Properties

Label 177.12.a.a.1.16
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 1.16
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+16.4601 q^{2} -243.000 q^{3} -1777.06 q^{4} -11170.6 q^{5} -3999.81 q^{6} -64730.0 q^{7} -62961.0 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+16.4601 q^{2} -243.000 q^{3} -1777.06 q^{4} -11170.6 q^{5} -3999.81 q^{6} -64730.0 q^{7} -62961.0 q^{8} +59049.0 q^{9} -183870. q^{10} -152927. q^{11} +431827. q^{12} +2.10945e6 q^{13} -1.06546e6 q^{14} +2.71446e6 q^{15} +2.60308e6 q^{16} -1.11407e7 q^{17} +971954. q^{18} -1.27705e7 q^{19} +1.98509e7 q^{20} +1.57294e7 q^{21} -2.51720e6 q^{22} +9.67947e6 q^{23} +1.52995e7 q^{24} +7.59546e7 q^{25} +3.47218e7 q^{26} -1.43489e7 q^{27} +1.15029e8 q^{28} +1.78369e8 q^{29} +4.46803e7 q^{30} +8.93963e7 q^{31} +1.71791e8 q^{32} +3.71614e7 q^{33} -1.83377e8 q^{34} +7.23074e8 q^{35} -1.04934e8 q^{36} -2.78055e8 q^{37} -2.10204e8 q^{38} -5.12596e8 q^{39} +7.03314e8 q^{40} +9.14877e8 q^{41} +2.58908e8 q^{42} -8.43912e7 q^{43} +2.71762e8 q^{44} -6.59614e8 q^{45} +1.59325e8 q^{46} -1.29951e9 q^{47} -6.32549e8 q^{48} +2.21265e9 q^{49} +1.25022e9 q^{50} +2.70719e9 q^{51} -3.74862e9 q^{52} +3.97563e9 q^{53} -2.36185e8 q^{54} +1.70829e9 q^{55} +4.07547e9 q^{56} +3.10324e9 q^{57} +2.93597e9 q^{58} +7.14924e8 q^{59} -4.82377e9 q^{60} -6.12831e9 q^{61} +1.47147e9 q^{62} -3.82224e9 q^{63} -2.50341e9 q^{64} -2.35638e10 q^{65} +6.11680e8 q^{66} +5.63319e9 q^{67} +1.97978e10 q^{68} -2.35211e9 q^{69} +1.19019e10 q^{70} -1.19880e10 q^{71} -3.71779e9 q^{72} -2.52775e10 q^{73} -4.57682e9 q^{74} -1.84570e10 q^{75} +2.26940e10 q^{76} +9.89899e9 q^{77} -8.43739e9 q^{78} +4.83860e10 q^{79} -2.90780e10 q^{80} +3.48678e9 q^{81} +1.50590e10 q^{82} -6.79955e10 q^{83} -2.79521e10 q^{84} +1.24449e11 q^{85} -1.38909e9 q^{86} -4.33436e10 q^{87} +9.62846e9 q^{88} +4.73407e10 q^{89} -1.08573e10 q^{90} -1.36545e11 q^{91} -1.72010e10 q^{92} -2.17233e10 q^{93} -2.13900e10 q^{94} +1.42655e11 q^{95} -4.17453e10 q^{96} -1.10243e11 q^{97} +3.64205e10 q^{98} -9.03021e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} + O(q^{10}) \) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} - 859751q^{10} + 579094q^{11} - 5606010q^{12} - 2018538q^{13} + 4157413q^{14} - 925344q^{15} + 20190274q^{16} - 13084493q^{17} - 4605822q^{18} + 9917231q^{19} + 10165633q^{20} + 24013017q^{21} - 89820518q^{22} - 63513223q^{23} + 28587735q^{24} + 218986852q^{25} - 77999532q^{26} - 373071582q^{27} - 444601862q^{28} + 81530981q^{29} + 208919493q^{30} - 408861231q^{31} - 26253128q^{32} - 140719842q^{33} - 508910076q^{34} - 75731421q^{35} + 1362260430q^{36} - 802381301q^{37} + 732704675q^{38} + 490504734q^{39} - 646130800q^{40} - 1354472849q^{41} - 1010251359q^{42} + 282952194q^{43} + 1846047996q^{44} + 224858592q^{45} + 9629305849q^{46} - 1196794197q^{47} - 4906236582q^{48} + 10889725683q^{49} - 6236232091q^{50} + 3179531799q^{51} - 1968200812q^{52} - 8276044236q^{53} + 1119214746q^{54} - 6672895076q^{55} + 2579741342q^{56} - 2409887133q^{57} - 9401656060q^{58} + 18588031774q^{59} - 2470248819q^{60} - 21181559029q^{61} - 6117706514q^{62} - 5835163131q^{63} + 42975855037q^{64} + 25680681860q^{65} + 21826385874q^{66} + 26234163394q^{67} + 19707344091q^{68} + 15433713189q^{69} + 129203099090q^{70} + 52088830406q^{71} - 6946819605q^{72} + 20943384867q^{73} + 41969200146q^{74} - 53213805036q^{75} + 223987219368q^{76} + 94604773153q^{77} + 18953886276q^{78} + 68965662774q^{79} + 218947784293q^{80} + 90656394426q^{81} + 11938614923q^{82} + 17947446393q^{83} + 108038252466q^{84} - 52849386709q^{85} + 384986147852q^{86} - 19812028383q^{87} - 49061112607q^{88} + 38570593981q^{89} - 50767436799q^{90} - 226268806999q^{91} - 79559686310q^{92} + 99353279133q^{93} - 16709400108q^{94} - 252795831501q^{95} + 6379510104q^{96} - 186894587836q^{97} - 252443311612q^{98} + 34194921606q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.4601 0.363721 0.181860 0.983324i \(-0.441788\pi\)
0.181860 + 0.983324i \(0.441788\pi\)
\(3\) −243.000 −0.577350
\(4\) −1777.06 −0.867707
\(5\) −11170.6 −1.59861 −0.799304 0.600926i \(-0.794799\pi\)
−0.799304 + 0.600926i \(0.794799\pi\)
\(6\) −3999.81 −0.209994
\(7\) −64730.0 −1.45568 −0.727841 0.685746i \(-0.759476\pi\)
−0.727841 + 0.685746i \(0.759476\pi\)
\(8\) −62961.0 −0.679324
\(9\) 59049.0 0.333333
\(10\) −183870. −0.581447
\(11\) −152927. −0.286303 −0.143151 0.989701i \(-0.545724\pi\)
−0.143151 + 0.989701i \(0.545724\pi\)
\(12\) 431827. 0.500971
\(13\) 2.10945e6 1.57572 0.787862 0.615852i \(-0.211188\pi\)
0.787862 + 0.615852i \(0.211188\pi\)
\(14\) −1.06546e6 −0.529462
\(15\) 2.71446e6 0.922957
\(16\) 2.60308e6 0.620623
\(17\) −1.11407e7 −1.90302 −0.951511 0.307614i \(-0.900469\pi\)
−0.951511 + 0.307614i \(0.900469\pi\)
\(18\) 971954. 0.121240
\(19\) −1.27705e7 −1.18322 −0.591608 0.806226i \(-0.701507\pi\)
−0.591608 + 0.806226i \(0.701507\pi\)
\(20\) 1.98509e7 1.38712
\(21\) 1.57294e7 0.840438
\(22\) −2.51720e6 −0.104134
\(23\) 9.67947e6 0.313580 0.156790 0.987632i \(-0.449885\pi\)
0.156790 + 0.987632i \(0.449885\pi\)
\(24\) 1.52995e7 0.392208
\(25\) 7.59546e7 1.55555
\(26\) 3.47218e7 0.573124
\(27\) −1.43489e7 −0.192450
\(28\) 1.15029e8 1.26311
\(29\) 1.78369e8 1.61484 0.807421 0.589976i \(-0.200863\pi\)
0.807421 + 0.589976i \(0.200863\pi\)
\(30\) 4.46803e7 0.335699
\(31\) 8.93963e7 0.560829 0.280414 0.959879i \(-0.409528\pi\)
0.280414 + 0.959879i \(0.409528\pi\)
\(32\) 1.71791e8 0.905057
\(33\) 3.71614e7 0.165297
\(34\) −1.83377e8 −0.692169
\(35\) 7.23074e8 2.32707
\(36\) −1.04934e8 −0.289236
\(37\) −2.78055e8 −0.659206 −0.329603 0.944120i \(-0.606915\pi\)
−0.329603 + 0.944120i \(0.606915\pi\)
\(38\) −2.10204e8 −0.430360
\(39\) −5.12596e8 −0.909745
\(40\) 7.03314e8 1.08597
\(41\) 9.14877e8 1.23325 0.616625 0.787257i \(-0.288499\pi\)
0.616625 + 0.787257i \(0.288499\pi\)
\(42\) 2.58908e8 0.305685
\(43\) −8.43912e7 −0.0875428 −0.0437714 0.999042i \(-0.513937\pi\)
−0.0437714 + 0.999042i \(0.513937\pi\)
\(44\) 2.71762e8 0.248427
\(45\) −6.59614e8 −0.532870
\(46\) 1.59325e8 0.114056
\(47\) −1.29951e9 −0.826495 −0.413247 0.910619i \(-0.635606\pi\)
−0.413247 + 0.910619i \(0.635606\pi\)
\(48\) −6.32549e8 −0.358317
\(49\) 2.21265e9 1.11901
\(50\) 1.25022e9 0.565786
\(51\) 2.70719e9 1.09871
\(52\) −3.74862e9 −1.36727
\(53\) 3.97563e9 1.30584 0.652918 0.757429i \(-0.273545\pi\)
0.652918 + 0.757429i \(0.273545\pi\)
\(54\) −2.36185e8 −0.0699981
\(55\) 1.70829e9 0.457686
\(56\) 4.07547e9 0.988879
\(57\) 3.10324e9 0.683130
\(58\) 2.93597e9 0.587351
\(59\) 7.14924e8 0.130189
\(60\) −4.82377e9 −0.800857
\(61\) −6.12831e9 −0.929023 −0.464512 0.885567i \(-0.653770\pi\)
−0.464512 + 0.885567i \(0.653770\pi\)
\(62\) 1.47147e9 0.203985
\(63\) −3.82224e9 −0.485227
\(64\) −2.50341e9 −0.291435
\(65\) −2.35638e10 −2.51897
\(66\) 6.11680e8 0.0601219
\(67\) 5.63319e9 0.509733 0.254867 0.966976i \(-0.417968\pi\)
0.254867 + 0.966976i \(0.417968\pi\)
\(68\) 1.97978e10 1.65127
\(69\) −2.35211e9 −0.181045
\(70\) 1.19019e10 0.846402
\(71\) −1.19880e10 −0.788543 −0.394272 0.918994i \(-0.629003\pi\)
−0.394272 + 0.918994i \(0.629003\pi\)
\(72\) −3.71779e9 −0.226441
\(73\) −2.52775e10 −1.42711 −0.713555 0.700599i \(-0.752916\pi\)
−0.713555 + 0.700599i \(0.752916\pi\)
\(74\) −4.57682e9 −0.239767
\(75\) −1.84570e10 −0.898098
\(76\) 2.26940e10 1.02668
\(77\) 9.89899e9 0.416766
\(78\) −8.43739e9 −0.330893
\(79\) 4.83860e10 1.76918 0.884588 0.466373i \(-0.154440\pi\)
0.884588 + 0.466373i \(0.154440\pi\)
\(80\) −2.90780e10 −0.992134
\(81\) 3.48678e9 0.111111
\(82\) 1.50590e10 0.448559
\(83\) −6.79955e10 −1.89474 −0.947372 0.320135i \(-0.896272\pi\)
−0.947372 + 0.320135i \(0.896272\pi\)
\(84\) −2.79521e10 −0.729255
\(85\) 1.24449e11 3.04219
\(86\) −1.38909e9 −0.0318411
\(87\) −4.33436e10 −0.932329
\(88\) 9.62846e9 0.194492
\(89\) 4.73407e10 0.898648 0.449324 0.893369i \(-0.351665\pi\)
0.449324 + 0.893369i \(0.351665\pi\)
\(90\) −1.08573e10 −0.193816
\(91\) −1.36545e11 −2.29375
\(92\) −1.72010e10 −0.272096
\(93\) −2.17233e10 −0.323795
\(94\) −2.13900e10 −0.300613
\(95\) 1.42655e11 1.89150
\(96\) −4.17453e10 −0.522535
\(97\) −1.10243e11 −1.30348 −0.651740 0.758442i \(-0.725961\pi\)
−0.651740 + 0.758442i \(0.725961\pi\)
\(98\) 3.64205e10 0.407007
\(99\) −9.03021e9 −0.0954343
\(100\) −1.34976e11 −1.34976
\(101\) 3.27761e10 0.310306 0.155153 0.987890i \(-0.450413\pi\)
0.155153 + 0.987890i \(0.450413\pi\)
\(102\) 4.45607e10 0.399624
\(103\) −5.94861e10 −0.505604 −0.252802 0.967518i \(-0.581352\pi\)
−0.252802 + 0.967518i \(0.581352\pi\)
\(104\) −1.32813e11 −1.07043
\(105\) −1.75707e11 −1.34353
\(106\) 6.54393e10 0.474960
\(107\) −1.17653e11 −0.810950 −0.405475 0.914106i \(-0.632894\pi\)
−0.405475 + 0.914106i \(0.632894\pi\)
\(108\) 2.54989e10 0.166990
\(109\) 1.58422e11 0.986208 0.493104 0.869970i \(-0.335862\pi\)
0.493104 + 0.869970i \(0.335862\pi\)
\(110\) 2.81187e10 0.166470
\(111\) 6.75673e10 0.380593
\(112\) −1.68498e11 −0.903430
\(113\) 3.71456e11 1.89660 0.948301 0.317373i \(-0.102801\pi\)
0.948301 + 0.317373i \(0.102801\pi\)
\(114\) 5.10796e10 0.248468
\(115\) −1.08126e11 −0.501292
\(116\) −3.16973e11 −1.40121
\(117\) 1.24561e11 0.525241
\(118\) 1.17677e10 0.0473524
\(119\) 7.21139e11 2.77020
\(120\) −1.70905e11 −0.626987
\(121\) −2.61925e11 −0.918031
\(122\) −1.00873e11 −0.337905
\(123\) −2.22315e11 −0.712018
\(124\) −1.58863e11 −0.486635
\(125\) −3.03020e11 −0.888108
\(126\) −6.29146e10 −0.176487
\(127\) 9.64316e10 0.259000 0.129500 0.991579i \(-0.458663\pi\)
0.129500 + 0.991579i \(0.458663\pi\)
\(128\) −3.93035e11 −1.01106
\(129\) 2.05071e10 0.0505429
\(130\) −3.87864e11 −0.916200
\(131\) 2.77574e11 0.628618 0.314309 0.949321i \(-0.398227\pi\)
0.314309 + 0.949321i \(0.398227\pi\)
\(132\) −6.60381e10 −0.143429
\(133\) 8.26636e11 1.72239
\(134\) 9.27230e10 0.185400
\(135\) 1.60286e11 0.307652
\(136\) 7.01431e11 1.29277
\(137\) 1.43914e9 0.00254766 0.00127383 0.999999i \(-0.499595\pi\)
0.00127383 + 0.999999i \(0.499595\pi\)
\(138\) −3.87160e10 −0.0658500
\(139\) 3.30415e11 0.540106 0.270053 0.962846i \(-0.412959\pi\)
0.270053 + 0.962846i \(0.412959\pi\)
\(140\) −1.28495e12 −2.01921
\(141\) 3.15780e11 0.477177
\(142\) −1.97324e11 −0.286809
\(143\) −3.22592e11 −0.451134
\(144\) 1.53709e11 0.206874
\(145\) −1.99249e12 −2.58150
\(146\) −4.16070e11 −0.519070
\(147\) −5.37674e11 −0.646061
\(148\) 4.94121e11 0.571998
\(149\) 7.88510e11 0.879595 0.439797 0.898097i \(-0.355050\pi\)
0.439797 + 0.898097i \(0.355050\pi\)
\(150\) −3.03804e11 −0.326657
\(151\) −3.06791e11 −0.318031 −0.159015 0.987276i \(-0.550832\pi\)
−0.159015 + 0.987276i \(0.550832\pi\)
\(152\) 8.04045e11 0.803786
\(153\) −6.57848e11 −0.634341
\(154\) 1.62939e11 0.151586
\(155\) −9.98612e11 −0.896546
\(156\) 9.10916e11 0.789392
\(157\) 1.19299e12 0.998137 0.499069 0.866562i \(-0.333676\pi\)
0.499069 + 0.866562i \(0.333676\pi\)
\(158\) 7.96440e11 0.643486
\(159\) −9.66078e11 −0.753925
\(160\) −1.91901e12 −1.44683
\(161\) −6.26552e11 −0.456473
\(162\) 5.73929e10 0.0404134
\(163\) −1.63583e12 −1.11354 −0.556770 0.830667i \(-0.687959\pi\)
−0.556770 + 0.830667i \(0.687959\pi\)
\(164\) −1.62579e12 −1.07010
\(165\) −4.15115e11 −0.264245
\(166\) −1.11921e12 −0.689158
\(167\) −6.56904e11 −0.391347 −0.195673 0.980669i \(-0.562689\pi\)
−0.195673 + 0.980669i \(0.562689\pi\)
\(168\) −9.90339e11 −0.570930
\(169\) 2.65761e12 1.48291
\(170\) 2.04844e12 1.10651
\(171\) −7.54086e11 −0.394405
\(172\) 1.49969e11 0.0759616
\(173\) −2.18034e12 −1.06972 −0.534861 0.844940i \(-0.679636\pi\)
−0.534861 + 0.844940i \(0.679636\pi\)
\(174\) −7.13441e11 −0.339107
\(175\) −4.91654e12 −2.26439
\(176\) −3.98083e11 −0.177686
\(177\) −1.73727e11 −0.0751646
\(178\) 7.79234e11 0.326857
\(179\) 7.00365e11 0.284861 0.142431 0.989805i \(-0.454508\pi\)
0.142431 + 0.989805i \(0.454508\pi\)
\(180\) 1.17218e12 0.462375
\(181\) 2.48114e12 0.949336 0.474668 0.880165i \(-0.342568\pi\)
0.474668 + 0.880165i \(0.342568\pi\)
\(182\) −2.24754e12 −0.834286
\(183\) 1.48918e12 0.536372
\(184\) −6.09429e11 −0.213022
\(185\) 3.10605e12 1.05381
\(186\) −3.57568e11 −0.117771
\(187\) 1.70372e12 0.544841
\(188\) 2.30931e12 0.717156
\(189\) 9.28805e11 0.280146
\(190\) 2.34811e12 0.687977
\(191\) 3.13043e12 0.891088 0.445544 0.895260i \(-0.353010\pi\)
0.445544 + 0.895260i \(0.353010\pi\)
\(192\) 6.08328e11 0.168260
\(193\) 6.73197e12 1.80958 0.904788 0.425863i \(-0.140029\pi\)
0.904788 + 0.425863i \(0.140029\pi\)
\(194\) −1.81460e12 −0.474103
\(195\) 5.72601e12 1.45433
\(196\) −3.93202e12 −0.970973
\(197\) 2.94797e11 0.0707879 0.0353940 0.999373i \(-0.488731\pi\)
0.0353940 + 0.999373i \(0.488731\pi\)
\(198\) −1.48638e11 −0.0347114
\(199\) 4.13612e12 0.939509 0.469755 0.882797i \(-0.344342\pi\)
0.469755 + 0.882797i \(0.344342\pi\)
\(200\) −4.78218e12 −1.05672
\(201\) −1.36886e12 −0.294295
\(202\) 5.39499e11 0.112865
\(203\) −1.15458e13 −2.35070
\(204\) −4.81086e12 −0.953359
\(205\) −1.02197e13 −1.97149
\(206\) −9.79148e11 −0.183899
\(207\) 5.71563e11 0.104527
\(208\) 5.49106e12 0.977931
\(209\) 1.95296e12 0.338758
\(210\) −2.89216e12 −0.488671
\(211\) −4.91072e12 −0.808336 −0.404168 0.914685i \(-0.632439\pi\)
−0.404168 + 0.914685i \(0.632439\pi\)
\(212\) −7.06495e12 −1.13308
\(213\) 2.91308e12 0.455266
\(214\) −1.93659e12 −0.294959
\(215\) 9.42702e11 0.139947
\(216\) 9.03422e11 0.130736
\(217\) −5.78663e12 −0.816389
\(218\) 2.60764e12 0.358704
\(219\) 6.14242e12 0.823943
\(220\) −3.03575e12 −0.397138
\(221\) −2.35007e13 −2.99864
\(222\) 1.11217e12 0.138429
\(223\) 6.81556e12 0.827608 0.413804 0.910366i \(-0.364200\pi\)
0.413804 + 0.910366i \(0.364200\pi\)
\(224\) −1.11200e13 −1.31748
\(225\) 4.48504e12 0.518517
\(226\) 6.11421e12 0.689833
\(227\) 1.77863e12 0.195859 0.0979297 0.995193i \(-0.468778\pi\)
0.0979297 + 0.995193i \(0.468778\pi\)
\(228\) −5.51465e12 −0.592757
\(229\) −9.79549e12 −1.02785 −0.513927 0.857834i \(-0.671810\pi\)
−0.513927 + 0.857834i \(0.671810\pi\)
\(230\) −1.77976e12 −0.182330
\(231\) −2.40546e12 −0.240620
\(232\) −1.12303e13 −1.09700
\(233\) 7.07640e11 0.0675079 0.0337540 0.999430i \(-0.489254\pi\)
0.0337540 + 0.999430i \(0.489254\pi\)
\(234\) 2.05028e12 0.191041
\(235\) 1.45163e13 1.32124
\(236\) −1.27047e12 −0.112966
\(237\) −1.17578e13 −1.02143
\(238\) 1.18700e13 1.00758
\(239\) 1.37934e13 1.14415 0.572074 0.820202i \(-0.306139\pi\)
0.572074 + 0.820202i \(0.306139\pi\)
\(240\) 7.06596e12 0.572809
\(241\) −1.47934e13 −1.17213 −0.586064 0.810264i \(-0.699323\pi\)
−0.586064 + 0.810264i \(0.699323\pi\)
\(242\) −4.31131e12 −0.333907
\(243\) −8.47289e11 −0.0641500
\(244\) 1.08904e13 0.806120
\(245\) −2.47167e13 −1.78886
\(246\) −3.65933e12 −0.258976
\(247\) −2.69387e13 −1.86442
\(248\) −5.62848e12 −0.380984
\(249\) 1.65229e13 1.09393
\(250\) −4.98774e12 −0.323023
\(251\) −2.11426e13 −1.33953 −0.669766 0.742573i \(-0.733605\pi\)
−0.669766 + 0.742573i \(0.733605\pi\)
\(252\) 6.79237e12 0.421035
\(253\) −1.48026e12 −0.0897788
\(254\) 1.58728e12 0.0942035
\(255\) −3.02410e13 −1.75641
\(256\) −1.34242e12 −0.0763077
\(257\) 1.68358e13 0.936704 0.468352 0.883542i \(-0.344848\pi\)
0.468352 + 0.883542i \(0.344848\pi\)
\(258\) 3.37549e11 0.0183835
\(259\) 1.79985e13 0.959594
\(260\) 4.18745e13 2.18573
\(261\) 1.05325e13 0.538280
\(262\) 4.56891e12 0.228642
\(263\) −9.58815e12 −0.469870 −0.234935 0.972011i \(-0.575488\pi\)
−0.234935 + 0.972011i \(0.575488\pi\)
\(264\) −2.33972e12 −0.112290
\(265\) −4.44102e13 −2.08752
\(266\) 1.36065e13 0.626467
\(267\) −1.15038e13 −0.518835
\(268\) −1.00105e13 −0.442299
\(269\) 7.58983e12 0.328545 0.164272 0.986415i \(-0.447472\pi\)
0.164272 + 0.986415i \(0.447472\pi\)
\(270\) 2.63833e12 0.111900
\(271\) 6.24707e12 0.259624 0.129812 0.991539i \(-0.458563\pi\)
0.129812 + 0.991539i \(0.458563\pi\)
\(272\) −2.90002e13 −1.18106
\(273\) 3.31803e13 1.32430
\(274\) 2.36885e10 0.000926635 0
\(275\) −1.16155e13 −0.445358
\(276\) 4.17985e12 0.157094
\(277\) 1.30447e13 0.480611 0.240306 0.970697i \(-0.422752\pi\)
0.240306 + 0.970697i \(0.422752\pi\)
\(278\) 5.43867e12 0.196448
\(279\) 5.27876e12 0.186943
\(280\) −4.55255e13 −1.58083
\(281\) 2.02178e13 0.688413 0.344207 0.938894i \(-0.388148\pi\)
0.344207 + 0.938894i \(0.388148\pi\)
\(282\) 5.19778e12 0.173559
\(283\) 5.92770e11 0.0194116 0.00970578 0.999953i \(-0.496911\pi\)
0.00970578 + 0.999953i \(0.496911\pi\)
\(284\) 2.13034e13 0.684225
\(285\) −3.46651e13 −1.09206
\(286\) −5.30991e12 −0.164087
\(287\) −5.92200e13 −1.79522
\(288\) 1.01441e13 0.301686
\(289\) 8.98436e13 2.62149
\(290\) −3.27966e13 −0.938945
\(291\) 2.67889e13 0.752565
\(292\) 4.49197e13 1.23831
\(293\) 1.59170e12 0.0430615 0.0215308 0.999768i \(-0.493146\pi\)
0.0215308 + 0.999768i \(0.493146\pi\)
\(294\) −8.85017e12 −0.234986
\(295\) −7.98615e12 −0.208121
\(296\) 1.75066e13 0.447814
\(297\) 2.19434e12 0.0550990
\(298\) 1.29790e13 0.319927
\(299\) 2.04183e13 0.494116
\(300\) 3.27992e13 0.779286
\(301\) 5.46264e12 0.127435
\(302\) −5.04982e12 −0.115674
\(303\) −7.96460e12 −0.179155
\(304\) −3.32427e13 −0.734331
\(305\) 6.84570e13 1.48515
\(306\) −1.08283e13 −0.230723
\(307\) 6.83866e12 0.143123 0.0715616 0.997436i \(-0.477202\pi\)
0.0715616 + 0.997436i \(0.477202\pi\)
\(308\) −1.75912e13 −0.361631
\(309\) 1.44551e13 0.291911
\(310\) −1.64373e13 −0.326092
\(311\) −1.84837e13 −0.360253 −0.180126 0.983643i \(-0.557651\pi\)
−0.180126 + 0.983643i \(0.557651\pi\)
\(312\) 3.22735e13 0.618011
\(313\) −8.66400e13 −1.63014 −0.815069 0.579364i \(-0.803301\pi\)
−0.815069 + 0.579364i \(0.803301\pi\)
\(314\) 1.96368e13 0.363043
\(315\) 4.26968e13 0.775689
\(316\) −8.59851e13 −1.53513
\(317\) 1.39730e13 0.245169 0.122584 0.992458i \(-0.460882\pi\)
0.122584 + 0.992458i \(0.460882\pi\)
\(318\) −1.59018e13 −0.274218
\(319\) −2.72775e13 −0.462334
\(320\) 2.79646e13 0.465891
\(321\) 2.85898e13 0.468202
\(322\) −1.03131e13 −0.166029
\(323\) 1.42273e14 2.25169
\(324\) −6.19624e12 −0.0964119
\(325\) 1.60222e14 2.45112
\(326\) −2.69259e13 −0.405017
\(327\) −3.84964e13 −0.569388
\(328\) −5.76016e13 −0.837776
\(329\) 8.41171e13 1.20311
\(330\) −6.83285e12 −0.0961115
\(331\) −8.87012e13 −1.22709 −0.613544 0.789661i \(-0.710257\pi\)
−0.613544 + 0.789661i \(0.710257\pi\)
\(332\) 1.20832e14 1.64408
\(333\) −1.64189e13 −0.219735
\(334\) −1.08127e13 −0.142341
\(335\) −6.29262e13 −0.814864
\(336\) 4.09449e13 0.521596
\(337\) −1.90009e13 −0.238128 −0.119064 0.992887i \(-0.537989\pi\)
−0.119064 + 0.992887i \(0.537989\pi\)
\(338\) 4.37445e13 0.539364
\(339\) −9.02638e13 −1.09500
\(340\) −2.21153e14 −2.63973
\(341\) −1.36711e13 −0.160567
\(342\) −1.24124e13 −0.143453
\(343\) −1.52324e13 −0.173241
\(344\) 5.31336e12 0.0594699
\(345\) 2.62745e13 0.289421
\(346\) −3.58887e13 −0.389080
\(347\) 1.24628e14 1.32985 0.664926 0.746910i \(-0.268463\pi\)
0.664926 + 0.746910i \(0.268463\pi\)
\(348\) 7.70244e13 0.808989
\(349\) 1.32777e14 1.37272 0.686360 0.727262i \(-0.259207\pi\)
0.686360 + 0.727262i \(0.259207\pi\)
\(350\) −8.09269e13 −0.823605
\(351\) −3.02683e13 −0.303248
\(352\) −2.62716e13 −0.259120
\(353\) 1.02923e14 0.999425 0.499712 0.866191i \(-0.333439\pi\)
0.499712 + 0.866191i \(0.333439\pi\)
\(354\) −2.85956e12 −0.0273389
\(355\) 1.33913e14 1.26057
\(356\) −8.41275e13 −0.779763
\(357\) −1.75237e14 −1.59937
\(358\) 1.15281e13 0.103610
\(359\) −3.37199e13 −0.298446 −0.149223 0.988804i \(-0.547677\pi\)
−0.149223 + 0.988804i \(0.547677\pi\)
\(360\) 4.15300e13 0.361991
\(361\) 4.65959e13 0.399998
\(362\) 4.08399e13 0.345293
\(363\) 6.36477e13 0.530025
\(364\) 2.42649e14 1.99031
\(365\) 2.82365e14 2.28139
\(366\) 2.45121e13 0.195090
\(367\) 1.19637e14 0.938002 0.469001 0.883198i \(-0.344614\pi\)
0.469001 + 0.883198i \(0.344614\pi\)
\(368\) 2.51964e13 0.194615
\(369\) 5.40226e13 0.411084
\(370\) 5.11259e13 0.383293
\(371\) −2.57343e14 −1.90088
\(372\) 3.86037e13 0.280959
\(373\) 1.00137e14 0.718115 0.359058 0.933315i \(-0.383098\pi\)
0.359058 + 0.933315i \(0.383098\pi\)
\(374\) 2.80434e13 0.198170
\(375\) 7.36338e13 0.512750
\(376\) 8.18182e13 0.561458
\(377\) 3.76260e14 2.54455
\(378\) 1.52882e13 0.101895
\(379\) −1.71675e14 −1.12770 −0.563848 0.825879i \(-0.690680\pi\)
−0.563848 + 0.825879i \(0.690680\pi\)
\(380\) −2.53506e14 −1.64127
\(381\) −2.34329e13 −0.149533
\(382\) 5.15273e13 0.324107
\(383\) 1.39642e14 0.865811 0.432906 0.901439i \(-0.357488\pi\)
0.432906 + 0.901439i \(0.357488\pi\)
\(384\) 9.55075e13 0.583735
\(385\) −1.10578e14 −0.666246
\(386\) 1.10809e14 0.658180
\(387\) −4.98322e12 −0.0291809
\(388\) 1.95908e14 1.13104
\(389\) −9.70423e13 −0.552381 −0.276190 0.961103i \(-0.589072\pi\)
−0.276190 + 0.961103i \(0.589072\pi\)
\(390\) 9.42508e13 0.528969
\(391\) −1.07836e14 −0.596750
\(392\) −1.39311e14 −0.760170
\(393\) −6.74506e13 −0.362933
\(394\) 4.85240e12 0.0257470
\(395\) −5.40502e14 −2.82822
\(396\) 1.60473e13 0.0828090
\(397\) −9.02498e13 −0.459302 −0.229651 0.973273i \(-0.573758\pi\)
−0.229651 + 0.973273i \(0.573758\pi\)
\(398\) 6.80810e13 0.341719
\(399\) −2.00873e14 −0.994420
\(400\) 1.97716e14 0.965411
\(401\) 2.30178e14 1.10859 0.554294 0.832321i \(-0.312988\pi\)
0.554294 + 0.832321i \(0.312988\pi\)
\(402\) −2.25317e13 −0.107041
\(403\) 1.88577e14 0.883712
\(404\) −5.82453e13 −0.269255
\(405\) −3.89495e13 −0.177623
\(406\) −1.90046e14 −0.854997
\(407\) 4.25222e13 0.188732
\(408\) −1.70448e14 −0.746380
\(409\) −9.68498e13 −0.418428 −0.209214 0.977870i \(-0.567090\pi\)
−0.209214 + 0.977870i \(0.567090\pi\)
\(410\) −1.68218e14 −0.717070
\(411\) −3.49712e11 −0.00147089
\(412\) 1.05711e14 0.438716
\(413\) −4.62771e13 −0.189514
\(414\) 9.40799e12 0.0380185
\(415\) 7.59552e14 3.02895
\(416\) 3.62385e14 1.42612
\(417\) −8.02909e13 −0.311830
\(418\) 3.21460e13 0.123213
\(419\) −3.76267e14 −1.42337 −0.711687 0.702496i \(-0.752069\pi\)
−0.711687 + 0.702496i \(0.752069\pi\)
\(420\) 3.12243e14 1.16579
\(421\) 4.89674e14 1.80449 0.902246 0.431221i \(-0.141917\pi\)
0.902246 + 0.431221i \(0.141917\pi\)
\(422\) −8.08310e13 −0.294008
\(423\) −7.67346e13 −0.275498
\(424\) −2.50310e14 −0.887085
\(425\) −8.46189e14 −2.96025
\(426\) 4.79497e13 0.165590
\(427\) 3.96686e14 1.35236
\(428\) 2.09078e14 0.703667
\(429\) 7.83899e13 0.260462
\(430\) 1.55170e13 0.0509015
\(431\) −1.98756e14 −0.643717 −0.321858 0.946788i \(-0.604308\pi\)
−0.321858 + 0.946788i \(0.604308\pi\)
\(432\) −3.73514e13 −0.119439
\(433\) −3.45187e14 −1.08986 −0.544931 0.838481i \(-0.683444\pi\)
−0.544931 + 0.838481i \(0.683444\pi\)
\(434\) −9.52486e13 −0.296937
\(435\) 4.84175e14 1.49043
\(436\) −2.81525e14 −0.855740
\(437\) −1.23612e14 −0.371033
\(438\) 1.01105e14 0.299685
\(439\) 2.91082e14 0.852041 0.426021 0.904714i \(-0.359915\pi\)
0.426021 + 0.904714i \(0.359915\pi\)
\(440\) −1.07556e14 −0.310917
\(441\) 1.30655e14 0.373003
\(442\) −3.86825e14 −1.09067
\(443\) 2.93730e14 0.817951 0.408976 0.912545i \(-0.365886\pi\)
0.408976 + 0.912545i \(0.365886\pi\)
\(444\) −1.20072e14 −0.330243
\(445\) −5.28825e14 −1.43659
\(446\) 1.12185e14 0.301018
\(447\) −1.91608e14 −0.507834
\(448\) 1.62046e14 0.424237
\(449\) −2.68378e14 −0.694053 −0.347026 0.937855i \(-0.612809\pi\)
−0.347026 + 0.937855i \(0.612809\pi\)
\(450\) 7.38244e13 0.188595
\(451\) −1.39910e14 −0.353083
\(452\) −6.60102e14 −1.64569
\(453\) 7.45502e13 0.183615
\(454\) 2.92765e13 0.0712381
\(455\) 1.52529e15 3.66682
\(456\) −1.95383e14 −0.464066
\(457\) −8.00988e14 −1.87969 −0.939847 0.341596i \(-0.889032\pi\)
−0.939847 + 0.341596i \(0.889032\pi\)
\(458\) −1.61235e14 −0.373851
\(459\) 1.59857e14 0.366237
\(460\) 1.92146e14 0.434974
\(461\) 5.13483e14 1.14861 0.574303 0.818643i \(-0.305273\pi\)
0.574303 + 0.818643i \(0.305273\pi\)
\(462\) −3.95941e13 −0.0875184
\(463\) −8.22013e14 −1.79549 −0.897746 0.440514i \(-0.854796\pi\)
−0.897746 + 0.440514i \(0.854796\pi\)
\(464\) 4.64309e14 1.00221
\(465\) 2.42663e14 0.517621
\(466\) 1.16478e13 0.0245540
\(467\) −8.62251e14 −1.79635 −0.898175 0.439638i \(-0.855107\pi\)
−0.898175 + 0.439638i \(0.855107\pi\)
\(468\) −2.21352e14 −0.455756
\(469\) −3.64636e14 −0.742009
\(470\) 2.38940e14 0.480563
\(471\) −2.89898e14 −0.576275
\(472\) −4.50124e13 −0.0884404
\(473\) 1.29057e13 0.0250638
\(474\) −1.93535e14 −0.371517
\(475\) −9.69980e14 −1.84055
\(476\) −1.28151e15 −2.40372
\(477\) 2.34757e14 0.435279
\(478\) 2.27041e14 0.416150
\(479\) −3.18332e14 −0.576813 −0.288406 0.957508i \(-0.593125\pi\)
−0.288406 + 0.957508i \(0.593125\pi\)
\(480\) 4.66320e14 0.835329
\(481\) −5.86542e14 −1.03873
\(482\) −2.43502e14 −0.426328
\(483\) 1.52252e14 0.263545
\(484\) 4.65457e14 0.796582
\(485\) 1.23148e15 2.08376
\(486\) −1.39465e13 −0.0233327
\(487\) −4.48030e13 −0.0741136 −0.0370568 0.999313i \(-0.511798\pi\)
−0.0370568 + 0.999313i \(0.511798\pi\)
\(488\) 3.85845e14 0.631108
\(489\) 3.97506e14 0.642903
\(490\) −4.06839e14 −0.650645
\(491\) 8.28198e14 1.30974 0.654871 0.755740i \(-0.272723\pi\)
0.654871 + 0.755740i \(0.272723\pi\)
\(492\) 3.95068e14 0.617823
\(493\) −1.98716e15 −3.07308
\(494\) −4.43415e14 −0.678129
\(495\) 1.00873e14 0.152562
\(496\) 2.32706e14 0.348063
\(497\) 7.75983e14 1.14787
\(498\) 2.71969e14 0.397885
\(499\) −6.53023e14 −0.944877 −0.472438 0.881364i \(-0.656626\pi\)
−0.472438 + 0.881364i \(0.656626\pi\)
\(500\) 5.38486e14 0.770618
\(501\) 1.59628e14 0.225944
\(502\) −3.48010e14 −0.487215
\(503\) 7.51686e14 1.04091 0.520455 0.853889i \(-0.325763\pi\)
0.520455 + 0.853889i \(0.325763\pi\)
\(504\) 2.40652e14 0.329626
\(505\) −3.66130e14 −0.496058
\(506\) −2.43652e13 −0.0326544
\(507\) −6.45799e14 −0.856157
\(508\) −1.71365e14 −0.224736
\(509\) −1.01096e15 −1.31156 −0.655780 0.754952i \(-0.727660\pi\)
−0.655780 + 0.754952i \(0.727660\pi\)
\(510\) −4.97771e14 −0.638842
\(511\) 1.63621e15 2.07742
\(512\) 7.82839e14 0.983304
\(513\) 1.83243e14 0.227710
\(514\) 2.77120e14 0.340698
\(515\) 6.64496e14 0.808263
\(516\) −3.64424e13 −0.0438564
\(517\) 1.98730e14 0.236628
\(518\) 2.96257e14 0.349024
\(519\) 5.29823e14 0.617604
\(520\) 1.48360e15 1.71119
\(521\) 1.53933e15 1.75681 0.878406 0.477915i \(-0.158607\pi\)
0.878406 + 0.477915i \(0.158607\pi\)
\(522\) 1.73366e14 0.195784
\(523\) −1.22582e15 −1.36983 −0.684914 0.728624i \(-0.740160\pi\)
−0.684914 + 0.728624i \(0.740160\pi\)
\(524\) −4.93267e14 −0.545457
\(525\) 1.19472e15 1.30734
\(526\) −1.57822e14 −0.170902
\(527\) −9.95939e14 −1.06727
\(528\) 9.67341e13 0.102587
\(529\) −8.59118e14 −0.901668
\(530\) −7.30998e14 −0.759275
\(531\) 4.22156e13 0.0433963
\(532\) −1.46899e15 −1.49453
\(533\) 1.92988e15 1.94326
\(534\) −1.89354e14 −0.188711
\(535\) 1.31426e15 1.29639
\(536\) −3.54671e14 −0.346274
\(537\) −1.70189e14 −0.164465
\(538\) 1.24929e14 0.119498
\(539\) −3.38375e14 −0.320376
\(540\) −2.84839e14 −0.266952
\(541\) −1.25060e15 −1.16020 −0.580101 0.814544i \(-0.696987\pi\)
−0.580101 + 0.814544i \(0.696987\pi\)
\(542\) 1.02827e14 0.0944307
\(543\) −6.02918e14 −0.548099
\(544\) −1.91388e15 −1.72234
\(545\) −1.76967e15 −1.57656
\(546\) 5.46152e14 0.481675
\(547\) 1.43464e14 0.125260 0.0626302 0.998037i \(-0.480051\pi\)
0.0626302 + 0.998037i \(0.480051\pi\)
\(548\) −2.55745e12 −0.00221062
\(549\) −3.61871e14 −0.309674
\(550\) −1.91193e14 −0.161986
\(551\) −2.27786e15 −1.91070
\(552\) 1.48091e14 0.122988
\(553\) −3.13203e15 −2.57536
\(554\) 2.14717e14 0.174808
\(555\) −7.54769e14 −0.608419
\(556\) −5.87169e14 −0.468654
\(557\) −6.32352e14 −0.499753 −0.249877 0.968278i \(-0.580390\pi\)
−0.249877 + 0.968278i \(0.580390\pi\)
\(558\) 8.68891e13 0.0679950
\(559\) −1.78019e14 −0.137943
\(560\) 1.88222e15 1.44423
\(561\) −4.14004e14 −0.314564
\(562\) 3.32787e14 0.250390
\(563\) 1.54714e15 1.15275 0.576373 0.817187i \(-0.304468\pi\)
0.576373 + 0.817187i \(0.304468\pi\)
\(564\) −5.61161e14 −0.414050
\(565\) −4.14940e15 −3.03192
\(566\) 9.75706e12 0.00706039
\(567\) −2.25700e14 −0.161742
\(568\) 7.54776e14 0.535676
\(569\) −1.14138e15 −0.802257 −0.401128 0.916022i \(-0.631382\pi\)
−0.401128 + 0.916022i \(0.631382\pi\)
\(570\) −5.70591e14 −0.397204
\(571\) 1.47138e15 1.01444 0.507218 0.861818i \(-0.330674\pi\)
0.507218 + 0.861818i \(0.330674\pi\)
\(572\) 5.73267e14 0.391452
\(573\) −7.60695e14 −0.514470
\(574\) −9.74768e14 −0.652959
\(575\) 7.35200e14 0.487790
\(576\) −1.47824e14 −0.0971450
\(577\) 4.32743e14 0.281684 0.140842 0.990032i \(-0.455019\pi\)
0.140842 + 0.990032i \(0.455019\pi\)
\(578\) 1.47884e15 0.953492
\(579\) −1.63587e15 −1.04476
\(580\) 3.54078e15 2.23999
\(581\) 4.40135e15 2.75814
\(582\) 4.40949e14 0.273723
\(583\) −6.07983e14 −0.373864
\(584\) 1.59149e15 0.969470
\(585\) −1.39142e15 −0.839656
\(586\) 2.61996e13 0.0156624
\(587\) 5.38665e14 0.319014 0.159507 0.987197i \(-0.449010\pi\)
0.159507 + 0.987197i \(0.449010\pi\)
\(588\) 9.55481e14 0.560592
\(589\) −1.14164e15 −0.663581
\(590\) −1.31453e14 −0.0756980
\(591\) −7.16357e13 −0.0408694
\(592\) −7.23800e14 −0.409118
\(593\) −2.95891e15 −1.65703 −0.828517 0.559965i \(-0.810815\pi\)
−0.828517 + 0.559965i \(0.810815\pi\)
\(594\) 3.61191e13 0.0200406
\(595\) −8.05556e15 −4.42846
\(596\) −1.40123e15 −0.763231
\(597\) −1.00508e15 −0.542426
\(598\) 3.36088e14 0.179720
\(599\) 3.63909e15 1.92817 0.964084 0.265598i \(-0.0855694\pi\)
0.964084 + 0.265598i \(0.0855694\pi\)
\(600\) 1.16207e15 0.610099
\(601\) −8.03333e13 −0.0417913 −0.0208957 0.999782i \(-0.506652\pi\)
−0.0208957 + 0.999782i \(0.506652\pi\)
\(602\) 8.99158e13 0.0463506
\(603\) 3.32634e14 0.169911
\(604\) 5.45187e14 0.275958
\(605\) 2.92586e15 1.46757
\(606\) −1.31098e14 −0.0651624
\(607\) −1.24715e15 −0.614302 −0.307151 0.951661i \(-0.599376\pi\)
−0.307151 + 0.951661i \(0.599376\pi\)
\(608\) −2.19386e15 −1.07088
\(609\) 2.80563e15 1.35717
\(610\) 1.12681e15 0.540178
\(611\) −2.74124e15 −1.30233
\(612\) 1.16904e15 0.550422
\(613\) −2.30183e15 −1.07409 −0.537045 0.843553i \(-0.680460\pi\)
−0.537045 + 0.843553i \(0.680460\pi\)
\(614\) 1.12565e14 0.0520569
\(615\) 2.48340e15 1.13824
\(616\) −6.23251e14 −0.283119
\(617\) −1.60231e15 −0.721405 −0.360702 0.932681i \(-0.617463\pi\)
−0.360702 + 0.932681i \(0.617463\pi\)
\(618\) 2.37933e14 0.106174
\(619\) 7.65945e14 0.338765 0.169383 0.985550i \(-0.445823\pi\)
0.169383 + 0.985550i \(0.445823\pi\)
\(620\) 1.77460e15 0.777940
\(621\) −1.38890e14 −0.0603485
\(622\) −3.04244e14 −0.131031
\(623\) −3.06436e15 −1.30815
\(624\) −1.33433e15 −0.564609
\(625\) −3.23803e14 −0.135813
\(626\) −1.42610e15 −0.592915
\(627\) −4.74570e14 −0.195582
\(628\) −2.12003e15 −0.866091
\(629\) 3.09773e15 1.25448
\(630\) 7.02795e14 0.282134
\(631\) 4.62791e15 1.84172 0.920860 0.389894i \(-0.127488\pi\)
0.920860 + 0.389894i \(0.127488\pi\)
\(632\) −3.04643e15 −1.20184
\(633\) 1.19331e15 0.466693
\(634\) 2.29998e14 0.0891729
\(635\) −1.07720e15 −0.414039
\(636\) 1.71678e15 0.654186
\(637\) 4.66747e15 1.76325
\(638\) −4.48991e14 −0.168160
\(639\) −7.07879e14 −0.262848
\(640\) 4.39044e15 1.61629
\(641\) −2.60535e15 −0.950926 −0.475463 0.879736i \(-0.657719\pi\)
−0.475463 + 0.879736i \(0.657719\pi\)
\(642\) 4.70591e14 0.170295
\(643\) −4.08506e15 −1.46568 −0.732839 0.680402i \(-0.761805\pi\)
−0.732839 + 0.680402i \(0.761805\pi\)
\(644\) 1.11342e15 0.396085
\(645\) −2.29077e14 −0.0807983
\(646\) 2.34183e15 0.818984
\(647\) 3.60941e15 1.25159 0.625797 0.779986i \(-0.284774\pi\)
0.625797 + 0.779986i \(0.284774\pi\)
\(648\) −2.19531e14 −0.0754804
\(649\) −1.09332e14 −0.0372734
\(650\) 2.63728e15 0.891523
\(651\) 1.40615e15 0.471342
\(652\) 2.90697e15 0.966227
\(653\) 6.85186e14 0.225832 0.112916 0.993605i \(-0.463981\pi\)
0.112916 + 0.993605i \(0.463981\pi\)
\(654\) −6.33656e14 −0.207098
\(655\) −3.10068e15 −1.00492
\(656\) 2.38150e15 0.765384
\(657\) −1.49261e15 −0.475704
\(658\) 1.38458e15 0.437597
\(659\) 6.08725e15 1.90788 0.953941 0.299995i \(-0.0969850\pi\)
0.953941 + 0.299995i \(0.0969850\pi\)
\(660\) 7.37687e14 0.229288
\(661\) −2.77767e15 −0.856193 −0.428097 0.903733i \(-0.640816\pi\)
−0.428097 + 0.903733i \(0.640816\pi\)
\(662\) −1.46003e15 −0.446317
\(663\) 5.71068e15 1.73126
\(664\) 4.28106e15 1.28714
\(665\) −9.23404e15 −2.75342
\(666\) −2.70256e14 −0.0799223
\(667\) 1.72651e15 0.506382
\(668\) 1.16736e15 0.339574
\(669\) −1.65618e15 −0.477820
\(670\) −1.03577e15 −0.296383
\(671\) 9.37187e14 0.265982
\(672\) 2.70217e15 0.760645
\(673\) 1.00569e15 0.280789 0.140395 0.990096i \(-0.455163\pi\)
0.140395 + 0.990096i \(0.455163\pi\)
\(674\) −3.12758e14 −0.0866121
\(675\) −1.08987e15 −0.299366
\(676\) −4.72274e15 −1.28673
\(677\) 3.18500e15 0.860740 0.430370 0.902652i \(-0.358383\pi\)
0.430370 + 0.902652i \(0.358383\pi\)
\(678\) −1.48575e15 −0.398275
\(679\) 7.13600e15 1.89745
\(680\) −7.83541e15 −2.06663
\(681\) −4.32208e14 −0.113079
\(682\) −2.25029e14 −0.0584015
\(683\) −3.93928e15 −1.01415 −0.507076 0.861901i \(-0.669274\pi\)
−0.507076 + 0.861901i \(0.669274\pi\)
\(684\) 1.34006e15 0.342228
\(685\) −1.60761e13 −0.00407271
\(686\) −2.50727e14 −0.0630114
\(687\) 2.38030e15 0.593431
\(688\) −2.19677e14 −0.0543311
\(689\) 8.38638e15 2.05764
\(690\) 4.32482e14 0.105268
\(691\) 4.91573e15 1.18702 0.593511 0.804826i \(-0.297741\pi\)
0.593511 + 0.804826i \(0.297741\pi\)
\(692\) 3.87461e15 0.928206
\(693\) 5.84526e14 0.138922
\(694\) 2.05139e15 0.483694
\(695\) −3.69094e15 −0.863418
\(696\) 2.72896e15 0.633353
\(697\) −1.01924e16 −2.34690
\(698\) 2.18552e15 0.499287
\(699\) −1.71956e14 −0.0389757
\(700\) 8.73702e15 1.96483
\(701\) −7.69250e14 −0.171640 −0.0858200 0.996311i \(-0.527351\pi\)
−0.0858200 + 0.996311i \(0.527351\pi\)
\(702\) −4.98219e14 −0.110298
\(703\) 3.55091e15 0.779982
\(704\) 3.82840e14 0.0834387
\(705\) −3.52746e15 −0.762820
\(706\) 1.69412e15 0.363511
\(707\) −2.12160e15 −0.451707
\(708\) 3.08723e14 0.0652209
\(709\) −6.83725e15 −1.43327 −0.716633 0.697450i \(-0.754318\pi\)
−0.716633 + 0.697450i \(0.754318\pi\)
\(710\) 2.20423e15 0.458496
\(711\) 2.85715e15 0.589725
\(712\) −2.98062e15 −0.610473
\(713\) 8.65309e14 0.175865
\(714\) −2.88442e15 −0.581725
\(715\) 3.60356e15 0.721187
\(716\) −1.24459e15 −0.247176
\(717\) −3.35179e15 −0.660574
\(718\) −5.55033e14 −0.108551
\(719\) −9.95468e15 −1.93205 −0.966025 0.258450i \(-0.916788\pi\)
−0.966025 + 0.258450i \(0.916788\pi\)
\(720\) −1.71703e15 −0.330711
\(721\) 3.85054e15 0.735999
\(722\) 7.66974e14 0.145488
\(723\) 3.59481e15 0.676729
\(724\) −4.40915e15 −0.823746
\(725\) 1.35479e16 2.51197
\(726\) 1.04765e15 0.192781
\(727\) −5.39737e15 −0.985696 −0.492848 0.870115i \(-0.664044\pi\)
−0.492848 + 0.870115i \(0.664044\pi\)
\(728\) 8.59699e15 1.55820
\(729\) 2.05891e14 0.0370370
\(730\) 4.64776e15 0.829789
\(731\) 9.40178e14 0.166596
\(732\) −2.64637e15 −0.465414
\(733\) 7.52773e14 0.131399 0.0656995 0.997839i \(-0.479072\pi\)
0.0656995 + 0.997839i \(0.479072\pi\)
\(734\) 1.96925e15 0.341171
\(735\) 6.00615e15 1.03280
\(736\) 1.66285e15 0.283808
\(737\) −8.61469e14 −0.145938
\(738\) 8.89218e14 0.149520
\(739\) 1.89513e15 0.316297 0.158149 0.987415i \(-0.449447\pi\)
0.158149 + 0.987415i \(0.449447\pi\)
\(740\) −5.51964e15 −0.914401
\(741\) 6.54611e15 1.07642
\(742\) −4.23589e15 −0.691390
\(743\) −4.47283e15 −0.724675 −0.362338 0.932047i \(-0.618021\pi\)
−0.362338 + 0.932047i \(0.618021\pi\)
\(744\) 1.36772e15 0.219961
\(745\) −8.80814e15 −1.40613
\(746\) 1.64826e15 0.261193
\(747\) −4.01506e15 −0.631581
\(748\) −3.02762e15 −0.472762
\(749\) 7.61571e15 1.18048
\(750\) 1.21202e15 0.186498
\(751\) −9.55724e15 −1.45987 −0.729933 0.683519i \(-0.760449\pi\)
−0.729933 + 0.683519i \(0.760449\pi\)
\(752\) −3.38272e15 −0.512942
\(753\) 5.13765e15 0.773379
\(754\) 6.19328e15 0.925504
\(755\) 3.42704e15 0.508407
\(756\) −1.65055e15 −0.243085
\(757\) −3.34017e15 −0.488361 −0.244181 0.969730i \(-0.578519\pi\)
−0.244181 + 0.969730i \(0.578519\pi\)
\(758\) −2.82580e15 −0.410166
\(759\) 3.59702e14 0.0518338
\(760\) −8.98168e15 −1.28494
\(761\) 1.20740e16 1.71489 0.857446 0.514574i \(-0.172050\pi\)
0.857446 + 0.514574i \(0.172050\pi\)
\(762\) −3.85708e14 −0.0543884
\(763\) −1.02546e16 −1.43561
\(764\) −5.56298e15 −0.773204
\(765\) 7.34857e15 1.01406
\(766\) 2.29853e15 0.314914
\(767\) 1.50810e15 0.205142
\(768\) 3.26208e14 0.0440563
\(769\) 9.51994e15 1.27655 0.638277 0.769807i \(-0.279647\pi\)
0.638277 + 0.769807i \(0.279647\pi\)
\(770\) −1.82013e15 −0.242327
\(771\) −4.09111e15 −0.540806
\(772\) −1.19631e16 −1.57018
\(773\) 4.88498e15 0.636614 0.318307 0.947988i \(-0.396886\pi\)
0.318307 + 0.947988i \(0.396886\pi\)
\(774\) −8.20243e13 −0.0106137
\(775\) 6.79006e15 0.872398
\(776\) 6.94098e15 0.885485
\(777\) −4.37364e15 −0.554022
\(778\) −1.59733e15 −0.200912
\(779\) −1.16835e16 −1.45920
\(780\) −1.01755e16 −1.26193
\(781\) 1.83329e15 0.225762
\(782\) −1.77500e15 −0.217050
\(783\) −2.55940e15 −0.310776
\(784\) 5.75971e15 0.694484
\(785\) −1.33265e16 −1.59563
\(786\) −1.11024e15 −0.132006
\(787\) −5.43878e14 −0.0642156 −0.0321078 0.999484i \(-0.510222\pi\)
−0.0321078 + 0.999484i \(0.510222\pi\)
\(788\) −5.23874e14 −0.0614232
\(789\) 2.32992e15 0.271280
\(790\) −8.89673e15 −1.02868
\(791\) −2.40444e16 −2.76085
\(792\) 5.68551e14 0.0648308
\(793\) −1.29273e16 −1.46388
\(794\) −1.48552e15 −0.167058
\(795\) 1.07917e16 1.20523
\(796\) −7.35015e15 −0.815219
\(797\) 7.21945e14 0.0795212 0.0397606 0.999209i \(-0.487340\pi\)
0.0397606 + 0.999209i \(0.487340\pi\)
\(798\) −3.30639e15 −0.361691
\(799\) 1.44774e16 1.57284
\(800\) 1.30483e16 1.40786
\(801\) 2.79542e15 0.299549
\(802\) 3.78876e15 0.403217
\(803\) 3.86562e15 0.408586
\(804\) 2.43256e15 0.255362
\(805\) 6.99897e15 0.729721
\(806\) 3.10400e15 0.321424
\(807\) −1.84433e15 −0.189685
\(808\) −2.06362e15 −0.210798
\(809\) −5.35902e15 −0.543711 −0.271856 0.962338i \(-0.587637\pi\)
−0.271856 + 0.962338i \(0.587637\pi\)
\(810\) −6.41114e14 −0.0646052
\(811\) −8.07892e15 −0.808609 −0.404305 0.914624i \(-0.632486\pi\)
−0.404305 + 0.914624i \(0.632486\pi\)
\(812\) 2.05177e16 2.03972
\(813\) −1.51804e15 −0.149894
\(814\) 6.99921e14 0.0686459
\(815\) 1.82732e16 1.78011
\(816\) 7.04705e15 0.681885
\(817\) 1.07772e15 0.103582
\(818\) −1.59416e15 −0.152191
\(819\) −8.06282e15 −0.764585
\(820\) 1.81611e16 1.71067
\(821\) −1.54990e16 −1.45017 −0.725083 0.688662i \(-0.758198\pi\)
−0.725083 + 0.688662i \(0.758198\pi\)
\(822\) −5.75630e12 −0.000534993 0
\(823\) 9.28888e15 0.857559 0.428780 0.903409i \(-0.358944\pi\)
0.428780 + 0.903409i \(0.358944\pi\)
\(824\) 3.74530e15 0.343469
\(825\) 2.82258e15 0.257128
\(826\) −7.61726e14 −0.0689300
\(827\) −2.77458e15 −0.249412 −0.124706 0.992194i \(-0.539799\pi\)
−0.124706 + 0.992194i \(0.539799\pi\)
\(828\) −1.01570e15 −0.0906985
\(829\) 6.35935e15 0.564108 0.282054 0.959398i \(-0.408984\pi\)
0.282054 + 0.959398i \(0.408984\pi\)
\(830\) 1.25023e16 1.10169
\(831\) −3.16985e15 −0.277481
\(832\) −5.28081e15 −0.459221
\(833\) −2.46505e16 −2.12950
\(834\) −1.32160e15 −0.113419
\(835\) 7.33803e15 0.625610
\(836\) −3.47054e15 −0.293943
\(837\) −1.28274e15 −0.107932
\(838\) −6.19340e15 −0.517711
\(839\) 1.15571e16 0.959748 0.479874 0.877337i \(-0.340682\pi\)
0.479874 + 0.877337i \(0.340682\pi\)
\(840\) 1.10627e16 0.912694
\(841\) 1.96149e16 1.60771
\(842\) 8.06009e15 0.656331
\(843\) −4.91293e15 −0.397456
\(844\) 8.72667e15 0.701399
\(845\) −2.96871e16 −2.37059
\(846\) −1.26306e15 −0.100204
\(847\) 1.69544e16 1.33636
\(848\) 1.03489e16 0.810432
\(849\) −1.44043e14 −0.0112073
\(850\) −1.39284e16 −1.07670
\(851\) −2.69142e15 −0.206714
\(852\) −5.17673e15 −0.395037
\(853\) −1.56446e16 −1.18617 −0.593083 0.805141i \(-0.702089\pi\)
−0.593083 + 0.805141i \(0.702089\pi\)
\(854\) 6.52949e15 0.491882
\(855\) 8.42361e15 0.630499
\(856\) 7.40758e15 0.550897
\(857\) −2.08939e16 −1.54392 −0.771960 0.635671i \(-0.780724\pi\)
−0.771960 + 0.635671i \(0.780724\pi\)
\(858\) 1.29031e15 0.0947356
\(859\) −1.69176e16 −1.23417 −0.617086 0.786896i \(-0.711687\pi\)
−0.617086 + 0.786896i \(0.711687\pi\)
\(860\) −1.67524e15 −0.121433
\(861\) 1.43905e16 1.03647
\(862\) −3.27154e15 −0.234133
\(863\) 1.88111e16 1.33769 0.668843 0.743404i \(-0.266790\pi\)
0.668843 + 0.743404i \(0.266790\pi\)
\(864\) −2.46502e15 −0.174178
\(865\) 2.43558e16 1.71007
\(866\) −5.68183e15 −0.396405
\(867\) −2.18320e16 −1.51352
\(868\) 1.02832e16 0.708386
\(869\) −7.39955e15 −0.506520
\(870\) 7.96958e15 0.542100
\(871\) 1.18829e16 0.803199
\(872\) −9.97438e15 −0.669955
\(873\) −6.50971e15 −0.434494
\(874\) −2.03467e15 −0.134952
\(875\) 1.96145e16 1.29280
\(876\) −1.09155e16 −0.714941
\(877\) −1.19995e16 −0.781029 −0.390514 0.920597i \(-0.627703\pi\)
−0.390514 + 0.920597i \(0.627703\pi\)
\(878\) 4.79124e15 0.309905
\(879\) −3.86783e14 −0.0248616
\(880\) 4.44683e15 0.284051
\(881\) 1.22001e16 0.774457 0.387229 0.921984i \(-0.373432\pi\)
0.387229 + 0.921984i \(0.373432\pi\)
\(882\) 2.15059e15 0.135669
\(883\) 1.42018e16 0.890349 0.445174 0.895444i \(-0.353142\pi\)
0.445174 + 0.895444i \(0.353142\pi\)
\(884\) 4.17623e16 2.60194
\(885\) 1.94063e15 0.120159
\(886\) 4.83483e15 0.297506
\(887\) 2.18442e16 1.33584 0.667921 0.744232i \(-0.267184\pi\)
0.667921 + 0.744232i \(0.267184\pi\)
\(888\) −4.25411e15 −0.258546
\(889\) −6.24202e15 −0.377021
\(890\) −8.70452e15 −0.522516
\(891\) −5.33225e14 −0.0318114
\(892\) −1.21117e16 −0.718122
\(893\) 1.65954e16 0.977921
\(894\) −3.15389e15 −0.184710
\(895\) −7.82352e15 −0.455381
\(896\) 2.54412e16 1.47178
\(897\) −4.96165e15 −0.285278
\(898\) −4.41754e15 −0.252441
\(899\) 1.59455e16 0.905650
\(900\) −7.97021e15 −0.449921
\(901\) −4.42913e16 −2.48504
\(902\) −2.30293e15 −0.128424
\(903\) −1.32742e15 −0.0735744
\(904\) −2.33873e16 −1.28841
\(905\) −2.77159e16 −1.51762
\(906\) 1.22711e15 0.0667847
\(907\) −3.97869e15 −0.215229 −0.107614 0.994193i \(-0.534321\pi\)
−0.107614 + 0.994193i \(0.534321\pi\)
\(908\) −3.16075e15 −0.169949
\(909\) 1.93540e15 0.103435
\(910\) 2.51064e16 1.33370
\(911\) −1.74764e16 −0.922785 −0.461393 0.887196i \(-0.652650\pi\)
−0.461393 + 0.887196i \(0.652650\pi\)
\(912\) 8.07798e15 0.423966
\(913\) 1.03984e16 0.542470
\(914\) −1.31844e16 −0.683683
\(915\) −1.66351e16 −0.857449
\(916\) 1.74072e16 0.891876
\(917\) −1.79674e16 −0.915069
\(918\) 2.63127e15 0.133208
\(919\) 6.80581e15 0.342487 0.171244 0.985229i \(-0.445221\pi\)
0.171244 + 0.985229i \(0.445221\pi\)
\(920\) 6.80770e15 0.340539
\(921\) −1.66179e15 −0.0826322
\(922\) 8.45199e15 0.417772
\(923\) −2.52880e16 −1.24253
\(924\) 4.27465e15 0.208788
\(925\) −2.11196e16 −1.02543
\(926\) −1.35304e16 −0.653058
\(927\) −3.51259e15 −0.168535
\(928\) 3.06422e16 1.46152
\(929\) 8.16106e15 0.386955 0.193477 0.981105i \(-0.438023\pi\)
0.193477 + 0.981105i \(0.438023\pi\)
\(930\) 3.99426e15 0.188270
\(931\) −2.82567e16 −1.32403
\(932\) −1.25752e15 −0.0585771
\(933\) 4.49155e15 0.207992
\(934\) −1.41928e16 −0.653370
\(935\) −1.90316e16 −0.870987
\(936\) −7.84247e15 −0.356809
\(937\) 2.62034e16 1.18520 0.592598 0.805499i \(-0.298102\pi\)
0.592598 + 0.805499i \(0.298102\pi\)
\(938\) −6.00196e15 −0.269884
\(939\) 2.10535e16 0.941161
\(940\) −2.57964e16 −1.14645
\(941\) −3.62454e16 −1.60144 −0.800718 0.599042i \(-0.795548\pi\)
−0.800718 + 0.599042i \(0.795548\pi\)
\(942\) −4.77175e15 −0.209603
\(943\) 8.85552e15 0.386723
\(944\) 1.86101e15 0.0807983
\(945\) −1.03753e16 −0.447844
\(946\) 2.12430e14 0.00911621
\(947\) 2.14057e16 0.913282 0.456641 0.889651i \(-0.349052\pi\)
0.456641 + 0.889651i \(0.349052\pi\)
\(948\) 2.08944e16 0.886306
\(949\) −5.33215e16 −2.24873
\(950\) −1.59660e16 −0.669447
\(951\) −3.39545e15 −0.141548
\(952\) −4.54036e16 −1.88186
\(953\) −1.55330e16 −0.640096 −0.320048 0.947401i \(-0.603699\pi\)
−0.320048 + 0.947401i \(0.603699\pi\)
\(954\) 3.86413e15 0.158320
\(955\) −3.49689e16 −1.42450
\(956\) −2.45117e16 −0.992785
\(957\) 6.62843e15 0.266928
\(958\) −5.23978e15 −0.209799
\(959\) −9.31558e13 −0.00370858
\(960\) −6.79540e15 −0.268982
\(961\) −1.74168e16 −0.685471
\(962\) −9.65455e15 −0.377806
\(963\) −6.94732e15 −0.270317
\(964\) 2.62889e16 1.01706
\(965\) −7.52002e16 −2.89280
\(966\) 2.50609e15 0.0958566
\(967\) −4.38978e16 −1.66954 −0.834771 0.550597i \(-0.814400\pi\)
−0.834771 + 0.550597i \(0.814400\pi\)
\(968\) 1.64911e16 0.623640
\(969\) −3.45723e16 −1.30001
\(970\) 2.02703e16 0.757905
\(971\) −2.75018e16 −1.02248 −0.511241 0.859437i \(-0.670814\pi\)
−0.511241 + 0.859437i \(0.670814\pi\)
\(972\) 1.50569e15 0.0556634
\(973\) −2.13878e16 −0.786222
\(974\) −7.37463e14 −0.0269566
\(975\) −3.89340e16 −1.41515
\(976\) −1.59525e16 −0.576573
\(977\) −1.21036e16 −0.435004 −0.217502 0.976060i \(-0.569791\pi\)
−0.217502 + 0.976060i \(0.569791\pi\)
\(978\) 6.54300e15 0.233837
\(979\) −7.23969e15 −0.257285
\(980\) 4.39231e16 1.55221
\(981\) 9.35464e15 0.328736
\(982\) 1.36322e16 0.476380
\(983\) 3.98310e15 0.138413 0.0692065 0.997602i \(-0.477953\pi\)
0.0692065 + 0.997602i \(0.477953\pi\)
\(984\) 1.39972e16 0.483690
\(985\) −3.29307e15 −0.113162
\(986\) −3.27088e16 −1.11774
\(987\) −2.04404e16 −0.694618
\(988\) 4.78719e16 1.61777
\(989\) −8.16862e14 −0.0274517
\(990\) 1.66038e15 0.0554900
\(991\) −3.26106e16 −1.08381 −0.541906 0.840439i \(-0.682297\pi\)
−0.541906 + 0.840439i \(0.682297\pi\)
\(992\) 1.53575e16 0.507582
\(993\) 2.15544e16 0.708460
\(994\) 1.27728e16 0.417503
\(995\) −4.62030e16 −1.50191
\(996\) −2.93623e16 −0.949212
\(997\) 3.71945e16 1.19579 0.597895 0.801574i \(-0.296004\pi\)
0.597895 + 0.801574i \(0.296004\pi\)
\(998\) −1.07488e16 −0.343671
\(999\) 3.98978e15 0.126864
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.1.16 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.a.1.16 26 1.1 even 1 trivial