e7 −0.285637
\(225\) 6.01577e7 0.352089
\(226\) −5.35004e7 −0.308303
\(227\) −3.33997e8 −1.89519 −0.947593 0.319479i \(-0.896492\pi\)
−0.947593 + 0.319479i \(0.896492\pi\)
\(228\) −2.79726e7 −0.156301
\(229\) −5.10139e7 −0.280714 −0.140357 0.990101i \(-0.544825\pi\)
−0.140357 + 0.990101i \(0.544825\pi\)
\(230\) 1.53884e8 0.833961
\(231\) 9.46237e6 0.0505077
\(232\) 1.19677e8 0.629218
\(233\) −1.21435e8 −0.628923 −0.314461 0.949270i \(-0.601824\pi\)
−0.314461 + 0.949270i \(0.601824\pi\)
\(234\) 5.20847e7 0.265739
\(235\) 1.12476e8 0.565354
\(236\) 1.47574e7 0.0730834
\(237\) −8.84409e7 −0.431553
\(238\) 7.24015e7 0.348119
\(239\) 2.41270e7 0.114317 0.0571585 0.998365i \(-0.481796\pi\)
0.0571585 + 0.998365i \(0.481796\pi\)
\(240\) 2.18981e7 0.102251
\(241\) −5.58221e7 −0.256890 −0.128445 0.991717i \(-0.540999\pi\)
−0.128445 + 0.991717i \(0.540999\pi\)
\(242\) −1.33597e8 −0.605958
\(243\) −1.43489e7 −0.0641500
\(244\) −2.41537e8 −1.06444
\(245\) −3.00384e8 −1.30496
\(246\) −1.00686e8 −0.431218
\(247\) −1.37480e8 −0.580498
\(248\) −3.42371e8 −1.42533
\(249\) −2.67873e8 −1.09959
\(250\) 1.32019e7 0.0534376
\(251\) 1.85785e8 0.741570 0.370785 0.928719i \(-0.379089\pi\)
0.370785 + 0.928719i \(0.379089\pi\)
\(252\) 1.42584e7 0.0561267
\(253\) 6.59703e7 0.256110
\(254\) −5.93301e7 −0.227173
\(255\) 3.84150e8 1.45081
\(256\) −2.82952e8 −1.05408
\(257\) −4.53248e8 −1.66560 −0.832799 0.553575i \(-0.813263\pi\)
−0.832799 + 0.553575i \(0.813263\pi\)
\(258\) −6.06611e7 −0.219908
\(259\) 3.80760e7 0.136177
\(260\) −2.74609e8 −0.968966
\(261\) −5.82593e7 −0.202826
\(262\) −2.20266e8 −0.756646
\(263\) 4.28480e8 1.45240 0.726199 0.687485i \(-0.241285\pi\)
0.726199 + 0.687485i \(0.241285\pi\)
\(264\) 5.20574e7 0.174128
\(265\) 7.16899e8 2.36645
\(266\) 2.94076e7 0.0958020
\(267\) 1.71425e7 0.0551169
\(268\) 4.20677e7 0.133499
\(269\) −2.65173e7 −0.0830609 −0.0415305 0.999137i \(-0.513223\pi\)
−0.0415305 + 0.999137i \(0.513223\pi\)
\(270\) −5.91130e7 −0.182772
\(271\) 1.36921e8 0.417905 0.208952 0.977926i \(-0.432995\pi\)
0.208952 + 0.977926i \(0.432995\pi\)
\(272\) 7.18305e7 0.216430
\(273\) 7.00775e7 0.208453
\(274\) 7.44427e7 0.218623
\(275\) 1.06246e8 0.308068
\(276\) 9.94077e7 0.284602
\(277\) 4.27728e8 1.20917 0.604586 0.796540i \(-0.293338\pi\)
0.604586 + 0.796540i \(0.293338\pi\)
\(278\) 1.76145e8 0.491715
\(279\) 1.66668e8 0.459450
\(280\) 1.63378e8 0.444776
\(281\) −3.52657e8 −0.948157 −0.474079 0.880482i \(-0.657219\pi\)
−0.474079 + 0.880482i \(0.657219\pi\)
\(282\) −5.67733e7 −0.150755
\(283\) 5.10806e8 1.33969 0.669844 0.742502i \(-0.266361\pi\)
0.669844 + 0.742502i \(0.266361\pi\)
\(284\) −2.03742e8 −0.527796
\(285\) 1.56032e8 0.399260
\(286\) 9.19878e7 0.232514
\(287\) −1.35468e8 −0.338261
\(288\) 1.28683e8 0.317429
\(289\) 8.49757e8 2.07087
\(290\) −2.40010e8 −0.577879
\(291\) −2.19477e8 −0.522112
\(292\) 5.44952e7 0.128091
\(293\) 1.53209e8 0.355835 0.177917 0.984045i \(-0.443064\pi\)
0.177917 + 0.984045i \(0.443064\pi\)
\(294\) 1.51622e8 0.347975
\(295\) −8.23172e7 −0.186687
\(296\) 2.09476e8 0.469476
\(297\) −2.53419e7 −0.0561295
\(298\) −2.23906e8 −0.490128
\(299\) 4.88570e8 1.05701
\(300\) 1.60097e8 0.342340
\(301\) −8.16165e7 −0.172503
\(302\) −2.68597e8 −0.561148
\(303\) −1.26844e8 −0.261952
\(304\) 2.91757e7 0.0595613
\(305\) 1.34730e9 2.71903
\(306\) −1.93904e8 −0.386867
\(307\) 7.83772e8 1.54598 0.772992 0.634415i \(-0.218759\pi\)
0.772992 + 0.634415i \(0.218759\pi\)
\(308\) 2.51820e7 0.0491093
\(309\) −1.66695e8 −0.321417
\(310\) 6.86621e8 1.30903
\(311\) −7.44813e7 −0.140406 −0.0702030 0.997533i \(-0.522365\pi\)
−0.0702030 + 0.997533i \(0.522365\pi\)
\(312\) 3.85532e8 0.718654
\(313\) 4.78847e8 0.882657 0.441328 0.897346i \(-0.354507\pi\)
0.441328 + 0.897346i \(0.354507\pi\)
\(314\) −4.79986e8 −0.874934
\(315\) −7.95337e7 −0.143372
\(316\) −2.35366e8 −0.419604
\(317\) −3.22619e8 −0.568829 −0.284415 0.958701i \(-0.591799\pi\)
−0.284415 + 0.958701i \(0.591799\pi\)
\(318\) −3.61863e8 −0.631029
\(319\) −1.02893e8 −0.177467
\(320\) 6.33946e8 1.08150
\(321\) 3.65913e7 0.0617462
\(322\) −1.04507e8 −0.174442
\(323\) 5.11819e8 0.845099
\(324\) −3.81865e7 −0.0623738
\(325\) 7.86846e8 1.27145
\(326\) −3.17249e8 −0.507152
\(327\) −2.83083e8 −0.447710
\(328\) −7.45282e8 −1.16617
\(329\) −7.63857e7 −0.118257
\(330\) −1.04401e8 −0.159920
\(331\) 1.02925e9 1.56000 0.779998 0.625782i \(-0.215220\pi\)
0.779998 + 0.625782i \(0.215220\pi\)
\(332\) −7.12885e8 −1.06914
\(333\) −1.01974e8 −0.151334
\(334\) 4.56294e8 0.670089
\(335\) −2.34654e8 −0.341014
\(336\) −1.48717e7 −0.0213881
\(337\) 3.61784e7 0.0514926 0.0257463 0.999669i \(-0.491804\pi\)
0.0257463 + 0.999669i \(0.491804\pi\)
\(338\) 2.11077e8 0.297326
\(339\) 1.92781e8 0.268760
\(340\) 1.02233e9 1.41064
\(341\) 2.94356e8 0.402006
\(342\) −7.87588e7 −0.106465
\(343\) 4.28169e8 0.572910
\(344\) −4.49015e8 −0.594711
\(345\) −5.54498e8 −0.726998
\(346\) −7.55400e8 −0.980417
\(347\) 1.13436e9 1.45746 0.728729 0.684802i \(-0.240111\pi\)
0.728729 + 0.684802i \(0.240111\pi\)
\(348\) −1.55044e8 −0.197210
\(349\) −3.76547e8 −0.474166 −0.237083 0.971489i \(-0.576191\pi\)
−0.237083 + 0.971489i \(0.576191\pi\)
\(350\) −1.68310e8 −0.209832
\(351\) −1.87680e8 −0.231655
\(352\) 2.27269e8 0.277742
\(353\) −1.14395e9 −1.38419 −0.692097 0.721805i \(-0.743313\pi\)
−0.692097 + 0.721805i \(0.743313\pi\)
\(354\) 4.15505e7 0.0497812
\(355\) 1.13648e9 1.34822
\(356\) 4.56210e7 0.0535908
\(357\) −2.60888e8 −0.303470
\(358\) 9.64257e8 1.11071
\(359\) −1.69002e9 −1.92780 −0.963900 0.266265i \(-0.914210\pi\)
−0.963900 + 0.266265i \(0.914210\pi\)
\(360\) −4.37556e8 −0.494282
\(361\) −6.85984e8 −0.767430
\(362\) −5.34177e8 −0.591841
\(363\) 4.81397e8 0.528239
\(364\) 1.86496e8 0.202682
\(365\) −3.03975e8 −0.327200
\(366\) −6.80064e8 −0.725047
\(367\) −1.11560e9 −1.17809 −0.589046 0.808100i \(-0.700496\pi\)
−0.589046 + 0.808100i \(0.700496\pi\)
\(368\) −1.03683e8 −0.108453
\(369\) 3.62808e8 0.375911
\(370\) −4.20102e8 −0.431170
\(371\) −4.86869e8 −0.494998
\(372\) 4.43551e8 0.446728
\(373\) −1.06668e9 −1.06427 −0.532134 0.846660i \(-0.678610\pi\)
−0.532134 + 0.846660i \(0.678610\pi\)
\(374\) −3.42457e8 −0.338497
\(375\) −4.75712e7 −0.0465837
\(376\) −4.20237e8 −0.407697
\(377\) −7.62015e8 −0.732435
\(378\) 4.01455e7 0.0382310
\(379\) 9.06012e7 0.0854864 0.0427432 0.999086i \(-0.486390\pi\)
0.0427432 + 0.999086i \(0.486390\pi\)
\(380\) 4.15245e8 0.388205
\(381\) 2.13787e8 0.198036
\(382\) 5.75691e8 0.528406
\(383\) 1.02770e9 0.934700 0.467350 0.884072i \(-0.345209\pi\)
0.467350 + 0.884072i \(0.345209\pi\)
\(384\) 2.90061e8 0.261414
\(385\) −1.40466e8 −0.125446
\(386\) 1.00933e9 0.893257
\(387\) 2.18583e8 0.191703
\(388\) −5.84091e8 −0.507656
\(389\) 1.62652e9 1.40099 0.700495 0.713657i \(-0.252962\pi\)
0.700495 + 0.713657i \(0.252962\pi\)
\(390\) −7.73181e8 −0.660017
\(391\) −1.81888e9 −1.53881
\(392\) 1.12231e9 0.941050
\(393\) 7.93696e8 0.659599
\(394\) 8.19184e8 0.674753
\(395\) 1.31288e9 1.07185
\(396\) −6.74419e7 −0.0545753
\(397\) −1.41716e8 −0.113672 −0.0568359 0.998384i \(-0.518101\pi\)
−0.0568359 + 0.998384i \(0.518101\pi\)
\(398\) −3.81780e8 −0.303545
\(399\) −1.05966e8 −0.0835145
\(400\) −1.66983e8 −0.130455
\(401\) −1.70702e9 −1.32200 −0.661002 0.750384i \(-0.729868\pi\)
−0.661002 + 0.750384i \(0.729868\pi\)
\(402\) 1.18444e8 0.0909333
\(403\) 2.17997e9 1.65914
\(404\) −3.37569e8 −0.254699
\(405\) 2.13005e8 0.159330
\(406\) 1.62999e8 0.120877
\(407\) −1.80099e8 −0.132413
\(408\) −1.43528e9 −1.04623
\(409\) −1.96299e9 −1.41869 −0.709345 0.704862i \(-0.751009\pi\)
−0.709345 + 0.704862i \(0.751009\pi\)
\(410\) 1.49466e9 1.07102
\(411\) −2.68243e8 −0.190582
\(412\) −4.43623e8 −0.312517
\(413\) 5.59043e7 0.0390499
\(414\) 2.79889e8 0.193858
\(415\) 3.97649e9 2.73106
\(416\) 1.68313e9 1.14628
\(417\) −6.34713e8 −0.428648
\(418\) −1.39097e8 −0.0931540
\(419\) 2.45844e9 1.63271 0.816357 0.577547i \(-0.195990\pi\)
0.816357 + 0.577547i \(0.195990\pi\)
\(420\) −2.11662e8 −0.139402
\(421\) 6.57370e8 0.429361 0.214681 0.976684i \(-0.431129\pi\)
0.214681 + 0.976684i \(0.431129\pi\)
\(422\) −1.08105e9 −0.700247
\(423\) 2.04574e8 0.131419
\(424\) −2.67852e9 −1.70653
\(425\) −2.92931e9 −1.85099
\(426\) −5.73649e8 −0.359511
\(427\) −9.14994e8 −0.568749
\(428\) 9.73799e7 0.0600366
\(429\) −3.31464e8 −0.202692
\(430\) 9.00495e8 0.546187
\(431\) −1.65546e9 −0.995976 −0.497988 0.867184i \(-0.665928\pi\)
−0.497988 + 0.867184i \(0.665928\pi\)
\(432\) 3.98289e7 0.0237687
\(433\) −1.96267e9 −1.16182 −0.580910 0.813968i \(-0.697303\pi\)
−0.580910 + 0.813968i \(0.697303\pi\)
\(434\) −4.66306e8 −0.273815
\(435\) 8.64841e8 0.503760
\(436\) −7.53365e8 −0.435314
\(437\) −7.38781e8 −0.423478
\(438\) 1.53435e8 0.0872498
\(439\) 6.65374e8 0.375353 0.187677 0.982231i \(-0.439904\pi\)
0.187677 + 0.982231i \(0.439904\pi\)
\(440\) −7.72775e8 −0.432483
\(441\) −5.46349e8 −0.303344
\(442\) −2.53621e9 −1.39703
\(443\) 2.45181e8 0.133991 0.0669953 0.997753i \(-0.478659\pi\)
0.0669953 + 0.997753i \(0.478659\pi\)
\(444\) −2.71382e8 −0.147143
\(445\) −2.54475e8 −0.136894
\(446\) 1.48201e9 0.791003
\(447\) 8.06813e8 0.427264
\(448\) −4.30533e8 −0.226221
\(449\) 1.56613e9 0.816516 0.408258 0.912867i \(-0.366136\pi\)
0.408258 + 0.912867i \(0.366136\pi\)
\(450\) 4.50763e8 0.233187
\(451\) 6.40762e8 0.328911
\(452\) 5.13044e8 0.261318
\(453\) 9.67851e8 0.489176
\(454\) −2.50265e9 −1.25517
\(455\) −1.04028e9 −0.517737
\(456\) −5.82974e8 −0.287920
\(457\) 2.77494e9 1.36002 0.680012 0.733201i \(-0.261974\pi\)
0.680012 + 0.733201i \(0.261974\pi\)
\(458\) −3.82249e8 −0.185916
\(459\) 6.98704e8 0.337247
\(460\) −1.47568e9 −0.706868
\(461\) 1.84492e9 0.877051 0.438525 0.898719i \(-0.355501\pi\)
0.438525 + 0.898719i \(0.355501\pi\)
\(462\) 7.09017e7 0.0334511
\(463\) −1.44653e9 −0.677322 −0.338661 0.940908i \(-0.609974\pi\)
−0.338661 + 0.940908i \(0.609974\pi\)
\(464\) 1.61713e8 0.0751506
\(465\) −2.47414e9 −1.14114
\(466\) −9.09914e8 −0.416533
\(467\) −1.95804e9 −0.889636 −0.444818 0.895621i \(-0.646732\pi\)
−0.444818 + 0.895621i \(0.646732\pi\)
\(468\) −4.99469e8 −0.225241
\(469\) 1.59361e8 0.0713309
\(470\) 8.42782e8 0.374432
\(471\) 1.72956e9 0.762715
\(472\) 3.07558e8 0.134626
\(473\) 3.86044e8 0.167735
\(474\) −6.62689e8 −0.285815
\(475\) −1.18981e9 −0.509390
\(476\) −6.94297e8 −0.295067
\(477\) 1.30392e9 0.550093
\(478\) 1.80784e8 0.0757116
\(479\) −3.02650e9 −1.25825 −0.629124 0.777305i \(-0.716586\pi\)
−0.629124 + 0.777305i \(0.716586\pi\)
\(480\) −1.91026e9 −0.788401
\(481\) −1.33379e9 −0.546488
\(482\) −4.18277e8 −0.170137
\(483\) 3.76577e8 0.152068
\(484\) 1.28113e9 0.513612
\(485\) 3.25807e9 1.29677
\(486\) −1.07517e8 −0.0424863
\(487\) −1.90649e9 −0.747967 −0.373983 0.927435i \(-0.622008\pi\)
−0.373983 + 0.927435i \(0.622008\pi\)
\(488\) −5.03385e9 −1.96079
\(489\) 1.14316e9 0.442105
\(490\) −2.25079e9 −0.864267
\(491\) 2.37779e9 0.906544 0.453272 0.891372i \(-0.350257\pi\)
0.453272 + 0.891372i \(0.350257\pi\)
\(492\) 9.65534e8 0.365502
\(493\) 2.83687e9 1.06629
\(494\) −1.03014e9 −0.384462
\(495\) 3.76192e8 0.139409
\(496\) −4.62629e8 −0.170234
\(497\) −7.71817e8 −0.282012
\(498\) −2.00718e9 −0.728254
\(499\) −1.25478e9 −0.452081 −0.226041 0.974118i \(-0.572578\pi\)
−0.226041 + 0.974118i \(0.572578\pi\)
\(500\) −1.26600e8 −0.0452939
\(501\) −1.64419e9 −0.584144
\(502\) 1.39209e9 0.491139
\(503\) −1.13501e9 −0.397660 −0.198830 0.980034i \(-0.563714\pi\)
−0.198830 + 0.980034i \(0.563714\pi\)
\(504\) 2.97158e8 0.103391
\(505\) 1.88297e9 0.650613
\(506\) 4.94317e8 0.169621
\(507\) −7.60586e8 −0.259191
\(508\) 5.68949e8 0.192553
\(509\) −4.34255e9 −1.45960 −0.729798 0.683663i \(-0.760386\pi\)
−0.729798 + 0.683663i \(0.760386\pi\)
\(510\) 2.87844e9 0.960864
\(511\) 2.06439e8 0.0684414
\(512\) −7.45063e8 −0.245329
\(513\) 2.83796e8 0.0928100
\(514\) −3.39620e9 −1.10312
\(515\) 2.47454e9 0.798305
\(516\) 5.81712e8 0.186395
\(517\) 3.61302e8 0.114988
\(518\) 2.85305e8 0.0901892
\(519\) 2.72197e9 0.854669
\(520\) −5.72311e9 −1.78493
\(521\) 1.60678e9 0.497764 0.248882 0.968534i \(-0.419937\pi\)
0.248882 + 0.968534i \(0.419937\pi\)
\(522\) −4.36538e8 −0.134331
\(523\) −1.33872e9 −0.409198 −0.204599 0.978846i \(-0.565589\pi\)
−0.204599 + 0.978846i \(0.565589\pi\)
\(524\) 2.11225e9 0.641336
\(525\) 6.06480e8 0.182919
\(526\) 3.21061e9 0.961917
\(527\) −8.11572e9 −2.41541
\(528\) 7.03426e7 0.0207970
\(529\) −7.79382e8 −0.228905
\(530\) 5.37174e9 1.56729
\(531\) −1.49721e8 −0.0433963
\(532\) −2.82006e8 −0.0812021
\(533\) 4.74542e9 1.35747
\(534\) 1.28449e8 0.0365037
\(535\) −5.43187e8 −0.153360
\(536\) 8.76728e8 0.245917
\(537\) −3.47456e9 −0.968255
\(538\) −1.98695e8 −0.0550109
\(539\) −9.64917e8 −0.265417
\(540\) 5.66867e8 0.154918
\(541\) 2.04184e9 0.554410 0.277205 0.960811i \(-0.410592\pi\)
0.277205 + 0.960811i \(0.410592\pi\)
\(542\) 1.02595e9 0.276777
\(543\) 1.92483e9 0.515932
\(544\) −6.26606e9 −1.66878
\(545\) 4.20228e9 1.11198
\(546\) 5.25092e8 0.138058
\(547\) 4.97747e9 1.30033 0.650164 0.759794i \(-0.274700\pi\)
0.650164 + 0.759794i \(0.274700\pi\)
\(548\) −7.13871e8 −0.185305
\(549\) 2.45051e9 0.632053
\(550\) 7.96101e8 0.204032
\(551\) 1.15226e9 0.293442
\(552\) 2.07175e9 0.524263
\(553\) −8.91616e8 −0.224202
\(554\) 3.20497e9 0.800830
\(555\) 1.51378e9 0.375869
\(556\) −1.68915e9 −0.416780
\(557\) 4.71080e9 1.15505 0.577526 0.816372i \(-0.304018\pi\)
0.577526 + 0.816372i \(0.304018\pi\)
\(558\) 1.24885e9 0.304292
\(559\) 2.85901e9 0.692267
\(560\) 2.20765e8 0.0531218
\(561\) 1.23399e9 0.295082
\(562\) −2.64247e9 −0.627961
\(563\) −4.24894e9 −1.00346 −0.501731 0.865024i \(-0.667303\pi\)
−0.501731 + 0.865024i \(0.667303\pi\)
\(564\) 5.44430e8 0.127781
\(565\) −2.86177e9 −0.667521
\(566\) 3.82748e9 0.887269
\(567\) −1.44658e8 −0.0333275
\(568\) −4.24616e9 −0.972249
\(569\) 5.26230e9 1.19752 0.598759 0.800929i \(-0.295661\pi\)
0.598759 + 0.800929i \(0.295661\pi\)
\(570\) 1.16915e9 0.264428
\(571\) −3.29668e9 −0.741056 −0.370528 0.928821i \(-0.620823\pi\)
−0.370528 + 0.928821i \(0.620823\pi\)
\(572\) −8.82120e8 −0.197080
\(573\) −2.07442e9 −0.460633
\(574\) −1.01507e9 −0.224029
\(575\) 4.22829e9 0.927529
\(576\) 1.15304e9 0.251401
\(577\) −3.50100e9 −0.758713 −0.379356 0.925251i \(-0.623855\pi\)
−0.379356 + 0.925251i \(0.623855\pi\)
\(578\) 6.36725e9 1.37153
\(579\) −3.63696e9 −0.778688
\(580\) 2.30159e9 0.489812
\(581\) −2.70056e9 −0.571265
\(582\) −1.64455e9 −0.345793
\(583\) 2.30288e9 0.481316
\(584\) 1.13573e9 0.235955
\(585\) 2.78605e9 0.575364
\(586\) 1.14800e9 0.235668
\(587\) −3.06924e9 −0.626322 −0.313161 0.949700i \(-0.601388\pi\)
−0.313161 + 0.949700i \(0.601388\pi\)
\(588\) −1.45399e9 −0.294945
\(589\) −3.29640e9 −0.664716
\(590\) −6.16805e8 −0.123642
\(591\) −2.95181e9 −0.588210
\(592\) 2.83055e8 0.0560718
\(593\) −9.06218e9 −1.78460 −0.892300 0.451442i \(-0.850910\pi\)
−0.892300 + 0.451442i \(0.850910\pi\)
\(594\) −1.89887e8 −0.0371743
\(595\) 3.87280e9 0.753730
\(596\) 2.14716e9 0.415434
\(597\) 1.37569e9 0.264612
\(598\) 3.66087e9 0.700052
\(599\) 5.53085e9 1.05147 0.525736 0.850648i \(-0.323790\pi\)
0.525736 + 0.850648i \(0.323790\pi\)
\(600\) 3.33656e9 0.630622
\(601\) 2.64790e9 0.497555 0.248777 0.968561i \(-0.419971\pi\)
0.248777 + 0.968561i \(0.419971\pi\)
\(602\) −6.11555e8 −0.114248
\(603\) −4.26797e8 −0.0792703
\(604\) 2.57573e9 0.475631
\(605\) −7.14619e9 −1.31199
\(606\) −9.50448e8 −0.173490
\(607\) 1.49005e9 0.270421 0.135210 0.990817i \(-0.456829\pi\)
0.135210 + 0.990817i \(0.456829\pi\)
\(608\) −2.54512e9 −0.459246
\(609\) −5.87341e8 −0.105373
\(610\) 1.00953e10 1.80080
\(611\) 2.67577e9 0.474575
\(612\) 1.85945e9 0.327910
\(613\) 6.01176e9 1.05412 0.527060 0.849828i \(-0.323294\pi\)
0.527060 + 0.849828i \(0.323294\pi\)
\(614\) 5.87282e9 1.02390
\(615\) −5.38577e9 −0.933652
\(616\) 5.24816e8 0.0904638
\(617\) 1.06578e10 1.82671 0.913356 0.407161i \(-0.133481\pi\)
0.913356 + 0.407161i \(0.133481\pi\)
\(618\) −1.24905e9 −0.212873
\(619\) −3.27736e9 −0.555401 −0.277700 0.960668i \(-0.589572\pi\)
−0.277700 + 0.960668i \(0.589572\pi\)
\(620\) −6.58438e9 −1.10954
\(621\) −1.00854e9 −0.168994
\(622\) −5.58090e8 −0.0929903
\(623\) 1.72822e8 0.0286346
\(624\) 5.20951e8 0.0858323
\(625\) −5.74076e9 −0.940567
\(626\) 3.58801e9 0.584580
\(627\) 5.01217e8 0.0812062
\(628\) 4.60285e9 0.741597
\(629\) 4.96552e9 0.795586
\(630\) −5.95948e8 −0.0949546
\(631\) −5.83790e9 −0.925026 −0.462513 0.886613i \(-0.653052\pi\)
−0.462513 + 0.886613i \(0.653052\pi\)
\(632\) −4.90524e9 −0.772949
\(633\) 3.89540e9 0.610434
\(634\) −2.41739e9 −0.376733
\(635\) −3.17361e9 −0.491864
\(636\) 3.47010e9 0.534862
\(637\) −7.14609e9 −1.09542
\(638\) −7.70978e8 −0.117536
\(639\) 2.06706e9 0.313401
\(640\) −4.30586e9 −0.649277
\(641\) −9.30439e9 −1.39536 −0.697678 0.716412i \(-0.745783\pi\)
−0.697678 + 0.716412i \(0.745783\pi\)
\(642\) 2.74180e8 0.0408943
\(643\) 7.27839e9 1.07969 0.539843 0.841766i \(-0.318484\pi\)
0.539843 + 0.841766i \(0.318484\pi\)
\(644\) 1.00218e9 0.147858
\(645\) −3.24480e9 −0.476134
\(646\) 3.83507e9 0.559705
\(647\) −8.75508e9 −1.27085 −0.635426 0.772162i \(-0.719176\pi\)
−0.635426 + 0.772162i \(0.719176\pi\)
\(648\) −7.95841e8 −0.114898
\(649\) −2.64425e8 −0.0379705
\(650\) 5.89585e9 0.842073
\(651\) 1.68027e9 0.238696
\(652\) 3.04227e9 0.429864
\(653\) 5.51167e8 0.0774617 0.0387308 0.999250i \(-0.487669\pi\)
0.0387308 + 0.999250i \(0.487669\pi\)
\(654\) −2.12115e9 −0.296517
\(655\) −1.17822e10 −1.63825
\(656\) −1.00706e9 −0.139281
\(657\) −5.52880e8 −0.0760592
\(658\) −5.72360e8 −0.0783210
\(659\) 2.23902e8 0.0304761 0.0152381 0.999884i \(-0.495149\pi\)
0.0152381 + 0.999884i \(0.495149\pi\)
\(660\) 1.00115e9 0.135549
\(661\) 1.42160e10 1.91458 0.957288 0.289137i \(-0.0933682\pi\)
0.957288 + 0.289137i \(0.0933682\pi\)
\(662\) 7.71221e9 1.03318
\(663\) 9.13885e9 1.21785
\(664\) −1.48572e10 −1.96946
\(665\) 1.57303e9 0.207426
\(666\) −7.64095e8 −0.100228
\(667\) −4.09486e9 −0.534317
\(668\) −4.37565e9 −0.567970
\(669\) −5.34019e9 −0.689550
\(670\) −1.75827e9 −0.225852
\(671\) 4.32789e9 0.553029
\(672\) 1.29732e9 0.164912
\(673\) −1.15239e10 −1.45730 −0.728648 0.684888i \(-0.759851\pi\)
−0.728648 + 0.684888i \(0.759851\pi\)
\(674\) 2.71085e8 0.0341033
\(675\) −1.62426e9 −0.203279
\(676\) −2.02414e9 −0.252015
\(677\) −1.31983e10 −1.63477 −0.817385 0.576092i \(-0.804577\pi\)
−0.817385 + 0.576092i \(0.804577\pi\)
\(678\) 1.44451e9 0.177999
\(679\) −2.21266e9 −0.271250
\(680\) 2.13063e10 2.59852
\(681\) 9.01792e9 1.09419
\(682\) 2.20561e9 0.266247
\(683\) −1.04211e10 −1.25153 −0.625766 0.780010i \(-0.715214\pi\)
−0.625766 + 0.780010i \(0.715214\pi\)
\(684\) 7.55260e8 0.0902403
\(685\) 3.98199e9 0.473351
\(686\) 3.20828e9 0.379436
\(687\) 1.37738e9 0.162071
\(688\) −6.06732e8 −0.0710292
\(689\) 1.70549e10 1.98647
\(690\) −4.15486e9 −0.481488
\(691\) 1.27099e10 1.46544 0.732721 0.680529i \(-0.238250\pi\)
0.732721 + 0.680529i \(0.238250\pi\)
\(692\) 7.24394e9 0.831005
\(693\) −2.55484e8 −0.0291607
\(694\) 8.49976e9 0.965269
\(695\) 9.42211e9 1.06464
\(696\) −3.23127e9 −0.363279
\(697\) −1.76665e10 −1.97622
\(698\) −2.82148e9 −0.314038
\(699\) 3.27874e9 0.363109
\(700\) 1.61401e9 0.177854
\(701\) −9.44240e9 −1.03531 −0.517654 0.855590i \(-0.673194\pi\)
−0.517654 + 0.855590i \(0.673194\pi\)
\(702\) −1.40629e9 −0.153424
\(703\) 2.01687e9 0.218944
\(704\) 2.03641e9 0.219968
\(705\) −3.03684e9 −0.326407
\(706\) −8.57167e9 −0.916745
\(707\) −1.27878e9 −0.136091
\(708\) −3.98451e8 −0.0421947
\(709\) 5.62595e9 0.592835 0.296417 0.955058i \(-0.404208\pi\)
0.296417 + 0.955058i \(0.404208\pi\)
\(710\) 8.51564e9 0.892921
\(711\) 2.38790e9 0.249157
\(712\) 9.50783e8 0.0987191
\(713\) 1.17146e10 1.21036
\(714\) −1.95484e9 −0.200987
\(715\) 4.92049e9 0.503427
\(716\) −9.24678e9 −0.941445
\(717\) −6.51429e8 −0.0660009
\(718\) −1.26634e10 −1.27677
\(719\) 1.77080e9 0.177672 0.0888360 0.996046i \(-0.471685\pi\)
0.0888360 + 0.996046i \(0.471685\pi\)
\(720\) −5.91248e8 −0.0590345
\(721\) −1.68054e9 −0.166984
\(722\) −5.14009e9 −0.508266
\(723\) 1.50720e9 0.148315
\(724\) 5.12251e9 0.501647
\(725\) −6.59480e9 −0.642715
\(726\) 3.60712e9 0.349850
\(727\) −2.42657e9 −0.234219 −0.117110 0.993119i \(-0.537363\pi\)
−0.117110 + 0.993119i \(0.537363\pi\)
\(728\) 3.88674e9 0.373359
\(729\) 3.87420e8 0.0370370
\(730\) −2.27769e9 −0.216703
\(731\) −1.06437e10 −1.00781
\(732\) 6.52150e9 0.614552
\(733\) 7.37181e9 0.691370 0.345685 0.938351i \(-0.387647\pi\)
0.345685 + 0.938351i \(0.387647\pi\)
\(734\) −8.35925e9 −0.780245
\(735\) 8.11038e9 0.753417
\(736\) 9.04471e9 0.836223
\(737\) −7.53774e8 −0.0693593
\(738\) 2.71853e9 0.248964
\(739\) −1.11711e6 −0.000101822 0 −5.09108e−5 1.00000i \(-0.500016\pi\)
−5.09108e−5 1.00000i \(0.500016\pi\)
\(740\) 4.02858e9 0.365461
\(741\) 3.71197e9 0.335151
\(742\) −3.64812e9 −0.327835
\(743\) −6.56798e9 −0.587450 −0.293725 0.955890i \(-0.594895\pi\)
−0.293725 + 0.955890i \(0.594895\pi\)
\(744\) 9.24401e9 0.822915
\(745\) −1.19769e10 −1.06120
\(746\) −7.99262e9 −0.704861
\(747\) 7.23257e9 0.634849
\(748\) 3.28401e9 0.286912
\(749\) 3.68895e8 0.0320787
\(750\) −3.56452e8 −0.0308522
\(751\) 5.66489e9 0.488036 0.244018 0.969771i \(-0.421534\pi\)
0.244018 + 0.969771i \(0.421534\pi\)
\(752\) −5.67846e8 −0.0486932
\(753\) −5.01619e9 −0.428146
\(754\) −5.70980e9 −0.485088
\(755\) −1.43674e10 −1.21497
\(756\) −3.84977e8 −0.0324047
\(757\) −8.32870e9 −0.697817 −0.348908 0.937157i \(-0.613448\pi\)
−0.348908 + 0.937157i \(0.613448\pi\)
\(758\) 6.78877e8 0.0566173
\(759\) −1.78120e9 −0.147865
\(760\) 8.65407e9 0.715110
\(761\) −1.51950e10 −1.24984 −0.624918 0.780690i \(-0.714868\pi\)
−0.624918 + 0.780690i \(0.714868\pi\)
\(762\) 1.60191e9 0.131159
\(763\) −2.85390e9 −0.232597
\(764\) −5.52061e9 −0.447878
\(765\) −1.03720e10 −0.837624
\(766\) 7.70060e9 0.619048
\(767\) −1.95831e9 −0.156711
\(768\) 7.63970e9 0.608572
\(769\) 5.04268e9 0.399871 0.199935 0.979809i \(-0.435927\pi\)
0.199935 + 0.979809i \(0.435927\pi\)
\(770\) −1.05251e9 −0.0830826
\(771\) 1.22377e10 0.961634
\(772\) −9.67898e9 −0.757128
\(773\) 8.09728e9 0.630537 0.315269 0.949002i \(-0.397905\pi\)
0.315269 + 0.949002i \(0.397905\pi\)
\(774\) 1.63785e9 0.126964
\(775\) 1.88664e10 1.45591
\(776\) −1.21730e10 −0.935149
\(777\) −1.02805e9 −0.0786216
\(778\) 1.21875e10 0.927870
\(779\) −7.17569e9 −0.543854
\(780\) 7.41445e9 0.559433
\(781\) 3.65067e9 0.274217
\(782\) −1.36289e10 −1.01915
\(783\) 1.57300e9 0.117102
\(784\) 1.51653e9 0.112394
\(785\) −2.56748e10 −1.89436
\(786\) 5.94718e9 0.436850
\(787\) −1.32475e10 −0.968773 −0.484387 0.874854i \(-0.660957\pi\)
−0.484387 + 0.874854i \(0.660957\pi\)
\(788\) −7.85560e9 −0.571923
\(789\) −1.15690e10 −0.838542
\(790\) 9.83741e9 0.709882
\(791\) 1.94352e9 0.139628
\(792\) −1.40555e9 −0.100533
\(793\) 3.20520e10 2.28244
\(794\) −1.06188e9 −0.0752843
\(795\) −1.93563e10 −1.36627
\(796\) 3.66110e9 0.257286
\(797\) −1.25069e10 −0.875076 −0.437538 0.899200i \(-0.644149\pi\)
−0.437538 + 0.899200i \(0.644149\pi\)
\(798\) −7.94006e8 −0.0553113
\(799\) −9.96151e9 −0.690894
\(800\) 1.45666e10 1.00587
\(801\) −4.62847e8 −0.0318217
\(802\) −1.27907e10 −0.875557
\(803\) −9.76451e8 −0.0665497
\(804\) −1.13583e9 −0.0770754
\(805\) −5.59017e9 −0.377693
\(806\) 1.63346e10 1.09884
\(807\) 7.15968e8 0.0479552
\(808\) −7.03524e9 −0.469179
\(809\) 2.07620e10 1.37864 0.689319 0.724458i \(-0.257910\pi\)
0.689319 + 0.724458i \(0.257910\pi\)
\(810\) 1.59605e9 0.105524
\(811\) −8.31574e8 −0.0547429 −0.0273715 0.999625i \(-0.508714\pi\)
−0.0273715 + 0.999625i \(0.508714\pi\)
\(812\) −1.56308e9 −0.102456
\(813\) −3.69687e9 −0.241277
\(814\) −1.34948e9 −0.0876964
\(815\) −1.69698e10 −1.09806
\(816\) −1.93942e9 −0.124956
\(817\) −4.32318e9 −0.277349
\(818\) −1.47088e10 −0.939592
\(819\) −1.89209e9 −0.120351
\(820\) −1.43331e10 −0.907800
\(821\) −7.46166e9 −0.470581 −0.235290 0.971925i \(-0.575604\pi\)
−0.235290 + 0.971925i \(0.575604\pi\)
\(822\) −2.00995e9 −0.126222
\(823\) −1.83123e10 −1.14510 −0.572550 0.819870i \(-0.694046\pi\)
−0.572550 + 0.819870i \(0.694046\pi\)
\(824\) −9.24551e9 −0.575686
\(825\) −2.86863e9 −0.177863
\(826\) 4.18892e8 0.0258626
\(827\) 2.57787e10 1.58487 0.792433 0.609960i \(-0.208814\pi\)
0.792433 + 0.609960i \(0.208814\pi\)
\(828\) −2.68401e9 −0.164315
\(829\) −2.26603e10 −1.38142 −0.690710 0.723132i \(-0.742702\pi\)
−0.690710 + 0.723132i \(0.742702\pi\)
\(830\) 2.97959e10 1.80877
\(831\) −1.15487e10 −0.698116
\(832\) 1.50815e10 0.907845
\(833\) 2.66038e10 1.59473
\(834\) −4.75592e9 −0.283892
\(835\) 2.44075e10 1.45084
\(836\) 1.33388e9 0.0789577
\(837\) −4.50004e9 −0.265264
\(838\) 1.84211e10 1.08134
\(839\) −9.57948e9 −0.559983 −0.279992 0.960002i \(-0.590332\pi\)
−0.279992 + 0.960002i \(0.590332\pi\)
\(840\) −4.41122e9 −0.256792
\(841\) −1.08632e10 −0.629755
\(842\) 4.92569e9 0.284364
\(843\) 9.52174e9 0.547419
\(844\) 1.03668e10 0.593532
\(845\) 1.12907e10 0.643756
\(846\) 1.53288e9 0.0870385
\(847\) 4.85320e9 0.274433
\(848\) −3.61935e9 −0.203819
\(849\) −1.37918e10 −0.773469
\(850\) −2.19494e10 −1.22590
\(851\) −7.16744e9 −0.398667
\(852\) 5.50103e9 0.304723
\(853\) 2.74400e10 1.51378 0.756889 0.653544i \(-0.226718\pi\)
0.756889 + 0.653544i \(0.226718\pi\)
\(854\) −6.85607e9 −0.376680
\(855\) −4.21286e9 −0.230513
\(856\) 2.02948e9 0.110593
\(857\) −1.29104e10 −0.700657 −0.350329 0.936627i \(-0.613930\pi\)
−0.350329 + 0.936627i \(0.613930\pi\)
\(858\) −2.48367e9 −0.134242
\(859\) 2.65615e10 1.42981 0.714903 0.699224i \(-0.246471\pi\)
0.714903 + 0.699224i \(0.246471\pi\)
\(860\) −8.63533e9 −0.462950
\(861\) 3.65765e9 0.195295
\(862\) −1.24044e10 −0.659631
\(863\) 1.29749e10 0.687173 0.343586 0.939121i \(-0.388358\pi\)
0.343586 + 0.939121i \(0.388358\pi\)
\(864\) −3.47444e9 −0.183268
\(865\) −4.04068e10 −2.12275
\(866\) −1.47063e10 −0.769469
\(867\) −2.29434e10 −1.19562
\(868\) 4.47166e9 0.232087
\(869\) 4.21732e9 0.218005
\(870\) 6.48027e9 0.333638
\(871\) −5.58238e9 −0.286257
\(872\) −1.57008e10 −0.801889
\(873\) 5.92588e9 0.301442
\(874\) −5.53570e9 −0.280468
\(875\) −4.79588e8 −0.0242014
\(876\) −1.47137e9 −0.0739533
\(877\) −1.08321e10 −0.542270 −0.271135 0.962541i \(-0.587399\pi\)
−0.271135 + 0.962541i \(0.587399\pi\)
\(878\) 4.98566e9 0.248595
\(879\) −4.13665e9 −0.205441
\(880\) −1.04421e9 −0.0516535
\(881\) 1.78817e10 0.881034 0.440517 0.897744i \(-0.354795\pi\)
0.440517 + 0.897744i \(0.354795\pi\)
\(882\) −4.09380e9 −0.200903
\(883\) 2.05428e9 0.100415 0.0502073 0.998739i \(-0.484012\pi\)
0.0502073 + 0.998739i \(0.484012\pi\)
\(884\) 2.43211e10 1.18413
\(885\) 2.22257e9 0.107784
\(886\) 1.83715e9 0.0887414
\(887\) 2.26672e10 1.09060 0.545299 0.838242i \(-0.316416\pi\)
0.545299 + 0.838242i \(0.316416\pi\)
\(888\) −5.65585e9 −0.271052
\(889\) 2.15530e9 0.102885
\(890\) −1.90679e9 −0.0906644
\(891\) 6.84230e8 0.0324064
\(892\) −1.42118e10 −0.670457
\(893\) −4.04611e9 −0.190133
\(894\) 6.04547e9 0.282975
\(895\) 5.15787e10 2.40486
\(896\) 2.92424e9 0.135811
\(897\) −1.31914e10 −0.610264
\(898\) 1.17350e10 0.540775
\(899\) −1.82710e10 −0.838695
\(900\) −4.32261e9 −0.197650
\(901\) −6.34929e10 −2.89193
\(902\) 4.80124e9 0.217836
\(903\) 2.20365e9 0.0995944
\(904\) 1.06923e10 0.481373
\(905\) −2.85735e10 −1.28142
\(906\) 7.25213e9 0.323979
\(907\) −3.69882e10 −1.64603 −0.823015 0.568020i \(-0.807710\pi\)
−0.823015 + 0.568020i \(0.807710\pi\)
\(908\) 2.39992e10 1.06389
\(909\) 3.42480e9 0.151238
\(910\) −7.79483e9 −0.342895
\(911\) −3.70589e10 −1.62397 −0.811986 0.583677i \(-0.801613\pi\)
−0.811986 + 0.583677i \(0.801613\pi\)
\(912\) −7.87745e8 −0.0343877
\(913\) 1.27736e10 0.555475
\(914\) 2.07927e10 0.900739
\(915\) −3.63771e10 −1.56983
\(916\) 3.66559e9 0.157583
\(917\) 8.00164e9 0.342678
\(918\) 5.23540e9 0.223358
\(919\) −3.58106e10 −1.52197 −0.760986 0.648768i \(-0.775285\pi\)
−0.760986 + 0.648768i \(0.775285\pi\)
\(920\) −3.07544e10 −1.30212
\(921\) −2.11618e10 −0.892575
\(922\) 1.38240e10 0.580867
\(923\) 2.70366e10 1.13174
\(924\) −6.79915e8 −0.0283532
\(925\) −1.15432e10 −0.479546
\(926\) −1.08389e10 −0.448587
\(927\) 4.50077e9 0.185570
\(928\) −1.41069e10 −0.579446
\(929\) −3.54484e10 −1.45058 −0.725291 0.688443i \(-0.758295\pi\)
−0.725291 + 0.688443i \(0.758295\pi\)
\(930\) −1.85388e10 −0.755772
\(931\) 1.08058e10 0.438867
\(932\) 8.72565e9 0.353055
\(933\) 2.01099e9 0.0810634
\(934\) −1.46716e10 −0.589202
\(935\) −1.83182e10 −0.732897
\(936\) −1.04094e10 −0.414915
\(937\) 3.58282e10 1.42278 0.711388 0.702800i \(-0.248067\pi\)
0.711388 + 0.702800i \(0.248067\pi\)
\(938\) 1.19410e9 0.0472421
\(939\) −1.29289e10 −0.509602
\(940\) −8.08189e9 −0.317370
\(941\) −5.81166e9 −0.227372 −0.113686 0.993517i \(-0.536266\pi\)
−0.113686 + 0.993517i \(0.536266\pi\)
\(942\) 1.29596e10 0.505143
\(943\) 2.55006e10 0.990284
\(944\) 4.15588e8 0.0160791
\(945\) 2.14741e9 0.0827758
\(946\) 2.89264e9 0.111090
\(947\) 3.32826e10 1.27348 0.636740 0.771079i \(-0.280283\pi\)
0.636740 + 0.771079i \(0.280283\pi\)
\(948\) 6.35489e9 0.242258
\(949\) −7.23151e9 −0.274661
\(950\) −8.91528e9 −0.337367
\(951\) 8.71070e9 0.328414
\(952\) −1.44698e10 −0.543541
\(953\) −1.73227e9 −0.0648322 −0.0324161 0.999474i \(-0.510320\pi\)
−0.0324161 + 0.999474i \(0.510320\pi\)
\(954\) 9.77030e9 0.364324
\(955\) 3.07941e10 1.14408
\(956\) −1.73364e9 −0.0641735
\(957\) 2.77811e9 0.102461
\(958\) −2.26776e10 −0.833332
\(959\) −2.70429e9 −0.0990122
\(960\) −1.71165e10 −0.624406
\(961\) 2.47572e10 0.899848
\(962\) −9.99415e9 −0.361937
\(963\) −9.87966e8 −0.0356492
\(964\) 4.01108e9 0.144209
\(965\) 5.39895e10 1.93403
\(966\) 2.82170e9 0.100714
\(967\) 3.35118e10 1.19180 0.595902 0.803057i \(-0.296795\pi\)
0.595902 + 0.803057i \(0.296795\pi\)
\(968\) 2.67000e10 0.946122
\(969\) −1.38191e10 −0.487918
\(970\) 2.44128e10 0.858848
\(971\) −2.03751e10 −0.714219 −0.357110 0.934063i \(-0.616238\pi\)
−0.357110 + 0.934063i \(0.616238\pi\)
\(972\) 1.03104e9 0.0360115
\(973\) −6.39885e9 −0.222693
\(974\) −1.42853e10 −0.495375
\(975\) −2.12448e10 −0.734070
\(976\) −6.80200e9 −0.234187
\(977\) 4.03720e10 1.38500 0.692498 0.721419i \(-0.256510\pi\)
0.692498 + 0.721419i \(0.256510\pi\)
\(978\) 8.56572e9 0.292805
\(979\) −8.17443e8 −0.0278431
\(980\) 2.15840e10 0.732556
\(981\) 7.64325e9 0.258486
\(982\) 1.78169e10 0.600400
\(983\) 1.10528e10 0.371139 0.185569 0.982631i \(-0.440587\pi\)
0.185569 + 0.982631i \(0.440587\pi\)
\(984\) 2.01226e10 0.673289
\(985\) 4.38187e10 1.46094
\(986\) 2.12567e10 0.706199
\(987\) 2.06241e9 0.0682756
\(988\) 9.87859e9 0.325871
\(989\) 1.53635e10 0.505014
\(990\) 2.81881e9 0.0923301
\(991\) 3.39895e10 1.10940 0.554699 0.832051i \(-0.312833\pi\)
0.554699 + 0.832051i \(0.312833\pi\)
\(992\) 4.03570e10 1.31259
\(993\) −2.77898e10 −0.900664
\(994\) −5.78324e9 −0.186775
\(995\) −2.04217e10 −0.657219
\(996\) 1.92479e10 0.617271
\(997\) 5.14554e10 1.64436 0.822182 0.569225i \(-0.192757\pi\)
0.822182 + 0.569225i \(0.192757\pi\)
\(998\) −9.40211e9 −0.299412
\(999\) 2.75330e9 0.0873725
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.1.11 17
3.2 odd 2 531.8.a.c.1.7 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.11 17 1.1 even 1 trivial
531.8.a.c.1.7 17 3.2 odd 2