Properties

Label 177.12.a.b.1.13
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $27$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(1\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-17.2441 q^{2} +243.000 q^{3} -1750.64 q^{4} +7994.91 q^{5} -4190.32 q^{6} +1325.62 q^{7} +65504.2 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-17.2441 q^{2} +243.000 q^{3} -1750.64 q^{4} +7994.91 q^{5} -4190.32 q^{6} +1325.62 q^{7} +65504.2 q^{8} +59049.0 q^{9} -137865. q^{10} +134920. q^{11} -425406. q^{12} -29851.6 q^{13} -22859.1 q^{14} +1.94276e6 q^{15} +2.45575e6 q^{16} -3.25895e6 q^{17} -1.01825e6 q^{18} -1.09243e7 q^{19} -1.39962e7 q^{20} +322125. q^{21} -2.32657e6 q^{22} +185019. q^{23} +1.59175e7 q^{24} +1.50905e7 q^{25} +514764. q^{26} +1.43489e7 q^{27} -2.32068e6 q^{28} +1.68886e7 q^{29} -3.35012e7 q^{30} -5.53827e7 q^{31} -1.76500e8 q^{32} +3.27855e7 q^{33} +5.61977e7 q^{34} +1.05982e7 q^{35} -1.03374e8 q^{36} -7.21695e8 q^{37} +1.88379e8 q^{38} -7.25393e6 q^{39} +5.23700e8 q^{40} -8.63631e8 q^{41} -5.55476e6 q^{42} +1.00149e9 q^{43} -2.36196e8 q^{44} +4.72092e8 q^{45} -3.19049e6 q^{46} +3.01812e9 q^{47} +5.96748e8 q^{48} -1.97557e9 q^{49} -2.60222e8 q^{50} -7.91925e8 q^{51} +5.22594e7 q^{52} +2.68649e9 q^{53} -2.47434e8 q^{54} +1.07867e9 q^{55} +8.68335e7 q^{56} -2.65460e9 q^{57} -2.91229e8 q^{58} -7.14924e8 q^{59} -3.40108e9 q^{60} -1.08583e10 q^{61} +9.55025e8 q^{62} +7.82764e7 q^{63} -1.98580e9 q^{64} -2.38661e8 q^{65} -5.65356e8 q^{66} +1.61605e10 q^{67} +5.70525e9 q^{68} +4.49597e7 q^{69} -1.82756e8 q^{70} +3.75065e9 q^{71} +3.86795e9 q^{72} -2.36110e10 q^{73} +1.24450e10 q^{74} +3.66699e9 q^{75} +1.91245e10 q^{76} +1.78852e8 q^{77} +1.25088e8 q^{78} -4.18117e10 q^{79} +1.96335e10 q^{80} +3.48678e9 q^{81} +1.48925e10 q^{82} +2.14444e10 q^{83} -5.63926e8 q^{84} -2.60550e10 q^{85} -1.72697e10 q^{86} +4.10393e9 q^{87} +8.83780e9 q^{88} +2.25760e10 q^{89} -8.14080e9 q^{90} -3.95718e7 q^{91} -3.23902e8 q^{92} -1.34580e10 q^{93} -5.20448e10 q^{94} -8.73387e10 q^{95} -4.28894e10 q^{96} -3.27724e10 q^{97} +3.40669e10 q^{98} +7.96687e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} - 383719q^{10} - 1816556q^{11} + 6352506q^{12} - 3951804q^{13} - 6207867q^{14} - 4176684q^{15} + 28295194q^{16} - 17723275q^{17} - 7558272q^{18} - 19573013q^{19} - 48468099q^{20} - 30758697q^{21} - 1729910q^{22} - 88593797q^{23} - 86458671q^{24} + 345714963q^{25} - 6676346q^{26} + 387420489q^{27} + 126954286q^{28} - 276632427q^{29} - 93243717q^{30} - 357680917q^{31} - 859842334q^{32} - 441423108q^{33} + 232730000q^{34} - 510315139q^{35} + 1543658958q^{36} - 660238257q^{37} - 2067286961q^{38} - 960288372q^{39} - 3388951110q^{40} - 1671147569q^{41} - 1508511681q^{42} - 1883107790q^{43} - 3895687630q^{44} - 1014934212q^{45} - 1720344243q^{46} - 5818572501q^{47} + 6875732142q^{48} - 18858180q^{49} - 21474519647q^{50} - 4306755825q^{51} - 42214560062q^{52} - 11444513368q^{53} - 1836660096q^{54} - 24401486484q^{55} - 50583585764q^{56} - 4756242159q^{57} - 45017395090q^{58} - 19302956073q^{59} - 11777748057q^{60} + 408637955q^{61} - 28543084070q^{62} - 7474363371q^{63} + 33067284293q^{64} - 21656714730q^{65} - 420368130q^{66} - 49803132690q^{67} - 16500749319q^{68} - 21528292671q^{69} - 45808890782q^{70} - 34127492216q^{71} - 21009457053q^{72} - 55734362153q^{73} - 40367816298q^{74} + 84008736009q^{75} - 14840406404q^{76} - 99723443615q^{77} - 1622352078q^{78} - 76484916442q^{79} + 93882788915q^{80} + 94143178827q^{81} + 52951239205q^{82} - 140433865655q^{83} + 30849891498q^{84} + 34329063335q^{85} + 175223869508q^{86} - 67221679761q^{87} + 268823645069q^{88} - 1191878597q^{89} - 22658223231q^{90} + 201632581559q^{91} - 206501888812q^{92} - 86916462831q^{93} + 319770144384q^{94} - 81387074885q^{95} - 208941687162q^{96} - 144896178730q^{97} + 135739195260q^{98} - 107265815244q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −17.2441 −0.381044 −0.190522 0.981683i \(-0.561018\pi\)
−0.190522 + 0.981683i \(0.561018\pi\)
\(3\) 243.000 0.577350
\(4\) −1750.64 −0.854805
\(5\) 7994.91 1.14414 0.572069 0.820205i \(-0.306141\pi\)
0.572069 + 0.820205i \(0.306141\pi\)
\(6\) −4190.32 −0.219996
\(7\) 1325.62 0.0298112 0.0149056 0.999889i \(-0.495255\pi\)
0.0149056 + 0.999889i \(0.495255\pi\)
\(8\) 65504.2 0.706763
\(9\) 59049.0 0.333333
\(10\) −137865. −0.435968
\(11\) 134920. 0.252590 0.126295 0.991993i \(-0.459691\pi\)
0.126295 + 0.991993i \(0.459691\pi\)
\(12\) −425406. −0.493522
\(13\) −29851.6 −0.0222987 −0.0111493 0.999938i \(-0.503549\pi\)
−0.0111493 + 0.999938i \(0.503549\pi\)
\(14\) −22859.1 −0.0113594
\(15\) 1.94276e6 0.660569
\(16\) 2.45575e6 0.585497
\(17\) −3.25895e6 −0.556684 −0.278342 0.960482i \(-0.589785\pi\)
−0.278342 + 0.960482i \(0.589785\pi\)
\(18\) −1.01825e6 −0.127015
\(19\) −1.09243e7 −1.01216 −0.506079 0.862487i \(-0.668905\pi\)
−0.506079 + 0.862487i \(0.668905\pi\)
\(20\) −1.39962e7 −0.978016
\(21\) 322125. 0.0172115
\(22\) −2.32657e6 −0.0962479
\(23\) 185019. 0.00599396 0.00299698 0.999996i \(-0.499046\pi\)
0.00299698 + 0.999996i \(0.499046\pi\)
\(24\) 1.59175e7 0.408050
\(25\) 1.50905e7 0.309054
\(26\) 514764. 0.00849678
\(27\) 1.43489e7 0.192450
\(28\) −2.32068e6 −0.0254828
\(29\) 1.68886e7 0.152899 0.0764495 0.997073i \(-0.475642\pi\)
0.0764495 + 0.997073i \(0.475642\pi\)
\(30\) −3.35012e7 −0.251706
\(31\) −5.53827e7 −0.347444 −0.173722 0.984795i \(-0.555579\pi\)
−0.173722 + 0.984795i \(0.555579\pi\)
\(32\) −1.76500e8 −0.929864
\(33\) 3.27855e7 0.145833
\(34\) 5.61977e7 0.212121
\(35\) 1.05982e7 0.0341081
\(36\) −1.03374e8 −0.284935
\(37\) −7.21695e8 −1.71098 −0.855489 0.517821i \(-0.826743\pi\)
−0.855489 + 0.517821i \(0.826743\pi\)
\(38\) 1.88379e8 0.385677
\(39\) −7.25393e6 −0.0128741
\(40\) 5.23700e8 0.808635
\(41\) −8.63631e8 −1.16417 −0.582086 0.813128i \(-0.697763\pi\)
−0.582086 + 0.813128i \(0.697763\pi\)
\(42\) −5.55476e6 −0.00655834
\(43\) 1.00149e9 1.03889 0.519444 0.854505i \(-0.326139\pi\)
0.519444 + 0.854505i \(0.326139\pi\)
\(44\) −2.36196e8 −0.215915
\(45\) 4.72092e8 0.381380
\(46\) −3.19049e6 −0.00228397
\(47\) 3.01812e9 1.91954 0.959772 0.280779i \(-0.0905929\pi\)
0.959772 + 0.280779i \(0.0905929\pi\)
\(48\) 5.96748e8 0.338037
\(49\) −1.97557e9 −0.999111
\(50\) −2.60222e8 −0.117763
\(51\) −7.91925e8 −0.321402
\(52\) 5.22594e7 0.0190610
\(53\) 2.68649e9 0.882405 0.441203 0.897407i \(-0.354552\pi\)
0.441203 + 0.897407i \(0.354552\pi\)
\(54\) −2.47434e8 −0.0733320
\(55\) 1.07867e9 0.288998
\(56\) 8.68335e7 0.0210694
\(57\) −2.65460e9 −0.584369
\(58\) −2.91229e8 −0.0582613
\(59\) −7.14924e8 −0.130189
\(60\) −3.40108e9 −0.564658
\(61\) −1.08583e10 −1.64607 −0.823035 0.567990i \(-0.807721\pi\)
−0.823035 + 0.567990i \(0.807721\pi\)
\(62\) 9.55025e8 0.132392
\(63\) 7.82764e7 0.00993706
\(64\) −1.98580e9 −0.231178
\(65\) −2.38661e8 −0.0255128
\(66\) −5.65356e8 −0.0555687
\(67\) 1.61605e10 1.46233 0.731163 0.682203i \(-0.238978\pi\)
0.731163 + 0.682203i \(0.238978\pi\)
\(68\) 5.70525e9 0.475856
\(69\) 4.49597e7 0.00346062
\(70\) −1.82756e8 −0.0129967
\(71\) 3.75065e9 0.246710 0.123355 0.992363i \(-0.460635\pi\)
0.123355 + 0.992363i \(0.460635\pi\)
\(72\) 3.86795e9 0.235588
\(73\) −2.36110e10 −1.33302 −0.666512 0.745495i \(-0.732213\pi\)
−0.666512 + 0.745495i \(0.732213\pi\)
\(74\) 1.24450e10 0.651959
\(75\) 3.66699e9 0.178432
\(76\) 1.91245e10 0.865197
\(77\) 1.78852e8 0.00752999
\(78\) 1.25088e8 0.00490562
\(79\) −4.18117e10 −1.52879 −0.764397 0.644746i \(-0.776963\pi\)
−0.764397 + 0.644746i \(0.776963\pi\)
\(80\) 1.96335e10 0.669890
\(81\) 3.48678e9 0.111111
\(82\) 1.48925e10 0.443601
\(83\) 2.14444e10 0.597564 0.298782 0.954321i \(-0.403420\pi\)
0.298782 + 0.954321i \(0.403420\pi\)
\(84\) −5.63926e8 −0.0147125
\(85\) −2.60550e10 −0.636924
\(86\) −1.72697e10 −0.395862
\(87\) 4.10393e9 0.0882762
\(88\) 8.83780e9 0.178521
\(89\) 2.25760e10 0.428550 0.214275 0.976773i \(-0.431261\pi\)
0.214275 + 0.976773i \(0.431261\pi\)
\(90\) −8.14080e9 −0.145323
\(91\) −3.95718e7 −0.000664749 0
\(92\) −3.23902e8 −0.00512367
\(93\) −1.34580e10 −0.200597
\(94\) −5.20448e10 −0.731432
\(95\) −8.73387e10 −1.15805
\(96\) −4.28894e10 −0.536857
\(97\) −3.27724e10 −0.387493 −0.193747 0.981052i \(-0.562064\pi\)
−0.193747 + 0.981052i \(0.562064\pi\)
\(98\) 3.40669e10 0.380706
\(99\) 7.96687e9 0.0841965
\(100\) −2.64181e10 −0.264181
\(101\) 8.33085e10 0.788718 0.394359 0.918957i \(-0.370967\pi\)
0.394359 + 0.918957i \(0.370967\pi\)
\(102\) 1.36560e10 0.122468
\(103\) 2.29703e11 1.95237 0.976187 0.216932i \(-0.0696051\pi\)
0.976187 + 0.216932i \(0.0696051\pi\)
\(104\) −1.95540e9 −0.0157599
\(105\) 2.57536e9 0.0196923
\(106\) −4.63261e10 −0.336236
\(107\) −1.56240e11 −1.07692 −0.538459 0.842652i \(-0.680993\pi\)
−0.538459 + 0.842652i \(0.680993\pi\)
\(108\) −2.51198e10 −0.164507
\(109\) −2.21150e11 −1.37670 −0.688352 0.725377i \(-0.741666\pi\)
−0.688352 + 0.725377i \(0.741666\pi\)
\(110\) −1.86007e10 −0.110121
\(111\) −1.75372e11 −0.987833
\(112\) 3.25539e9 0.0174544
\(113\) 3.63850e11 1.85777 0.928883 0.370373i \(-0.120770\pi\)
0.928883 + 0.370373i \(0.120770\pi\)
\(114\) 4.57762e10 0.222671
\(115\) 1.47921e9 0.00685793
\(116\) −2.95659e10 −0.130699
\(117\) −1.76271e9 −0.00743289
\(118\) 1.23282e10 0.0496078
\(119\) −4.32012e9 −0.0165954
\(120\) 1.27259e11 0.466866
\(121\) −2.67108e11 −0.936198
\(122\) 1.87242e11 0.627226
\(123\) −2.09862e11 −0.672135
\(124\) 9.69552e10 0.296997
\(125\) −2.69729e11 −0.790539
\(126\) −1.34981e9 −0.00378646
\(127\) −3.83293e11 −1.02946 −0.514732 0.857351i \(-0.672108\pi\)
−0.514732 + 0.857351i \(0.672108\pi\)
\(128\) 3.95715e11 1.01795
\(129\) 2.43361e11 0.599802
\(130\) 4.11549e9 0.00972149
\(131\) 4.10464e11 0.929571 0.464785 0.885423i \(-0.346131\pi\)
0.464785 + 0.885423i \(0.346131\pi\)
\(132\) −5.73956e10 −0.124659
\(133\) −1.44814e10 −0.0301736
\(134\) −2.78674e11 −0.557211
\(135\) 1.14718e11 0.220190
\(136\) −2.13475e11 −0.393444
\(137\) 2.08833e11 0.369689 0.184845 0.982768i \(-0.440822\pi\)
0.184845 + 0.982768i \(0.440822\pi\)
\(138\) −7.75290e8 −0.00131865
\(139\) 4.49732e11 0.735145 0.367572 0.929995i \(-0.380189\pi\)
0.367572 + 0.929995i \(0.380189\pi\)
\(140\) −1.85536e10 −0.0291558
\(141\) 7.33403e11 1.10825
\(142\) −6.46767e10 −0.0940073
\(143\) −4.02756e9 −0.00563241
\(144\) 1.45010e11 0.195166
\(145\) 1.35023e11 0.174938
\(146\) 4.07150e11 0.507941
\(147\) −4.80063e11 −0.576837
\(148\) 1.26343e12 1.46255
\(149\) 1.77907e10 0.0198458 0.00992288 0.999951i \(-0.496841\pi\)
0.00992288 + 0.999951i \(0.496841\pi\)
\(150\) −6.32340e10 −0.0679906
\(151\) −1.59845e12 −1.65701 −0.828506 0.559980i \(-0.810809\pi\)
−0.828506 + 0.559980i \(0.810809\pi\)
\(152\) −7.15586e11 −0.715356
\(153\) −1.92438e11 −0.185561
\(154\) −3.08414e9 −0.00286926
\(155\) −4.42780e11 −0.397524
\(156\) 1.26990e10 0.0110049
\(157\) −1.19670e12 −1.00124 −0.500619 0.865668i \(-0.666894\pi\)
−0.500619 + 0.865668i \(0.666894\pi\)
\(158\) 7.21005e11 0.582538
\(159\) 6.52817e11 0.509457
\(160\) −1.41110e12 −1.06389
\(161\) 2.45265e8 0.000178687 0
\(162\) −6.01265e10 −0.0423383
\(163\) −1.67271e12 −1.13865 −0.569324 0.822113i \(-0.692795\pi\)
−0.569324 + 0.822113i \(0.692795\pi\)
\(164\) 1.51191e12 0.995140
\(165\) 2.62117e11 0.166853
\(166\) −3.69790e11 −0.227699
\(167\) −1.38696e12 −0.826274 −0.413137 0.910669i \(-0.635567\pi\)
−0.413137 + 0.910669i \(0.635567\pi\)
\(168\) 2.11005e10 0.0121644
\(169\) −1.79127e12 −0.999503
\(170\) 4.49295e11 0.242696
\(171\) −6.45068e11 −0.337386
\(172\) −1.75324e12 −0.888047
\(173\) −3.50693e12 −1.72058 −0.860288 0.509808i \(-0.829717\pi\)
−0.860288 + 0.509808i \(0.829717\pi\)
\(174\) −7.07685e10 −0.0336372
\(175\) 2.00042e10 0.00921325
\(176\) 3.31329e11 0.147890
\(177\) −1.73727e11 −0.0751646
\(178\) −3.89302e11 −0.163297
\(179\) −4.28803e12 −1.74408 −0.872039 0.489436i \(-0.837203\pi\)
−0.872039 + 0.489436i \(0.837203\pi\)
\(180\) −8.26463e11 −0.326005
\(181\) −1.32012e12 −0.505105 −0.252553 0.967583i \(-0.581270\pi\)
−0.252553 + 0.967583i \(0.581270\pi\)
\(182\) 6.82380e8 0.000253299 0
\(183\) −2.63857e12 −0.950359
\(184\) 1.21195e10 0.00423631
\(185\) −5.76989e12 −1.95760
\(186\) 2.32071e11 0.0764363
\(187\) −4.39696e11 −0.140613
\(188\) −5.28364e12 −1.64084
\(189\) 1.90212e10 0.00573716
\(190\) 1.50608e12 0.441268
\(191\) 4.44766e11 0.126604 0.0633021 0.997994i \(-0.479837\pi\)
0.0633021 + 0.997994i \(0.479837\pi\)
\(192\) −4.82550e11 −0.133471
\(193\) −4.40420e12 −1.18386 −0.591932 0.805988i \(-0.701635\pi\)
−0.591932 + 0.805988i \(0.701635\pi\)
\(194\) 5.65131e11 0.147652
\(195\) −5.79946e10 −0.0147298
\(196\) 3.45851e12 0.854045
\(197\) −1.08074e12 −0.259511 −0.129756 0.991546i \(-0.541419\pi\)
−0.129756 + 0.991546i \(0.541419\pi\)
\(198\) −1.37382e11 −0.0320826
\(199\) 1.76499e12 0.400914 0.200457 0.979703i \(-0.435757\pi\)
0.200457 + 0.979703i \(0.435757\pi\)
\(200\) 9.88491e11 0.218428
\(201\) 3.92701e12 0.844274
\(202\) −1.43658e12 −0.300536
\(203\) 2.23878e10 0.00455810
\(204\) 1.38638e12 0.274736
\(205\) −6.90465e12 −1.33197
\(206\) −3.96103e12 −0.743941
\(207\) 1.09252e10 0.00199799
\(208\) −7.33081e10 −0.0130558
\(209\) −1.47390e12 −0.255660
\(210\) −4.44098e10 −0.00750366
\(211\) −9.48437e12 −1.56119 −0.780593 0.625039i \(-0.785083\pi\)
−0.780593 + 0.625039i \(0.785083\pi\)
\(212\) −4.70308e12 −0.754285
\(213\) 9.11409e11 0.142438
\(214\) 2.69422e12 0.410353
\(215\) 8.00680e12 1.18863
\(216\) 9.39913e11 0.136017
\(217\) −7.34163e10 −0.0103577
\(218\) 3.81353e12 0.524585
\(219\) −5.73746e12 −0.769621
\(220\) −1.88837e12 −0.247037
\(221\) 9.72848e10 0.0124133
\(222\) 3.02413e12 0.376408
\(223\) 1.52602e13 1.85303 0.926514 0.376260i \(-0.122790\pi\)
0.926514 + 0.376260i \(0.122790\pi\)
\(224\) −2.33971e11 −0.0277203
\(225\) 8.91079e11 0.103018
\(226\) −6.27427e12 −0.707892
\(227\) 1.23239e12 0.135709 0.0678543 0.997695i \(-0.478385\pi\)
0.0678543 + 0.997695i \(0.478385\pi\)
\(228\) 4.64725e12 0.499522
\(229\) 9.34468e11 0.0980549 0.0490275 0.998797i \(-0.484388\pi\)
0.0490275 + 0.998797i \(0.484388\pi\)
\(230\) −2.55077e10 −0.00261317
\(231\) 4.34610e10 0.00434744
\(232\) 1.10627e12 0.108063
\(233\) −1.10618e13 −1.05528 −0.527641 0.849468i \(-0.676923\pi\)
−0.527641 + 0.849468i \(0.676923\pi\)
\(234\) 3.03963e10 0.00283226
\(235\) 2.41296e13 2.19623
\(236\) 1.25158e12 0.111286
\(237\) −1.01602e13 −0.882649
\(238\) 7.44966e10 0.00632359
\(239\) −1.79204e13 −1.48648 −0.743240 0.669024i \(-0.766712\pi\)
−0.743240 + 0.669024i \(0.766712\pi\)
\(240\) 4.77095e12 0.386761
\(241\) −9.45537e11 −0.0749177 −0.0374589 0.999298i \(-0.511926\pi\)
−0.0374589 + 0.999298i \(0.511926\pi\)
\(242\) 4.60604e12 0.356733
\(243\) 8.47289e11 0.0641500
\(244\) 1.90090e13 1.40707
\(245\) −1.57945e13 −1.14312
\(246\) 3.61889e12 0.256113
\(247\) 3.26107e11 0.0225698
\(248\) −3.62780e12 −0.245561
\(249\) 5.21099e12 0.345004
\(250\) 4.65124e12 0.301230
\(251\) −1.19285e13 −0.755752 −0.377876 0.925856i \(-0.623345\pi\)
−0.377876 + 0.925856i \(0.623345\pi\)
\(252\) −1.37034e11 −0.00849425
\(253\) 2.49627e10 0.00151401
\(254\) 6.60955e12 0.392271
\(255\) −6.33137e12 −0.367728
\(256\) −2.75683e12 −0.156707
\(257\) 1.72401e13 0.959196 0.479598 0.877488i \(-0.340783\pi\)
0.479598 + 0.877488i \(0.340783\pi\)
\(258\) −4.19655e12 −0.228551
\(259\) −9.56692e11 −0.0510063
\(260\) 4.17809e11 0.0218084
\(261\) 9.97254e11 0.0509663
\(262\) −7.07807e12 −0.354208
\(263\) −1.30175e13 −0.637928 −0.318964 0.947767i \(-0.603335\pi\)
−0.318964 + 0.947767i \(0.603335\pi\)
\(264\) 2.14758e12 0.103069
\(265\) 2.14783e13 1.00959
\(266\) 2.49719e11 0.0114975
\(267\) 5.48596e12 0.247423
\(268\) −2.82913e13 −1.25000
\(269\) 2.39837e13 1.03819 0.519096 0.854716i \(-0.326269\pi\)
0.519096 + 0.854716i \(0.326269\pi\)
\(270\) −1.97821e12 −0.0839020
\(271\) 8.58896e12 0.356952 0.178476 0.983944i \(-0.442883\pi\)
0.178476 + 0.983944i \(0.442883\pi\)
\(272\) −8.00318e12 −0.325937
\(273\) −9.61594e9 −0.000383793 0
\(274\) −3.60114e12 −0.140868
\(275\) 2.03601e12 0.0780637
\(276\) −7.87083e10 −0.00295815
\(277\) 3.88767e13 1.43235 0.716177 0.697919i \(-0.245890\pi\)
0.716177 + 0.697919i \(0.245890\pi\)
\(278\) −7.75523e12 −0.280123
\(279\) −3.27029e12 −0.115815
\(280\) 6.94226e11 0.0241064
\(281\) −5.40461e13 −1.84026 −0.920130 0.391612i \(-0.871917\pi\)
−0.920130 + 0.391612i \(0.871917\pi\)
\(282\) −1.26469e13 −0.422292
\(283\) −3.02124e13 −0.989371 −0.494686 0.869072i \(-0.664717\pi\)
−0.494686 + 0.869072i \(0.664717\pi\)
\(284\) −6.56605e12 −0.210889
\(285\) −2.12233e13 −0.668600
\(286\) 6.94517e10 0.00214620
\(287\) −1.14484e12 −0.0347053
\(288\) −1.04221e13 −0.309955
\(289\) −2.36511e13 −0.690103
\(290\) −2.32835e12 −0.0666590
\(291\) −7.96370e12 −0.223719
\(292\) 4.13343e13 1.13948
\(293\) 5.03877e13 1.36318 0.681589 0.731735i \(-0.261289\pi\)
0.681589 + 0.731735i \(0.261289\pi\)
\(294\) 8.27826e12 0.219801
\(295\) −5.71576e12 −0.148954
\(296\) −4.72740e13 −1.20926
\(297\) 1.93595e12 0.0486109
\(298\) −3.06784e11 −0.00756211
\(299\) −5.52312e9 −0.000133657 0
\(300\) −6.41959e12 −0.152525
\(301\) 1.32759e12 0.0309705
\(302\) 2.75638e13 0.631395
\(303\) 2.02440e13 0.455366
\(304\) −2.68273e13 −0.592615
\(305\) −8.68113e13 −1.88333
\(306\) 3.31842e12 0.0707071
\(307\) −4.20907e13 −0.880897 −0.440448 0.897778i \(-0.645181\pi\)
−0.440448 + 0.897778i \(0.645181\pi\)
\(308\) −3.13106e11 −0.00643668
\(309\) 5.58179e13 1.12720
\(310\) 7.63534e12 0.151474
\(311\) 9.40628e12 0.183331 0.0916655 0.995790i \(-0.470781\pi\)
0.0916655 + 0.995790i \(0.470781\pi\)
\(312\) −4.75163e11 −0.00909897
\(313\) 7.43970e13 1.39979 0.699893 0.714248i \(-0.253231\pi\)
0.699893 + 0.714248i \(0.253231\pi\)
\(314\) 2.06360e13 0.381516
\(315\) 6.25813e11 0.0113694
\(316\) 7.31973e13 1.30682
\(317\) 2.30400e13 0.404257 0.202128 0.979359i \(-0.435214\pi\)
0.202128 + 0.979359i \(0.435214\pi\)
\(318\) −1.12573e13 −0.194126
\(319\) 2.27860e12 0.0386207
\(320\) −1.58763e13 −0.264499
\(321\) −3.79664e13 −0.621758
\(322\) −4.22937e9 −6.80877e−5 0
\(323\) 3.56017e13 0.563452
\(324\) −6.10411e12 −0.0949784
\(325\) −4.50475e11 −0.00689148
\(326\) 2.88444e13 0.433875
\(327\) −5.37394e13 −0.794840
\(328\) −5.65714e13 −0.822793
\(329\) 4.00087e12 0.0572239
\(330\) −4.51997e12 −0.0635783
\(331\) −1.02273e13 −0.141483 −0.0707417 0.997495i \(-0.522537\pi\)
−0.0707417 + 0.997495i \(0.522537\pi\)
\(332\) −3.75415e13 −0.510801
\(333\) −4.26154e13 −0.570326
\(334\) 2.39169e13 0.314847
\(335\) 1.29202e14 1.67310
\(336\) 7.91060e11 0.0100773
\(337\) −2.55314e13 −0.319971 −0.159986 0.987119i \(-0.551145\pi\)
−0.159986 + 0.987119i \(0.551145\pi\)
\(338\) 3.08888e13 0.380855
\(339\) 8.84156e13 1.07258
\(340\) 4.56130e13 0.544446
\(341\) −7.47222e12 −0.0877608
\(342\) 1.11236e13 0.128559
\(343\) −5.24003e12 −0.0595959
\(344\) 6.56015e13 0.734248
\(345\) 3.59449e11 0.00395943
\(346\) 6.04739e13 0.655616
\(347\) 9.22758e13 0.984636 0.492318 0.870415i \(-0.336150\pi\)
0.492318 + 0.870415i \(0.336150\pi\)
\(348\) −7.18450e12 −0.0754590
\(349\) 8.18322e13 0.846027 0.423013 0.906123i \(-0.360972\pi\)
0.423013 + 0.906123i \(0.360972\pi\)
\(350\) −3.44955e11 −0.00351066
\(351\) −4.28337e11 −0.00429138
\(352\) −2.38133e13 −0.234874
\(353\) 1.64332e14 1.59573 0.797867 0.602833i \(-0.205961\pi\)
0.797867 + 0.602833i \(0.205961\pi\)
\(354\) 2.99576e12 0.0286411
\(355\) 2.99862e13 0.282270
\(356\) −3.95224e13 −0.366327
\(357\) −1.04979e12 −0.00958136
\(358\) 7.39432e13 0.664571
\(359\) 4.41523e13 0.390781 0.195391 0.980725i \(-0.437403\pi\)
0.195391 + 0.980725i \(0.437403\pi\)
\(360\) 3.09240e13 0.269545
\(361\) 2.84965e12 0.0244625
\(362\) 2.27643e13 0.192468
\(363\) −6.49073e13 −0.540514
\(364\) 6.92760e10 0.000568231 0
\(365\) −1.88767e14 −1.52516
\(366\) 4.54998e13 0.362129
\(367\) −8.52900e13 −0.668705 −0.334353 0.942448i \(-0.608518\pi\)
−0.334353 + 0.942448i \(0.608518\pi\)
\(368\) 4.54362e11 0.00350945
\(369\) −5.09965e13 −0.388057
\(370\) 9.94966e13 0.745931
\(371\) 3.56126e12 0.0263055
\(372\) 2.35601e13 0.171471
\(373\) 2.26293e14 1.62283 0.811414 0.584471i \(-0.198698\pi\)
0.811414 + 0.584471i \(0.198698\pi\)
\(374\) 7.58217e12 0.0535796
\(375\) −6.55442e13 −0.456418
\(376\) 1.97699e14 1.35666
\(377\) −5.04151e11 −0.00340944
\(378\) −3.28003e11 −0.00218611
\(379\) 2.21395e13 0.145429 0.0727147 0.997353i \(-0.476834\pi\)
0.0727147 + 0.997353i \(0.476834\pi\)
\(380\) 1.52899e14 0.989906
\(381\) −9.31403e13 −0.594361
\(382\) −7.66960e12 −0.0482419
\(383\) −2.57233e14 −1.59490 −0.797451 0.603384i \(-0.793819\pi\)
−0.797451 + 0.603384i \(0.793819\pi\)
\(384\) 9.61587e13 0.587715
\(385\) 1.42991e12 0.00861536
\(386\) 7.59465e13 0.451105
\(387\) 5.91368e13 0.346296
\(388\) 5.73728e13 0.331231
\(389\) 1.48946e14 0.847825 0.423913 0.905703i \(-0.360656\pi\)
0.423913 + 0.905703i \(0.360656\pi\)
\(390\) 1.00006e12 0.00561271
\(391\) −6.02969e11 −0.00333674
\(392\) −1.29408e14 −0.706135
\(393\) 9.97426e13 0.536688
\(394\) 1.86364e13 0.0988854
\(395\) −3.34281e14 −1.74915
\(396\) −1.39471e13 −0.0719716
\(397\) −3.16206e14 −1.60925 −0.804623 0.593786i \(-0.797632\pi\)
−0.804623 + 0.593786i \(0.797632\pi\)
\(398\) −3.04357e13 −0.152766
\(399\) −3.51899e12 −0.0174207
\(400\) 3.70585e13 0.180950
\(401\) 1.69636e14 0.817004 0.408502 0.912757i \(-0.366051\pi\)
0.408502 + 0.912757i \(0.366051\pi\)
\(402\) −6.77177e13 −0.321706
\(403\) 1.65326e12 0.00774754
\(404\) −1.45843e14 −0.674200
\(405\) 2.78765e13 0.127127
\(406\) −3.86058e11 −0.00173684
\(407\) −9.73709e13 −0.432175
\(408\) −5.18744e13 −0.227155
\(409\) −1.55395e14 −0.671366 −0.335683 0.941975i \(-0.608967\pi\)
−0.335683 + 0.941975i \(0.608967\pi\)
\(410\) 1.19065e14 0.507541
\(411\) 5.07465e13 0.213440
\(412\) −4.02128e14 −1.66890
\(413\) −9.47717e11 −0.00388109
\(414\) −1.88395e11 −0.000761322 0
\(415\) 1.71446e14 0.683696
\(416\) 5.26879e12 0.0207347
\(417\) 1.09285e14 0.424436
\(418\) 2.54161e13 0.0974180
\(419\) 5.61588e13 0.212442 0.106221 0.994343i \(-0.466125\pi\)
0.106221 + 0.994343i \(0.466125\pi\)
\(420\) −4.50854e12 −0.0168331
\(421\) 4.27803e13 0.157649 0.0788247 0.996888i \(-0.474883\pi\)
0.0788247 + 0.996888i \(0.474883\pi\)
\(422\) 1.63549e14 0.594882
\(423\) 1.78217e14 0.639848
\(424\) 1.75976e14 0.623652
\(425\) −4.91792e13 −0.172045
\(426\) −1.57164e13 −0.0542752
\(427\) −1.43940e13 −0.0490713
\(428\) 2.73521e14 0.920554
\(429\) −9.78698e11 −0.00325187
\(430\) −1.38070e14 −0.452922
\(431\) −2.22098e14 −0.719317 −0.359658 0.933084i \(-0.617107\pi\)
−0.359658 + 0.933084i \(0.617107\pi\)
\(432\) 3.52374e13 0.112679
\(433\) 4.37273e14 1.38061 0.690303 0.723521i \(-0.257477\pi\)
0.690303 + 0.723521i \(0.257477\pi\)
\(434\) 1.26600e12 0.00394675
\(435\) 3.28105e13 0.101000
\(436\) 3.87154e14 1.17681
\(437\) −2.02120e12 −0.00606683
\(438\) 9.89374e13 0.293260
\(439\) −1.67284e14 −0.489667 −0.244833 0.969565i \(-0.578733\pi\)
−0.244833 + 0.969565i \(0.578733\pi\)
\(440\) 7.06574e13 0.204253
\(441\) −1.16655e14 −0.333037
\(442\) −1.67759e12 −0.00473002
\(443\) −5.50257e14 −1.53231 −0.766153 0.642659i \(-0.777831\pi\)
−0.766153 + 0.642659i \(0.777831\pi\)
\(444\) 3.07013e14 0.844405
\(445\) 1.80493e14 0.490321
\(446\) −2.63148e14 −0.706086
\(447\) 4.32313e12 0.0114579
\(448\) −2.63241e12 −0.00689168
\(449\) 1.21006e14 0.312933 0.156467 0.987683i \(-0.449990\pi\)
0.156467 + 0.987683i \(0.449990\pi\)
\(450\) −1.53659e13 −0.0392544
\(451\) −1.16521e14 −0.294058
\(452\) −6.36971e14 −1.58803
\(453\) −3.88423e14 −0.956677
\(454\) −2.12515e13 −0.0517110
\(455\) −3.16373e11 −0.000760566 0
\(456\) −1.73887e14 −0.413011
\(457\) 6.51012e14 1.52774 0.763871 0.645369i \(-0.223296\pi\)
0.763871 + 0.645369i \(0.223296\pi\)
\(458\) −1.61141e13 −0.0373633
\(459\) −4.67624e13 −0.107134
\(460\) −2.58957e12 −0.00586219
\(461\) 1.38009e14 0.308710 0.154355 0.988015i \(-0.450670\pi\)
0.154355 + 0.988015i \(0.450670\pi\)
\(462\) −7.49446e11 −0.00165657
\(463\) −1.78732e14 −0.390396 −0.195198 0.980764i \(-0.562535\pi\)
−0.195198 + 0.980764i \(0.562535\pi\)
\(464\) 4.14742e13 0.0895219
\(465\) −1.07596e14 −0.229511
\(466\) 1.90751e14 0.402109
\(467\) 6.46209e14 1.34626 0.673132 0.739523i \(-0.264949\pi\)
0.673132 + 0.739523i \(0.264949\pi\)
\(468\) 3.08586e12 0.00635367
\(469\) 2.14227e13 0.0435937
\(470\) −4.16093e14 −0.836859
\(471\) −2.90798e14 −0.578065
\(472\) −4.68305e13 −0.0920127
\(473\) 1.35120e14 0.262412
\(474\) 1.75204e14 0.336328
\(475\) −1.64853e14 −0.312811
\(476\) 7.56299e12 0.0141858
\(477\) 1.58635e14 0.294135
\(478\) 3.09021e14 0.566415
\(479\) −2.06190e14 −0.373613 −0.186806 0.982397i \(-0.559814\pi\)
−0.186806 + 0.982397i \(0.559814\pi\)
\(480\) −3.42897e14 −0.614239
\(481\) 2.15437e13 0.0381525
\(482\) 1.63049e13 0.0285470
\(483\) 5.95994e10 0.000103165 0
\(484\) 4.67611e14 0.800267
\(485\) −2.62013e14 −0.443346
\(486\) −1.46107e13 −0.0244440
\(487\) −1.41182e14 −0.233545 −0.116772 0.993159i \(-0.537255\pi\)
−0.116772 + 0.993159i \(0.537255\pi\)
\(488\) −7.11265e14 −1.16338
\(489\) −4.06469e14 −0.657398
\(490\) 2.72362e14 0.435580
\(491\) −2.31067e14 −0.365417 −0.182709 0.983167i \(-0.558487\pi\)
−0.182709 + 0.983167i \(0.558487\pi\)
\(492\) 3.67393e14 0.574544
\(493\) −5.50391e13 −0.0851164
\(494\) −5.62342e12 −0.00860008
\(495\) 6.36944e13 0.0963325
\(496\) −1.36006e14 −0.203427
\(497\) 4.97193e12 0.00735471
\(498\) −8.98589e13 −0.131462
\(499\) 5.09795e14 0.737637 0.368818 0.929501i \(-0.379762\pi\)
0.368818 + 0.929501i \(0.379762\pi\)
\(500\) 4.72199e14 0.675757
\(501\) −3.37032e14 −0.477050
\(502\) 2.05696e14 0.287975
\(503\) −2.71402e14 −0.375828 −0.187914 0.982185i \(-0.560173\pi\)
−0.187914 + 0.982185i \(0.560173\pi\)
\(504\) 5.12743e12 0.00702315
\(505\) 6.66044e14 0.902402
\(506\) −4.30460e11 −0.000576906 0
\(507\) −4.35278e14 −0.577063
\(508\) 6.71009e14 0.879990
\(509\) −1.29361e15 −1.67825 −0.839123 0.543941i \(-0.816931\pi\)
−0.839123 + 0.543941i \(0.816931\pi\)
\(510\) 1.09179e14 0.140121
\(511\) −3.12991e13 −0.0397390
\(512\) −7.62885e14 −0.958240
\(513\) −1.56751e14 −0.194790
\(514\) −2.97290e14 −0.365496
\(515\) 1.83646e15 2.23379
\(516\) −4.26038e14 −0.512714
\(517\) 4.07204e14 0.484857
\(518\) 1.64973e13 0.0194357
\(519\) −8.52185e14 −0.993376
\(520\) −1.56333e13 −0.0180315
\(521\) 6.01854e14 0.686884 0.343442 0.939174i \(-0.388407\pi\)
0.343442 + 0.939174i \(0.388407\pi\)
\(522\) −1.71968e13 −0.0194204
\(523\) −1.06743e15 −1.19284 −0.596418 0.802674i \(-0.703410\pi\)
−0.596418 + 0.802674i \(0.703410\pi\)
\(524\) −7.18574e14 −0.794602
\(525\) 4.86103e12 0.00531927
\(526\) 2.24475e14 0.243079
\(527\) 1.80490e14 0.193417
\(528\) 8.05130e13 0.0853846
\(529\) −9.52776e14 −0.999964
\(530\) −3.70373e14 −0.384700
\(531\) −4.22156e13 −0.0433963
\(532\) 2.53518e13 0.0257926
\(533\) 2.57807e13 0.0259595
\(534\) −9.46005e13 −0.0942793
\(535\) −1.24913e15 −1.23214
\(536\) 1.05858e15 1.03352
\(537\) −1.04199e15 −1.00694
\(538\) −4.13577e14 −0.395598
\(539\) −2.66543e14 −0.252365
\(540\) −2.00830e14 −0.188219
\(541\) 2.76164e14 0.256202 0.128101 0.991761i \(-0.459112\pi\)
0.128101 + 0.991761i \(0.459112\pi\)
\(542\) −1.48109e14 −0.136014
\(543\) −3.20790e14 −0.291623
\(544\) 5.75204e14 0.517640
\(545\) −1.76807e15 −1.57514
\(546\) 1.65818e11 0.000146242 0
\(547\) −3.59061e14 −0.313500 −0.156750 0.987638i \(-0.550102\pi\)
−0.156750 + 0.987638i \(0.550102\pi\)
\(548\) −3.65592e14 −0.316012
\(549\) −6.41173e14 −0.548690
\(550\) −3.51091e13 −0.0297457
\(551\) −1.84496e14 −0.154758
\(552\) 2.94505e12 0.00244584
\(553\) −5.54263e13 −0.0455751
\(554\) −6.70393e14 −0.545790
\(555\) −1.40208e15 −1.13022
\(556\) −7.87320e14 −0.628405
\(557\) 3.36952e14 0.266296 0.133148 0.991096i \(-0.457491\pi\)
0.133148 + 0.991096i \(0.457491\pi\)
\(558\) 5.63933e13 0.0441305
\(559\) −2.98960e13 −0.0231658
\(560\) 2.60266e13 0.0199702
\(561\) −1.06846e14 −0.0811827
\(562\) 9.31976e14 0.701221
\(563\) 2.16086e15 1.61001 0.805007 0.593265i \(-0.202162\pi\)
0.805007 + 0.593265i \(0.202162\pi\)
\(564\) −1.28393e15 −0.947338
\(565\) 2.90895e15 2.12554
\(566\) 5.20985e14 0.376994
\(567\) 4.62214e12 0.00331235
\(568\) 2.45683e14 0.174365
\(569\) −8.33757e14 −0.586033 −0.293016 0.956107i \(-0.594659\pi\)
−0.293016 + 0.956107i \(0.594659\pi\)
\(570\) 3.65977e14 0.254766
\(571\) −1.16917e15 −0.806080 −0.403040 0.915182i \(-0.632046\pi\)
−0.403040 + 0.915182i \(0.632046\pi\)
\(572\) 7.05082e12 0.00481461
\(573\) 1.08078e14 0.0730950
\(574\) 1.97418e13 0.0132243
\(575\) 2.79204e12 0.00185246
\(576\) −1.17260e14 −0.0770592
\(577\) −8.63090e14 −0.561810 −0.280905 0.959736i \(-0.590635\pi\)
−0.280905 + 0.959736i \(0.590635\pi\)
\(578\) 4.07843e14 0.262960
\(579\) −1.07022e15 −0.683504
\(580\) −2.36376e14 −0.149538
\(581\) 2.84271e13 0.0178141
\(582\) 1.37327e14 0.0852470
\(583\) 3.62461e14 0.222886
\(584\) −1.54662e15 −0.942132
\(585\) −1.40927e13 −0.00850425
\(586\) −8.68890e14 −0.519431
\(587\) −1.15770e15 −0.685627 −0.342813 0.939404i \(-0.611380\pi\)
−0.342813 + 0.939404i \(0.611380\pi\)
\(588\) 8.40419e14 0.493083
\(589\) 6.05016e14 0.351668
\(590\) 9.85631e13 0.0567582
\(591\) −2.62620e14 −0.149829
\(592\) −1.77230e15 −1.00177
\(593\) 3.40899e14 0.190908 0.0954542 0.995434i \(-0.469570\pi\)
0.0954542 + 0.995434i \(0.469570\pi\)
\(594\) −3.33837e13 −0.0185229
\(595\) −3.45390e13 −0.0189874
\(596\) −3.11451e13 −0.0169643
\(597\) 4.28893e14 0.231468
\(598\) 9.52412e10 5.09294e−5 0
\(599\) 6.59440e14 0.349404 0.174702 0.984621i \(-0.444104\pi\)
0.174702 + 0.984621i \(0.444104\pi\)
\(600\) 2.40203e14 0.126109
\(601\) −7.80789e14 −0.406185 −0.203093 0.979160i \(-0.565099\pi\)
−0.203093 + 0.979160i \(0.565099\pi\)
\(602\) −2.28931e13 −0.0118011
\(603\) 9.54263e14 0.487442
\(604\) 2.79831e15 1.41642
\(605\) −2.13551e15 −1.07114
\(606\) −3.49089e14 −0.173515
\(607\) 3.21551e15 1.58384 0.791921 0.610623i \(-0.209081\pi\)
0.791921 + 0.610623i \(0.209081\pi\)
\(608\) 1.92813e15 0.941168
\(609\) 5.44024e12 0.00263162
\(610\) 1.49698e15 0.717633
\(611\) −9.00956e13 −0.0428033
\(612\) 3.36889e14 0.158619
\(613\) 3.43995e15 1.60517 0.802583 0.596541i \(-0.203459\pi\)
0.802583 + 0.596541i \(0.203459\pi\)
\(614\) 7.25816e14 0.335661
\(615\) −1.67783e15 −0.769015
\(616\) 1.17155e13 0.00532192
\(617\) −2.37363e15 −1.06867 −0.534337 0.845272i \(-0.679439\pi\)
−0.534337 + 0.845272i \(0.679439\pi\)
\(618\) −9.62530e14 −0.429515
\(619\) −5.35918e14 −0.237028 −0.118514 0.992952i \(-0.537813\pi\)
−0.118514 + 0.992952i \(0.537813\pi\)
\(620\) 7.75149e14 0.339806
\(621\) 2.65483e12 0.00115354
\(622\) −1.62203e14 −0.0698573
\(623\) 2.99271e13 0.0127756
\(624\) −1.78139e13 −0.00753777
\(625\) −2.89330e15 −1.21354
\(626\) −1.28291e15 −0.533381
\(627\) −3.58158e14 −0.147606
\(628\) 2.09499e15 0.855863
\(629\) 2.35197e15 0.952474
\(630\) −1.07916e13 −0.00433224
\(631\) 1.90985e15 0.760044 0.380022 0.924978i \(-0.375916\pi\)
0.380022 + 0.924978i \(0.375916\pi\)
\(632\) −2.73884e15 −1.08049
\(633\) −2.30470e15 −0.901352
\(634\) −3.97305e14 −0.154040
\(635\) −3.06440e15 −1.17785
\(636\) −1.14285e15 −0.435487
\(637\) 5.89739e13 0.0222788
\(638\) −3.92925e13 −0.0147162
\(639\) 2.21472e14 0.0822365
\(640\) 3.16371e15 1.16468
\(641\) 4.57415e15 1.66952 0.834760 0.550614i \(-0.185606\pi\)
0.834760 + 0.550614i \(0.185606\pi\)
\(642\) 6.54696e14 0.236918
\(643\) −6.28032e14 −0.225331 −0.112666 0.993633i \(-0.535939\pi\)
−0.112666 + 0.993633i \(0.535939\pi\)
\(644\) −4.29371e11 −0.000152743 0
\(645\) 1.94565e15 0.686257
\(646\) −6.13919e14 −0.214700
\(647\) −4.79908e15 −1.66412 −0.832060 0.554686i \(-0.812838\pi\)
−0.832060 + 0.554686i \(0.812838\pi\)
\(648\) 2.28399e14 0.0785292
\(649\) −9.64573e13 −0.0328844
\(650\) 7.76804e12 0.00262596
\(651\) −1.78402e13 −0.00598003
\(652\) 2.92832e15 0.973322
\(653\) −3.50364e15 −1.15478 −0.577388 0.816470i \(-0.695928\pi\)
−0.577388 + 0.816470i \(0.695928\pi\)
\(654\) 9.26687e14 0.302869
\(655\) 3.28162e15 1.06356
\(656\) −2.12086e15 −0.681619
\(657\) −1.39420e15 −0.444341
\(658\) −6.89915e13 −0.0218048
\(659\) 3.69398e15 1.15778 0.578889 0.815407i \(-0.303487\pi\)
0.578889 + 0.815407i \(0.303487\pi\)
\(660\) −4.58873e14 −0.142627
\(661\) −3.57572e15 −1.10219 −0.551093 0.834444i \(-0.685789\pi\)
−0.551093 + 0.834444i \(0.685789\pi\)
\(662\) 1.76360e14 0.0539114
\(663\) 2.36402e13 0.00716683
\(664\) 1.40470e15 0.422336
\(665\) −1.15778e14 −0.0345228
\(666\) 7.34864e14 0.217320
\(667\) 3.12472e12 0.000916471 0
\(668\) 2.42807e15 0.706303
\(669\) 3.70822e15 1.06985
\(670\) −2.22797e15 −0.637527
\(671\) −1.46500e15 −0.415780
\(672\) −5.68550e13 −0.0160043
\(673\) −1.82280e15 −0.508927 −0.254464 0.967082i \(-0.581899\pi\)
−0.254464 + 0.967082i \(0.581899\pi\)
\(674\) 4.40267e14 0.121923
\(675\) 2.16532e14 0.0594774
\(676\) 3.13587e15 0.854380
\(677\) −3.90103e15 −1.05424 −0.527122 0.849789i \(-0.676729\pi\)
−0.527122 + 0.849789i \(0.676729\pi\)
\(678\) −1.52465e15 −0.408701
\(679\) −4.34437e13 −0.0115516
\(680\) −1.70671e15 −0.450154
\(681\) 2.99472e14 0.0783514
\(682\) 1.28852e14 0.0334407
\(683\) −1.94562e15 −0.500891 −0.250446 0.968131i \(-0.580577\pi\)
−0.250446 + 0.968131i \(0.580577\pi\)
\(684\) 1.12928e15 0.288399
\(685\) 1.66960e15 0.422976
\(686\) 9.03596e13 0.0227087
\(687\) 2.27076e14 0.0566120
\(688\) 2.45940e15 0.608266
\(689\) −8.01960e13 −0.0196765
\(690\) −6.19837e12 −0.00150872
\(691\) −6.12024e15 −1.47788 −0.738941 0.673771i \(-0.764674\pi\)
−0.738941 + 0.673771i \(0.764674\pi\)
\(692\) 6.13938e15 1.47076
\(693\) 1.05610e13 0.00251000
\(694\) −1.59121e15 −0.375190
\(695\) 3.59557e15 0.841107
\(696\) 2.68824e14 0.0623904
\(697\) 2.81453e15 0.648075
\(698\) −1.41112e15 −0.322374
\(699\) −2.68802e15 −0.609267
\(700\) −3.50203e13 −0.00787553
\(701\) 6.24454e15 1.39332 0.696661 0.717401i \(-0.254668\pi\)
0.696661 + 0.717401i \(0.254668\pi\)
\(702\) 7.38629e12 0.00163521
\(703\) 7.88400e15 1.73178
\(704\) −2.67924e14 −0.0583931
\(705\) 5.86349e15 1.26799
\(706\) −2.83375e15 −0.608046
\(707\) 1.10435e14 0.0235126
\(708\) 3.04133e14 0.0642511
\(709\) 1.48232e15 0.310734 0.155367 0.987857i \(-0.450344\pi\)
0.155367 + 0.987857i \(0.450344\pi\)
\(710\) −5.17084e14 −0.107557
\(711\) −2.46894e15 −0.509598
\(712\) 1.47882e15 0.302883
\(713\) −1.02469e13 −0.00208257
\(714\) 1.81027e13 0.00365092
\(715\) −3.22000e13 −0.00644426
\(716\) 7.50680e15 1.49085
\(717\) −4.35466e15 −0.858220
\(718\) −7.61366e14 −0.148905
\(719\) −8.47515e15 −1.64490 −0.822448 0.568840i \(-0.807393\pi\)
−0.822448 + 0.568840i \(0.807393\pi\)
\(720\) 1.15934e15 0.223297
\(721\) 3.04499e14 0.0582026
\(722\) −4.91396e13 −0.00932131
\(723\) −2.29765e14 −0.0432538
\(724\) 2.31106e15 0.431767
\(725\) 2.54857e14 0.0472540
\(726\) 1.11927e15 0.205960
\(727\) 1.05823e16 1.93260 0.966301 0.257415i \(-0.0828706\pi\)
0.966301 + 0.257415i \(0.0828706\pi\)
\(728\) −2.59212e12 −0.000469820 0
\(729\) 2.05891e14 0.0370370
\(730\) 3.25513e15 0.581155
\(731\) −3.26380e15 −0.578332
\(732\) 4.61919e15 0.812372
\(733\) 6.21466e15 1.08479 0.542395 0.840124i \(-0.317518\pi\)
0.542395 + 0.840124i \(0.317518\pi\)
\(734\) 1.47075e15 0.254806
\(735\) −3.83806e15 −0.659982
\(736\) −3.26559e13 −0.00557357
\(737\) 2.18037e15 0.369368
\(738\) 8.79389e14 0.147867
\(739\) −9.46484e14 −0.157968 −0.0789840 0.996876i \(-0.525168\pi\)
−0.0789840 + 0.996876i \(0.525168\pi\)
\(740\) 1.01010e16 1.67336
\(741\) 7.92440e13 0.0130307
\(742\) −6.14108e13 −0.0100236
\(743\) 2.09825e15 0.339953 0.169977 0.985448i \(-0.445631\pi\)
0.169977 + 0.985448i \(0.445631\pi\)
\(744\) −8.81555e14 −0.141775
\(745\) 1.42235e14 0.0227063
\(746\) −3.90222e15 −0.618370
\(747\) 1.26627e15 0.199188
\(748\) 7.69751e14 0.120196
\(749\) −2.07115e14 −0.0321042
\(750\) 1.13025e15 0.173915
\(751\) 5.77633e14 0.0882332 0.0441166 0.999026i \(-0.485953\pi\)
0.0441166 + 0.999026i \(0.485953\pi\)
\(752\) 7.41175e15 1.12389
\(753\) −2.89862e15 −0.436333
\(754\) 8.69363e12 0.00129915
\(755\) −1.27795e16 −1.89585
\(756\) −3.32992e13 −0.00490416
\(757\) 5.59338e15 0.817800 0.408900 0.912579i \(-0.365913\pi\)
0.408900 + 0.912579i \(0.365913\pi\)
\(758\) −3.81776e14 −0.0554150
\(759\) 6.06595e12 0.000874116 0
\(760\) −5.72105e15 −0.818466
\(761\) −1.23287e16 −1.75107 −0.875533 0.483158i \(-0.839490\pi\)
−0.875533 + 0.483158i \(0.839490\pi\)
\(762\) 1.60612e15 0.226478
\(763\) −2.93160e14 −0.0410412
\(764\) −7.78626e14 −0.108222
\(765\) −1.53852e15 −0.212308
\(766\) 4.43576e15 0.607728
\(767\) 2.13416e13 0.00290304
\(768\) −6.69909e14 −0.0904751
\(769\) 7.07708e15 0.948985 0.474492 0.880260i \(-0.342632\pi\)
0.474492 + 0.880260i \(0.342632\pi\)
\(770\) −2.46574e13 −0.00328283
\(771\) 4.18934e15 0.553792
\(772\) 7.71017e15 1.01197
\(773\) 7.41041e15 0.965728 0.482864 0.875695i \(-0.339597\pi\)
0.482864 + 0.875695i \(0.339597\pi\)
\(774\) −1.01976e15 −0.131954
\(775\) −8.35753e14 −0.107379
\(776\) −2.14673e15 −0.273866
\(777\) −2.32476e14 −0.0294485
\(778\) −2.56844e15 −0.323059
\(779\) 9.43454e15 1.17832
\(780\) 1.01528e14 0.0125911
\(781\) 5.06037e14 0.0623163
\(782\) 1.03977e13 0.00127145
\(783\) 2.42333e14 0.0294254
\(784\) −4.85151e15 −0.584977
\(785\) −9.56751e15 −1.14555
\(786\) −1.71997e15 −0.204502
\(787\) −1.46049e16 −1.72440 −0.862200 0.506568i \(-0.830914\pi\)
−0.862200 + 0.506568i \(0.830914\pi\)
\(788\) 1.89199e15 0.221832
\(789\) −3.16326e15 −0.368308
\(790\) 5.76437e15 0.666504
\(791\) 4.82326e14 0.0553822
\(792\) 5.21863e14 0.0595070
\(793\) 3.24138e14 0.0367052
\(794\) 5.45269e15 0.613194
\(795\) 5.21922e15 0.582890
\(796\) −3.08987e15 −0.342703
\(797\) 1.41948e16 1.56353 0.781767 0.623571i \(-0.214319\pi\)
0.781767 + 0.623571i \(0.214319\pi\)
\(798\) 6.06817e13 0.00663808
\(799\) −9.83590e15 −1.06858
\(800\) −2.66347e15 −0.287378
\(801\) 1.33309e15 0.142850
\(802\) −2.92522e15 −0.311315
\(803\) −3.18558e15 −0.336708
\(804\) −6.87478e15 −0.721690
\(805\) 1.96087e12 0.000204443 0
\(806\) −2.85090e13 −0.00295216
\(807\) 5.82803e15 0.599401
\(808\) 5.45705e15 0.557436
\(809\) −1.91599e15 −0.194391 −0.0971956 0.995265i \(-0.530987\pi\)
−0.0971956 + 0.995265i \(0.530987\pi\)
\(810\) −4.80706e14 −0.0484409
\(811\) 1.00041e16 1.00130 0.500648 0.865651i \(-0.333095\pi\)
0.500648 + 0.865651i \(0.333095\pi\)
\(812\) −3.91930e13 −0.00389629
\(813\) 2.08712e15 0.206086
\(814\) 1.67907e15 0.164678
\(815\) −1.33732e16 −1.30277
\(816\) −1.94477e15 −0.188180
\(817\) −1.09405e16 −1.05152
\(818\) 2.67965e15 0.255820
\(819\) −2.33667e12 −0.000221583 0
\(820\) 1.20876e16 1.13858
\(821\) −1.46063e16 −1.36663 −0.683317 0.730122i \(-0.739463\pi\)
−0.683317 + 0.730122i \(0.739463\pi\)
\(822\) −8.75078e14 −0.0813302
\(823\) 5.76089e15 0.531852 0.265926 0.963993i \(-0.414322\pi\)
0.265926 + 0.963993i \(0.414322\pi\)
\(824\) 1.50465e16 1.37987
\(825\) 4.94749e14 0.0450701
\(826\) 1.63425e13 0.00147887
\(827\) 1.07854e16 0.969519 0.484760 0.874647i \(-0.338907\pi\)
0.484760 + 0.874647i \(0.338907\pi\)
\(828\) −1.91261e13 −0.00170789
\(829\) −1.84718e15 −0.163854 −0.0819272 0.996638i \(-0.526108\pi\)
−0.0819272 + 0.996638i \(0.526108\pi\)
\(830\) −2.95644e15 −0.260519
\(831\) 9.44703e15 0.826970
\(832\) 5.92793e13 0.00515495
\(833\) 6.43828e15 0.556189
\(834\) −1.88452e15 −0.161729
\(835\) −1.10886e16 −0.945372
\(836\) 2.58027e15 0.218540
\(837\) −7.94681e14 −0.0668656
\(838\) −9.68408e14 −0.0809499
\(839\) −1.40672e16 −1.16820 −0.584100 0.811682i \(-0.698552\pi\)
−0.584100 + 0.811682i \(0.698552\pi\)
\(840\) 1.68697e14 0.0139178
\(841\) −1.19153e16 −0.976622
\(842\) −7.37708e14 −0.0600714
\(843\) −1.31332e16 −1.06247
\(844\) 1.66037e16 1.33451
\(845\) −1.43210e16 −1.14357
\(846\) −3.07319e15 −0.243811
\(847\) −3.54084e14 −0.0279092
\(848\) 6.59736e15 0.516646
\(849\) −7.34161e15 −0.571214
\(850\) 8.48051e14 0.0655568
\(851\) −1.33528e14 −0.0102555
\(852\) −1.59555e15 −0.121757
\(853\) 2.39241e16 1.81391 0.906954 0.421230i \(-0.138401\pi\)
0.906954 + 0.421230i \(0.138401\pi\)
\(854\) 2.48211e14 0.0186983
\(855\) −5.15726e15 −0.386016
\(856\) −1.02344e16 −0.761125
\(857\) 2.19672e16 1.62323 0.811614 0.584194i \(-0.198589\pi\)
0.811614 + 0.584194i \(0.198589\pi\)
\(858\) 1.68768e13 0.00123911
\(859\) −2.63625e14 −0.0192320 −0.00961601 0.999954i \(-0.503061\pi\)
−0.00961601 + 0.999954i \(0.503061\pi\)
\(860\) −1.40170e16 −1.01605
\(861\) −2.78197e14 −0.0200371
\(862\) 3.82988e15 0.274092
\(863\) −5.79310e14 −0.0411957 −0.0205978 0.999788i \(-0.506557\pi\)
−0.0205978 + 0.999788i \(0.506557\pi\)
\(864\) −2.53258e15 −0.178952
\(865\) −2.80376e16 −1.96858
\(866\) −7.54039e15 −0.526072
\(867\) −5.74723e15 −0.398431
\(868\) 1.28526e14 0.00885383
\(869\) −5.64122e15 −0.386157
\(870\) −5.65788e14 −0.0384856
\(871\) −4.82417e14 −0.0326079
\(872\) −1.44862e16 −0.973004
\(873\) −1.93518e15 −0.129164
\(874\) 3.48538e13 0.00231173
\(875\) −3.57558e14 −0.0235669
\(876\) 1.00442e16 0.657876
\(877\) −1.32387e16 −0.861680 −0.430840 0.902428i \(-0.641783\pi\)
−0.430840 + 0.902428i \(0.641783\pi\)
\(878\) 2.88467e15 0.186585
\(879\) 1.22442e16 0.787031
\(880\) 2.64895e15 0.169207
\(881\) 2.12452e16 1.34863 0.674317 0.738442i \(-0.264438\pi\)
0.674317 + 0.738442i \(0.264438\pi\)
\(882\) 2.01162e15 0.126902
\(883\) 6.50488e15 0.407807 0.203904 0.978991i \(-0.434637\pi\)
0.203904 + 0.978991i \(0.434637\pi\)
\(884\) −1.70311e14 −0.0106110
\(885\) −1.38893e15 −0.0859987
\(886\) 9.48869e15 0.583876
\(887\) −1.40683e15 −0.0860326 −0.0430163 0.999074i \(-0.513697\pi\)
−0.0430163 + 0.999074i \(0.513697\pi\)
\(888\) −1.14876e16 −0.698164
\(889\) −5.08101e14 −0.0306895
\(890\) −3.11244e15 −0.186834
\(891\) 4.70436e14 0.0280655
\(892\) −2.67150e16 −1.58398
\(893\) −3.29708e16 −1.94288
\(894\) −7.45485e13 −0.00436599
\(895\) −3.42824e16 −1.99547
\(896\) 5.24567e14 0.0303464
\(897\) −1.34212e12 −7.71671e−5 0
\(898\) −2.08664e15 −0.119242
\(899\) −9.35336e14 −0.0531238
\(900\) −1.55996e15 −0.0880602
\(901\) −8.75514e15 −0.491221
\(902\) 2.00930e15 0.112049
\(903\) 3.22604e14 0.0178808
\(904\) 2.38337e16 1.31300
\(905\) −1.05543e16 −0.577911
\(906\) 6.69801e15 0.364536
\(907\) 4.31251e15 0.233287 0.116643 0.993174i \(-0.462787\pi\)
0.116643 + 0.993174i \(0.462787\pi\)
\(908\) −2.15748e15 −0.116004
\(909\) 4.91928e15 0.262906
\(910\) 5.45557e12 0.000289809 0
\(911\) −7.59414e14 −0.0400984 −0.0200492 0.999799i \(-0.506382\pi\)
−0.0200492 + 0.999799i \(0.506382\pi\)
\(912\) −6.51904e15 −0.342146
\(913\) 2.89327e15 0.150939
\(914\) −1.12261e16 −0.582138
\(915\) −2.10951e16 −1.08734
\(916\) −1.63592e15 −0.0838179
\(917\) 5.44118e14 0.0277116
\(918\) 8.06375e14 0.0408228
\(919\) 2.69528e16 1.35634 0.678170 0.734905i \(-0.262773\pi\)
0.678170 + 0.734905i \(0.262773\pi\)
\(920\) 9.68946e13 0.00484693
\(921\) −1.02280e16 −0.508586
\(922\) −2.37983e15 −0.117632
\(923\) −1.11963e14 −0.00550129
\(924\) −7.60846e13 −0.00371622
\(925\) −1.08907e16 −0.528784
\(926\) 3.08207e15 0.148758
\(927\) 1.35638e16 0.650791
\(928\) −2.98083e15 −0.142175
\(929\) −2.52655e16 −1.19796 −0.598980 0.800764i \(-0.704427\pi\)
−0.598980 + 0.800764i \(0.704427\pi\)
\(930\) 1.85539e15 0.0874538
\(931\) 2.15817e16 1.01126
\(932\) 1.93652e16 0.902060
\(933\) 2.28573e15 0.105846
\(934\) −1.11433e16 −0.512986
\(935\) −3.51533e15 −0.160880
\(936\) −1.15465e14 −0.00525329
\(937\) 7.21858e15 0.326500 0.163250 0.986585i \(-0.447802\pi\)
0.163250 + 0.986585i \(0.447802\pi\)
\(938\) −3.69415e14 −0.0166111
\(939\) 1.80785e16 0.808167
\(940\) −4.22423e16 −1.87734
\(941\) −1.23022e16 −0.543551 −0.271775 0.962361i \(-0.587611\pi\)
−0.271775 + 0.962361i \(0.587611\pi\)
\(942\) 5.01455e15 0.220268
\(943\) −1.59788e14 −0.00697800
\(944\) −1.75568e15 −0.0762252
\(945\) 1.52073e14 0.00656411
\(946\) −2.33003e15 −0.0999907
\(947\) −1.82694e16 −0.779470 −0.389735 0.920927i \(-0.627433\pi\)
−0.389735 + 0.920927i \(0.627433\pi\)
\(948\) 1.77869e16 0.754493
\(949\) 7.04824e14 0.0297246
\(950\) 2.84274e15 0.119195
\(951\) 5.59873e15 0.233398
\(952\) −2.82986e14 −0.0117290
\(953\) −4.67260e15 −0.192552 −0.0962759 0.995355i \(-0.530693\pi\)
−0.0962759 + 0.995355i \(0.530693\pi\)
\(954\) −2.73551e15 −0.112079
\(955\) 3.55587e15 0.144853
\(956\) 3.13722e16 1.27065
\(957\) 5.53701e14 0.0222977
\(958\) 3.55556e15 0.142363
\(959\) 2.76833e14 0.0110209
\(960\) −3.85794e15 −0.152709
\(961\) −2.23412e16 −0.879283
\(962\) −3.71502e14 −0.0145378
\(963\) −9.22583e15 −0.358972
\(964\) 1.65530e15 0.0640401
\(965\) −3.52112e16 −1.35450
\(966\) −1.02774e12 −3.93105e−5 0
\(967\) −3.47991e16 −1.32349 −0.661747 0.749727i \(-0.730185\pi\)
−0.661747 + 0.749727i \(0.730185\pi\)
\(968\) −1.74967e16 −0.661671
\(969\) 8.65121e15 0.325309
\(970\) 4.51818e15 0.168935
\(971\) 1.97264e16 0.733402 0.366701 0.930339i \(-0.380487\pi\)
0.366701 + 0.930339i \(0.380487\pi\)
\(972\) −1.48330e15 −0.0548358
\(973\) 5.96173e14 0.0219155
\(974\) 2.43456e15 0.0889909
\(975\) −1.09465e14 −0.00397880
\(976\) −2.66653e16 −0.963769
\(977\) 3.00929e16 1.08154 0.540771 0.841170i \(-0.318133\pi\)
0.540771 + 0.841170i \(0.318133\pi\)
\(978\) 7.00919e15 0.250498
\(979\) 3.04594e15 0.108247
\(980\) 2.76505e16 0.977147
\(981\) −1.30587e16 −0.458901
\(982\) 3.98454e15 0.139240
\(983\) 4.40527e16 1.53084 0.765418 0.643533i \(-0.222532\pi\)
0.765418 + 0.643533i \(0.222532\pi\)
\(984\) −1.37468e16 −0.475040
\(985\) −8.64041e15 −0.296917
\(986\) 9.49100e14 0.0324331
\(987\) 9.72212e14 0.0330382
\(988\) −5.70896e14 −0.0192927
\(989\) 1.85294e14 0.00622706
\(990\) −1.09835e15 −0.0367070
\(991\) 3.88270e14 0.0129041 0.00645206 0.999979i \(-0.497946\pi\)
0.00645206 + 0.999979i \(0.497946\pi\)
\(992\) 9.77503e15 0.323076
\(993\) −2.48522e15 −0.0816855
\(994\) −8.57365e13 −0.00280247
\(995\) 1.41110e16 0.458701
\(996\) −9.12258e15 −0.294911
\(997\) 5.91431e15 0.190143 0.0950715 0.995470i \(-0.469692\pi\)
0.0950715 + 0.995470i \(0.469692\pi\)
\(998\) −8.79096e15 −0.281072
\(999\) −1.03555e16 −0.329278
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 177.12.a.b.1.13 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.b.1.13 27 1.1 even 1 trivial