Properties

Label 177.12.a.a.1.22
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $26$
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: \(26\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.22
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+60.2204 q^{2} -243.000 q^{3} +1578.50 q^{4} -10580.7 q^{5} -14633.6 q^{6} +72271.5 q^{7} -28273.6 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+60.2204 q^{2} -243.000 q^{3} +1578.50 q^{4} -10580.7 q^{5} -14633.6 q^{6} +72271.5 q^{7} -28273.6 q^{8} +59049.0 q^{9} -637177. q^{10} +271871. q^{11} -383575. q^{12} -546888. q^{13} +4.35222e6 q^{14} +2.57112e6 q^{15} -4.93541e6 q^{16} +17551.1 q^{17} +3.55596e6 q^{18} -1.54579e6 q^{19} -1.67017e7 q^{20} -1.75620e7 q^{21} +1.63722e7 q^{22} +2.86475e7 q^{23} +6.87048e6 q^{24} +6.31240e7 q^{25} -3.29338e7 q^{26} -1.43489e7 q^{27} +1.14080e8 q^{28} +1.81331e8 q^{29} +1.54834e8 q^{30} -2.76482e8 q^{31} -2.39308e8 q^{32} -6.60647e7 q^{33} +1.05694e6 q^{34} -7.64686e8 q^{35} +9.32087e7 q^{36} +4.29304e8 q^{37} -9.30883e7 q^{38} +1.32894e8 q^{39} +2.99156e8 q^{40} +7.55351e7 q^{41} -1.05759e9 q^{42} +1.28422e9 q^{43} +4.29148e8 q^{44} -6.24782e8 q^{45} +1.72516e9 q^{46} -4.08831e8 q^{47} +1.19931e9 q^{48} +3.24585e9 q^{49} +3.80135e9 q^{50} -4.26492e6 q^{51} -8.63261e8 q^{52} -5.10167e9 q^{53} -8.64097e8 q^{54} -2.87660e9 q^{55} -2.04338e9 q^{56} +3.75628e8 q^{57} +1.09199e10 q^{58} +7.14924e8 q^{59} +4.05851e9 q^{60} -5.06312e9 q^{61} -1.66499e10 q^{62} +4.26756e9 q^{63} -4.30352e9 q^{64} +5.78648e9 q^{65} -3.97844e9 q^{66} -2.02454e10 q^{67} +2.77044e7 q^{68} -6.96134e9 q^{69} -4.60497e10 q^{70} -5.57130e9 q^{71} -1.66953e9 q^{72} -1.58821e10 q^{73} +2.58529e10 q^{74} -1.53391e10 q^{75} -2.44003e9 q^{76} +1.96486e10 q^{77} +8.00291e9 q^{78} -1.93324e10 q^{79} +5.22203e10 q^{80} +3.48678e9 q^{81} +4.54875e9 q^{82} -6.53551e10 q^{83} -2.77216e10 q^{84} -1.85704e8 q^{85} +7.73365e10 q^{86} -4.40635e10 q^{87} -7.68678e9 q^{88} -4.74435e10 q^{89} -3.76246e10 q^{90} -3.95244e10 q^{91} +4.52200e10 q^{92} +6.71852e10 q^{93} -2.46200e10 q^{94} +1.63556e10 q^{95} +5.81519e10 q^{96} -1.44690e11 q^{97} +1.95466e11 q^{98} +1.60537e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} + O(q^{10}) \) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} - 859751q^{10} + 579094q^{11} - 5606010q^{12} - 2018538q^{13} + 4157413q^{14} - 925344q^{15} + 20190274q^{16} - 13084493q^{17} - 4605822q^{18} + 9917231q^{19} + 10165633q^{20} + 24013017q^{21} - 89820518q^{22} - 63513223q^{23} + 28587735q^{24} + 218986852q^{25} - 77999532q^{26} - 373071582q^{27} - 444601862q^{28} + 81530981q^{29} + 208919493q^{30} - 408861231q^{31} - 26253128q^{32} - 140719842q^{33} - 508910076q^{34} - 75731421q^{35} + 1362260430q^{36} - 802381301q^{37} + 732704675q^{38} + 490504734q^{39} - 646130800q^{40} - 1354472849q^{41} - 1010251359q^{42} + 282952194q^{43} + 1846047996q^{44} + 224858592q^{45} + 9629305849q^{46} - 1196794197q^{47} - 4906236582q^{48} + 10889725683q^{49} - 6236232091q^{50} + 3179531799q^{51} - 1968200812q^{52} - 8276044236q^{53} + 1119214746q^{54} - 6672895076q^{55} + 2579741342q^{56} - 2409887133q^{57} - 9401656060q^{58} + 18588031774q^{59} - 2470248819q^{60} - 21181559029q^{61} - 6117706514q^{62} - 5835163131q^{63} + 42975855037q^{64} + 25680681860q^{65} + 21826385874q^{66} + 26234163394q^{67} + 19707344091q^{68} + 15433713189q^{69} + 129203099090q^{70} + 52088830406q^{71} - 6946819605q^{72} + 20943384867q^{73} + 41969200146q^{74} - 53213805036q^{75} + 223987219368q^{76} + 94604773153q^{77} + 18953886276q^{78} + 68965662774q^{79} + 218947784293q^{80} + 90656394426q^{81} + 11938614923q^{82} + 17947446393q^{83} + 108038252466q^{84} - 52849386709q^{85} + 384986147852q^{86} - 19812028383q^{87} - 49061112607q^{88} + 38570593981q^{89} - 50767436799q^{90} - 226268806999q^{91} - 79559686310q^{92} + 99353279133q^{93} - 16709400108q^{94} - 252795831501q^{95} + 6379510104q^{96} - 186894587836q^{97} - 252443311612q^{98} + 34194921606q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 60.2204 1.33070 0.665348 0.746533i \(-0.268283\pi\)
0.665348 + 0.746533i \(0.268283\pi\)
\(3\) −243.000 −0.577350
\(4\) 1578.50 0.770751
\(5\) −10580.7 −1.51419 −0.757096 0.653303i \(-0.773383\pi\)
−0.757096 + 0.653303i \(0.773383\pi\)
\(6\) −14633.6 −0.768278
\(7\) 72271.5 1.62528 0.812640 0.582766i \(-0.198030\pi\)
0.812640 + 0.582766i \(0.198030\pi\)
\(8\) −28273.6 −0.305061
\(9\) 59049.0 0.333333
\(10\) −637177. −2.01493
\(11\) 271871. 0.508983 0.254492 0.967075i \(-0.418092\pi\)
0.254492 + 0.967075i \(0.418092\pi\)
\(12\) −383575. −0.444993
\(13\) −546888. −0.408517 −0.204258 0.978917i \(-0.565478\pi\)
−0.204258 + 0.978917i \(0.565478\pi\)
\(14\) 4.35222e6 2.16275
\(15\) 2.57112e6 0.874219
\(16\) −4.93541e6 −1.17669
\(17\) 17551.1 0.00299803 0.00149901 0.999999i \(-0.499523\pi\)
0.00149901 + 0.999999i \(0.499523\pi\)
\(18\) 3.55596e6 0.443565
\(19\) −1.54579e6 −0.143221 −0.0716105 0.997433i \(-0.522814\pi\)
−0.0716105 + 0.997433i \(0.522814\pi\)
\(20\) −1.67017e7 −1.16707
\(21\) −1.75620e7 −0.938356
\(22\) 1.63722e7 0.677302
\(23\) 2.86475e7 0.928076 0.464038 0.885815i \(-0.346400\pi\)
0.464038 + 0.885815i \(0.346400\pi\)
\(24\) 6.87048e6 0.176127
\(25\) 6.31240e7 1.29278
\(26\) −3.29338e7 −0.543611
\(27\) −1.43489e7 −0.192450
\(28\) 1.14080e8 1.25269
\(29\) 1.81331e8 1.64166 0.820832 0.571170i \(-0.193510\pi\)
0.820832 + 0.571170i \(0.193510\pi\)
\(30\) 1.54834e8 1.16332
\(31\) −2.76482e8 −1.73452 −0.867258 0.497859i \(-0.834120\pi\)
−0.867258 + 0.497859i \(0.834120\pi\)
\(32\) −2.39308e8 −1.26076
\(33\) −6.60647e7 −0.293862
\(34\) 1.05694e6 0.00398946
\(35\) −7.64686e8 −2.46099
\(36\) 9.32087e7 0.256917
\(37\) 4.29304e8 1.01778 0.508892 0.860831i \(-0.330055\pi\)
0.508892 + 0.860831i \(0.330055\pi\)
\(38\) −9.30883e7 −0.190584
\(39\) 1.32894e8 0.235857
\(40\) 2.99156e8 0.461920
\(41\) 7.55351e7 0.101821 0.0509105 0.998703i \(-0.483788\pi\)
0.0509105 + 0.998703i \(0.483788\pi\)
\(42\) −1.05759e9 −1.24867
\(43\) 1.28422e9 1.33218 0.666092 0.745870i \(-0.267966\pi\)
0.666092 + 0.745870i \(0.267966\pi\)
\(44\) 4.29148e8 0.392299
\(45\) −6.24782e8 −0.504731
\(46\) 1.72516e9 1.23499
\(47\) −4.08831e8 −0.260020 −0.130010 0.991513i \(-0.541501\pi\)
−0.130010 + 0.991513i \(0.541501\pi\)
\(48\) 1.19931e9 0.679365
\(49\) 3.24585e9 1.64153
\(50\) 3.80135e9 1.72030
\(51\) −4.26492e6 −0.00173091
\(52\) −8.63261e8 −0.314865
\(53\) −5.10167e9 −1.67569 −0.837847 0.545905i \(-0.816186\pi\)
−0.837847 + 0.545905i \(0.816186\pi\)
\(54\) −8.64097e8 −0.256093
\(55\) −2.87660e9 −0.770699
\(56\) −2.04338e9 −0.495809
\(57\) 3.75628e8 0.0826887
\(58\) 1.09199e10 2.18455
\(59\) 7.14924e8 0.130189
\(60\) 4.05851e9 0.673806
\(61\) −5.06312e9 −0.767546 −0.383773 0.923427i \(-0.625375\pi\)
−0.383773 + 0.923427i \(0.625375\pi\)
\(62\) −1.66499e10 −2.30811
\(63\) 4.26756e9 0.541760
\(64\) −4.30352e9 −0.500995
\(65\) 5.78648e9 0.618573
\(66\) −3.97844e9 −0.391040
\(67\) −2.02454e10 −1.83195 −0.915976 0.401233i \(-0.868582\pi\)
−0.915976 + 0.401233i \(0.868582\pi\)
\(68\) 2.77044e7 0.00231073
\(69\) −6.96134e9 −0.535825
\(70\) −4.60497e10 −3.27482
\(71\) −5.57130e9 −0.366468 −0.183234 0.983069i \(-0.558657\pi\)
−0.183234 + 0.983069i \(0.558657\pi\)
\(72\) −1.66953e9 −0.101687
\(73\) −1.58821e10 −0.896672 −0.448336 0.893865i \(-0.647983\pi\)
−0.448336 + 0.893865i \(0.647983\pi\)
\(74\) 2.58529e10 1.35436
\(75\) −1.53391e10 −0.746386
\(76\) −2.44003e9 −0.110388
\(77\) 1.96486e10 0.827240
\(78\) 8.00291e9 0.313854
\(79\) −1.93324e10 −0.706865 −0.353432 0.935460i \(-0.614986\pi\)
−0.353432 + 0.935460i \(0.614986\pi\)
\(80\) 5.22203e10 1.78174
\(81\) 3.48678e9 0.111111
\(82\) 4.54875e9 0.135493
\(83\) −6.53551e10 −1.82117 −0.910584 0.413324i \(-0.864368\pi\)
−0.910584 + 0.413324i \(0.864368\pi\)
\(84\) −2.77216e10 −0.723239
\(85\) −1.85704e8 −0.00453959
\(86\) 7.73365e10 1.77273
\(87\) −4.40635e10 −0.947815
\(88\) −7.68678e9 −0.155271
\(89\) −4.74435e10 −0.900600 −0.450300 0.892877i \(-0.648683\pi\)
−0.450300 + 0.892877i \(0.648683\pi\)
\(90\) −3.76246e10 −0.671643
\(91\) −3.95244e10 −0.663954
\(92\) 4.52200e10 0.715316
\(93\) 6.71852e10 1.00142
\(94\) −2.46200e10 −0.346007
\(95\) 1.63556e10 0.216864
\(96\) 5.81519e10 0.727901
\(97\) −1.44690e11 −1.71078 −0.855392 0.517981i \(-0.826684\pi\)
−0.855392 + 0.517981i \(0.826684\pi\)
\(98\) 1.95466e11 2.18438
\(99\) 1.60537e10 0.169661
\(100\) 9.96411e10 0.996411
\(101\) −1.14979e10 −0.108856 −0.0544279 0.998518i \(-0.517334\pi\)
−0.0544279 + 0.998518i \(0.517334\pi\)
\(102\) −2.56835e8 −0.00230332
\(103\) −2.72025e10 −0.231209 −0.115604 0.993295i \(-0.536881\pi\)
−0.115604 + 0.993295i \(0.536881\pi\)
\(104\) 1.54625e10 0.124622
\(105\) 1.85819e11 1.42085
\(106\) −3.07224e11 −2.22984
\(107\) −2.12775e11 −1.46659 −0.733296 0.679909i \(-0.762019\pi\)
−0.733296 + 0.679909i \(0.762019\pi\)
\(108\) −2.26497e10 −0.148331
\(109\) 2.20305e11 1.37145 0.685723 0.727863i \(-0.259486\pi\)
0.685723 + 0.727863i \(0.259486\pi\)
\(110\) −1.73230e11 −1.02557
\(111\) −1.04321e11 −0.587617
\(112\) −3.56690e11 −1.91246
\(113\) 3.47152e10 0.177251 0.0886254 0.996065i \(-0.471753\pi\)
0.0886254 + 0.996065i \(0.471753\pi\)
\(114\) 2.26205e10 0.110033
\(115\) −3.03112e11 −1.40529
\(116\) 2.86231e11 1.26531
\(117\) −3.22932e10 −0.136172
\(118\) 4.30530e10 0.173242
\(119\) 1.26845e9 0.00487263
\(120\) −7.26948e10 −0.266690
\(121\) −2.11398e11 −0.740936
\(122\) −3.04903e11 −1.02137
\(123\) −1.83550e10 −0.0587864
\(124\) −4.36427e11 −1.33688
\(125\) −1.51261e11 −0.443324
\(126\) 2.56994e11 0.720918
\(127\) −5.88133e11 −1.57963 −0.789815 0.613346i \(-0.789823\pi\)
−0.789815 + 0.613346i \(0.789823\pi\)
\(128\) 2.30944e11 0.594089
\(129\) −3.12066e11 −0.769137
\(130\) 3.48464e11 0.823132
\(131\) 2.67992e11 0.606919 0.303459 0.952844i \(-0.401858\pi\)
0.303459 + 0.952844i \(0.401858\pi\)
\(132\) −1.04283e11 −0.226494
\(133\) −1.11717e11 −0.232774
\(134\) −1.21918e12 −2.43777
\(135\) 1.51822e11 0.291406
\(136\) −4.96233e8 −0.000914580 0
\(137\) 8.69217e11 1.53874 0.769370 0.638804i \(-0.220570\pi\)
0.769370 + 0.638804i \(0.220570\pi\)
\(138\) −4.19215e11 −0.713020
\(139\) −2.13982e11 −0.349781 −0.174891 0.984588i \(-0.555957\pi\)
−0.174891 + 0.984588i \(0.555957\pi\)
\(140\) −1.20706e12 −1.89681
\(141\) 9.93460e10 0.150122
\(142\) −3.35506e11 −0.487657
\(143\) −1.48683e11 −0.207928
\(144\) −2.91431e11 −0.392231
\(145\) −1.91862e12 −2.48579
\(146\) −9.56430e11 −1.19320
\(147\) −7.88741e11 −0.947740
\(148\) 6.77655e11 0.784457
\(149\) −1.02330e12 −1.14150 −0.570751 0.821123i \(-0.693348\pi\)
−0.570751 + 0.821123i \(0.693348\pi\)
\(150\) −9.23728e11 −0.993213
\(151\) 8.02192e11 0.831582 0.415791 0.909460i \(-0.363505\pi\)
0.415791 + 0.909460i \(0.363505\pi\)
\(152\) 4.37051e10 0.0436911
\(153\) 1.03638e9 0.000999343 0
\(154\) 1.18324e12 1.10080
\(155\) 2.92539e12 2.62639
\(156\) 2.09772e11 0.181787
\(157\) 4.16854e11 0.348767 0.174384 0.984678i \(-0.444207\pi\)
0.174384 + 0.984678i \(0.444207\pi\)
\(158\) −1.16420e12 −0.940622
\(159\) 1.23970e12 0.967462
\(160\) 2.53206e12 1.90903
\(161\) 2.07040e12 1.50838
\(162\) 2.09976e11 0.147855
\(163\) 2.08094e12 1.41654 0.708269 0.705942i \(-0.249476\pi\)
0.708269 + 0.705942i \(0.249476\pi\)
\(164\) 1.19232e11 0.0784786
\(165\) 6.99014e11 0.444963
\(166\) −3.93571e12 −2.42342
\(167\) 1.59680e12 0.951286 0.475643 0.879638i \(-0.342215\pi\)
0.475643 + 0.879638i \(0.342215\pi\)
\(168\) 4.96540e11 0.286255
\(169\) −1.49307e12 −0.833114
\(170\) −1.11832e10 −0.00604081
\(171\) −9.12776e10 −0.0477403
\(172\) 2.02714e12 1.02678
\(173\) 1.26910e12 0.622650 0.311325 0.950304i \(-0.399227\pi\)
0.311325 + 0.950304i \(0.399227\pi\)
\(174\) −2.65352e12 −1.26125
\(175\) 4.56207e12 2.10113
\(176\) −1.34180e12 −0.598917
\(177\) −1.73727e11 −0.0751646
\(178\) −2.85707e12 −1.19842
\(179\) 2.19909e12 0.894439 0.447220 0.894424i \(-0.352414\pi\)
0.447220 + 0.894424i \(0.352414\pi\)
\(180\) −9.86218e11 −0.389022
\(181\) 2.31193e12 0.884590 0.442295 0.896870i \(-0.354164\pi\)
0.442295 + 0.896870i \(0.354164\pi\)
\(182\) −2.38018e12 −0.883520
\(183\) 1.23034e12 0.443143
\(184\) −8.09968e11 −0.283119
\(185\) −4.54235e12 −1.54112
\(186\) 4.04592e12 1.33259
\(187\) 4.77164e9 0.00152595
\(188\) −6.45340e11 −0.200410
\(189\) −1.03702e12 −0.312785
\(190\) 9.84944e11 0.288580
\(191\) −2.82113e12 −0.803044 −0.401522 0.915849i \(-0.631519\pi\)
−0.401522 + 0.915849i \(0.631519\pi\)
\(192\) 1.04575e12 0.289250
\(193\) −8.99959e10 −0.0241912 −0.0120956 0.999927i \(-0.503850\pi\)
−0.0120956 + 0.999927i \(0.503850\pi\)
\(194\) −8.71332e12 −2.27653
\(195\) −1.40611e12 −0.357133
\(196\) 5.12357e12 1.26521
\(197\) −1.49426e12 −0.358807 −0.179403 0.983776i \(-0.557417\pi\)
−0.179403 + 0.983776i \(0.557417\pi\)
\(198\) 9.66762e11 0.225767
\(199\) 3.35067e12 0.761096 0.380548 0.924761i \(-0.375735\pi\)
0.380548 + 0.924761i \(0.375735\pi\)
\(200\) −1.78474e12 −0.394376
\(201\) 4.91962e12 1.05768
\(202\) −6.92410e11 −0.144854
\(203\) 1.31051e13 2.66816
\(204\) −6.73217e9 −0.00133410
\(205\) −7.99217e11 −0.154177
\(206\) −1.63815e12 −0.307669
\(207\) 1.69161e12 0.309359
\(208\) 2.69912e12 0.480699
\(209\) −4.20257e11 −0.0728971
\(210\) 1.11901e13 1.89072
\(211\) −2.00542e12 −0.330105 −0.165052 0.986285i \(-0.552779\pi\)
−0.165052 + 0.986285i \(0.552779\pi\)
\(212\) −8.05297e12 −1.29154
\(213\) 1.35383e12 0.211580
\(214\) −1.28134e13 −1.95159
\(215\) −1.35880e13 −2.01718
\(216\) 4.05695e11 0.0587089
\(217\) −1.99818e13 −2.81907
\(218\) 1.32669e13 1.82498
\(219\) 3.85936e12 0.517694
\(220\) −4.54071e12 −0.594017
\(221\) −9.59849e9 −0.00122474
\(222\) −6.28224e12 −0.781940
\(223\) −5.22181e12 −0.634080 −0.317040 0.948412i \(-0.602689\pi\)
−0.317040 + 0.948412i \(0.602689\pi\)
\(224\) −1.72952e13 −2.04909
\(225\) 3.72741e12 0.430926
\(226\) 2.09056e12 0.235867
\(227\) 2.44041e12 0.268733 0.134366 0.990932i \(-0.457100\pi\)
0.134366 + 0.990932i \(0.457100\pi\)
\(228\) 5.92928e11 0.0637324
\(229\) −1.37439e13 −1.44216 −0.721081 0.692850i \(-0.756355\pi\)
−0.721081 + 0.692850i \(0.756355\pi\)
\(230\) −1.82535e13 −1.87001
\(231\) −4.77460e12 −0.477607
\(232\) −5.12689e12 −0.500807
\(233\) 1.11387e13 1.06262 0.531308 0.847179i \(-0.321701\pi\)
0.531308 + 0.847179i \(0.321701\pi\)
\(234\) −1.94471e12 −0.181204
\(235\) 4.32574e12 0.393720
\(236\) 1.12851e12 0.100343
\(237\) 4.69777e12 0.408108
\(238\) 7.63863e10 0.00648399
\(239\) −1.36418e12 −0.113157 −0.0565787 0.998398i \(-0.518019\pi\)
−0.0565787 + 0.998398i \(0.518019\pi\)
\(240\) −1.26895e13 −1.02869
\(241\) −1.82710e13 −1.44766 −0.723831 0.689977i \(-0.757621\pi\)
−0.723831 + 0.689977i \(0.757621\pi\)
\(242\) −1.27305e13 −0.985961
\(243\) −8.47289e11 −0.0641500
\(244\) −7.99213e12 −0.591587
\(245\) −3.43435e13 −2.48560
\(246\) −1.10535e12 −0.0782268
\(247\) 8.45375e11 0.0585081
\(248\) 7.81715e12 0.529132
\(249\) 1.58813e13 1.05145
\(250\) −9.10898e12 −0.589929
\(251\) −4.99770e12 −0.316639 −0.158320 0.987388i \(-0.550608\pi\)
−0.158320 + 0.987388i \(0.550608\pi\)
\(252\) 6.73634e12 0.417562
\(253\) 7.78843e12 0.472375
\(254\) −3.54176e13 −2.10201
\(255\) 4.51260e10 0.00262093
\(256\) 2.27211e13 1.29155
\(257\) −2.11473e12 −0.117658 −0.0588291 0.998268i \(-0.518737\pi\)
−0.0588291 + 0.998268i \(0.518737\pi\)
\(258\) −1.87928e13 −1.02349
\(259\) 3.10264e13 1.65418
\(260\) 9.13394e12 0.476766
\(261\) 1.07074e13 0.547221
\(262\) 1.61386e13 0.807624
\(263\) −2.27554e13 −1.11514 −0.557568 0.830131i \(-0.688266\pi\)
−0.557568 + 0.830131i \(0.688266\pi\)
\(264\) 1.86789e12 0.0896456
\(265\) 5.39794e13 2.53732
\(266\) −6.72764e12 −0.309752
\(267\) 1.15288e13 0.519961
\(268\) −3.19573e13 −1.41198
\(269\) −7.23012e12 −0.312974 −0.156487 0.987680i \(-0.550017\pi\)
−0.156487 + 0.987680i \(0.550017\pi\)
\(270\) 9.14279e12 0.387773
\(271\) −2.87461e13 −1.19467 −0.597334 0.801992i \(-0.703773\pi\)
−0.597334 + 0.801992i \(0.703773\pi\)
\(272\) −8.66220e10 −0.00352776
\(273\) 9.60443e12 0.383334
\(274\) 5.23446e13 2.04759
\(275\) 1.71616e13 0.658003
\(276\) −1.09885e13 −0.412988
\(277\) 2.37373e13 0.874566 0.437283 0.899324i \(-0.355941\pi\)
0.437283 + 0.899324i \(0.355941\pi\)
\(278\) −1.28861e13 −0.465453
\(279\) −1.63260e13 −0.578172
\(280\) 2.16204e13 0.750750
\(281\) −1.69968e13 −0.578737 −0.289369 0.957218i \(-0.593445\pi\)
−0.289369 + 0.957218i \(0.593445\pi\)
\(282\) 5.98266e12 0.199767
\(283\) −1.11630e13 −0.365556 −0.182778 0.983154i \(-0.558509\pi\)
−0.182778 + 0.983154i \(0.558509\pi\)
\(284\) −8.79429e12 −0.282456
\(285\) −3.97442e12 −0.125207
\(286\) −8.95375e12 −0.276689
\(287\) 5.45904e12 0.165488
\(288\) −1.41309e13 −0.420254
\(289\) −3.42716e13 −0.999991
\(290\) −1.15540e14 −3.30784
\(291\) 3.51598e13 0.987722
\(292\) −2.50699e13 −0.691111
\(293\) −9.31969e11 −0.0252133 −0.0126066 0.999921i \(-0.504013\pi\)
−0.0126066 + 0.999921i \(0.504013\pi\)
\(294\) −4.74983e13 −1.26115
\(295\) −7.56443e12 −0.197131
\(296\) −1.21380e13 −0.310485
\(297\) −3.90105e12 −0.0979539
\(298\) −6.16233e13 −1.51899
\(299\) −1.56670e13 −0.379134
\(300\) −2.42128e13 −0.575278
\(301\) 9.28128e13 2.16517
\(302\) 4.83083e13 1.10658
\(303\) 2.79399e12 0.0628480
\(304\) 7.62913e12 0.168527
\(305\) 5.35716e13 1.16221
\(306\) 6.24110e10 0.00132982
\(307\) −4.57224e13 −0.956902 −0.478451 0.878114i \(-0.658802\pi\)
−0.478451 + 0.878114i \(0.658802\pi\)
\(308\) 3.10152e13 0.637596
\(309\) 6.61022e12 0.133489
\(310\) 1.76168e14 3.49493
\(311\) 3.92854e13 0.765682 0.382841 0.923814i \(-0.374946\pi\)
0.382841 + 0.923814i \(0.374946\pi\)
\(312\) −3.75738e12 −0.0719507
\(313\) −3.49347e13 −0.657299 −0.328650 0.944452i \(-0.606594\pi\)
−0.328650 + 0.944452i \(0.606594\pi\)
\(314\) 2.51031e13 0.464103
\(315\) −4.51540e13 −0.820329
\(316\) −3.05161e13 −0.544817
\(317\) −9.26365e13 −1.62538 −0.812692 0.582693i \(-0.801999\pi\)
−0.812692 + 0.582693i \(0.801999\pi\)
\(318\) 7.46555e13 1.28740
\(319\) 4.92988e13 0.835579
\(320\) 4.55344e13 0.758603
\(321\) 5.17043e13 0.846738
\(322\) 1.24680e14 2.00720
\(323\) −2.71304e10 −0.000429381 0
\(324\) 5.50388e12 0.0856390
\(325\) −3.45217e13 −0.528122
\(326\) 1.25315e14 1.88498
\(327\) −5.35341e13 −0.791804
\(328\) −2.13565e12 −0.0310616
\(329\) −2.95469e13 −0.422605
\(330\) 4.20949e13 0.592110
\(331\) 5.60914e13 0.775966 0.387983 0.921667i \(-0.373172\pi\)
0.387983 + 0.921667i \(0.373172\pi\)
\(332\) −1.03163e14 −1.40367
\(333\) 2.53500e13 0.339261
\(334\) 9.61602e13 1.26587
\(335\) 2.14211e14 2.77393
\(336\) 8.66756e13 1.10416
\(337\) −9.34576e13 −1.17125 −0.585626 0.810582i \(-0.699151\pi\)
−0.585626 + 0.810582i \(0.699151\pi\)
\(338\) −8.99136e13 −1.10862
\(339\) −8.43579e12 −0.102336
\(340\) −2.93133e11 −0.00349889
\(341\) −7.51676e13 −0.882839
\(342\) −5.49677e12 −0.0635279
\(343\) 9.16780e13 1.04267
\(344\) −3.63096e13 −0.406397
\(345\) 7.36562e13 0.811342
\(346\) 7.64260e13 0.828557
\(347\) 1.10432e14 1.17838 0.589188 0.807996i \(-0.299448\pi\)
0.589188 + 0.807996i \(0.299448\pi\)
\(348\) −6.95542e13 −0.730529
\(349\) −1.84292e14 −1.90532 −0.952659 0.304040i \(-0.901664\pi\)
−0.952659 + 0.304040i \(0.901664\pi\)
\(350\) 2.74730e14 2.79596
\(351\) 7.84724e12 0.0786190
\(352\) −6.50610e13 −0.641706
\(353\) −9.33906e12 −0.0906864 −0.0453432 0.998971i \(-0.514438\pi\)
−0.0453432 + 0.998971i \(0.514438\pi\)
\(354\) −1.04619e13 −0.100021
\(355\) 5.89485e13 0.554903
\(356\) −7.48895e13 −0.694138
\(357\) −3.08232e11 −0.00281322
\(358\) 1.32430e14 1.19023
\(359\) 7.66828e13 0.678701 0.339350 0.940660i \(-0.389793\pi\)
0.339350 + 0.940660i \(0.389793\pi\)
\(360\) 1.76648e13 0.153973
\(361\) −1.14101e14 −0.979488
\(362\) 1.39225e14 1.17712
\(363\) 5.13696e13 0.427780
\(364\) −6.23892e13 −0.511743
\(365\) 1.68045e14 1.35773
\(366\) 7.40915e13 0.589688
\(367\) −1.15789e14 −0.907829 −0.453914 0.891045i \(-0.649973\pi\)
−0.453914 + 0.891045i \(0.649973\pi\)
\(368\) −1.41387e14 −1.09206
\(369\) 4.46027e12 0.0339403
\(370\) −2.73542e14 −2.05076
\(371\) −3.68705e14 −2.72347
\(372\) 1.06052e14 0.771848
\(373\) 4.15227e13 0.297774 0.148887 0.988854i \(-0.452431\pi\)
0.148887 + 0.988854i \(0.452431\pi\)
\(374\) 2.87350e11 0.00203057
\(375\) 3.67563e13 0.255953
\(376\) 1.15591e13 0.0793217
\(377\) −9.91679e13 −0.670647
\(378\) −6.24496e13 −0.416222
\(379\) 1.74723e14 1.14772 0.573858 0.818955i \(-0.305446\pi\)
0.573858 + 0.818955i \(0.305446\pi\)
\(380\) 2.58174e13 0.167148
\(381\) 1.42916e14 0.911999
\(382\) −1.69890e14 −1.06861
\(383\) 1.90325e14 1.18005 0.590027 0.807384i \(-0.299117\pi\)
0.590027 + 0.807384i \(0.299117\pi\)
\(384\) −5.61193e13 −0.342997
\(385\) −2.07896e14 −1.25260
\(386\) −5.41959e12 −0.0321911
\(387\) 7.58321e13 0.444061
\(388\) −2.28394e14 −1.31859
\(389\) 8.37280e13 0.476593 0.238297 0.971192i \(-0.423411\pi\)
0.238297 + 0.971192i \(0.423411\pi\)
\(390\) −8.46768e13 −0.475235
\(391\) 5.02795e11 0.00278240
\(392\) −9.17718e13 −0.500767
\(393\) −6.51222e13 −0.350405
\(394\) −8.99847e13 −0.477463
\(395\) 2.04551e14 1.07033
\(396\) 2.53408e13 0.130766
\(397\) 9.06523e13 0.461351 0.230675 0.973031i \(-0.425906\pi\)
0.230675 + 0.973031i \(0.425906\pi\)
\(398\) 2.01779e14 1.01279
\(399\) 2.71472e13 0.134392
\(400\) −3.11543e14 −1.52121
\(401\) 1.91294e14 0.921312 0.460656 0.887579i \(-0.347614\pi\)
0.460656 + 0.887579i \(0.347614\pi\)
\(402\) 2.96262e14 1.40745
\(403\) 1.51205e14 0.708578
\(404\) −1.81494e13 −0.0839008
\(405\) −3.68928e13 −0.168244
\(406\) 7.89195e14 3.55051
\(407\) 1.16715e14 0.518034
\(408\) 1.20585e11 0.000528033 0
\(409\) 3.07393e14 1.32805 0.664026 0.747709i \(-0.268846\pi\)
0.664026 + 0.747709i \(0.268846\pi\)
\(410\) −4.81292e13 −0.205162
\(411\) −2.11220e14 −0.888392
\(412\) −4.29391e13 −0.178205
\(413\) 5.16687e13 0.211593
\(414\) 1.01869e14 0.411662
\(415\) 6.91506e14 2.75760
\(416\) 1.30875e14 0.515042
\(417\) 5.19977e13 0.201946
\(418\) −2.53080e13 −0.0970038
\(419\) 8.82048e13 0.333669 0.166834 0.985985i \(-0.446646\pi\)
0.166834 + 0.985985i \(0.446646\pi\)
\(420\) 2.93315e14 1.09512
\(421\) −7.66855e13 −0.282593 −0.141297 0.989967i \(-0.545127\pi\)
−0.141297 + 0.989967i \(0.545127\pi\)
\(422\) −1.20767e14 −0.439269
\(423\) −2.41411e13 −0.0866732
\(424\) 1.44242e14 0.511188
\(425\) 1.10790e12 0.00387579
\(426\) 8.15280e13 0.281549
\(427\) −3.65920e14 −1.24748
\(428\) −3.35865e14 −1.13038
\(429\) 3.61300e13 0.120047
\(430\) −8.18277e14 −2.68426
\(431\) 4.40817e14 1.42769 0.713845 0.700304i \(-0.246952\pi\)
0.713845 + 0.700304i \(0.246952\pi\)
\(432\) 7.08178e13 0.226455
\(433\) −2.03364e14 −0.642082 −0.321041 0.947065i \(-0.604033\pi\)
−0.321041 + 0.947065i \(0.604033\pi\)
\(434\) −1.20331e15 −3.75133
\(435\) 4.66225e14 1.43517
\(436\) 3.47751e14 1.05704
\(437\) −4.42831e13 −0.132920
\(438\) 2.32412e14 0.688893
\(439\) 2.71140e14 0.793668 0.396834 0.917890i \(-0.370109\pi\)
0.396834 + 0.917890i \(0.370109\pi\)
\(440\) 8.13318e13 0.235110
\(441\) 1.91664e14 0.547178
\(442\) −5.78025e11 −0.00162976
\(443\) −3.26496e14 −0.909195 −0.454597 0.890697i \(-0.650217\pi\)
−0.454597 + 0.890697i \(0.650217\pi\)
\(444\) −1.64670e14 −0.452907
\(445\) 5.01987e14 1.36368
\(446\) −3.14459e14 −0.843768
\(447\) 2.48661e14 0.659046
\(448\) −3.11022e14 −0.814257
\(449\) 5.36119e14 1.38646 0.693228 0.720719i \(-0.256188\pi\)
0.693228 + 0.720719i \(0.256188\pi\)
\(450\) 2.24466e14 0.573432
\(451\) 2.05358e13 0.0518252
\(452\) 5.47979e13 0.136616
\(453\) −1.94933e14 −0.480114
\(454\) 1.46962e14 0.357601
\(455\) 4.18198e14 1.00535
\(456\) −1.06203e13 −0.0252251
\(457\) −7.62845e14 −1.79018 −0.895091 0.445883i \(-0.852890\pi\)
−0.895091 + 0.445883i \(0.852890\pi\)
\(458\) −8.27662e14 −1.91908
\(459\) −2.51839e11 −0.000576971 0
\(460\) −4.78461e14 −1.08313
\(461\) 2.64452e14 0.591551 0.295775 0.955257i \(-0.404422\pi\)
0.295775 + 0.955257i \(0.404422\pi\)
\(462\) −2.87528e14 −0.635550
\(463\) −6.38847e14 −1.39541 −0.697704 0.716386i \(-0.745795\pi\)
−0.697704 + 0.716386i \(0.745795\pi\)
\(464\) −8.94945e14 −1.93174
\(465\) −7.10870e14 −1.51635
\(466\) 6.70776e14 1.41402
\(467\) 2.37097e14 0.493950 0.246975 0.969022i \(-0.420564\pi\)
0.246975 + 0.969022i \(0.420564\pi\)
\(468\) −5.09747e13 −0.104955
\(469\) −1.46316e15 −2.97743
\(470\) 2.60498e14 0.523921
\(471\) −1.01295e14 −0.201361
\(472\) −2.02135e13 −0.0397155
\(473\) 3.49143e14 0.678059
\(474\) 2.82901e14 0.543068
\(475\) −9.75766e13 −0.185153
\(476\) 2.00224e12 0.00375559
\(477\) −3.01248e14 −0.558565
\(478\) −8.21514e13 −0.150578
\(479\) −2.41888e14 −0.438297 −0.219149 0.975691i \(-0.570328\pi\)
−0.219149 + 0.975691i \(0.570328\pi\)
\(480\) −6.15290e14 −1.10218
\(481\) −2.34781e14 −0.415781
\(482\) −1.10028e15 −1.92640
\(483\) −5.03107e14 −0.870865
\(484\) −3.33691e14 −0.571077
\(485\) 1.53093e15 2.59046
\(486\) −5.10241e13 −0.0853642
\(487\) −2.22716e14 −0.368418 −0.184209 0.982887i \(-0.558972\pi\)
−0.184209 + 0.982887i \(0.558972\pi\)
\(488\) 1.43153e14 0.234148
\(489\) −5.05669e14 −0.817839
\(490\) −2.06818e15 −3.30757
\(491\) −5.53939e14 −0.876019 −0.438010 0.898970i \(-0.644316\pi\)
−0.438010 + 0.898970i \(0.644316\pi\)
\(492\) −2.89734e13 −0.0453097
\(493\) 3.18257e12 0.00492175
\(494\) 5.09089e13 0.0778565
\(495\) −1.69860e14 −0.256900
\(496\) 1.36455e15 2.04099
\(497\) −4.02647e14 −0.595613
\(498\) 9.56378e14 1.39916
\(499\) −3.19570e13 −0.0462395 −0.0231197 0.999733i \(-0.507360\pi\)
−0.0231197 + 0.999733i \(0.507360\pi\)
\(500\) −2.38765e14 −0.341692
\(501\) −3.88023e14 −0.549225
\(502\) −3.00963e14 −0.421350
\(503\) −6.92782e14 −0.959340 −0.479670 0.877449i \(-0.659244\pi\)
−0.479670 + 0.877449i \(0.659244\pi\)
\(504\) −1.20659e14 −0.165270
\(505\) 1.21657e14 0.164829
\(506\) 4.69022e14 0.628588
\(507\) 3.62817e14 0.480999
\(508\) −9.28367e14 −1.21750
\(509\) 1.40246e15 1.81946 0.909729 0.415203i \(-0.136289\pi\)
0.909729 + 0.415203i \(0.136289\pi\)
\(510\) 2.71751e12 0.00348767
\(511\) −1.14783e15 −1.45734
\(512\) 8.95303e14 1.12457
\(513\) 2.21804e13 0.0275629
\(514\) −1.27350e14 −0.156567
\(515\) 2.87823e14 0.350095
\(516\) −4.92596e14 −0.592813
\(517\) −1.11149e14 −0.132346
\(518\) 1.86843e15 2.20121
\(519\) −3.08392e14 −0.359487
\(520\) −1.63604e14 −0.188702
\(521\) −1.57947e15 −1.80262 −0.901312 0.433170i \(-0.857395\pi\)
−0.901312 + 0.433170i \(0.857395\pi\)
\(522\) 6.44807e14 0.728185
\(523\) 4.67432e14 0.522347 0.261173 0.965292i \(-0.415891\pi\)
0.261173 + 0.965292i \(0.415891\pi\)
\(524\) 4.23026e14 0.467783
\(525\) −1.10858e15 −1.21309
\(526\) −1.37034e15 −1.48391
\(527\) −4.85257e12 −0.00520013
\(528\) 3.26057e14 0.345785
\(529\) −1.32131e14 −0.138675
\(530\) 3.25066e15 3.37641
\(531\) 4.22156e13 0.0433963
\(532\) −1.76345e14 −0.179411
\(533\) −4.13092e13 −0.0415956
\(534\) 6.94267e14 0.691910
\(535\) 2.25132e15 2.22070
\(536\) 5.72409e14 0.558856
\(537\) −5.34378e14 −0.516405
\(538\) −4.35401e14 −0.416473
\(539\) 8.82453e14 0.835513
\(540\) 2.39651e14 0.224602
\(541\) −2.81135e14 −0.260813 −0.130406 0.991461i \(-0.541628\pi\)
−0.130406 + 0.991461i \(0.541628\pi\)
\(542\) −1.73110e15 −1.58974
\(543\) −5.61798e14 −0.510718
\(544\) −4.20013e12 −0.00377980
\(545\) −2.33099e15 −2.07663
\(546\) 5.78383e14 0.510101
\(547\) 1.18965e15 1.03870 0.519349 0.854562i \(-0.326174\pi\)
0.519349 + 0.854562i \(0.326174\pi\)
\(548\) 1.37206e15 1.18599
\(549\) −2.98972e14 −0.255849
\(550\) 1.03348e15 0.875601
\(551\) −2.80301e14 −0.235121
\(552\) 1.96822e14 0.163459
\(553\) −1.39718e15 −1.14885
\(554\) 1.42947e15 1.16378
\(555\) 1.10379e15 0.889766
\(556\) −3.37771e14 −0.269594
\(557\) 3.74465e14 0.295943 0.147971 0.988992i \(-0.452726\pi\)
0.147971 + 0.988992i \(0.452726\pi\)
\(558\) −9.83159e14 −0.769371
\(559\) −7.02326e14 −0.544219
\(560\) 3.77404e15 2.89583
\(561\) −1.15951e12 −0.000881005 0
\(562\) −1.02355e15 −0.770123
\(563\) −3.61189e13 −0.0269115 −0.0134558 0.999909i \(-0.504283\pi\)
−0.0134558 + 0.999909i \(0.504283\pi\)
\(564\) 1.56818e14 0.115707
\(565\) −3.67312e14 −0.268392
\(566\) −6.72238e14 −0.486444
\(567\) 2.51995e14 0.180587
\(568\) 1.57521e14 0.111795
\(569\) −1.41723e15 −0.996144 −0.498072 0.867136i \(-0.665959\pi\)
−0.498072 + 0.867136i \(0.665959\pi\)
\(570\) −2.39341e14 −0.166612
\(571\) −5.84698e14 −0.403119 −0.201559 0.979476i \(-0.564601\pi\)
−0.201559 + 0.979476i \(0.564601\pi\)
\(572\) −2.34696e14 −0.160261
\(573\) 6.85534e14 0.463638
\(574\) 3.28745e14 0.220214
\(575\) 1.80834e15 1.19980
\(576\) −2.54118e14 −0.166998
\(577\) −1.82207e15 −1.18604 −0.593020 0.805188i \(-0.702064\pi\)
−0.593020 + 0.805188i \(0.702064\pi\)
\(578\) −2.06385e15 −1.33068
\(579\) 2.18690e13 0.0139668
\(580\) −3.02854e15 −1.91593
\(581\) −4.72331e15 −2.95991
\(582\) 2.11734e15 1.31436
\(583\) −1.38700e15 −0.852900
\(584\) 4.49045e14 0.273539
\(585\) 3.41686e14 0.206191
\(586\) −5.61235e13 −0.0335512
\(587\) 2.12988e15 1.26138 0.630690 0.776035i \(-0.282772\pi\)
0.630690 + 0.776035i \(0.282772\pi\)
\(588\) −1.24503e15 −0.730472
\(589\) 4.27385e14 0.248419
\(590\) −4.55533e14 −0.262321
\(591\) 3.63104e14 0.207157
\(592\) −2.11879e15 −1.19762
\(593\) 3.36610e15 1.88507 0.942533 0.334113i \(-0.108437\pi\)
0.942533 + 0.334113i \(0.108437\pi\)
\(594\) −2.34923e14 −0.130347
\(595\) −1.34211e13 −0.00737810
\(596\) −1.61527e15 −0.879814
\(597\) −8.14212e14 −0.439419
\(598\) −9.43471e14 −0.504512
\(599\) −1.26182e15 −0.668575 −0.334287 0.942471i \(-0.608496\pi\)
−0.334287 + 0.942471i \(0.608496\pi\)
\(600\) 4.33692e14 0.227693
\(601\) 3.33140e15 1.73307 0.866537 0.499114i \(-0.166341\pi\)
0.866537 + 0.499114i \(0.166341\pi\)
\(602\) 5.58923e15 2.88118
\(603\) −1.19547e15 −0.610651
\(604\) 1.26626e15 0.640943
\(605\) 2.23674e15 1.12192
\(606\) 1.68256e14 0.0836315
\(607\) 9.30726e14 0.458442 0.229221 0.973374i \(-0.426382\pi\)
0.229221 + 0.973374i \(0.426382\pi\)
\(608\) 3.69921e14 0.180567
\(609\) −3.18454e15 −1.54046
\(610\) 3.22610e15 1.54655
\(611\) 2.23585e14 0.106222
\(612\) 1.63592e12 0.000770244 0
\(613\) −2.95946e15 −1.38096 −0.690478 0.723353i \(-0.742600\pi\)
−0.690478 + 0.723353i \(0.742600\pi\)
\(614\) −2.75342e15 −1.27335
\(615\) 1.94210e14 0.0890139
\(616\) −5.55535e14 −0.252358
\(617\) 3.82949e15 1.72414 0.862070 0.506789i \(-0.169168\pi\)
0.862070 + 0.506789i \(0.169168\pi\)
\(618\) 3.98070e14 0.177633
\(619\) −2.08118e15 −0.920474 −0.460237 0.887796i \(-0.652236\pi\)
−0.460237 + 0.887796i \(0.652236\pi\)
\(620\) 4.61772e15 2.02429
\(621\) −4.11060e14 −0.178608
\(622\) 2.36578e15 1.01889
\(623\) −3.42882e15 −1.46373
\(624\) −6.55885e14 −0.277532
\(625\) −1.48178e15 −0.621502
\(626\) −2.10378e15 −0.874665
\(627\) 1.02122e14 0.0420872
\(628\) 6.58003e14 0.268813
\(629\) 7.53476e12 0.00305134
\(630\) −2.71919e15 −1.09161
\(631\) −4.91710e15 −1.95681 −0.978403 0.206708i \(-0.933725\pi\)
−0.978403 + 0.206708i \(0.933725\pi\)
\(632\) 5.46596e14 0.215636
\(633\) 4.87317e14 0.190586
\(634\) −5.57861e15 −2.16289
\(635\) 6.22289e15 2.39186
\(636\) 1.95687e15 0.745673
\(637\) −1.77511e15 −0.670594
\(638\) 2.96879e15 1.11190
\(639\) −3.28980e14 −0.122156
\(640\) −2.44356e15 −0.899565
\(641\) −4.55696e15 −1.66324 −0.831621 0.555343i \(-0.812587\pi\)
−0.831621 + 0.555343i \(0.812587\pi\)
\(642\) 3.11365e15 1.12675
\(643\) 6.14757e14 0.220568 0.110284 0.993900i \(-0.464824\pi\)
0.110284 + 0.993900i \(0.464824\pi\)
\(644\) 3.26812e15 1.16259
\(645\) 3.30189e15 1.16462
\(646\) −1.63380e12 −0.000571375 0
\(647\) 2.37438e15 0.823337 0.411668 0.911334i \(-0.364946\pi\)
0.411668 + 0.911334i \(0.364946\pi\)
\(648\) −9.85839e13 −0.0338956
\(649\) 1.94367e14 0.0662640
\(650\) −2.07891e15 −0.702769
\(651\) 4.85558e15 1.62759
\(652\) 3.28477e15 1.09180
\(653\) −3.85797e15 −1.27156 −0.635779 0.771871i \(-0.719321\pi\)
−0.635779 + 0.771871i \(0.719321\pi\)
\(654\) −3.22385e15 −1.05365
\(655\) −2.83556e15 −0.918992
\(656\) −3.72797e14 −0.119812
\(657\) −9.37825e14 −0.298891
\(658\) −1.77933e15 −0.562358
\(659\) −3.32746e15 −1.04290 −0.521450 0.853282i \(-0.674609\pi\)
−0.521450 + 0.853282i \(0.674609\pi\)
\(660\) 1.10339e15 0.342956
\(661\) 4.49012e15 1.38404 0.692022 0.721876i \(-0.256720\pi\)
0.692022 + 0.721876i \(0.256720\pi\)
\(662\) 3.37785e15 1.03257
\(663\) 2.33243e12 0.000707106 0
\(664\) 1.84782e15 0.555567
\(665\) 1.18205e15 0.352465
\(666\) 1.52659e15 0.451453
\(667\) 5.19469e15 1.52359
\(668\) 2.52055e15 0.733205
\(669\) 1.26890e15 0.366086
\(670\) 1.28999e16 3.69125
\(671\) −1.37652e15 −0.390668
\(672\) 4.20273e15 1.18304
\(673\) −2.57597e15 −0.719213 −0.359606 0.933104i \(-0.617089\pi\)
−0.359606 + 0.933104i \(0.617089\pi\)
\(674\) −5.62806e15 −1.55858
\(675\) −9.05760e14 −0.248795
\(676\) −2.35682e15 −0.642124
\(677\) −3.48738e15 −0.942457 −0.471229 0.882011i \(-0.656189\pi\)
−0.471229 + 0.882011i \(0.656189\pi\)
\(678\) −5.08007e14 −0.136178
\(679\) −1.04570e16 −2.78050
\(680\) 5.25051e12 0.00138485
\(681\) −5.93020e14 −0.155153
\(682\) −4.52662e15 −1.17479
\(683\) 7.56829e15 1.94843 0.974213 0.225629i \(-0.0724439\pi\)
0.974213 + 0.225629i \(0.0724439\pi\)
\(684\) −1.44081e14 −0.0367959
\(685\) −9.19696e15 −2.32995
\(686\) 5.52089e15 1.38748
\(687\) 3.33976e15 0.832633
\(688\) −6.33817e15 −1.56757
\(689\) 2.79004e15 0.684549
\(690\) 4.43560e15 1.07965
\(691\) −2.98790e15 −0.721502 −0.360751 0.932662i \(-0.617480\pi\)
−0.360751 + 0.932662i \(0.617480\pi\)
\(692\) 2.00328e15 0.479908
\(693\) 1.16023e15 0.275747
\(694\) 6.65027e15 1.56806
\(695\) 2.26409e15 0.529637
\(696\) 1.24583e15 0.289141
\(697\) 1.32572e12 0.000305262 0
\(698\) −1.10982e16 −2.53540
\(699\) −2.70670e15 −0.613501
\(700\) 7.20121e15 1.61945
\(701\) −5.12968e15 −1.14457 −0.572283 0.820056i \(-0.693942\pi\)
−0.572283 + 0.820056i \(0.693942\pi\)
\(702\) 4.72564e14 0.104618
\(703\) −6.63615e14 −0.145768
\(704\) −1.17000e15 −0.254998
\(705\) −1.05115e15 −0.227314
\(706\) −5.62402e14 −0.120676
\(707\) −8.30972e14 −0.176921
\(708\) −2.74227e14 −0.0579332
\(709\) 2.21160e15 0.463609 0.231805 0.972762i \(-0.425537\pi\)
0.231805 + 0.972762i \(0.425537\pi\)
\(710\) 3.54991e15 0.738407
\(711\) −1.14156e15 −0.235622
\(712\) 1.34140e15 0.274737
\(713\) −7.92053e15 −1.60976
\(714\) −1.85619e13 −0.00374353
\(715\) 1.57318e15 0.314843
\(716\) 3.47126e15 0.689390
\(717\) 3.31495e14 0.0653314
\(718\) 4.61787e15 0.903144
\(719\) 4.96793e15 0.964198 0.482099 0.876117i \(-0.339874\pi\)
0.482099 + 0.876117i \(0.339874\pi\)
\(720\) 3.08356e15 0.593914
\(721\) −1.96597e15 −0.375779
\(722\) −6.87120e15 −1.30340
\(723\) 4.43984e15 0.835809
\(724\) 3.64937e15 0.681799
\(725\) 1.14464e16 2.12231
\(726\) 3.09350e15 0.569245
\(727\) 5.58242e15 1.01949 0.509745 0.860325i \(-0.329740\pi\)
0.509745 + 0.860325i \(0.329740\pi\)
\(728\) 1.11750e15 0.202546
\(729\) 2.05891e14 0.0370370
\(730\) 1.01197e16 1.80673
\(731\) 2.25396e13 0.00399392
\(732\) 1.94209e15 0.341553
\(733\) −3.22472e15 −0.562886 −0.281443 0.959578i \(-0.590813\pi\)
−0.281443 + 0.959578i \(0.590813\pi\)
\(734\) −6.97286e15 −1.20804
\(735\) 8.34547e15 1.43506
\(736\) −6.85558e15 −1.17008
\(737\) −5.50413e15 −0.932433
\(738\) 2.68599e14 0.0451643
\(739\) −4.12810e15 −0.688979 −0.344489 0.938790i \(-0.611948\pi\)
−0.344489 + 0.938790i \(0.611948\pi\)
\(740\) −7.17010e15 −1.18782
\(741\) −2.05426e14 −0.0337797
\(742\) −2.22036e16 −3.62411
\(743\) −2.39676e15 −0.388317 −0.194158 0.980970i \(-0.562198\pi\)
−0.194158 + 0.980970i \(0.562198\pi\)
\(744\) −1.89957e15 −0.305495
\(745\) 1.08272e16 1.72845
\(746\) 2.50052e15 0.396247
\(747\) −3.85915e15 −0.607056
\(748\) 7.53203e12 0.00117612
\(749\) −1.53776e16 −2.38362
\(750\) 2.21348e15 0.340596
\(751\) −4.56496e15 −0.697296 −0.348648 0.937254i \(-0.613359\pi\)
−0.348648 + 0.937254i \(0.613359\pi\)
\(752\) 2.01775e15 0.305963
\(753\) 1.21444e15 0.182812
\(754\) −5.97193e15 −0.892427
\(755\) −8.48778e15 −1.25917
\(756\) −1.63693e15 −0.241080
\(757\) 1.25124e16 1.82941 0.914706 0.404119i \(-0.132422\pi\)
0.914706 + 0.404119i \(0.132422\pi\)
\(758\) 1.05219e16 1.52726
\(759\) −1.89259e15 −0.272726
\(760\) −4.62433e14 −0.0661567
\(761\) 3.98552e15 0.566070 0.283035 0.959110i \(-0.408659\pi\)
0.283035 + 0.959110i \(0.408659\pi\)
\(762\) 8.60648e15 1.21359
\(763\) 1.59218e16 2.22898
\(764\) −4.45315e15 −0.618947
\(765\) −1.09656e13 −0.00151320
\(766\) 1.14614e16 1.57029
\(767\) −3.90983e14 −0.0531843
\(768\) −5.52123e15 −0.745675
\(769\) −6.39979e15 −0.858165 −0.429082 0.903265i \(-0.641163\pi\)
−0.429082 + 0.903265i \(0.641163\pi\)
\(770\) −1.25196e16 −1.66683
\(771\) 5.13879e14 0.0679300
\(772\) −1.42058e14 −0.0186454
\(773\) −3.99226e14 −0.0520274 −0.0260137 0.999662i \(-0.508281\pi\)
−0.0260137 + 0.999662i \(0.508281\pi\)
\(774\) 4.56664e15 0.590910
\(775\) −1.74527e16 −2.24235
\(776\) 4.09092e15 0.521893
\(777\) −7.53943e15 −0.955042
\(778\) 5.04213e15 0.634200
\(779\) −1.16762e14 −0.0145829
\(780\) −2.21955e15 −0.275261
\(781\) −1.51468e15 −0.186526
\(782\) 3.02785e13 0.00370252
\(783\) −2.60191e15 −0.315938
\(784\) −1.60196e16 −1.93158
\(785\) −4.41062e15 −0.528101
\(786\) −3.92168e15 −0.466282
\(787\) 9.03929e15 1.06727 0.533634 0.845716i \(-0.320826\pi\)
0.533634 + 0.845716i \(0.320826\pi\)
\(788\) −2.35868e15 −0.276551
\(789\) 5.52957e15 0.643825
\(790\) 1.23181e16 1.42428
\(791\) 2.50892e15 0.288082
\(792\) −4.53896e14 −0.0517569
\(793\) 2.76896e15 0.313555
\(794\) 5.45912e15 0.613917
\(795\) −1.31170e16 −1.46492
\(796\) 5.28902e15 0.586616
\(797\) −1.25066e15 −0.137759 −0.0688793 0.997625i \(-0.521942\pi\)
−0.0688793 + 0.997625i \(0.521942\pi\)
\(798\) 1.63482e15 0.178835
\(799\) −7.17545e12 −0.000779546 0
\(800\) −1.51061e16 −1.62989
\(801\) −2.80149e15 −0.300200
\(802\) 1.15198e16 1.22599
\(803\) −4.31790e15 −0.456391
\(804\) 7.76562e15 0.815207
\(805\) −2.19064e16 −2.28398
\(806\) 9.10562e15 0.942902
\(807\) 1.75692e15 0.180696
\(808\) 3.25088e14 0.0332076
\(809\) 1.61191e16 1.63540 0.817699 0.575645i \(-0.195249\pi\)
0.817699 + 0.575645i \(0.195249\pi\)
\(810\) −2.22170e15 −0.223881
\(811\) 1.28888e16 1.29003 0.645013 0.764171i \(-0.276852\pi\)
0.645013 + 0.764171i \(0.276852\pi\)
\(812\) 2.06864e16 2.05649
\(813\) 6.98530e15 0.689742
\(814\) 7.02865e15 0.689346
\(815\) −2.20179e16 −2.14491
\(816\) 2.10491e13 0.00203675
\(817\) −1.98514e15 −0.190797
\(818\) 1.85113e16 1.76723
\(819\) −2.33388e15 −0.221318
\(820\) −1.26156e15 −0.118832
\(821\) −4.50064e14 −0.0421101 −0.0210551 0.999778i \(-0.506703\pi\)
−0.0210551 + 0.999778i \(0.506703\pi\)
\(822\) −1.27197e16 −1.18218
\(823\) 4.42512e15 0.408532 0.204266 0.978915i \(-0.434519\pi\)
0.204266 + 0.978915i \(0.434519\pi\)
\(824\) 7.69113e14 0.0705327
\(825\) −4.17027e15 −0.379898
\(826\) 3.11151e15 0.281566
\(827\) −1.31229e16 −1.17964 −0.589821 0.807534i \(-0.700802\pi\)
−0.589821 + 0.807534i \(0.700802\pi\)
\(828\) 2.67020e15 0.238439
\(829\) 1.94702e16 1.72711 0.863556 0.504253i \(-0.168232\pi\)
0.863556 + 0.504253i \(0.168232\pi\)
\(830\) 4.16428e16 3.66953
\(831\) −5.76816e15 −0.504931
\(832\) 2.35354e15 0.204665
\(833\) 5.69683e13 0.00492136
\(834\) 3.13133e15 0.268729
\(835\) −1.68954e16 −1.44043
\(836\) −6.63375e14 −0.0561855
\(837\) 3.96722e15 0.333808
\(838\) 5.31173e15 0.444011
\(839\) −5.38401e15 −0.447111 −0.223555 0.974691i \(-0.571766\pi\)
−0.223555 + 0.974691i \(0.571766\pi\)
\(840\) −5.25377e15 −0.433446
\(841\) 2.06806e16 1.69506
\(842\) −4.61803e15 −0.376046
\(843\) 4.13021e15 0.334134
\(844\) −3.16555e15 −0.254429
\(845\) 1.57978e16 1.26150
\(846\) −1.45379e15 −0.115336
\(847\) −1.52780e16 −1.20423
\(848\) 2.51788e16 1.97178
\(849\) 2.71260e15 0.211054
\(850\) 6.67180e13 0.00515749
\(851\) 1.22985e16 0.944580
\(852\) 2.13701e15 0.163076
\(853\) 2.47846e16 1.87916 0.939578 0.342335i \(-0.111218\pi\)
0.939578 + 0.342335i \(0.111218\pi\)
\(854\) −2.20358e16 −1.66001
\(855\) 9.65784e14 0.0722881
\(856\) 6.01591e15 0.447400
\(857\) −4.78471e14 −0.0353558 −0.0176779 0.999844i \(-0.505627\pi\)
−0.0176779 + 0.999844i \(0.505627\pi\)
\(858\) 2.17576e15 0.159746
\(859\) −1.55690e16 −1.13579 −0.567897 0.823100i \(-0.692243\pi\)
−0.567897 + 0.823100i \(0.692243\pi\)
\(860\) −2.14487e16 −1.55475
\(861\) −1.32655e15 −0.0955443
\(862\) 2.65462e16 1.89982
\(863\) 9.85390e15 0.700727 0.350364 0.936614i \(-0.386058\pi\)
0.350364 + 0.936614i \(0.386058\pi\)
\(864\) 3.43381e15 0.242634
\(865\) −1.34281e16 −0.942811
\(866\) −1.22467e16 −0.854416
\(867\) 8.32800e15 0.577345
\(868\) −3.15413e16 −2.17280
\(869\) −5.25592e15 −0.359782
\(870\) 2.80763e16 1.90978
\(871\) 1.10719e16 0.748383
\(872\) −6.22881e15 −0.418374
\(873\) −8.54382e15 −0.570261
\(874\) −2.66675e15 −0.176876
\(875\) −1.09318e16 −0.720525
\(876\) 6.09200e15 0.399013
\(877\) −5.55455e15 −0.361536 −0.180768 0.983526i \(-0.557858\pi\)
−0.180768 + 0.983526i \(0.557858\pi\)
\(878\) 1.63282e16 1.05613
\(879\) 2.26468e14 0.0145569
\(880\) 1.41972e16 0.906876
\(881\) 6.25867e15 0.397297 0.198648 0.980071i \(-0.436345\pi\)
0.198648 + 0.980071i \(0.436345\pi\)
\(882\) 1.15421e16 0.728127
\(883\) −7.69702e15 −0.482546 −0.241273 0.970457i \(-0.577565\pi\)
−0.241273 + 0.970457i \(0.577565\pi\)
\(884\) −1.51512e13 −0.000943973 0
\(885\) 1.83816e15 0.113814
\(886\) −1.96617e16 −1.20986
\(887\) −1.58445e16 −0.968941 −0.484471 0.874807i \(-0.660988\pi\)
−0.484471 + 0.874807i \(0.660988\pi\)
\(888\) 2.94952e15 0.179259
\(889\) −4.25053e16 −2.56734
\(890\) 3.02299e16 1.81464
\(891\) 9.47956e14 0.0565537
\(892\) −8.24261e15 −0.488718
\(893\) 6.31969e14 0.0372403
\(894\) 1.49745e16 0.876990
\(895\) −2.32680e16 −1.35435
\(896\) 1.66907e16 0.965560
\(897\) 3.80707e15 0.218893
\(898\) 3.22853e16 1.84495
\(899\) −5.01350e16 −2.84749
\(900\) 5.88371e15 0.332137
\(901\) −8.95399e13 −0.00502378
\(902\) 1.23667e15 0.0689635
\(903\) −2.25535e16 −1.25006
\(904\) −9.81523e14 −0.0540722
\(905\) −2.44619e16 −1.33944
\(906\) −1.17389e16 −0.638886
\(907\) −2.40161e16 −1.29916 −0.649580 0.760293i \(-0.725055\pi\)
−0.649580 + 0.760293i \(0.725055\pi\)
\(908\) 3.85218e15 0.207126
\(909\) −6.78941e14 −0.0362853
\(910\) 2.51840e16 1.33782
\(911\) 1.15571e16 0.610235 0.305118 0.952315i \(-0.401304\pi\)
0.305118 + 0.952315i \(0.401304\pi\)
\(912\) −1.85388e15 −0.0972993
\(913\) −1.77682e16 −0.926944
\(914\) −4.59389e16 −2.38219
\(915\) −1.30179e16 −0.671004
\(916\) −2.16947e16 −1.11155
\(917\) 1.93682e16 0.986413
\(918\) −1.51659e13 −0.000767772 0
\(919\) −1.41751e16 −0.713331 −0.356665 0.934232i \(-0.616086\pi\)
−0.356665 + 0.934232i \(0.616086\pi\)
\(920\) 8.57006e15 0.428697
\(921\) 1.11105e16 0.552468
\(922\) 1.59254e16 0.787174
\(923\) 3.04688e15 0.149708
\(924\) −7.53669e15 −0.368116
\(925\) 2.70994e16 1.31577
\(926\) −3.84716e16 −1.85686
\(927\) −1.60628e15 −0.0770696
\(928\) −4.33941e16 −2.06975
\(929\) −1.41176e16 −0.669381 −0.334690 0.942328i \(-0.608632\pi\)
−0.334690 + 0.942328i \(0.608632\pi\)
\(930\) −4.28089e16 −2.01780
\(931\) −5.01741e15 −0.235102
\(932\) 1.75824e16 0.819012
\(933\) −9.54634e15 −0.442067
\(934\) 1.42781e16 0.657297
\(935\) −5.04875e13 −0.00231058
\(936\) 9.13044e14 0.0415408
\(937\) 2.30681e16 1.04338 0.521691 0.853135i \(-0.325301\pi\)
0.521691 + 0.853135i \(0.325301\pi\)
\(938\) −8.81123e16 −3.96206
\(939\) 8.48913e15 0.379492
\(940\) 6.82817e15 0.303460
\(941\) 2.13136e16 0.941704 0.470852 0.882212i \(-0.343947\pi\)
0.470852 + 0.882212i \(0.343947\pi\)
\(942\) −6.10005e15 −0.267950
\(943\) 2.16389e15 0.0944976
\(944\) −3.52845e15 −0.153192
\(945\) 1.09724e16 0.473617
\(946\) 2.10256e16 0.902290
\(947\) 6.27878e15 0.267886 0.133943 0.990989i \(-0.457236\pi\)
0.133943 + 0.990989i \(0.457236\pi\)
\(948\) 7.41542e15 0.314550
\(949\) 8.68575e15 0.366305
\(950\) −5.87611e15 −0.246382
\(951\) 2.25107e16 0.938416
\(952\) −3.58635e13 −0.00148645
\(953\) −4.84286e15 −0.199568 −0.0997839 0.995009i \(-0.531815\pi\)
−0.0997839 + 0.995009i \(0.531815\pi\)
\(954\) −1.81413e16 −0.743280
\(955\) 2.98496e16 1.21596
\(956\) −2.15335e15 −0.0872162
\(957\) −1.19796e16 −0.482422
\(958\) −1.45666e16 −0.583241
\(959\) 6.28197e16 2.50088
\(960\) −1.10649e16 −0.437980
\(961\) 5.10340e16 2.00854
\(962\) −1.41386e16 −0.553278
\(963\) −1.25641e16 −0.488864
\(964\) −2.88407e16 −1.11579
\(965\) 9.52224e14 0.0366301
\(966\) −3.02973e16 −1.15886
\(967\) 5.51501e15 0.209749 0.104875 0.994485i \(-0.466556\pi\)
0.104875 + 0.994485i \(0.466556\pi\)
\(968\) 5.97697e15 0.226030
\(969\) 6.59269e12 0.000247903 0
\(970\) 9.21934e16 3.44711
\(971\) −2.93418e16 −1.09089 −0.545444 0.838147i \(-0.683639\pi\)
−0.545444 + 0.838147i \(0.683639\pi\)
\(972\) −1.33744e15 −0.0494437
\(973\) −1.54648e16 −0.568493
\(974\) −1.34120e16 −0.490253
\(975\) 8.38878e15 0.304911
\(976\) 2.49886e16 0.903166
\(977\) −3.77338e16 −1.35616 −0.678080 0.734988i \(-0.737188\pi\)
−0.678080 + 0.734988i \(0.737188\pi\)
\(978\) −3.04516e16 −1.08829
\(979\) −1.28985e16 −0.458390
\(980\) −5.42111e16 −1.91578
\(981\) 1.30088e16 0.457148
\(982\) −3.33584e16 −1.16571
\(983\) −1.48891e16 −0.517399 −0.258699 0.965958i \(-0.583294\pi\)
−0.258699 + 0.965958i \(0.583294\pi\)
\(984\) 5.18962e14 0.0179334
\(985\) 1.58103e16 0.543303
\(986\) 1.91656e14 0.00654936
\(987\) 7.17989e15 0.243991
\(988\) 1.33442e15 0.0450952
\(989\) 3.67898e16 1.23637
\(990\) −1.02291e16 −0.341855
\(991\) 5.16898e16 1.71791 0.858954 0.512053i \(-0.171115\pi\)
0.858954 + 0.512053i \(0.171115\pi\)
\(992\) 6.61645e16 2.18681
\(993\) −1.36302e16 −0.448004
\(994\) −2.42476e16 −0.792579
\(995\) −3.54525e16 −1.15245
\(996\) 2.50686e16 0.810408
\(997\) −3.77168e16 −1.21258 −0.606291 0.795243i \(-0.707344\pi\)
−0.606291 + 0.795243i \(0.707344\pi\)
\(998\) −1.92446e15 −0.0615307
\(999\) −6.16004e15 −0.195872
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.a.1.22 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.a.1.22 26 1.1 even 1 trivial