Properties

Label 177.8.a.a.1.10
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-3.09726\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.09726 q^{2} -27.0000 q^{3} -118.407 q^{4} -156.435 q^{5} -83.6261 q^{6} +11.3597 q^{7} -763.187 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+3.09726 q^{2} -27.0000 q^{3} -118.407 q^{4} -156.435 q^{5} -83.6261 q^{6} +11.3597 q^{7} -763.187 q^{8} +729.000 q^{9} -484.520 q^{10} +4296.19 q^{11} +3196.99 q^{12} +1741.86 q^{13} +35.1841 q^{14} +4223.74 q^{15} +12792.3 q^{16} -4764.38 q^{17} +2257.91 q^{18} +18278.8 q^{19} +18523.0 q^{20} -306.713 q^{21} +13306.4 q^{22} +82004.5 q^{23} +20606.1 q^{24} -53653.2 q^{25} +5395.00 q^{26} -19683.0 q^{27} -1345.07 q^{28} -157425. q^{29} +13082.0 q^{30} +37778.8 q^{31} +137309. q^{32} -115997. q^{33} -14756.5 q^{34} -1777.05 q^{35} -86318.7 q^{36} +450442. q^{37} +56614.3 q^{38} -47030.2 q^{39} +119389. q^{40} -437505. q^{41} -949.970 q^{42} -738856. q^{43} -508699. q^{44} -114041. q^{45} +253989. q^{46} +869122. q^{47} -345392. q^{48} -823414. q^{49} -166178. q^{50} +128638. q^{51} -206248. q^{52} -991560. q^{53} -60963.4 q^{54} -672074. q^{55} -8669.60 q^{56} -493528. q^{57} -487588. q^{58} +205379. q^{59} -500120. q^{60} -1.30896e6 q^{61} +117011. q^{62} +8281.24 q^{63} -1.21213e6 q^{64} -272487. q^{65} -359274. q^{66} +2.85617e6 q^{67} +564135. q^{68} -2.21412e6 q^{69} -5504.01 q^{70} -4.56945e6 q^{71} -556364. q^{72} +937053. q^{73} +1.39514e6 q^{74} +1.44864e6 q^{75} -2.16434e6 q^{76} +48803.6 q^{77} -145665. q^{78} +6.61566e6 q^{79} -2.00116e6 q^{80} +531441. q^{81} -1.35507e6 q^{82} -3.65210e6 q^{83} +36316.9 q^{84} +745314. q^{85} -2.28843e6 q^{86} +4.25049e6 q^{87} -3.27880e6 q^{88} -5.71107e6 q^{89} -353215. q^{90} +19787.0 q^{91} -9.70990e6 q^{92} -1.02003e6 q^{93} +2.69190e6 q^{94} -2.85944e6 q^{95} -3.70735e6 q^{96} -1.38687e7 q^{97} -2.55033e6 q^{98} +3.13193e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.09726 0.273762 0.136881 0.990587i \(-0.456292\pi\)
0.136881 + 0.990587i \(0.456292\pi\)
\(3\) −27.0000 −0.577350
\(4\) −118.407 −0.925054
\(5\) −156.435 −0.559678 −0.279839 0.960047i \(-0.590281\pi\)
−0.279839 + 0.960047i \(0.590281\pi\)
\(6\) −83.6261 −0.158057
\(7\) 11.3597 0.0125177 0.00625885 0.999980i \(-0.498008\pi\)
0.00625885 + 0.999980i \(0.498008\pi\)
\(8\) −763.187 −0.527007
\(9\) 729.000 0.333333
\(10\) −484.520 −0.153219
\(11\) 4296.19 0.973217 0.486608 0.873620i \(-0.338234\pi\)
0.486608 + 0.873620i \(0.338234\pi\)
\(12\) 3196.99 0.534080
\(13\) 1741.86 0.219893 0.109947 0.993938i \(-0.464932\pi\)
0.109947 + 0.993938i \(0.464932\pi\)
\(14\) 35.1841 0.00342687
\(15\) 4223.74 0.323130
\(16\) 12792.3 0.780780
\(17\) −4764.38 −0.235199 −0.117599 0.993061i \(-0.537520\pi\)
−0.117599 + 0.993061i \(0.537520\pi\)
\(18\) 2257.91 0.0912540
\(19\) 18278.8 0.611379 0.305689 0.952131i \(-0.401113\pi\)
0.305689 + 0.952131i \(0.401113\pi\)
\(20\) 18523.0 0.517732
\(21\) −306.713 −0.00722710
\(22\) 13306.4 0.266430
\(23\) 82004.5 1.40537 0.702684 0.711502i \(-0.251985\pi\)
0.702684 + 0.711502i \(0.251985\pi\)
\(24\) 20606.1 0.304268
\(25\) −53653.2 −0.686761
\(26\) 5395.00 0.0601984
\(27\) −19683.0 −0.192450
\(28\) −1345.07 −0.0115796
\(29\) −157425. −1.19862 −0.599311 0.800517i \(-0.704559\pi\)
−0.599311 + 0.800517i \(0.704559\pi\)
\(30\) 13082.0 0.0884608
\(31\) 37778.8 0.227762 0.113881 0.993494i \(-0.463672\pi\)
0.113881 + 0.993494i \(0.463672\pi\)
\(32\) 137309. 0.740755
\(33\) −115997. −0.561887
\(34\) −14756.5 −0.0643885
\(35\) −1777.05 −0.00700588
\(36\) −86318.7 −0.308351
\(37\) 450442. 1.46195 0.730975 0.682404i \(-0.239066\pi\)
0.730975 + 0.682404i \(0.239066\pi\)
\(38\) 56614.3 0.167372
\(39\) −47030.2 −0.126955
\(40\) 119389. 0.294954
\(41\) −437505. −0.991378 −0.495689 0.868500i \(-0.665084\pi\)
−0.495689 + 0.868500i \(0.665084\pi\)
\(42\) −949.970 −0.00197851
\(43\) −738856. −1.41716 −0.708582 0.705629i \(-0.750665\pi\)
−0.708582 + 0.705629i \(0.750665\pi\)
\(44\) −508699. −0.900278
\(45\) −114041. −0.186559
\(46\) 253989. 0.384736
\(47\) 869122. 1.22106 0.610532 0.791992i \(-0.290956\pi\)
0.610532 + 0.791992i \(0.290956\pi\)
\(48\) −345392. −0.450783
\(49\) −823414. −0.999843
\(50\) −166178. −0.188009
\(51\) 128638. 0.135792
\(52\) −206248. −0.203413
\(53\) −991560. −0.914858 −0.457429 0.889246i \(-0.651230\pi\)
−0.457429 + 0.889246i \(0.651230\pi\)
\(54\) −60963.4 −0.0526855
\(55\) −672074. −0.544688
\(56\) −8669.60 −0.00659691
\(57\) −493528. −0.352980
\(58\) −487588. −0.328137
\(59\) 205379. 0.130189
\(60\) −500120. −0.298913
\(61\) −1.30896e6 −0.738368 −0.369184 0.929356i \(-0.620363\pi\)
−0.369184 + 0.929356i \(0.620363\pi\)
\(62\) 117011. 0.0623527
\(63\) 8281.24 0.00417257
\(64\) −1.21213e6 −0.577989
\(65\) −272487. −0.123069
\(66\) −359274. −0.153823
\(67\) 2.85617e6 1.16017 0.580085 0.814556i \(-0.303019\pi\)
0.580085 + 0.814556i \(0.303019\pi\)
\(68\) 564135. 0.217572
\(69\) −2.21412e6 −0.811390
\(70\) −5504.01 −0.00191794
\(71\) −4.56945e6 −1.51516 −0.757582 0.652740i \(-0.773619\pi\)
−0.757582 + 0.652740i \(0.773619\pi\)
\(72\) −556364. −0.175669
\(73\) 937053. 0.281925 0.140963 0.990015i \(-0.454980\pi\)
0.140963 + 0.990015i \(0.454980\pi\)
\(74\) 1.39514e6 0.400226
\(75\) 1.44864e6 0.396502
\(76\) −2.16434e6 −0.565559
\(77\) 48803.6 0.0121824
\(78\) −145665. −0.0347555
\(79\) 6.61566e6 1.50966 0.754829 0.655922i \(-0.227720\pi\)
0.754829 + 0.655922i \(0.227720\pi\)
\(80\) −2.00116e6 −0.436985
\(81\) 531441. 0.111111
\(82\) −1.35507e6 −0.271402
\(83\) −3.65210e6 −0.701082 −0.350541 0.936547i \(-0.614002\pi\)
−0.350541 + 0.936547i \(0.614002\pi\)
\(84\) 36316.9 0.00668546
\(85\) 745314. 0.131635
\(86\) −2.28843e6 −0.387966
\(87\) 4.25049e6 0.692024
\(88\) −3.27880e6 −0.512892
\(89\) −5.71107e6 −0.858722 −0.429361 0.903133i \(-0.641261\pi\)
−0.429361 + 0.903133i \(0.641261\pi\)
\(90\) −353215. −0.0510728
\(91\) 19787.0 0.00275256
\(92\) −9.70990e6 −1.30004
\(93\) −1.02003e6 −0.131499
\(94\) 2.69190e6 0.334281
\(95\) −2.85944e6 −0.342175
\(96\) −3.70735e6 −0.427675
\(97\) −1.38687e7 −1.54289 −0.771446 0.636295i \(-0.780466\pi\)
−0.771446 + 0.636295i \(0.780466\pi\)
\(98\) −2.55033e6 −0.273719
\(99\) 3.13193e6 0.324406
\(100\) 6.35291e6 0.635291
\(101\) 1.27095e7 1.22745 0.613727 0.789518i \(-0.289670\pi\)
0.613727 + 0.789518i \(0.289670\pi\)
\(102\) 398426. 0.0371747
\(103\) 1.41595e7 1.27679 0.638394 0.769710i \(-0.279599\pi\)
0.638394 + 0.769710i \(0.279599\pi\)
\(104\) −1.32937e6 −0.115885
\(105\) 47980.5 0.00404485
\(106\) −3.07112e6 −0.250453
\(107\) −7.44830e6 −0.587779 −0.293889 0.955839i \(-0.594950\pi\)
−0.293889 + 0.955839i \(0.594950\pi\)
\(108\) 2.33060e6 0.178027
\(109\) −2.45904e7 −1.81875 −0.909374 0.415980i \(-0.863439\pi\)
−0.909374 + 0.415980i \(0.863439\pi\)
\(110\) −2.08159e6 −0.149115
\(111\) −1.21619e7 −0.844057
\(112\) 145317. 0.00977357
\(113\) 5.72325e6 0.373137 0.186569 0.982442i \(-0.440263\pi\)
0.186569 + 0.982442i \(0.440263\pi\)
\(114\) −1.52859e6 −0.0966325
\(115\) −1.28283e7 −0.786553
\(116\) 1.86403e7 1.10879
\(117\) 1.26982e6 0.0732977
\(118\) 636113. 0.0356408
\(119\) −54122.0 −0.00294415
\(120\) −3.22350e6 −0.170292
\(121\) −1.02989e6 −0.0528495
\(122\) −4.05420e6 −0.202137
\(123\) 1.18126e7 0.572372
\(124\) −4.47327e6 −0.210693
\(125\) 2.06147e7 0.944043
\(126\) 25649.2 0.00114229
\(127\) −1.29403e7 −0.560574 −0.280287 0.959916i \(-0.590430\pi\)
−0.280287 + 0.959916i \(0.590430\pi\)
\(128\) −2.13299e7 −0.898986
\(129\) 1.99491e7 0.818200
\(130\) −843965. −0.0336917
\(131\) −4.66242e7 −1.81201 −0.906007 0.423263i \(-0.860885\pi\)
−0.906007 + 0.423263i \(0.860885\pi\)
\(132\) 1.37349e7 0.519776
\(133\) 207642. 0.00765306
\(134\) 8.84630e6 0.317611
\(135\) 3.07910e6 0.107710
\(136\) 3.63611e6 0.123951
\(137\) −2.07610e7 −0.689803 −0.344902 0.938639i \(-0.612088\pi\)
−0.344902 + 0.938639i \(0.612088\pi\)
\(138\) −6.85772e6 −0.222128
\(139\) −1.80615e6 −0.0570430 −0.0285215 0.999593i \(-0.509080\pi\)
−0.0285215 + 0.999593i \(0.509080\pi\)
\(140\) 210416. 0.00648082
\(141\) −2.34663e7 −0.704981
\(142\) −1.41528e7 −0.414794
\(143\) 7.48337e6 0.214004
\(144\) 9.32558e6 0.260260
\(145\) 2.46268e7 0.670842
\(146\) 2.90230e6 0.0771805
\(147\) 2.22322e7 0.577260
\(148\) −5.33354e7 −1.35238
\(149\) 7.35804e7 1.82226 0.911130 0.412119i \(-0.135211\pi\)
0.911130 + 0.412119i \(0.135211\pi\)
\(150\) 4.48681e6 0.108547
\(151\) 2.57500e6 0.0608636 0.0304318 0.999537i \(-0.490312\pi\)
0.0304318 + 0.999537i \(0.490312\pi\)
\(152\) −1.39502e7 −0.322201
\(153\) −3.47323e6 −0.0783996
\(154\) 151158. 0.00333509
\(155\) −5.90991e6 −0.127474
\(156\) 5.56871e6 0.117441
\(157\) −2.53317e7 −0.522414 −0.261207 0.965283i \(-0.584121\pi\)
−0.261207 + 0.965283i \(0.584121\pi\)
\(158\) 2.04904e7 0.413287
\(159\) 2.67721e7 0.528193
\(160\) −2.14799e7 −0.414584
\(161\) 931548. 0.0175920
\(162\) 1.64601e6 0.0304180
\(163\) −1.12501e6 −0.0203470 −0.0101735 0.999948i \(-0.503238\pi\)
−0.0101735 + 0.999948i \(0.503238\pi\)
\(164\) 5.18036e7 0.917078
\(165\) 1.81460e7 0.314476
\(166\) −1.13115e7 −0.191930
\(167\) −5.17400e7 −0.859645 −0.429823 0.902913i \(-0.641424\pi\)
−0.429823 + 0.902913i \(0.641424\pi\)
\(168\) 234079. 0.00380873
\(169\) −5.97144e7 −0.951647
\(170\) 2.30843e6 0.0360368
\(171\) 1.33253e7 0.203793
\(172\) 8.74857e7 1.31095
\(173\) −5.17329e7 −0.759636 −0.379818 0.925061i \(-0.624013\pi\)
−0.379818 + 0.925061i \(0.624013\pi\)
\(174\) 1.31649e7 0.189450
\(175\) −609485. −0.00859667
\(176\) 5.49582e7 0.759868
\(177\) −5.54523e6 −0.0751646
\(178\) −1.76887e7 −0.235086
\(179\) 4.40517e7 0.574086 0.287043 0.957918i \(-0.407328\pi\)
0.287043 + 0.957918i \(0.407328\pi\)
\(180\) 1.35032e7 0.172577
\(181\) 8.76285e7 1.09843 0.549213 0.835683i \(-0.314928\pi\)
0.549213 + 0.835683i \(0.314928\pi\)
\(182\) 61285.7 0.000753545 0
\(183\) 3.53420e7 0.426297
\(184\) −6.25848e7 −0.740638
\(185\) −7.04647e7 −0.818221
\(186\) −3.15929e6 −0.0359993
\(187\) −2.04687e7 −0.228899
\(188\) −1.02910e8 −1.12955
\(189\) −223593. −0.00240903
\(190\) −8.85644e6 −0.0936746
\(191\) 1.82917e8 1.89949 0.949744 0.313028i \(-0.101344\pi\)
0.949744 + 0.313028i \(0.101344\pi\)
\(192\) 3.27276e7 0.333702
\(193\) −8.82731e7 −0.883848 −0.441924 0.897053i \(-0.645704\pi\)
−0.441924 + 0.897053i \(0.645704\pi\)
\(194\) −4.29551e7 −0.422385
\(195\) 7.35716e6 0.0710541
\(196\) 9.74979e7 0.924909
\(197\) 1.70359e8 1.58757 0.793785 0.608198i \(-0.208107\pi\)
0.793785 + 0.608198i \(0.208107\pi\)
\(198\) 9.70040e6 0.0888099
\(199\) 9.62750e7 0.866020 0.433010 0.901389i \(-0.357451\pi\)
0.433010 + 0.901389i \(0.357451\pi\)
\(200\) 4.09474e7 0.361928
\(201\) −7.71165e7 −0.669825
\(202\) 3.93648e7 0.336030
\(203\) −1.78831e6 −0.0150040
\(204\) −1.52317e7 −0.125615
\(205\) 6.84409e7 0.554852
\(206\) 4.38558e7 0.349536
\(207\) 5.97813e7 0.468456
\(208\) 2.22824e7 0.171688
\(209\) 7.85293e7 0.595004
\(210\) 148608. 0.00110733
\(211\) −2.06758e8 −1.51521 −0.757605 0.652713i \(-0.773631\pi\)
−0.757605 + 0.652713i \(0.773631\pi\)
\(212\) 1.17408e8 0.846293
\(213\) 1.23375e8 0.874780
\(214\) −2.30693e7 −0.160912
\(215\) 1.15583e8 0.793155
\(216\) 1.50218e7 0.101423
\(217\) 429156. 0.00285106
\(218\) −7.61629e7 −0.497904
\(219\) −2.53004e7 −0.162770
\(220\) 7.95782e7 0.503866
\(221\) −8.29888e6 −0.0517186
\(222\) −3.76687e7 −0.231071
\(223\) −4.61080e7 −0.278426 −0.139213 0.990262i \(-0.544457\pi\)
−0.139213 + 0.990262i \(0.544457\pi\)
\(224\) 1.55979e6 0.00927255
\(225\) −3.91132e7 −0.228920
\(226\) 1.77264e7 0.102151
\(227\) 1.52563e8 0.865683 0.432841 0.901470i \(-0.357511\pi\)
0.432841 + 0.901470i \(0.357511\pi\)
\(228\) 5.84371e7 0.326525
\(229\) −2.88554e8 −1.58782 −0.793912 0.608033i \(-0.791959\pi\)
−0.793912 + 0.608033i \(0.791959\pi\)
\(230\) −3.97328e7 −0.215328
\(231\) −1.31770e6 −0.00703353
\(232\) 1.20145e8 0.631682
\(233\) 3.41845e7 0.177045 0.0885224 0.996074i \(-0.471786\pi\)
0.0885224 + 0.996074i \(0.471786\pi\)
\(234\) 3.93296e6 0.0200661
\(235\) −1.35961e8 −0.683402
\(236\) −2.43183e7 −0.120432
\(237\) −1.78623e8 −0.871601
\(238\) −167630. −0.000805996 0
\(239\) 1.22378e8 0.579843 0.289922 0.957050i \(-0.406371\pi\)
0.289922 + 0.957050i \(0.406371\pi\)
\(240\) 5.40313e7 0.252293
\(241\) −1.04747e8 −0.482040 −0.241020 0.970520i \(-0.577482\pi\)
−0.241020 + 0.970520i \(0.577482\pi\)
\(242\) −3.18983e6 −0.0144682
\(243\) −1.43489e7 −0.0641500
\(244\) 1.54990e8 0.683030
\(245\) 1.28811e8 0.559590
\(246\) 3.65868e7 0.156694
\(247\) 3.18391e7 0.134438
\(248\) −2.88323e7 −0.120032
\(249\) 9.86066e7 0.404770
\(250\) 6.38491e7 0.258443
\(251\) 1.01531e8 0.405268 0.202634 0.979255i \(-0.435050\pi\)
0.202634 + 0.979255i \(0.435050\pi\)
\(252\) −980556. −0.00385985
\(253\) 3.52307e8 1.36773
\(254\) −4.00797e7 −0.153464
\(255\) −2.01235e7 −0.0759998
\(256\) 8.90886e7 0.331881
\(257\) 2.43506e8 0.894836 0.447418 0.894325i \(-0.352344\pi\)
0.447418 + 0.894325i \(0.352344\pi\)
\(258\) 6.17876e7 0.223992
\(259\) 5.11689e6 0.0183003
\(260\) 3.22644e7 0.113846
\(261\) −1.14763e8 −0.399540
\(262\) −1.44407e8 −0.496061
\(263\) −5.30485e8 −1.79816 −0.899079 0.437787i \(-0.855762\pi\)
−0.899079 + 0.437787i \(0.855762\pi\)
\(264\) 8.85276e7 0.296118
\(265\) 1.55114e8 0.512026
\(266\) 643123. 0.00209512
\(267\) 1.54199e8 0.495784
\(268\) −3.38190e8 −1.07322
\(269\) −1.66365e8 −0.521110 −0.260555 0.965459i \(-0.583906\pi\)
−0.260555 + 0.965459i \(0.583906\pi\)
\(270\) 9.53680e6 0.0294869
\(271\) −5.31604e8 −1.62254 −0.811271 0.584670i \(-0.801224\pi\)
−0.811271 + 0.584670i \(0.801224\pi\)
\(272\) −6.09473e7 −0.183638
\(273\) −534250. −0.00158919
\(274\) −6.43022e7 −0.188842
\(275\) −2.30504e8 −0.668367
\(276\) 2.62167e8 0.750579
\(277\) −3.03386e8 −0.857661 −0.428831 0.903385i \(-0.641074\pi\)
−0.428831 + 0.903385i \(0.641074\pi\)
\(278\) −5.59412e6 −0.0156162
\(279\) 2.75407e7 0.0759208
\(280\) 1.35623e6 0.00369215
\(281\) 6.31916e7 0.169898 0.0849489 0.996385i \(-0.472927\pi\)
0.0849489 + 0.996385i \(0.472927\pi\)
\(282\) −7.26813e7 −0.192997
\(283\) −2.67112e8 −0.700552 −0.350276 0.936647i \(-0.613912\pi\)
−0.350276 + 0.936647i \(0.613912\pi\)
\(284\) 5.41054e8 1.40161
\(285\) 7.72049e7 0.197555
\(286\) 2.31780e7 0.0585861
\(287\) −4.96993e6 −0.0124098
\(288\) 1.00098e8 0.246918
\(289\) −3.87639e8 −0.944682
\(290\) 7.62757e7 0.183651
\(291\) 3.74455e8 0.890789
\(292\) −1.10954e8 −0.260796
\(293\) −4.43690e8 −1.03049 −0.515244 0.857043i \(-0.672299\pi\)
−0.515244 + 0.857043i \(0.672299\pi\)
\(294\) 6.88589e7 0.158032
\(295\) −3.21284e7 −0.0728638
\(296\) −3.43771e8 −0.770458
\(297\) −8.45620e7 −0.187296
\(298\) 2.27898e8 0.498866
\(299\) 1.42840e8 0.309031
\(300\) −1.71529e8 −0.366785
\(301\) −8.39320e6 −0.0177396
\(302\) 7.97544e6 0.0166621
\(303\) −3.43158e8 −0.708671
\(304\) 2.33828e8 0.477352
\(305\) 2.04767e8 0.413248
\(306\) −1.07575e7 −0.0214628
\(307\) −4.86871e8 −0.960351 −0.480175 0.877173i \(-0.659427\pi\)
−0.480175 + 0.877173i \(0.659427\pi\)
\(308\) −5.77868e6 −0.0112694
\(309\) −3.82307e8 −0.737153
\(310\) −1.83046e7 −0.0348974
\(311\) −5.91749e8 −1.11552 −0.557758 0.830003i \(-0.688338\pi\)
−0.557758 + 0.830003i \(0.688338\pi\)
\(312\) 3.58929e7 0.0669063
\(313\) −6.83534e8 −1.25995 −0.629977 0.776614i \(-0.716936\pi\)
−0.629977 + 0.776614i \(0.716936\pi\)
\(314\) −7.84589e7 −0.143017
\(315\) −1.29547e6 −0.00233529
\(316\) −7.83340e8 −1.39651
\(317\) −3.17450e8 −0.559716 −0.279858 0.960041i \(-0.590287\pi\)
−0.279858 + 0.960041i \(0.590287\pi\)
\(318\) 8.29204e7 0.144599
\(319\) −6.76330e8 −1.16652
\(320\) 1.89619e8 0.323488
\(321\) 2.01104e8 0.339354
\(322\) 2.88525e6 0.00481602
\(323\) −8.70872e7 −0.143796
\(324\) −6.29263e7 −0.102784
\(325\) −9.34563e7 −0.151014
\(326\) −3.48446e6 −0.00557024
\(327\) 6.63940e8 1.05005
\(328\) 3.33898e8 0.522463
\(329\) 9.87298e6 0.0152849
\(330\) 5.62029e7 0.0860915
\(331\) 1.88196e8 0.285242 0.142621 0.989777i \(-0.454447\pi\)
0.142621 + 0.989777i \(0.454447\pi\)
\(332\) 4.32434e8 0.648539
\(333\) 3.28372e8 0.487317
\(334\) −1.60253e8 −0.235338
\(335\) −4.46804e8 −0.649321
\(336\) −3.92356e6 −0.00564277
\(337\) −7.47824e8 −1.06438 −0.532188 0.846626i \(-0.678630\pi\)
−0.532188 + 0.846626i \(0.678630\pi\)
\(338\) −1.84951e8 −0.260525
\(339\) −1.54528e8 −0.215431
\(340\) −8.82504e7 −0.121770
\(341\) 1.62305e8 0.221662
\(342\) 4.12718e7 0.0557908
\(343\) −1.87090e7 −0.0250334
\(344\) 5.63885e8 0.746855
\(345\) 3.46365e8 0.454117
\(346\) −1.60230e8 −0.207959
\(347\) −1.23827e8 −0.159097 −0.0795483 0.996831i \(-0.525348\pi\)
−0.0795483 + 0.996831i \(0.525348\pi\)
\(348\) −5.03287e8 −0.640160
\(349\) −1.33403e9 −1.67987 −0.839935 0.542688i \(-0.817407\pi\)
−0.839935 + 0.542688i \(0.817407\pi\)
\(350\) −1.88774e6 −0.00235344
\(351\) −3.42850e7 −0.0423184
\(352\) 5.89907e8 0.720915
\(353\) 1.33244e9 1.61226 0.806132 0.591735i \(-0.201557\pi\)
0.806132 + 0.591735i \(0.201557\pi\)
\(354\) −1.71751e7 −0.0205772
\(355\) 7.14820e8 0.848003
\(356\) 6.76231e8 0.794365
\(357\) 1.46129e6 0.00169980
\(358\) 1.36440e8 0.157163
\(359\) 7.56062e8 0.862435 0.431218 0.902248i \(-0.358084\pi\)
0.431218 + 0.902248i \(0.358084\pi\)
\(360\) 8.70346e7 0.0983180
\(361\) −5.59757e8 −0.626216
\(362\) 2.71409e8 0.300707
\(363\) 2.78070e7 0.0305127
\(364\) −2.34292e6 −0.00254626
\(365\) −1.46588e8 −0.157787
\(366\) 1.09463e8 0.116704
\(367\) −1.10105e9 −1.16272 −0.581358 0.813648i \(-0.697479\pi\)
−0.581358 + 0.813648i \(0.697479\pi\)
\(368\) 1.04903e9 1.09728
\(369\) −3.18941e8 −0.330459
\(370\) −2.18248e8 −0.223998
\(371\) −1.12639e7 −0.0114519
\(372\) 1.20778e8 0.121643
\(373\) −1.72328e9 −1.71940 −0.859698 0.510803i \(-0.829348\pi\)
−0.859698 + 0.510803i \(0.829348\pi\)
\(374\) −6.33969e7 −0.0626639
\(375\) −5.56596e8 −0.545043
\(376\) −6.63303e8 −0.643509
\(377\) −2.74213e8 −0.263569
\(378\) −692528. −0.000659502 0
\(379\) 3.29484e8 0.310883 0.155441 0.987845i \(-0.450320\pi\)
0.155441 + 0.987845i \(0.450320\pi\)
\(380\) 3.38578e8 0.316531
\(381\) 3.49389e8 0.323647
\(382\) 5.66541e8 0.520008
\(383\) −6.36350e8 −0.578762 −0.289381 0.957214i \(-0.593449\pi\)
−0.289381 + 0.957214i \(0.593449\pi\)
\(384\) 5.75906e8 0.519030
\(385\) −7.63457e6 −0.00681824
\(386\) −2.73405e8 −0.241964
\(387\) −5.38626e8 −0.472388
\(388\) 1.64215e9 1.42726
\(389\) −9.45771e8 −0.814634 −0.407317 0.913287i \(-0.633536\pi\)
−0.407317 + 0.913287i \(0.633536\pi\)
\(390\) 2.27871e7 0.0194519
\(391\) −3.90700e8 −0.330541
\(392\) 6.28419e8 0.526924
\(393\) 1.25885e9 1.04617
\(394\) 5.27647e8 0.434617
\(395\) −1.03492e9 −0.844922
\(396\) −3.70842e8 −0.300093
\(397\) 1.20371e9 0.965506 0.482753 0.875756i \(-0.339637\pi\)
0.482753 + 0.875756i \(0.339637\pi\)
\(398\) 2.98189e8 0.237083
\(399\) −5.60634e6 −0.00441850
\(400\) −6.86348e8 −0.536209
\(401\) 1.24141e9 0.961409 0.480705 0.876883i \(-0.340381\pi\)
0.480705 + 0.876883i \(0.340381\pi\)
\(402\) −2.38850e8 −0.183373
\(403\) 6.58053e7 0.0500834
\(404\) −1.50490e9 −1.13546
\(405\) −8.31358e7 −0.0621864
\(406\) −5.53887e6 −0.00410752
\(407\) 1.93518e9 1.42279
\(408\) −9.81750e7 −0.0715633
\(409\) −1.31447e9 −0.949990 −0.474995 0.879988i \(-0.657550\pi\)
−0.474995 + 0.879988i \(0.657550\pi\)
\(410\) 2.11980e8 0.151897
\(411\) 5.60546e8 0.398258
\(412\) −1.67659e9 −1.18110
\(413\) 2.33305e6 0.00162967
\(414\) 1.85158e8 0.128245
\(415\) 5.71315e8 0.392380
\(416\) 2.39173e8 0.162887
\(417\) 4.87661e7 0.0329338
\(418\) 2.43226e8 0.162890
\(419\) −1.30216e9 −0.864796 −0.432398 0.901683i \(-0.642333\pi\)
−0.432398 + 0.901683i \(0.642333\pi\)
\(420\) −5.68122e6 −0.00374170
\(421\) 2.34962e9 1.53465 0.767327 0.641256i \(-0.221587\pi\)
0.767327 + 0.641256i \(0.221587\pi\)
\(422\) −6.40383e8 −0.414807
\(423\) 6.33590e8 0.407021
\(424\) 7.56746e8 0.482136
\(425\) 2.55624e8 0.161525
\(426\) 3.82125e8 0.239482
\(427\) −1.48694e7 −0.00924267
\(428\) 8.81930e8 0.543727
\(429\) −2.02051e8 −0.123555
\(430\) 3.57990e8 0.217136
\(431\) −1.37817e9 −0.829147 −0.414573 0.910016i \(-0.636069\pi\)
−0.414573 + 0.910016i \(0.636069\pi\)
\(432\) −2.51791e8 −0.150261
\(433\) −1.93157e9 −1.14341 −0.571705 0.820459i \(-0.693718\pi\)
−0.571705 + 0.820459i \(0.693718\pi\)
\(434\) 1.32921e6 0.000780512 0
\(435\) −6.64924e8 −0.387311
\(436\) 2.91167e9 1.68244
\(437\) 1.49894e9 0.859212
\(438\) −7.83621e7 −0.0445602
\(439\) 2.84522e9 1.60505 0.802526 0.596617i \(-0.203489\pi\)
0.802526 + 0.596617i \(0.203489\pi\)
\(440\) 5.12918e8 0.287054
\(441\) −6.00269e8 −0.333281
\(442\) −2.57038e7 −0.0141586
\(443\) 3.69126e8 0.201726 0.100863 0.994900i \(-0.467840\pi\)
0.100863 + 0.994900i \(0.467840\pi\)
\(444\) 1.44006e9 0.780799
\(445\) 8.93410e8 0.480608
\(446\) −1.42809e8 −0.0762224
\(447\) −1.98667e9 −1.05208
\(448\) −1.37695e7 −0.00723510
\(449\) −3.21750e8 −0.167748 −0.0838739 0.996476i \(-0.526729\pi\)
−0.0838739 + 0.996476i \(0.526729\pi\)
\(450\) −1.21144e8 −0.0626697
\(451\) −1.87961e9 −0.964825
\(452\) −6.77673e8 −0.345172
\(453\) −6.95249e7 −0.0351396
\(454\) 4.72528e8 0.236991
\(455\) −3.09538e6 −0.00154054
\(456\) 3.76654e8 0.186023
\(457\) 3.67556e9 1.80143 0.900714 0.434413i \(-0.143044\pi\)
0.900714 + 0.434413i \(0.143044\pi\)
\(458\) −8.93727e8 −0.434686
\(459\) 9.37772e7 0.0452640
\(460\) 1.51897e9 0.727604
\(461\) −8.20164e8 −0.389895 −0.194947 0.980814i \(-0.562454\pi\)
−0.194947 + 0.980814i \(0.562454\pi\)
\(462\) −4.08125e6 −0.00192551
\(463\) −1.84327e9 −0.863089 −0.431544 0.902092i \(-0.642031\pi\)
−0.431544 + 0.902092i \(0.642031\pi\)
\(464\) −2.01383e9 −0.935859
\(465\) 1.59568e8 0.0735969
\(466\) 1.05878e8 0.0484682
\(467\) 7.89693e8 0.358797 0.179399 0.983776i \(-0.442585\pi\)
0.179399 + 0.983776i \(0.442585\pi\)
\(468\) −1.50355e8 −0.0678043
\(469\) 3.24453e7 0.0145227
\(470\) −4.21107e8 −0.187090
\(471\) 6.83955e8 0.301616
\(472\) −1.56743e8 −0.0686104
\(473\) −3.17427e9 −1.37921
\(474\) −5.53242e8 −0.238611
\(475\) −9.80717e8 −0.419871
\(476\) 6.40842e6 0.00272350
\(477\) −7.22848e8 −0.304953
\(478\) 3.79037e8 0.158739
\(479\) 3.05692e9 1.27089 0.635447 0.772144i \(-0.280816\pi\)
0.635447 + 0.772144i \(0.280816\pi\)
\(480\) 5.79958e8 0.239360
\(481\) 7.84606e8 0.321473
\(482\) −3.24430e8 −0.131964
\(483\) −2.51518e7 −0.0101567
\(484\) 1.21946e8 0.0488887
\(485\) 2.16955e9 0.863522
\(486\) −4.44424e7 −0.0175618
\(487\) 2.29199e9 0.899209 0.449605 0.893228i \(-0.351565\pi\)
0.449605 + 0.893228i \(0.351565\pi\)
\(488\) 9.98983e8 0.389125
\(489\) 3.03753e7 0.0117473
\(490\) 3.98960e8 0.153195
\(491\) −2.91988e9 −1.11322 −0.556608 0.830775i \(-0.687897\pi\)
−0.556608 + 0.830775i \(0.687897\pi\)
\(492\) −1.39870e9 −0.529476
\(493\) 7.50034e8 0.281914
\(494\) 9.86142e7 0.0368040
\(495\) −4.89942e8 −0.181563
\(496\) 4.83277e8 0.177832
\(497\) −5.19077e7 −0.0189664
\(498\) 3.05411e8 0.110811
\(499\) 2.09323e9 0.754163 0.377082 0.926180i \(-0.376928\pi\)
0.377082 + 0.926180i \(0.376928\pi\)
\(500\) −2.44092e9 −0.873291
\(501\) 1.39698e9 0.496316
\(502\) 3.14469e8 0.110947
\(503\) 2.06891e9 0.724860 0.362430 0.932011i \(-0.381947\pi\)
0.362430 + 0.932011i \(0.381947\pi\)
\(504\) −6.32014e6 −0.00219897
\(505\) −1.98821e9 −0.686979
\(506\) 1.09119e9 0.374432
\(507\) 1.61229e9 0.549434
\(508\) 1.53223e9 0.518561
\(509\) −3.44598e9 −1.15824 −0.579122 0.815241i \(-0.696605\pi\)
−0.579122 + 0.815241i \(0.696605\pi\)
\(510\) −6.23277e7 −0.0208059
\(511\) 1.06447e7 0.00352906
\(512\) 3.00615e9 0.989843
\(513\) −3.59782e8 −0.117660
\(514\) 7.54202e8 0.244972
\(515\) −2.21504e9 −0.714589
\(516\) −2.36211e9 −0.756879
\(517\) 3.73392e9 1.18836
\(518\) 1.58484e7 0.00500992
\(519\) 1.39679e9 0.438576
\(520\) 2.07959e8 0.0648583
\(521\) 3.18767e9 0.987509 0.493755 0.869601i \(-0.335624\pi\)
0.493755 + 0.869601i \(0.335624\pi\)
\(522\) −3.55452e8 −0.109379
\(523\) 4.05354e9 1.23902 0.619510 0.784988i \(-0.287331\pi\)
0.619510 + 0.784988i \(0.287331\pi\)
\(524\) 5.52062e9 1.67621
\(525\) 1.64561e7 0.00496329
\(526\) −1.64305e9 −0.492267
\(527\) −1.79992e8 −0.0535694
\(528\) −1.48387e9 −0.438710
\(529\) 3.31991e9 0.975059
\(530\) 4.80430e8 0.140173
\(531\) 1.49721e8 0.0433963
\(532\) −2.45863e7 −0.00707950
\(533\) −7.62072e8 −0.217997
\(534\) 4.77595e8 0.135727
\(535\) 1.16517e9 0.328967
\(536\) −2.17979e9 −0.611418
\(537\) −1.18939e9 −0.331448
\(538\) −5.15277e8 −0.142660
\(539\) −3.53755e9 −0.973064
\(540\) −3.64587e8 −0.0996376
\(541\) −1.94039e9 −0.526865 −0.263432 0.964678i \(-0.584855\pi\)
−0.263432 + 0.964678i \(0.584855\pi\)
\(542\) −1.64652e9 −0.444190
\(543\) −2.36597e9 −0.634176
\(544\) −6.54192e8 −0.174225
\(545\) 3.84679e9 1.01791
\(546\) −1.65471e6 −0.000435060 0
\(547\) 3.90843e9 1.02105 0.510525 0.859863i \(-0.329451\pi\)
0.510525 + 0.859863i \(0.329451\pi\)
\(548\) 2.45824e9 0.638106
\(549\) −9.54233e8 −0.246123
\(550\) −7.13933e8 −0.182974
\(551\) −2.87755e9 −0.732812
\(552\) 1.68979e9 0.427608
\(553\) 7.51521e7 0.0188974
\(554\) −9.39665e8 −0.234795
\(555\) 1.90255e9 0.472400
\(556\) 2.13861e8 0.0527678
\(557\) −1.37471e9 −0.337069 −0.168534 0.985696i \(-0.553903\pi\)
−0.168534 + 0.985696i \(0.553903\pi\)
\(558\) 8.53009e7 0.0207842
\(559\) −1.28698e9 −0.311625
\(560\) −2.27326e7 −0.00547005
\(561\) 5.52655e8 0.132155
\(562\) 1.95721e8 0.0465115
\(563\) −7.49339e9 −1.76970 −0.884849 0.465879i \(-0.845738\pi\)
−0.884849 + 0.465879i \(0.845738\pi\)
\(564\) 2.77857e9 0.652146
\(565\) −8.95316e8 −0.208837
\(566\) −8.27315e8 −0.191785
\(567\) 6.03702e6 0.00139086
\(568\) 3.48734e9 0.798501
\(569\) 4.64326e9 1.05665 0.528324 0.849043i \(-0.322821\pi\)
0.528324 + 0.849043i \(0.322821\pi\)
\(570\) 2.39124e8 0.0540830
\(571\) −7.07238e9 −1.58979 −0.794894 0.606748i \(-0.792474\pi\)
−0.794894 + 0.606748i \(0.792474\pi\)
\(572\) −8.86083e8 −0.197965
\(573\) −4.93875e9 −1.09667
\(574\) −1.53932e7 −0.00339733
\(575\) −4.39980e9 −0.965152
\(576\) −8.83644e8 −0.192663
\(577\) 7.54152e8 0.163434 0.0817172 0.996656i \(-0.473960\pi\)
0.0817172 + 0.996656i \(0.473960\pi\)
\(578\) −1.20062e9 −0.258618
\(579\) 2.38337e9 0.510290
\(580\) −2.91599e9 −0.620565
\(581\) −4.14868e7 −0.00877594
\(582\) 1.15979e9 0.243864
\(583\) −4.25994e9 −0.890355
\(584\) −7.15147e8 −0.148577
\(585\) −1.98643e8 −0.0410231
\(586\) −1.37423e9 −0.282109
\(587\) 4.19667e9 0.856389 0.428195 0.903687i \(-0.359150\pi\)
0.428195 + 0.903687i \(0.359150\pi\)
\(588\) −2.63244e9 −0.533997
\(589\) 6.90551e8 0.139249
\(590\) −9.95101e7 −0.0199474
\(591\) −4.59969e9 −0.916584
\(592\) 5.76218e9 1.14146
\(593\) −9.07288e8 −0.178671 −0.0893355 0.996002i \(-0.528474\pi\)
−0.0893355 + 0.996002i \(0.528474\pi\)
\(594\) −2.61911e8 −0.0512744
\(595\) 8.46656e6 0.00164777
\(596\) −8.71244e9 −1.68569
\(597\) −2.59943e9 −0.499997
\(598\) 4.42414e8 0.0846009
\(599\) −7.00263e9 −1.33127 −0.665637 0.746276i \(-0.731840\pi\)
−0.665637 + 0.746276i \(0.731840\pi\)
\(600\) −1.10558e9 −0.208959
\(601\) 3.01689e7 0.00566890 0.00283445 0.999996i \(-0.499098\pi\)
0.00283445 + 0.999996i \(0.499098\pi\)
\(602\) −2.59959e7 −0.00485644
\(603\) 2.08215e9 0.386723
\(604\) −3.04898e8 −0.0563021
\(605\) 1.61110e8 0.0295787
\(606\) −1.06285e9 −0.194007
\(607\) −1.46131e9 −0.265205 −0.132602 0.991169i \(-0.542333\pi\)
−0.132602 + 0.991169i \(0.542333\pi\)
\(608\) 2.50985e9 0.452882
\(609\) 4.82844e7 0.00866255
\(610\) 6.34218e8 0.113132
\(611\) 1.51389e9 0.268503
\(612\) 4.11255e8 0.0725239
\(613\) −5.73275e9 −1.00520 −0.502599 0.864520i \(-0.667623\pi\)
−0.502599 + 0.864520i \(0.667623\pi\)
\(614\) −1.50797e9 −0.262908
\(615\) −1.84790e9 −0.320344
\(616\) −3.72463e7 −0.00642023
\(617\) 3.48905e9 0.598010 0.299005 0.954251i \(-0.403345\pi\)
0.299005 + 0.954251i \(0.403345\pi\)
\(618\) −1.18411e9 −0.201805
\(619\) −5.02284e9 −0.851200 −0.425600 0.904911i \(-0.639937\pi\)
−0.425600 + 0.904911i \(0.639937\pi\)
\(620\) 6.99775e8 0.117920
\(621\) −1.61409e9 −0.270463
\(622\) −1.83280e9 −0.305386
\(623\) −6.48762e7 −0.0107492
\(624\) −6.01625e8 −0.0991242
\(625\) 9.66804e8 0.158401
\(626\) −2.11708e9 −0.344928
\(627\) −2.12029e9 −0.343526
\(628\) 2.99945e9 0.483262
\(629\) −2.14607e9 −0.343849
\(630\) −4.01242e6 −0.000639315 0
\(631\) −7.21168e9 −1.14270 −0.571352 0.820705i \(-0.693581\pi\)
−0.571352 + 0.820705i \(0.693581\pi\)
\(632\) −5.04899e9 −0.795600
\(633\) 5.58246e9 0.874807
\(634\) −9.83226e8 −0.153229
\(635\) 2.02432e9 0.313741
\(636\) −3.17001e9 −0.488608
\(637\) −1.43427e9 −0.219859
\(638\) −2.09477e9 −0.319348
\(639\) −3.33113e9 −0.505055
\(640\) 3.33673e9 0.503143
\(641\) 1.16522e10 1.74746 0.873728 0.486415i \(-0.161696\pi\)
0.873728 + 0.486415i \(0.161696\pi\)
\(642\) 6.22872e8 0.0929023
\(643\) −1.68105e9 −0.249369 −0.124684 0.992196i \(-0.539792\pi\)
−0.124684 + 0.992196i \(0.539792\pi\)
\(644\) −1.10302e8 −0.0162735
\(645\) −3.12073e9 −0.457928
\(646\) −2.69732e8 −0.0393658
\(647\) 3.46626e9 0.503148 0.251574 0.967838i \(-0.419052\pi\)
0.251574 + 0.967838i \(0.419052\pi\)
\(648\) −4.05589e8 −0.0585563
\(649\) 8.82348e8 0.126702
\(650\) −2.89459e8 −0.0413419
\(651\) −1.15872e7 −0.00164606
\(652\) 1.33209e8 0.0188221
\(653\) 8.92600e9 1.25447 0.627236 0.778830i \(-0.284186\pi\)
0.627236 + 0.778830i \(0.284186\pi\)
\(654\) 2.05640e9 0.287465
\(655\) 7.29364e9 1.01414
\(656\) −5.59669e9 −0.774048
\(657\) 6.83112e8 0.0939751
\(658\) 3.05792e7 0.00418443
\(659\) 1.71451e8 0.0233368 0.0116684 0.999932i \(-0.496286\pi\)
0.0116684 + 0.999932i \(0.496286\pi\)
\(660\) −2.14861e9 −0.290907
\(661\) 1.10119e10 1.48305 0.741526 0.670924i \(-0.234102\pi\)
0.741526 + 0.670924i \(0.234102\pi\)
\(662\) 5.82894e8 0.0780884
\(663\) 2.24070e8 0.0298597
\(664\) 2.78723e9 0.369475
\(665\) −3.24825e7 −0.00428325
\(666\) 1.01705e9 0.133409
\(667\) −1.29096e10 −1.68450
\(668\) 6.12638e9 0.795218
\(669\) 1.24492e9 0.160749
\(670\) −1.38387e9 −0.177760
\(671\) −5.62356e9 −0.718592
\(672\) −4.21144e7 −0.00535351
\(673\) −5.18178e9 −0.655279 −0.327640 0.944803i \(-0.606253\pi\)
−0.327640 + 0.944803i \(0.606253\pi\)
\(674\) −2.31621e9 −0.291386
\(675\) 1.05606e9 0.132167
\(676\) 7.07061e9 0.880325
\(677\) −8.09065e9 −1.00213 −0.501064 0.865410i \(-0.667058\pi\)
−0.501064 + 0.865410i \(0.667058\pi\)
\(678\) −4.78614e8 −0.0589768
\(679\) −1.57545e8 −0.0193135
\(680\) −5.68814e8 −0.0693728
\(681\) −4.11920e9 −0.499802
\(682\) 5.02701e8 0.0606827
\(683\) 1.31475e9 0.157896 0.0789478 0.996879i \(-0.474844\pi\)
0.0789478 + 0.996879i \(0.474844\pi\)
\(684\) −1.57780e9 −0.188520
\(685\) 3.24773e9 0.386068
\(686\) −5.79466e7 −0.00685321
\(687\) 7.79095e9 0.916731
\(688\) −9.45166e9 −1.10649
\(689\) −1.72716e9 −0.201171
\(690\) 1.07278e9 0.124320
\(691\) 9.51511e9 1.09709 0.548543 0.836123i \(-0.315183\pi\)
0.548543 + 0.836123i \(0.315183\pi\)
\(692\) 6.12553e9 0.702704
\(693\) 3.55778e7 0.00406081
\(694\) −3.83524e8 −0.0435546
\(695\) 2.82545e8 0.0319257
\(696\) −3.24392e9 −0.364702
\(697\) 2.08444e9 0.233171
\(698\) −4.13183e9 −0.459884
\(699\) −9.22981e8 −0.102217
\(700\) 7.21673e7 0.00795238
\(701\) −1.01482e10 −1.11270 −0.556349 0.830949i \(-0.687798\pi\)
−0.556349 + 0.830949i \(0.687798\pi\)
\(702\) −1.06190e8 −0.0115852
\(703\) 8.23354e9 0.893805
\(704\) −5.20755e9 −0.562509
\(705\) 3.67094e9 0.394562
\(706\) 4.12692e9 0.441377
\(707\) 1.44377e8 0.0153649
\(708\) 6.56594e8 0.0695313
\(709\) 1.71054e10 1.80248 0.901241 0.433319i \(-0.142658\pi\)
0.901241 + 0.433319i \(0.142658\pi\)
\(710\) 2.21399e9 0.232151
\(711\) 4.82282e9 0.503219
\(712\) 4.35862e9 0.452553
\(713\) 3.09803e9 0.320090
\(714\) 4.52601e6 0.000465342 0
\(715\) −1.17066e9 −0.119773
\(716\) −5.21602e9 −0.531060
\(717\) −3.30421e9 −0.334773
\(718\) 2.34172e9 0.236102
\(719\) 4.37544e9 0.439006 0.219503 0.975612i \(-0.429556\pi\)
0.219503 + 0.975612i \(0.429556\pi\)
\(720\) −1.45885e9 −0.145662
\(721\) 1.60848e8 0.0159824
\(722\) −1.73371e9 −0.171434
\(723\) 2.82818e9 0.278306
\(724\) −1.03758e10 −1.01610
\(725\) 8.44638e9 0.823166
\(726\) 8.61255e7 0.00835322
\(727\) −1.22749e10 −1.18481 −0.592405 0.805641i \(-0.701821\pi\)
−0.592405 + 0.805641i \(0.701821\pi\)
\(728\) −1.51012e7 −0.00145062
\(729\) 3.87420e8 0.0370370
\(730\) −4.54020e8 −0.0431962
\(731\) 3.52019e9 0.333315
\(732\) −4.18474e9 −0.394348
\(733\) −7.53477e9 −0.706653 −0.353327 0.935500i \(-0.614950\pi\)
−0.353327 + 0.935500i \(0.614950\pi\)
\(734\) −3.41023e9 −0.318308
\(735\) −3.47788e9 −0.323079
\(736\) 1.12600e10 1.04103
\(737\) 1.22706e10 1.12910
\(738\) −9.87844e8 −0.0904672
\(739\) 1.70023e10 1.54972 0.774858 0.632136i \(-0.217821\pi\)
0.774858 + 0.632136i \(0.217821\pi\)
\(740\) 8.34351e9 0.756899
\(741\) −8.59657e8 −0.0776178
\(742\) −3.48871e7 −0.00313510
\(743\) −1.25008e10 −1.11809 −0.559045 0.829137i \(-0.688832\pi\)
−0.559045 + 0.829137i \(0.688832\pi\)
\(744\) 7.78472e8 0.0693007
\(745\) −1.15105e10 −1.01988
\(746\) −5.33747e9 −0.470705
\(747\) −2.66238e9 −0.233694
\(748\) 2.42364e9 0.211744
\(749\) −8.46106e7 −0.00735764
\(750\) −1.72393e9 −0.149212
\(751\) −6.82394e9 −0.587889 −0.293945 0.955822i \(-0.594968\pi\)
−0.293945 + 0.955822i \(0.594968\pi\)
\(752\) 1.11181e10 0.953382
\(753\) −2.74135e9 −0.233981
\(754\) −8.49311e8 −0.0721551
\(755\) −4.02819e8 −0.0340640
\(756\) 2.64750e7 0.00222849
\(757\) 1.76261e10 1.47680 0.738400 0.674363i \(-0.235582\pi\)
0.738400 + 0.674363i \(0.235582\pi\)
\(758\) 1.02050e9 0.0851079
\(759\) −9.51229e9 −0.789658
\(760\) 2.18229e9 0.180329
\(761\) −8.48847e9 −0.698205 −0.349103 0.937084i \(-0.613514\pi\)
−0.349103 + 0.937084i \(0.613514\pi\)
\(762\) 1.08215e9 0.0886024
\(763\) −2.79340e8 −0.0227665
\(764\) −2.16586e10 −1.75713
\(765\) 5.43334e8 0.0438785
\(766\) −1.97094e9 −0.158443
\(767\) 3.57741e8 0.0286276
\(768\) −2.40539e9 −0.191612
\(769\) 1.91757e10 1.52058 0.760288 0.649586i \(-0.225058\pi\)
0.760288 + 0.649586i \(0.225058\pi\)
\(770\) −2.36463e7 −0.00186657
\(771\) −6.57466e9 −0.516634
\(772\) 1.04521e10 0.817607
\(773\) −7.92188e8 −0.0616879 −0.0308440 0.999524i \(-0.509819\pi\)
−0.0308440 + 0.999524i \(0.509819\pi\)
\(774\) −1.66827e9 −0.129322
\(775\) −2.02695e9 −0.156418
\(776\) 1.05844e10 0.813114
\(777\) −1.38156e8 −0.0105657
\(778\) −2.92930e9 −0.223016
\(779\) −7.99707e9 −0.606108
\(780\) −8.71139e8 −0.0657289
\(781\) −1.96312e10 −1.47458
\(782\) −1.21010e9 −0.0904895
\(783\) 3.09861e9 0.230675
\(784\) −1.05334e10 −0.780658
\(785\) 3.96275e9 0.292384
\(786\) 3.89900e9 0.286401
\(787\) 2.00728e10 1.46790 0.733950 0.679203i \(-0.237675\pi\)
0.733950 + 0.679203i \(0.237675\pi\)
\(788\) −2.01717e10 −1.46859
\(789\) 1.43231e10 1.03817
\(790\) −3.20542e9 −0.231307
\(791\) 6.50146e7 0.00467082
\(792\) −2.39025e9 −0.170964
\(793\) −2.28003e9 −0.162362
\(794\) 3.72821e9 0.264319
\(795\) −4.18809e9 −0.295618
\(796\) −1.13996e10 −0.801115
\(797\) 1.65749e10 1.15971 0.579853 0.814721i \(-0.303110\pi\)
0.579853 + 0.814721i \(0.303110\pi\)
\(798\) −1.73643e7 −0.00120962
\(799\) −4.14082e9 −0.287192
\(800\) −7.36707e9 −0.508721
\(801\) −4.16337e9 −0.286241
\(802\) 3.84496e9 0.263197
\(803\) 4.02576e9 0.274374
\(804\) 9.13113e9 0.619624
\(805\) −1.45726e8 −0.00984584
\(806\) 2.03817e8 0.0137109
\(807\) 4.49186e9 0.300863
\(808\) −9.69977e9 −0.646877
\(809\) −7.11284e9 −0.472305 −0.236153 0.971716i \(-0.575887\pi\)
−0.236153 + 0.971716i \(0.575887\pi\)
\(810\) −2.57494e8 −0.0170243
\(811\) 1.80923e9 0.119102 0.0595511 0.998225i \(-0.481033\pi\)
0.0595511 + 0.998225i \(0.481033\pi\)
\(812\) 2.11748e8 0.0138795
\(813\) 1.43533e10 0.936775
\(814\) 5.99378e9 0.389507
\(815\) 1.75991e8 0.0113878
\(816\) 1.64558e9 0.106024
\(817\) −1.35054e10 −0.866424
\(818\) −4.07126e9 −0.260071
\(819\) 1.44248e7 0.000917519 0
\(820\) −8.10388e9 −0.513268
\(821\) 1.91315e10 1.20655 0.603277 0.797531i \(-0.293861\pi\)
0.603277 + 0.797531i \(0.293861\pi\)
\(822\) 1.73616e9 0.109028
\(823\) −1.55173e10 −0.970325 −0.485163 0.874424i \(-0.661240\pi\)
−0.485163 + 0.874424i \(0.661240\pi\)
\(824\) −1.08064e10 −0.672876
\(825\) 6.22362e9 0.385882
\(826\) 7.22607e6 0.000446141 0
\(827\) 2.67848e10 1.64672 0.823360 0.567519i \(-0.192097\pi\)
0.823360 + 0.567519i \(0.192097\pi\)
\(828\) −7.07852e9 −0.433347
\(829\) 1.50178e10 0.915515 0.457757 0.889077i \(-0.348653\pi\)
0.457757 + 0.889077i \(0.348653\pi\)
\(830\) 1.76951e9 0.107419
\(831\) 8.19141e9 0.495171
\(832\) −2.11136e9 −0.127096
\(833\) 3.92305e9 0.235162
\(834\) 1.51041e8 0.00901602
\(835\) 8.09394e9 0.481124
\(836\) −9.29842e9 −0.550411
\(837\) −7.43600e8 −0.0438329
\(838\) −4.03312e9 −0.236748
\(839\) 1.78661e10 1.04439 0.522197 0.852825i \(-0.325113\pi\)
0.522197 + 0.852825i \(0.325113\pi\)
\(840\) −3.66181e7 −0.00213166
\(841\) 7.53290e9 0.436693
\(842\) 7.27739e9 0.420130
\(843\) −1.70617e9 −0.0980905
\(844\) 2.44815e10 1.40165
\(845\) 9.34141e9 0.532616
\(846\) 1.96239e9 0.111427
\(847\) −1.16992e7 −0.000661555 0
\(848\) −1.26843e10 −0.714303
\(849\) 7.21201e9 0.404464
\(850\) 7.91735e8 0.0442195
\(851\) 3.69382e10 2.05458
\(852\) −1.46085e10 −0.809219
\(853\) −1.05414e10 −0.581538 −0.290769 0.956793i \(-0.593911\pi\)
−0.290769 + 0.956793i \(0.593911\pi\)
\(854\) −4.60546e7 −0.00253029
\(855\) −2.08453e9 −0.114058
\(856\) 5.68445e9 0.309763
\(857\) −1.68053e10 −0.912037 −0.456018 0.889970i \(-0.650725\pi\)
−0.456018 + 0.889970i \(0.650725\pi\)
\(858\) −6.25805e8 −0.0338247
\(859\) −3.46312e9 −0.186420 −0.0932099 0.995646i \(-0.529713\pi\)
−0.0932099 + 0.995646i \(0.529713\pi\)
\(860\) −1.36858e10 −0.733712
\(861\) 1.34188e8 0.00716479
\(862\) −4.26855e9 −0.226989
\(863\) 1.26642e10 0.670716 0.335358 0.942091i \(-0.391143\pi\)
0.335358 + 0.942091i \(0.391143\pi\)
\(864\) −2.70266e9 −0.142558
\(865\) 8.09282e9 0.425151
\(866\) −5.98257e9 −0.313022
\(867\) 1.04663e10 0.545412
\(868\) −5.08151e7 −0.00263739
\(869\) 2.84222e10 1.46922
\(870\) −2.05944e9 −0.106031
\(871\) 4.97504e9 0.255113
\(872\) 1.87671e10 0.958492
\(873\) −1.01103e10 −0.514297
\(874\) 4.64263e9 0.235220
\(875\) 2.34177e8 0.0118172
\(876\) 2.99575e9 0.150571
\(877\) −1.35643e10 −0.679044 −0.339522 0.940598i \(-0.610265\pi\)
−0.339522 + 0.940598i \(0.610265\pi\)
\(878\) 8.81238e9 0.439403
\(879\) 1.19796e10 0.594953
\(880\) −8.59737e9 −0.425281
\(881\) 3.52328e10 1.73593 0.867963 0.496630i \(-0.165429\pi\)
0.867963 + 0.496630i \(0.165429\pi\)
\(882\) −1.85919e9 −0.0912397
\(883\) 1.37573e10 0.672469 0.336234 0.941778i \(-0.390847\pi\)
0.336234 + 0.941778i \(0.390847\pi\)
\(884\) 9.82645e8 0.0478425
\(885\) 8.67467e8 0.0420680
\(886\) 1.14328e9 0.0552248
\(887\) −2.09679e10 −1.00884 −0.504420 0.863459i \(-0.668294\pi\)
−0.504420 + 0.863459i \(0.668294\pi\)
\(888\) 9.28183e9 0.444824
\(889\) −1.46999e8 −0.00701709
\(890\) 2.76713e9 0.131572
\(891\) 2.28317e9 0.108135
\(892\) 5.45951e9 0.257559
\(893\) 1.58865e10 0.746532
\(894\) −6.15325e9 −0.288020
\(895\) −6.89121e9 −0.321303
\(896\) −2.42301e8 −0.0112532
\(897\) −3.85669e9 −0.178419
\(898\) −9.96545e8 −0.0459230
\(899\) −5.94734e9 −0.273001
\(900\) 4.63127e9 0.211764
\(901\) 4.72417e9 0.215173
\(902\) −5.82163e9 −0.264133
\(903\) 2.26616e8 0.0102420
\(904\) −4.36792e9 −0.196646
\(905\) −1.37081e10 −0.614764
\(906\) −2.15337e8 −0.00961989
\(907\) −6.58180e9 −0.292900 −0.146450 0.989218i \(-0.546785\pi\)
−0.146450 + 0.989218i \(0.546785\pi\)
\(908\) −1.80645e10 −0.800803
\(909\) 9.26526e9 0.409151
\(910\) −9.58721e6 −0.000421743 0
\(911\) 2.97498e10 1.30368 0.651839 0.758357i \(-0.273998\pi\)
0.651839 + 0.758357i \(0.273998\pi\)
\(912\) −6.31336e9 −0.275600
\(913\) −1.56901e10 −0.682305
\(914\) 1.13842e10 0.493163
\(915\) −5.52871e9 −0.238589
\(916\) 3.41668e10 1.46882
\(917\) −5.29638e8 −0.0226822
\(918\) 2.90453e8 0.0123916
\(919\) −9.95637e9 −0.423152 −0.211576 0.977361i \(-0.567860\pi\)
−0.211576 + 0.977361i \(0.567860\pi\)
\(920\) 9.79043e9 0.414519
\(921\) 1.31455e10 0.554459
\(922\) −2.54027e9 −0.106738
\(923\) −7.95934e9 −0.333174
\(924\) 1.56024e8 0.00650640
\(925\) −2.41676e10 −1.00401
\(926\) −5.70909e9 −0.236281
\(927\) 1.03223e10 0.425596
\(928\) −2.16160e10 −0.887884
\(929\) −6.90890e9 −0.282718 −0.141359 0.989958i \(-0.545147\pi\)
−0.141359 + 0.989958i \(0.545147\pi\)
\(930\) 4.94223e8 0.0201480
\(931\) −1.50510e10 −0.611283
\(932\) −4.04768e9 −0.163776
\(933\) 1.59772e10 0.644044
\(934\) 2.44589e9 0.0982251
\(935\) 3.20201e9 0.128110
\(936\) −9.69108e8 −0.0386284
\(937\) −6.04161e9 −0.239919 −0.119959 0.992779i \(-0.538276\pi\)
−0.119959 + 0.992779i \(0.538276\pi\)
\(938\) 1.00492e8 0.00397575
\(939\) 1.84554e10 0.727435
\(940\) 1.60987e10 0.632184
\(941\) −1.95376e10 −0.764376 −0.382188 0.924085i \(-0.624829\pi\)
−0.382188 + 0.924085i \(0.624829\pi\)
\(942\) 2.11839e9 0.0825710
\(943\) −3.58773e10 −1.39325
\(944\) 2.62727e9 0.101649
\(945\) 3.49778e7 0.00134828
\(946\) −9.83154e9 −0.377575
\(947\) −2.08163e10 −0.796486 −0.398243 0.917280i \(-0.630380\pi\)
−0.398243 + 0.917280i \(0.630380\pi\)
\(948\) 2.11502e10 0.806278
\(949\) 1.63222e9 0.0619934
\(950\) −3.03754e9 −0.114945
\(951\) 8.57115e9 0.323152
\(952\) 4.13052e7 0.00155159
\(953\) −2.45125e10 −0.917409 −0.458705 0.888589i \(-0.651686\pi\)
−0.458705 + 0.888589i \(0.651686\pi\)
\(954\) −2.23885e9 −0.0834845
\(955\) −2.86145e10 −1.06310
\(956\) −1.44904e10 −0.536387
\(957\) 1.82609e10 0.673490
\(958\) 9.46808e9 0.347923
\(959\) −2.35839e8 −0.00863475
\(960\) −5.11972e9 −0.186766
\(961\) −2.60854e10 −0.948124
\(962\) 2.43013e9 0.0880070
\(963\) −5.42981e9 −0.195926
\(964\) 1.24028e10 0.445913
\(965\) 1.38090e10 0.494670
\(966\) −7.79017e7 −0.00278053
\(967\) −9.86699e9 −0.350907 −0.175454 0.984488i \(-0.556139\pi\)
−0.175454 + 0.984488i \(0.556139\pi\)
\(968\) 7.85997e8 0.0278521
\(969\) 2.35135e9 0.0830204
\(970\) 6.71967e9 0.236400
\(971\) 1.19828e10 0.420039 0.210020 0.977697i \(-0.432647\pi\)
0.210020 + 0.977697i \(0.432647\pi\)
\(972\) 1.69901e9 0.0593423
\(973\) −2.05174e7 −0.000714047 0
\(974\) 7.09889e9 0.246169
\(975\) 2.52332e9 0.0871879
\(976\) −1.67446e10 −0.576503
\(977\) 7.54352e9 0.258787 0.129394 0.991593i \(-0.458697\pi\)
0.129394 + 0.991593i \(0.458697\pi\)
\(978\) 9.40804e7 0.00321598
\(979\) −2.45359e10 −0.835723
\(980\) −1.52521e10 −0.517651
\(981\) −1.79264e10 −0.606249
\(982\) −9.04363e9 −0.304756
\(983\) 2.03133e10 0.682091 0.341045 0.940047i \(-0.389219\pi\)
0.341045 + 0.940047i \(0.389219\pi\)
\(984\) −9.01525e9 −0.301644
\(985\) −2.66500e10 −0.888528
\(986\) 2.32305e9 0.0771774
\(987\) −2.66571e8 −0.00882474
\(988\) −3.76998e9 −0.124362
\(989\) −6.05895e10 −1.99164
\(990\) −1.51748e9 −0.0497049
\(991\) −3.47760e10 −1.13507 −0.567534 0.823350i \(-0.692103\pi\)
−0.567534 + 0.823350i \(0.692103\pi\)
\(992\) 5.18737e9 0.168716
\(993\) −5.08130e9 −0.164685
\(994\) −1.60772e8 −0.00519227
\(995\) −1.50608e10 −0.484692
\(996\) −1.16757e10 −0.374434
\(997\) −1.31923e10 −0.421586 −0.210793 0.977531i \(-0.567605\pi\)
−0.210793 + 0.977531i \(0.567605\pi\)
\(998\) 6.48329e9 0.206461
\(999\) −8.86604e9 −0.281352
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.a.1.10 16
3.2 odd 2 531.8.a.b.1.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.10 16 1.1 even 1 trivial
531.8.a.b.1.7 16 3.2 odd 2