Properties

Label 177.8.a.c.1.4
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-13.8304\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-13.8304 q^{2} -27.0000 q^{3} +63.2803 q^{4} -149.469 q^{5} +373.421 q^{6} +14.2725 q^{7} +895.100 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-13.8304 q^{2} -27.0000 q^{3} +63.2803 q^{4} -149.469 q^{5} +373.421 q^{6} +14.2725 q^{7} +895.100 q^{8} +729.000 q^{9} +2067.22 q^{10} +531.760 q^{11} -1708.57 q^{12} +4168.02 q^{13} -197.395 q^{14} +4035.66 q^{15} -20479.5 q^{16} +4562.28 q^{17} -10082.4 q^{18} +31839.8 q^{19} -9458.44 q^{20} -385.358 q^{21} -7354.45 q^{22} -56030.4 q^{23} -24167.7 q^{24} -55784.0 q^{25} -57645.4 q^{26} -19683.0 q^{27} +903.168 q^{28} +52043.8 q^{29} -55814.9 q^{30} -44036.0 q^{31} +168667. q^{32} -14357.5 q^{33} -63098.2 q^{34} -2133.30 q^{35} +46131.3 q^{36} +11309.5 q^{37} -440358. q^{38} -112536. q^{39} -133790. q^{40} -455543. q^{41} +5329.65 q^{42} +177174. q^{43} +33649.9 q^{44} -108963. q^{45} +774923. q^{46} -1.27260e6 q^{47} +552946. q^{48} -823339. q^{49} +771516. q^{50} -123182. q^{51} +263753. q^{52} +274388. q^{53} +272224. q^{54} -79481.6 q^{55} +12775.3 q^{56} -859675. q^{57} -719787. q^{58} -205379. q^{59} +255378. q^{60} -152764. q^{61} +609036. q^{62} +10404.7 q^{63} +288642. q^{64} -622989. q^{65} +198570. q^{66} +2.06545e6 q^{67} +288702. q^{68} +1.51282e6 q^{69} +29504.4 q^{70} -1.79110e6 q^{71} +652528. q^{72} +3.32985e6 q^{73} -156414. q^{74} +1.50617e6 q^{75} +2.01483e6 q^{76} +7589.54 q^{77} +1.55643e6 q^{78} +6.43667e6 q^{79} +3.06105e6 q^{80} +531441. q^{81} +6.30035e6 q^{82} -1.15501e6 q^{83} -24385.5 q^{84} -681919. q^{85} -2.45039e6 q^{86} -1.40518e6 q^{87} +475978. q^{88} +3.69123e6 q^{89} +1.50700e6 q^{90} +59488.0 q^{91} -3.54562e6 q^{92} +1.18897e6 q^{93} +1.76005e7 q^{94} -4.75907e6 q^{95} -4.55400e6 q^{96} +6.33530e6 q^{97} +1.13871e7 q^{98} +387653. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −13.8304 −1.22245 −0.611224 0.791458i \(-0.709322\pi\)
−0.611224 + 0.791458i \(0.709322\pi\)
\(3\) −27.0000 −0.577350
\(4\) 63.2803 0.494377
\(5\) −149.469 −0.534757 −0.267378 0.963592i \(-0.586157\pi\)
−0.267378 + 0.963592i \(0.586157\pi\)
\(6\) 373.421 0.705780
\(7\) 14.2725 0.0157274 0.00786370 0.999969i \(-0.497497\pi\)
0.00786370 + 0.999969i \(0.497497\pi\)
\(8\) 895.100 0.618097
\(9\) 729.000 0.333333
\(10\) 2067.22 0.653712
\(11\) 531.760 0.120459 0.0602297 0.998185i \(-0.480817\pi\)
0.0602297 + 0.998185i \(0.480817\pi\)
\(12\) −1708.57 −0.285429
\(13\) 4168.02 0.526172 0.263086 0.964772i \(-0.415260\pi\)
0.263086 + 0.964772i \(0.415260\pi\)
\(14\) −197.395 −0.0192259
\(15\) 4035.66 0.308742
\(16\) −20479.5 −1.24997
\(17\) 4562.28 0.225222 0.112611 0.993639i \(-0.464079\pi\)
0.112611 + 0.993639i \(0.464079\pi\)
\(18\) −10082.4 −0.407482
\(19\) 31839.8 1.06496 0.532480 0.846443i \(-0.321260\pi\)
0.532480 + 0.846443i \(0.321260\pi\)
\(20\) −9458.44 −0.264372
\(21\) −385.358 −0.00908022
\(22\) −7354.45 −0.147255
\(23\) −56030.4 −0.960231 −0.480116 0.877205i \(-0.659405\pi\)
−0.480116 + 0.877205i \(0.659405\pi\)
\(24\) −24167.7 −0.356859
\(25\) −55784.0 −0.714035
\(26\) −57645.4 −0.643217
\(27\) −19683.0 −0.192450
\(28\) 903.168 0.00777527
\(29\) 52043.8 0.396256 0.198128 0.980176i \(-0.436514\pi\)
0.198128 + 0.980176i \(0.436514\pi\)
\(30\) −55814.9 −0.377421
\(31\) −44036.0 −0.265486 −0.132743 0.991150i \(-0.542379\pi\)
−0.132743 + 0.991150i \(0.542379\pi\)
\(32\) 168667. 0.909923
\(33\) −14357.5 −0.0695473
\(34\) −63098.2 −0.275322
\(35\) −2133.30 −0.00841033
\(36\) 46131.3 0.164792
\(37\) 11309.5 0.0367059 0.0183529 0.999832i \(-0.494158\pi\)
0.0183529 + 0.999832i \(0.494158\pi\)
\(38\) −440358. −1.30186
\(39\) −112536. −0.303785
\(40\) −133790. −0.330531
\(41\) −455543. −1.03225 −0.516126 0.856513i \(-0.672626\pi\)
−0.516126 + 0.856513i \(0.672626\pi\)
\(42\) 5329.65 0.0111001
\(43\) 177174. 0.339829 0.169915 0.985459i \(-0.445651\pi\)
0.169915 + 0.985459i \(0.445651\pi\)
\(44\) 33649.9 0.0595524
\(45\) −108963. −0.178252
\(46\) 774923. 1.17383
\(47\) −1.27260e6 −1.78792 −0.893960 0.448147i \(-0.852084\pi\)
−0.893960 + 0.448147i \(0.852084\pi\)
\(48\) 552946. 0.721670
\(49\) −823339. −0.999753
\(50\) 771516. 0.872871
\(51\) −123182. −0.130032
\(52\) 263753. 0.260127
\(53\) 274388. 0.253162 0.126581 0.991956i \(-0.459600\pi\)
0.126581 + 0.991956i \(0.459600\pi\)
\(54\) 272224. 0.235260
\(55\) −79481.6 −0.0644165
\(56\) 12775.3 0.00972106
\(57\) −859675. −0.614854
\(58\) −719787. −0.484402
\(59\) −205379. −0.130189
\(60\) 255378. 0.152635
\(61\) −152764. −0.0861719 −0.0430860 0.999071i \(-0.513719\pi\)
−0.0430860 + 0.999071i \(0.513719\pi\)
\(62\) 609036. 0.324543
\(63\) 10404.7 0.00524247
\(64\) 288642. 0.137635
\(65\) −622989. −0.281374
\(66\) 198570. 0.0850179
\(67\) 2.06545e6 0.838984 0.419492 0.907759i \(-0.362208\pi\)
0.419492 + 0.907759i \(0.362208\pi\)
\(68\) 288702. 0.111345
\(69\) 1.51282e6 0.554390
\(70\) 29504.4 0.0102812
\(71\) −1.79110e6 −0.593904 −0.296952 0.954892i \(-0.595970\pi\)
−0.296952 + 0.954892i \(0.595970\pi\)
\(72\) 652528. 0.206032
\(73\) 3.32985e6 1.00183 0.500916 0.865496i \(-0.332996\pi\)
0.500916 + 0.865496i \(0.332996\pi\)
\(74\) −156414. −0.0448710
\(75\) 1.50617e6 0.412249
\(76\) 2.01483e6 0.526492
\(77\) 7589.54 0.00189451
\(78\) 1.55643e6 0.371362
\(79\) 6.43667e6 1.46881 0.734406 0.678710i \(-0.237461\pi\)
0.734406 + 0.678710i \(0.237461\pi\)
\(80\) 3.06105e6 0.668429
\(81\) 531441. 0.111111
\(82\) 6.30035e6 1.26187
\(83\) −1.15501e6 −0.221724 −0.110862 0.993836i \(-0.535361\pi\)
−0.110862 + 0.993836i \(0.535361\pi\)
\(84\) −24385.5 −0.00448905
\(85\) −681919. −0.120439
\(86\) −2.45039e6 −0.415423
\(87\) −1.40518e6 −0.228778
\(88\) 475978. 0.0744556
\(89\) 3.69123e6 0.555016 0.277508 0.960723i \(-0.410491\pi\)
0.277508 + 0.960723i \(0.410491\pi\)
\(90\) 1.50700e6 0.217904
\(91\) 59488.0 0.00827532
\(92\) −3.54562e6 −0.474717
\(93\) 1.18897e6 0.153279
\(94\) 1.76005e7 2.18564
\(95\) −4.75907e6 −0.569494
\(96\) −4.55400e6 −0.525344
\(97\) 6.33530e6 0.704800 0.352400 0.935849i \(-0.385366\pi\)
0.352400 + 0.935849i \(0.385366\pi\)
\(98\) 1.13871e7 1.22214
\(99\) 387653. 0.0401531
\(100\) −3.53003e6 −0.353003
\(101\) 8.28710e6 0.800346 0.400173 0.916440i \(-0.368950\pi\)
0.400173 + 0.916440i \(0.368950\pi\)
\(102\) 1.70365e6 0.158957
\(103\) 1.33969e7 1.20802 0.604008 0.796979i \(-0.293570\pi\)
0.604008 + 0.796979i \(0.293570\pi\)
\(104\) 3.73079e6 0.325225
\(105\) 57599.0 0.00485571
\(106\) −3.79490e6 −0.309478
\(107\) 7.64708e6 0.603466 0.301733 0.953393i \(-0.402435\pi\)
0.301733 + 0.953393i \(0.402435\pi\)
\(108\) −1.24555e6 −0.0951430
\(109\) 8.75164e6 0.647287 0.323644 0.946179i \(-0.395092\pi\)
0.323644 + 0.946179i \(0.395092\pi\)
\(110\) 1.09926e6 0.0787458
\(111\) −305355. −0.0211922
\(112\) −292293. −0.0196588
\(113\) −1.76375e7 −1.14990 −0.574952 0.818187i \(-0.694979\pi\)
−0.574952 + 0.818187i \(0.694979\pi\)
\(114\) 1.18897e7 0.751627
\(115\) 8.37480e6 0.513490
\(116\) 3.29335e6 0.195900
\(117\) 3.03848e6 0.175391
\(118\) 2.84048e6 0.159149
\(119\) 65115.1 0.00354215
\(120\) 3.61232e6 0.190832
\(121\) −1.92044e7 −0.985490
\(122\) 2.11278e6 0.105341
\(123\) 1.22997e7 0.595971
\(124\) −2.78661e6 −0.131250
\(125\) 2.00152e7 0.916592
\(126\) −143901. −0.00640864
\(127\) −3.15626e7 −1.36729 −0.683643 0.729817i \(-0.739605\pi\)
−0.683643 + 0.729817i \(0.739605\pi\)
\(128\) −2.55814e7 −1.07817
\(129\) −4.78370e6 −0.196201
\(130\) 8.61620e6 0.343965
\(131\) −1.54251e7 −0.599485 −0.299742 0.954020i \(-0.596901\pi\)
−0.299742 + 0.954020i \(0.596901\pi\)
\(132\) −908547. −0.0343826
\(133\) 454434. 0.0167490
\(134\) −2.85661e7 −1.02561
\(135\) 2.94200e6 0.102914
\(136\) 4.08370e6 0.139209
\(137\) 2.73563e6 0.0908940 0.0454470 0.998967i \(-0.485529\pi\)
0.0454470 + 0.998967i \(0.485529\pi\)
\(138\) −2.09229e7 −0.677712
\(139\) 2.21140e7 0.698418 0.349209 0.937045i \(-0.386450\pi\)
0.349209 + 0.937045i \(0.386450\pi\)
\(140\) −134996. −0.00415788
\(141\) 3.43601e7 1.03226
\(142\) 2.47717e7 0.726016
\(143\) 2.21638e6 0.0633824
\(144\) −1.49295e7 −0.416656
\(145\) −7.77893e6 −0.211900
\(146\) −4.60532e7 −1.22469
\(147\) 2.22302e7 0.577207
\(148\) 715666. 0.0181466
\(149\) 4.34797e6 0.107680 0.0538399 0.998550i \(-0.482854\pi\)
0.0538399 + 0.998550i \(0.482854\pi\)
\(150\) −2.08309e7 −0.503952
\(151\) 642629. 0.0151894 0.00759470 0.999971i \(-0.497583\pi\)
0.00759470 + 0.999971i \(0.497583\pi\)
\(152\) 2.84998e7 0.658248
\(153\) 3.32590e6 0.0750739
\(154\) −104966. −0.00231594
\(155\) 6.58202e6 0.141970
\(156\) −7.12134e6 −0.150185
\(157\) 4.54519e7 0.937354 0.468677 0.883370i \(-0.344731\pi\)
0.468677 + 0.883370i \(0.344731\pi\)
\(158\) −8.90218e7 −1.79555
\(159\) −7.40847e6 −0.146163
\(160\) −2.52105e7 −0.486587
\(161\) −799693. −0.0151019
\(162\) −7.35005e6 −0.135827
\(163\) 9.12208e7 1.64982 0.824911 0.565262i \(-0.191225\pi\)
0.824911 + 0.565262i \(0.191225\pi\)
\(164\) −2.88269e7 −0.510322
\(165\) 2.14600e6 0.0371909
\(166\) 1.59743e7 0.271045
\(167\) −6.56493e7 −1.09074 −0.545371 0.838194i \(-0.683611\pi\)
−0.545371 + 0.838194i \(0.683611\pi\)
\(168\) −344934. −0.00561246
\(169\) −4.53762e7 −0.723143
\(170\) 9.43122e6 0.147230
\(171\) 2.32112e7 0.354986
\(172\) 1.12116e7 0.168004
\(173\) −1.20346e8 −1.76714 −0.883568 0.468303i \(-0.844866\pi\)
−0.883568 + 0.468303i \(0.844866\pi\)
\(174\) 1.94342e7 0.279670
\(175\) −796177. −0.0112299
\(176\) −1.08902e7 −0.150571
\(177\) 5.54523e6 0.0751646
\(178\) −5.10512e7 −0.678478
\(179\) −1.33167e7 −0.173544 −0.0867722 0.996228i \(-0.527655\pi\)
−0.0867722 + 0.996228i \(0.527655\pi\)
\(180\) −6.89520e6 −0.0881238
\(181\) 1.20262e7 0.150748 0.0753741 0.997155i \(-0.475985\pi\)
0.0753741 + 0.997155i \(0.475985\pi\)
\(182\) −822744. −0.0101161
\(183\) 4.12462e6 0.0497514
\(184\) −5.01528e7 −0.593516
\(185\) −1.69041e6 −0.0196287
\(186\) −1.64440e7 −0.187375
\(187\) 2.42603e6 0.0271301
\(188\) −8.05302e7 −0.883907
\(189\) −280926. −0.00302674
\(190\) 6.58198e7 0.696176
\(191\) −2.96173e6 −0.0307559 −0.0153780 0.999882i \(-0.504895\pi\)
−0.0153780 + 0.999882i \(0.504895\pi\)
\(192\) −7.79333e6 −0.0794636
\(193\) −8.88812e6 −0.0889937 −0.0444968 0.999010i \(-0.514168\pi\)
−0.0444968 + 0.999010i \(0.514168\pi\)
\(194\) −8.76198e7 −0.861581
\(195\) 1.68207e7 0.162451
\(196\) −5.21012e7 −0.494255
\(197\) 5.08009e7 0.473412 0.236706 0.971581i \(-0.423932\pi\)
0.236706 + 0.971581i \(0.423932\pi\)
\(198\) −5.36140e6 −0.0490851
\(199\) 3.98854e7 0.358780 0.179390 0.983778i \(-0.442588\pi\)
0.179390 + 0.983778i \(0.442588\pi\)
\(200\) −4.99323e7 −0.441343
\(201\) −5.57673e7 −0.484388
\(202\) −1.14614e8 −0.978381
\(203\) 742795. 0.00623208
\(204\) −7.79496e6 −0.0642848
\(205\) 6.80896e7 0.552004
\(206\) −1.85284e8 −1.47674
\(207\) −4.08461e7 −0.320077
\(208\) −8.53588e7 −0.657698
\(209\) 1.69311e7 0.128284
\(210\) −796618. −0.00593585
\(211\) 7.97162e7 0.584195 0.292098 0.956389i \(-0.405647\pi\)
0.292098 + 0.956389i \(0.405647\pi\)
\(212\) 1.73633e7 0.125158
\(213\) 4.83597e7 0.342890
\(214\) −1.05762e8 −0.737705
\(215\) −2.64820e7 −0.181726
\(216\) −1.76183e7 −0.118953
\(217\) −628504. −0.00417541
\(218\) −1.21039e8 −0.791274
\(219\) −8.99061e7 −0.578408
\(220\) −5.02962e6 −0.0318460
\(221\) 1.90156e7 0.118505
\(222\) 4.22319e6 0.0259063
\(223\) 1.35288e8 0.816947 0.408473 0.912770i \(-0.366061\pi\)
0.408473 + 0.912770i \(0.366061\pi\)
\(224\) 2.40730e6 0.0143107
\(225\) −4.06665e7 −0.238012
\(226\) 2.43934e8 1.40570
\(227\) −1.97246e8 −1.11923 −0.559613 0.828754i \(-0.689050\pi\)
−0.559613 + 0.828754i \(0.689050\pi\)
\(228\) −5.44005e7 −0.303970
\(229\) 1.35002e8 0.742875 0.371438 0.928458i \(-0.378865\pi\)
0.371438 + 0.928458i \(0.378865\pi\)
\(230\) −1.15827e8 −0.627715
\(231\) −204918. −0.00109380
\(232\) 4.65844e7 0.244925
\(233\) 1.36315e8 0.705988 0.352994 0.935626i \(-0.385164\pi\)
0.352994 + 0.935626i \(0.385164\pi\)
\(234\) −4.20235e7 −0.214406
\(235\) 1.90214e8 0.956102
\(236\) −1.29964e7 −0.0643624
\(237\) −1.73790e8 −0.848019
\(238\) −900569. −0.00433010
\(239\) 1.31494e8 0.623038 0.311519 0.950240i \(-0.399162\pi\)
0.311519 + 0.950240i \(0.399162\pi\)
\(240\) −8.26483e7 −0.385918
\(241\) 2.49648e8 1.14887 0.574433 0.818552i \(-0.305223\pi\)
0.574433 + 0.818552i \(0.305223\pi\)
\(242\) 2.65605e8 1.20471
\(243\) −1.43489e7 −0.0641500
\(244\) −9.66693e6 −0.0426014
\(245\) 1.23064e8 0.534624
\(246\) −1.70109e8 −0.728543
\(247\) 1.32709e8 0.560351
\(248\) −3.94166e7 −0.164096
\(249\) 3.11852e7 0.128012
\(250\) −2.76819e8 −1.12048
\(251\) 3.53873e7 0.141250 0.0706252 0.997503i \(-0.477501\pi\)
0.0706252 + 0.997503i \(0.477501\pi\)
\(252\) 658410. 0.00259176
\(253\) −2.97947e7 −0.115669
\(254\) 4.36523e8 1.67143
\(255\) 1.84118e7 0.0695354
\(256\) 3.16855e8 1.18038
\(257\) 4.79092e7 0.176057 0.0880285 0.996118i \(-0.471943\pi\)
0.0880285 + 0.996118i \(0.471943\pi\)
\(258\) 6.61606e7 0.239845
\(259\) 161414. 0.000577288 0
\(260\) −3.94229e7 −0.139105
\(261\) 3.79399e7 0.132085
\(262\) 2.13335e8 0.732838
\(263\) 4.03661e8 1.36827 0.684135 0.729355i \(-0.260180\pi\)
0.684135 + 0.729355i \(0.260180\pi\)
\(264\) −1.28514e7 −0.0429870
\(265\) −4.10125e7 −0.135380
\(266\) −6.28501e6 −0.0204748
\(267\) −9.96631e7 −0.320439
\(268\) 1.30703e8 0.414775
\(269\) 1.59050e8 0.498196 0.249098 0.968478i \(-0.419866\pi\)
0.249098 + 0.968478i \(0.419866\pi\)
\(270\) −4.06890e7 −0.125807
\(271\) −2.63878e8 −0.805399 −0.402700 0.915332i \(-0.631928\pi\)
−0.402700 + 0.915332i \(0.631928\pi\)
\(272\) −9.34331e7 −0.281520
\(273\) −1.60618e6 −0.00477776
\(274\) −3.78349e7 −0.111113
\(275\) −2.96637e7 −0.0860123
\(276\) 9.57317e7 0.274078
\(277\) 5.07781e8 1.43548 0.717740 0.696311i \(-0.245177\pi\)
0.717740 + 0.696311i \(0.245177\pi\)
\(278\) −3.05846e8 −0.853779
\(279\) −3.21022e7 −0.0884954
\(280\) −1.90951e6 −0.00519840
\(281\) 4.01914e8 1.08059 0.540295 0.841476i \(-0.318313\pi\)
0.540295 + 0.841476i \(0.318313\pi\)
\(282\) −4.75214e8 −1.26188
\(283\) −1.41538e7 −0.0371210 −0.0185605 0.999828i \(-0.505908\pi\)
−0.0185605 + 0.999828i \(0.505908\pi\)
\(284\) −1.13341e8 −0.293612
\(285\) 1.28495e8 0.328797
\(286\) −3.06535e7 −0.0774816
\(287\) −6.50174e6 −0.0162346
\(288\) 1.22958e8 0.303308
\(289\) −3.89524e8 −0.949275
\(290\) 1.07586e8 0.259037
\(291\) −1.71053e8 −0.406917
\(292\) 2.10714e8 0.495283
\(293\) −5.93471e7 −0.137836 −0.0689180 0.997622i \(-0.521955\pi\)
−0.0689180 + 0.997622i \(0.521955\pi\)
\(294\) −3.07452e8 −0.705606
\(295\) 3.06978e7 0.0696194
\(296\) 1.01231e7 0.0226878
\(297\) −1.04666e7 −0.0231824
\(298\) −6.01342e7 −0.131633
\(299\) −2.33535e8 −0.505247
\(300\) 9.53108e7 0.203806
\(301\) 2.52872e6 0.00534463
\(302\) −8.88782e6 −0.0185682
\(303\) −2.23752e8 −0.462080
\(304\) −6.52063e8 −1.33117
\(305\) 2.28334e7 0.0460810
\(306\) −4.59986e7 −0.0917739
\(307\) 2.64952e8 0.522617 0.261308 0.965255i \(-0.415846\pi\)
0.261308 + 0.965255i \(0.415846\pi\)
\(308\) 480268. 0.000936605 0
\(309\) −3.61715e8 −0.697448
\(310\) −9.10320e7 −0.173551
\(311\) 2.51765e8 0.474607 0.237304 0.971436i \(-0.423736\pi\)
0.237304 + 0.971436i \(0.423736\pi\)
\(312\) −1.00731e8 −0.187769
\(313\) 1.78674e8 0.329348 0.164674 0.986348i \(-0.447343\pi\)
0.164674 + 0.986348i \(0.447343\pi\)
\(314\) −6.28619e8 −1.14587
\(315\) −1.55517e6 −0.00280344
\(316\) 4.07314e8 0.726147
\(317\) 7.62555e8 1.34451 0.672254 0.740320i \(-0.265326\pi\)
0.672254 + 0.740320i \(0.265326\pi\)
\(318\) 1.02462e8 0.178677
\(319\) 2.76748e7 0.0477328
\(320\) −4.31430e7 −0.0736013
\(321\) −2.06471e8 −0.348411
\(322\) 1.10601e7 0.0184613
\(323\) 1.45262e8 0.239852
\(324\) 3.36297e7 0.0549308
\(325\) −2.32509e8 −0.375705
\(326\) −1.26162e9 −2.01682
\(327\) −2.36294e8 −0.373711
\(328\) −4.07757e8 −0.638032
\(329\) −1.81631e7 −0.0281193
\(330\) −2.96801e7 −0.0454639
\(331\) 3.94113e8 0.597342 0.298671 0.954356i \(-0.403457\pi\)
0.298671 + 0.954356i \(0.403457\pi\)
\(332\) −7.30893e7 −0.109615
\(333\) 8.24459e6 0.0122353
\(334\) 9.07957e8 1.33338
\(335\) −3.08721e8 −0.448652
\(336\) 7.89192e6 0.0113500
\(337\) 1.50464e8 0.214155 0.107077 0.994251i \(-0.465851\pi\)
0.107077 + 0.994251i \(0.465851\pi\)
\(338\) 6.27571e8 0.884004
\(339\) 4.76212e8 0.663898
\(340\) −4.31520e7 −0.0595422
\(341\) −2.34166e7 −0.0319803
\(342\) −3.21021e8 −0.433952
\(343\) −2.35051e7 −0.0314509
\(344\) 1.58589e8 0.210047
\(345\) −2.26120e8 −0.296464
\(346\) 1.66443e9 2.16023
\(347\) 1.12645e9 1.44730 0.723651 0.690166i \(-0.242462\pi\)
0.723651 + 0.690166i \(0.242462\pi\)
\(348\) −8.89203e7 −0.113103
\(349\) −2.40653e8 −0.303042 −0.151521 0.988454i \(-0.548417\pi\)
−0.151521 + 0.988454i \(0.548417\pi\)
\(350\) 1.10115e7 0.0137280
\(351\) −8.20391e7 −0.101262
\(352\) 8.96902e7 0.109609
\(353\) −2.62604e8 −0.317754 −0.158877 0.987298i \(-0.550787\pi\)
−0.158877 + 0.987298i \(0.550787\pi\)
\(354\) −7.66929e7 −0.0918848
\(355\) 2.67714e8 0.317594
\(356\) 2.33582e8 0.274387
\(357\) −1.75811e6 −0.00204506
\(358\) 1.84175e8 0.212149
\(359\) 1.31180e7 0.0149636 0.00748179 0.999972i \(-0.497618\pi\)
0.00748179 + 0.999972i \(0.497618\pi\)
\(360\) −9.75327e7 −0.110177
\(361\) 1.19902e8 0.134138
\(362\) −1.66327e8 −0.184282
\(363\) 5.18519e8 0.568973
\(364\) 3.76442e6 0.00409113
\(365\) −4.97710e8 −0.535737
\(366\) −5.70452e7 −0.0608184
\(367\) 2.85308e8 0.301288 0.150644 0.988588i \(-0.451865\pi\)
0.150644 + 0.988588i \(0.451865\pi\)
\(368\) 1.14747e9 1.20026
\(369\) −3.32091e8 −0.344084
\(370\) 2.33791e7 0.0239951
\(371\) 3.91620e6 0.00398159
\(372\) 7.52385e7 0.0757774
\(373\) −1.76282e8 −0.175884 −0.0879420 0.996126i \(-0.528029\pi\)
−0.0879420 + 0.996126i \(0.528029\pi\)
\(374\) −3.35531e7 −0.0331651
\(375\) −5.40412e8 −0.529194
\(376\) −1.13910e9 −1.10511
\(377\) 2.16919e8 0.208499
\(378\) 3.88532e6 0.00370003
\(379\) −1.69178e9 −1.59627 −0.798137 0.602477i \(-0.794181\pi\)
−0.798137 + 0.602477i \(0.794181\pi\)
\(380\) −3.01155e8 −0.281545
\(381\) 8.52189e8 0.789402
\(382\) 4.09620e7 0.0375975
\(383\) 2.05814e9 1.87189 0.935944 0.352149i \(-0.114549\pi\)
0.935944 + 0.352149i \(0.114549\pi\)
\(384\) 6.90698e8 0.622485
\(385\) −1.13440e6 −0.00101310
\(386\) 1.22926e8 0.108790
\(387\) 1.29160e8 0.113276
\(388\) 4.00900e8 0.348437
\(389\) 1.26822e9 1.09237 0.546187 0.837663i \(-0.316079\pi\)
0.546187 + 0.837663i \(0.316079\pi\)
\(390\) −2.32637e8 −0.198588
\(391\) −2.55626e8 −0.216265
\(392\) −7.36971e8 −0.617944
\(393\) 4.16477e8 0.346113
\(394\) −7.02597e8 −0.578722
\(395\) −9.62082e8 −0.785457
\(396\) 2.45308e7 0.0198508
\(397\) 1.90127e9 1.52502 0.762512 0.646975i \(-0.223966\pi\)
0.762512 + 0.646975i \(0.223966\pi\)
\(398\) −5.51632e8 −0.438590
\(399\) −1.22697e7 −0.00967006
\(400\) 1.14243e9 0.892522
\(401\) −1.68338e9 −1.30370 −0.651849 0.758349i \(-0.726007\pi\)
−0.651849 + 0.758349i \(0.726007\pi\)
\(402\) 7.71284e8 0.592138
\(403\) −1.83543e8 −0.139691
\(404\) 5.24410e8 0.395673
\(405\) −7.94340e7 −0.0594174
\(406\) −1.02732e7 −0.00761838
\(407\) 6.01391e6 0.00442157
\(408\) −1.10260e8 −0.0803723
\(409\) 1.07416e8 0.0776317 0.0388159 0.999246i \(-0.487641\pi\)
0.0388159 + 0.999246i \(0.487641\pi\)
\(410\) −9.41707e8 −0.674795
\(411\) −7.38619e7 −0.0524776
\(412\) 8.47757e8 0.597215
\(413\) −2.93127e6 −0.00204753
\(414\) 5.64919e8 0.391277
\(415\) 1.72638e8 0.118568
\(416\) 7.03006e8 0.478776
\(417\) −5.97078e8 −0.403232
\(418\) −2.34164e8 −0.156821
\(419\) 2.36422e8 0.157014 0.0785071 0.996914i \(-0.474985\pi\)
0.0785071 + 0.996914i \(0.474985\pi\)
\(420\) 3.64488e6 0.00240055
\(421\) −1.71085e9 −1.11744 −0.558722 0.829355i \(-0.688708\pi\)
−0.558722 + 0.829355i \(0.688708\pi\)
\(422\) −1.10251e9 −0.714148
\(423\) −9.27722e8 −0.595973
\(424\) 2.45605e8 0.156479
\(425\) −2.54502e8 −0.160816
\(426\) −6.68835e8 −0.419165
\(427\) −2.18032e6 −0.00135526
\(428\) 4.83910e8 0.298340
\(429\) −5.98423e7 −0.0365938
\(430\) 3.66258e8 0.222150
\(431\) 6.46052e8 0.388685 0.194342 0.980934i \(-0.437743\pi\)
0.194342 + 0.980934i \(0.437743\pi\)
\(432\) 4.03098e8 0.240557
\(433\) 2.52351e9 1.49382 0.746908 0.664927i \(-0.231537\pi\)
0.746908 + 0.664927i \(0.231537\pi\)
\(434\) 8.69247e6 0.00510422
\(435\) 2.10031e8 0.122341
\(436\) 5.53807e8 0.320004
\(437\) −1.78400e9 −1.02261
\(438\) 1.24344e9 0.707074
\(439\) −1.77473e8 −0.100117 −0.0500584 0.998746i \(-0.515941\pi\)
−0.0500584 + 0.998746i \(0.515941\pi\)
\(440\) −7.11440e7 −0.0398156
\(441\) −6.00214e8 −0.333251
\(442\) −2.62994e8 −0.144867
\(443\) 1.83781e9 1.00436 0.502178 0.864764i \(-0.332532\pi\)
0.502178 + 0.864764i \(0.332532\pi\)
\(444\) −1.93230e7 −0.0104769
\(445\) −5.51724e8 −0.296798
\(446\) −1.87110e9 −0.998674
\(447\) −1.17395e8 −0.0621690
\(448\) 4.11964e6 0.00216464
\(449\) −8.93570e8 −0.465872 −0.232936 0.972492i \(-0.574833\pi\)
−0.232936 + 0.972492i \(0.574833\pi\)
\(450\) 5.62435e8 0.290957
\(451\) −2.42239e8 −0.124345
\(452\) −1.11610e9 −0.568487
\(453\) −1.73510e7 −0.00876961
\(454\) 2.72800e9 1.36820
\(455\) −8.89161e6 −0.00442528
\(456\) −7.69495e8 −0.380040
\(457\) −5.88094e8 −0.288231 −0.144115 0.989561i \(-0.546034\pi\)
−0.144115 + 0.989561i \(0.546034\pi\)
\(458\) −1.86713e9 −0.908126
\(459\) −8.97993e7 −0.0433440
\(460\) 5.29960e8 0.253858
\(461\) −2.65315e9 −1.26127 −0.630636 0.776079i \(-0.717206\pi\)
−0.630636 + 0.776079i \(0.717206\pi\)
\(462\) 2.83409e6 0.00133711
\(463\) 1.10539e9 0.517587 0.258793 0.965933i \(-0.416675\pi\)
0.258793 + 0.965933i \(0.416675\pi\)
\(464\) −1.06583e9 −0.495307
\(465\) −1.77714e8 −0.0819667
\(466\) −1.88529e9 −0.863033
\(467\) 6.70125e8 0.304472 0.152236 0.988344i \(-0.451353\pi\)
0.152236 + 0.988344i \(0.451353\pi\)
\(468\) 1.92276e8 0.0867091
\(469\) 2.94792e7 0.0131950
\(470\) −2.63073e9 −1.16878
\(471\) −1.22720e9 −0.541182
\(472\) −1.83835e8 −0.0804694
\(473\) 9.42140e7 0.0409356
\(474\) 2.40359e9 1.03666
\(475\) −1.77615e9 −0.760418
\(476\) 4.12050e6 0.00175116
\(477\) 2.00029e8 0.0843875
\(478\) −1.81862e9 −0.761631
\(479\) −9.38891e7 −0.0390338 −0.0195169 0.999810i \(-0.506213\pi\)
−0.0195169 + 0.999810i \(0.506213\pi\)
\(480\) 6.80683e8 0.280931
\(481\) 4.71380e7 0.0193136
\(482\) −3.45274e9 −1.40443
\(483\) 2.15917e7 0.00871911
\(484\) −1.21526e9 −0.487204
\(485\) −9.46931e8 −0.376896
\(486\) 1.98451e8 0.0784200
\(487\) 1.37204e9 0.538288 0.269144 0.963100i \(-0.413259\pi\)
0.269144 + 0.963100i \(0.413259\pi\)
\(488\) −1.36739e8 −0.0532626
\(489\) −2.46296e9 −0.952525
\(490\) −1.70202e9 −0.653550
\(491\) 3.03791e9 1.15822 0.579108 0.815251i \(-0.303401\pi\)
0.579108 + 0.815251i \(0.303401\pi\)
\(492\) 7.78326e8 0.294635
\(493\) 2.37438e8 0.0892455
\(494\) −1.83542e9 −0.685000
\(495\) −5.79421e7 −0.0214722
\(496\) 9.01834e8 0.331849
\(497\) −2.55635e7 −0.00934056
\(498\) −4.31305e8 −0.156488
\(499\) −3.44240e9 −1.24025 −0.620126 0.784502i \(-0.712919\pi\)
−0.620126 + 0.784502i \(0.712919\pi\)
\(500\) 1.26657e9 0.453142
\(501\) 1.77253e9 0.629741
\(502\) −4.89421e8 −0.172671
\(503\) 1.03977e8 0.0364291 0.0182146 0.999834i \(-0.494202\pi\)
0.0182146 + 0.999834i \(0.494202\pi\)
\(504\) 9.31321e6 0.00324035
\(505\) −1.23866e9 −0.427990
\(506\) 4.12073e8 0.141399
\(507\) 1.22516e9 0.417507
\(508\) −1.99729e9 −0.675955
\(509\) −9.41820e8 −0.316560 −0.158280 0.987394i \(-0.550595\pi\)
−0.158280 + 0.987394i \(0.550595\pi\)
\(510\) −2.54643e8 −0.0850034
\(511\) 4.75253e7 0.0157562
\(512\) −1.10782e9 −0.364774
\(513\) −6.26703e8 −0.204951
\(514\) −6.62604e8 −0.215220
\(515\) −2.00241e9 −0.645994
\(516\) −3.02714e8 −0.0969971
\(517\) −6.76715e8 −0.215372
\(518\) −2.23243e6 −0.000705704 0
\(519\) 3.24934e9 1.02026
\(520\) −5.57638e8 −0.173916
\(521\) 9.49236e8 0.294064 0.147032 0.989132i \(-0.453028\pi\)
0.147032 + 0.989132i \(0.453028\pi\)
\(522\) −5.24725e8 −0.161467
\(523\) −3.88735e9 −1.18822 −0.594111 0.804383i \(-0.702496\pi\)
−0.594111 + 0.804383i \(0.702496\pi\)
\(524\) −9.76104e8 −0.296372
\(525\) 2.14968e7 0.00648360
\(526\) −5.58280e9 −1.67264
\(527\) −2.00904e8 −0.0597933
\(528\) 2.94034e8 0.0869319
\(529\) −2.65425e8 −0.0779555
\(530\) 5.67220e8 0.165495
\(531\) −1.49721e8 −0.0433963
\(532\) 2.87567e7 0.00828034
\(533\) −1.89871e9 −0.543142
\(534\) 1.37838e9 0.391719
\(535\) −1.14300e9 −0.322707
\(536\) 1.84879e9 0.518574
\(537\) 3.59550e8 0.100196
\(538\) −2.19972e9 −0.609018
\(539\) −4.37819e8 −0.120430
\(540\) 1.86171e8 0.0508783
\(541\) −2.42183e9 −0.657588 −0.328794 0.944402i \(-0.606642\pi\)
−0.328794 + 0.944402i \(0.606642\pi\)
\(542\) 3.64955e9 0.984558
\(543\) −3.24707e8 −0.0870345
\(544\) 7.69505e8 0.204935
\(545\) −1.30810e9 −0.346141
\(546\) 2.22141e7 0.00584055
\(547\) 5.83350e9 1.52396 0.761980 0.647600i \(-0.224227\pi\)
0.761980 + 0.647600i \(0.224227\pi\)
\(548\) 1.73111e8 0.0449359
\(549\) −1.11365e8 −0.0287240
\(550\) 4.10261e8 0.105146
\(551\) 1.65706e9 0.421996
\(552\) 1.35413e9 0.342667
\(553\) 9.18674e7 0.0231006
\(554\) −7.02282e9 −1.75480
\(555\) 4.56412e7 0.0113326
\(556\) 1.39938e9 0.345282
\(557\) 2.32049e9 0.568967 0.284484 0.958681i \(-0.408178\pi\)
0.284484 + 0.958681i \(0.408178\pi\)
\(558\) 4.43987e8 0.108181
\(559\) 7.38465e8 0.178809
\(560\) 4.36888e7 0.0105126
\(561\) −6.55029e7 −0.0156636
\(562\) −5.55863e9 −1.32096
\(563\) −4.30898e9 −1.01764 −0.508821 0.860873i \(-0.669918\pi\)
−0.508821 + 0.860873i \(0.669918\pi\)
\(564\) 2.17432e9 0.510324
\(565\) 2.63626e9 0.614919
\(566\) 1.95753e8 0.0453785
\(567\) 7.58499e6 0.00174749
\(568\) −1.60322e9 −0.367090
\(569\) 5.58255e9 1.27040 0.635198 0.772349i \(-0.280918\pi\)
0.635198 + 0.772349i \(0.280918\pi\)
\(570\) −1.77714e9 −0.401938
\(571\) 5.36498e9 1.20598 0.602992 0.797747i \(-0.293975\pi\)
0.602992 + 0.797747i \(0.293975\pi\)
\(572\) 1.40253e8 0.0313348
\(573\) 7.99668e7 0.0177570
\(574\) 8.99217e7 0.0198460
\(575\) 3.12560e9 0.685639
\(576\) 2.10420e8 0.0458784
\(577\) 4.50669e9 0.976658 0.488329 0.872660i \(-0.337607\pi\)
0.488329 + 0.872660i \(0.337607\pi\)
\(578\) 5.38728e9 1.16044
\(579\) 2.39979e8 0.0513805
\(580\) −4.92253e8 −0.104759
\(581\) −1.64849e7 −0.00348714
\(582\) 2.36573e9 0.497434
\(583\) 1.45908e8 0.0304958
\(584\) 2.98055e9 0.619230
\(585\) −4.54159e8 −0.0937913
\(586\) 8.20794e8 0.168497
\(587\) 4.94446e9 1.00899 0.504494 0.863415i \(-0.331679\pi\)
0.504494 + 0.863415i \(0.331679\pi\)
\(588\) 1.40673e9 0.285358
\(589\) −1.40210e9 −0.282732
\(590\) −4.24563e8 −0.0851060
\(591\) −1.37162e9 −0.273325
\(592\) −2.31612e8 −0.0458812
\(593\) 7.08369e9 1.39498 0.697490 0.716594i \(-0.254300\pi\)
0.697490 + 0.716594i \(0.254300\pi\)
\(594\) 1.44758e8 0.0283393
\(595\) −9.73269e6 −0.00189419
\(596\) 2.75141e8 0.0532345
\(597\) −1.07691e9 −0.207142
\(598\) 3.22989e9 0.617638
\(599\) 6.53780e9 1.24290 0.621452 0.783452i \(-0.286543\pi\)
0.621452 + 0.783452i \(0.286543\pi\)
\(600\) 1.34817e9 0.254810
\(601\) −5.66921e9 −1.06528 −0.532638 0.846343i \(-0.678799\pi\)
−0.532638 + 0.846343i \(0.678799\pi\)
\(602\) −3.49732e7 −0.00653353
\(603\) 1.50572e9 0.279661
\(604\) 4.06657e7 0.00750930
\(605\) 2.87046e9 0.526997
\(606\) 3.09458e9 0.564868
\(607\) −3.43054e8 −0.0622590 −0.0311295 0.999515i \(-0.509910\pi\)
−0.0311295 + 0.999515i \(0.509910\pi\)
\(608\) 5.37032e9 0.969031
\(609\) −2.00555e7 −0.00359809
\(610\) −3.15796e8 −0.0563316
\(611\) −5.30420e9 −0.940753
\(612\) 2.10464e8 0.0371149
\(613\) −1.95826e9 −0.343367 −0.171683 0.985152i \(-0.554921\pi\)
−0.171683 + 0.985152i \(0.554921\pi\)
\(614\) −3.66440e9 −0.638872
\(615\) −1.83842e9 −0.318699
\(616\) 6.79340e6 0.00117099
\(617\) −1.99932e9 −0.342676 −0.171338 0.985212i \(-0.554809\pi\)
−0.171338 + 0.985212i \(0.554809\pi\)
\(618\) 5.00267e9 0.852593
\(619\) 1.57292e9 0.266556 0.133278 0.991079i \(-0.457450\pi\)
0.133278 + 0.991079i \(0.457450\pi\)
\(620\) 4.16512e8 0.0701870
\(621\) 1.10285e9 0.184797
\(622\) −3.48202e9 −0.580182
\(623\) 5.26830e7 0.00872896
\(624\) 2.30469e9 0.379722
\(625\) 1.36647e9 0.223882
\(626\) −2.47113e9 −0.402611
\(627\) −4.57140e8 −0.0740650
\(628\) 2.87621e9 0.463406
\(629\) 5.15969e7 0.00826697
\(630\) 2.15087e7 0.00342706
\(631\) 3.48635e9 0.552419 0.276209 0.961097i \(-0.410922\pi\)
0.276209 + 0.961097i \(0.410922\pi\)
\(632\) 5.76146e9 0.907869
\(633\) −2.15234e9 −0.337285
\(634\) −1.05464e10 −1.64359
\(635\) 4.71762e9 0.731165
\(636\) −4.68810e8 −0.0722599
\(637\) −3.43169e9 −0.526042
\(638\) −3.82753e8 −0.0583508
\(639\) −1.30571e9 −0.197968
\(640\) 3.82362e9 0.576561
\(641\) 3.27300e9 0.490843 0.245422 0.969416i \(-0.421074\pi\)
0.245422 + 0.969416i \(0.421074\pi\)
\(642\) 2.85558e9 0.425914
\(643\) 1.07236e10 1.59075 0.795373 0.606120i \(-0.207275\pi\)
0.795373 + 0.606120i \(0.207275\pi\)
\(644\) −5.06048e7 −0.00746606
\(645\) 7.15015e8 0.104920
\(646\) −2.00903e9 −0.293206
\(647\) 1.13384e10 1.64584 0.822919 0.568159i \(-0.192344\pi\)
0.822919 + 0.568159i \(0.192344\pi\)
\(648\) 4.75693e8 0.0686775
\(649\) −1.09212e8 −0.0156825
\(650\) 3.21569e9 0.459280
\(651\) 1.69696e7 0.00241067
\(652\) 5.77248e9 0.815635
\(653\) 2.78140e9 0.390901 0.195451 0.980714i \(-0.437383\pi\)
0.195451 + 0.980714i \(0.437383\pi\)
\(654\) 3.26805e9 0.456842
\(655\) 2.30557e9 0.320578
\(656\) 9.32928e9 1.29028
\(657\) 2.42746e9 0.333944
\(658\) 2.51204e8 0.0343744
\(659\) 8.83123e9 1.20205 0.601025 0.799230i \(-0.294759\pi\)
0.601025 + 0.799230i \(0.294759\pi\)
\(660\) 1.35800e8 0.0183863
\(661\) −5.82485e9 −0.784476 −0.392238 0.919864i \(-0.628299\pi\)
−0.392238 + 0.919864i \(0.628299\pi\)
\(662\) −5.45075e9 −0.730219
\(663\) −5.13422e8 −0.0684191
\(664\) −1.03385e9 −0.137047
\(665\) −6.79238e7 −0.00895666
\(666\) −1.14026e8 −0.0149570
\(667\) −2.91603e9 −0.380497
\(668\) −4.15431e9 −0.539239
\(669\) −3.65279e9 −0.471664
\(670\) 4.26974e9 0.548454
\(671\) −8.12335e7 −0.0103802
\(672\) −6.49970e7 −0.00826230
\(673\) 3.42844e9 0.433554 0.216777 0.976221i \(-0.430446\pi\)
0.216777 + 0.976221i \(0.430446\pi\)
\(674\) −2.08098e9 −0.261793
\(675\) 1.09800e9 0.137416
\(676\) −2.87142e9 −0.357506
\(677\) 8.10222e9 1.00356 0.501781 0.864995i \(-0.332678\pi\)
0.501781 + 0.864995i \(0.332678\pi\)
\(678\) −6.58621e9 −0.811580
\(679\) 9.04206e7 0.0110847
\(680\) −6.10386e8 −0.0744429
\(681\) 5.32565e9 0.646186
\(682\) 3.23861e8 0.0390943
\(683\) 1.95901e9 0.235269 0.117634 0.993057i \(-0.462469\pi\)
0.117634 + 0.993057i \(0.462469\pi\)
\(684\) 1.46881e9 0.175497
\(685\) −4.08892e8 −0.0486061
\(686\) 3.25086e8 0.0384471
\(687\) −3.64505e9 −0.428899
\(688\) −3.62844e9 −0.424776
\(689\) 1.14365e9 0.133207
\(690\) 3.12733e9 0.362411
\(691\) −7.12139e9 −0.821091 −0.410546 0.911840i \(-0.634662\pi\)
−0.410546 + 0.911840i \(0.634662\pi\)
\(692\) −7.61552e9 −0.873632
\(693\) 5.53277e6 0.000631505 0
\(694\) −1.55793e10 −1.76925
\(695\) −3.30536e9 −0.373484
\(696\) −1.25778e9 −0.141407
\(697\) −2.07831e9 −0.232486
\(698\) 3.32833e9 0.370453
\(699\) −3.68050e9 −0.407602
\(700\) −5.03823e7 −0.00555182
\(701\) −9.01916e9 −0.988902 −0.494451 0.869206i \(-0.664631\pi\)
−0.494451 + 0.869206i \(0.664631\pi\)
\(702\) 1.13463e9 0.123787
\(703\) 3.60091e8 0.0390903
\(704\) 1.53488e8 0.0165794
\(705\) −5.13577e9 −0.552006
\(706\) 3.63193e9 0.388437
\(707\) 1.18278e8 0.0125874
\(708\) 3.50904e8 0.0371597
\(709\) −1.63143e10 −1.71912 −0.859560 0.511034i \(-0.829263\pi\)
−0.859560 + 0.511034i \(0.829263\pi\)
\(710\) −3.70260e9 −0.388242
\(711\) 4.69233e9 0.489604
\(712\) 3.30402e9 0.343054
\(713\) 2.46735e9 0.254928
\(714\) 2.43154e7 0.00249998
\(715\) −3.31280e8 −0.0338941
\(716\) −8.42684e8 −0.0857964
\(717\) −3.55035e9 −0.359711
\(718\) −1.81427e8 −0.0182922
\(719\) 6.12136e9 0.614181 0.307091 0.951680i \(-0.400645\pi\)
0.307091 + 0.951680i \(0.400645\pi\)
\(720\) 2.23150e9 0.222810
\(721\) 1.91207e8 0.0189989
\(722\) −1.65829e9 −0.163976
\(723\) −6.74051e9 −0.663298
\(724\) 7.61019e8 0.0745265
\(725\) −2.90321e9 −0.282941
\(726\) −7.17133e9 −0.695539
\(727\) 1.56648e9 0.151201 0.0756003 0.997138i \(-0.475913\pi\)
0.0756003 + 0.997138i \(0.475913\pi\)
\(728\) 5.32477e7 0.00511495
\(729\) 3.87420e8 0.0370370
\(730\) 6.88353e9 0.654910
\(731\) 8.08318e8 0.0765370
\(732\) 2.61007e8 0.0245959
\(733\) −4.77128e9 −0.447477 −0.223739 0.974649i \(-0.571826\pi\)
−0.223739 + 0.974649i \(0.571826\pi\)
\(734\) −3.94592e9 −0.368309
\(735\) −3.32272e9 −0.308665
\(736\) −9.45046e9 −0.873737
\(737\) 1.09833e9 0.101064
\(738\) 4.59295e9 0.420625
\(739\) 8.82555e8 0.0804427 0.0402213 0.999191i \(-0.487194\pi\)
0.0402213 + 0.999191i \(0.487194\pi\)
\(740\) −1.06970e8 −0.00970399
\(741\) −3.58314e9 −0.323519
\(742\) −5.41627e7 −0.00486728
\(743\) 1.56677e9 0.140134 0.0700672 0.997542i \(-0.477679\pi\)
0.0700672 + 0.997542i \(0.477679\pi\)
\(744\) 1.06425e9 0.0947410
\(745\) −6.49886e8 −0.0575825
\(746\) 2.43805e9 0.215009
\(747\) −8.42002e8 −0.0739079
\(748\) 1.53520e8 0.0134125
\(749\) 1.09143e8 0.00949095
\(750\) 7.47412e9 0.646912
\(751\) 5.02135e9 0.432594 0.216297 0.976328i \(-0.430602\pi\)
0.216297 + 0.976328i \(0.430602\pi\)
\(752\) 2.60621e10 2.23484
\(753\) −9.55458e8 −0.0815510
\(754\) −3.00008e9 −0.254879
\(755\) −9.60531e7 −0.00812263
\(756\) −1.77771e7 −0.00149635
\(757\) −2.94352e9 −0.246622 −0.123311 0.992368i \(-0.539351\pi\)
−0.123311 + 0.992368i \(0.539351\pi\)
\(758\) 2.33980e10 1.95136
\(759\) 8.04456e8 0.0667815
\(760\) −4.25984e9 −0.352002
\(761\) −1.87159e10 −1.53945 −0.769724 0.638376i \(-0.779606\pi\)
−0.769724 + 0.638376i \(0.779606\pi\)
\(762\) −1.17861e10 −0.965003
\(763\) 1.24908e8 0.0101801
\(764\) −1.87419e8 −0.0152050
\(765\) −4.97119e8 −0.0401463
\(766\) −2.84650e10 −2.28828
\(767\) −8.56023e8 −0.0685017
\(768\) −8.55509e9 −0.681491
\(769\) −1.08476e10 −0.860187 −0.430093 0.902784i \(-0.641519\pi\)
−0.430093 + 0.902784i \(0.641519\pi\)
\(770\) 1.56892e7 0.00123847
\(771\) −1.29355e9 −0.101647
\(772\) −5.62443e8 −0.0439965
\(773\) −2.59403e9 −0.201998 −0.100999 0.994887i \(-0.532204\pi\)
−0.100999 + 0.994887i \(0.532204\pi\)
\(774\) −1.78634e9 −0.138474
\(775\) 2.45651e9 0.189567
\(776\) 5.67073e9 0.435635
\(777\) −4.35818e6 −0.000333297 0
\(778\) −1.75400e10 −1.33537
\(779\) −1.45044e10 −1.09931
\(780\) 1.06442e9 0.0803122
\(781\) −9.52435e8 −0.0715413
\(782\) 3.53541e9 0.264373
\(783\) −1.02438e9 −0.0762595
\(784\) 1.68616e10 1.24966
\(785\) −6.79366e9 −0.501256
\(786\) −5.76005e9 −0.423104
\(787\) −2.06990e10 −1.51369 −0.756846 0.653593i \(-0.773261\pi\)
−0.756846 + 0.653593i \(0.773261\pi\)
\(788\) 3.21470e9 0.234044
\(789\) −1.08989e10 −0.789971
\(790\) 1.33060e10 0.960180
\(791\) −2.51731e8 −0.0180850
\(792\) 3.46988e8 0.0248185
\(793\) −6.36721e8 −0.0453412
\(794\) −2.62953e10 −1.86426
\(795\) 1.10734e9 0.0781619
\(796\) 2.52396e9 0.177373
\(797\) −6.06784e9 −0.424551 −0.212275 0.977210i \(-0.568087\pi\)
−0.212275 + 0.977210i \(0.568087\pi\)
\(798\) 1.69695e8 0.0118211
\(799\) −5.80594e9 −0.402679
\(800\) −9.40891e9 −0.649718
\(801\) 2.69090e9 0.185005
\(802\) 2.32819e10 1.59370
\(803\) 1.77068e9 0.120680
\(804\) −3.52897e9 −0.239470
\(805\) 1.19529e8 0.00807586
\(806\) 2.53847e9 0.170765
\(807\) −4.29434e9 −0.287633
\(808\) 7.41778e9 0.494691
\(809\) −2.20534e10 −1.46438 −0.732192 0.681098i \(-0.761503\pi\)
−0.732192 + 0.681098i \(0.761503\pi\)
\(810\) 1.09860e9 0.0726346
\(811\) 9.91610e9 0.652782 0.326391 0.945235i \(-0.394167\pi\)
0.326391 + 0.945235i \(0.394167\pi\)
\(812\) 4.70043e7 0.00308100
\(813\) 7.12472e9 0.464998
\(814\) −8.31749e7 −0.00540514
\(815\) −1.36347e10 −0.882253
\(816\) 2.52269e9 0.162536
\(817\) 5.64119e9 0.361904
\(818\) −1.48561e9 −0.0949007
\(819\) 4.33668e7 0.00275844
\(820\) 4.30873e9 0.272898
\(821\) −4.09876e9 −0.258495 −0.129247 0.991612i \(-0.541256\pi\)
−0.129247 + 0.991612i \(0.541256\pi\)
\(822\) 1.02154e9 0.0641512
\(823\) −6.76087e9 −0.422769 −0.211384 0.977403i \(-0.567797\pi\)
−0.211384 + 0.977403i \(0.567797\pi\)
\(824\) 1.19915e10 0.746671
\(825\) 8.00919e8 0.0496592
\(826\) 4.05407e7 0.00250300
\(827\) 9.41705e8 0.0578956 0.0289478 0.999581i \(-0.490784\pi\)
0.0289478 + 0.999581i \(0.490784\pi\)
\(828\) −2.58476e9 −0.158239
\(829\) 1.36525e10 0.832284 0.416142 0.909300i \(-0.363382\pi\)
0.416142 + 0.909300i \(0.363382\pi\)
\(830\) −2.38766e9 −0.144943
\(831\) −1.37101e10 −0.828775
\(832\) 1.20306e9 0.0724197
\(833\) −3.75630e9 −0.225166
\(834\) 8.25783e9 0.492930
\(835\) 9.81253e9 0.583282
\(836\) 1.07141e9 0.0634209
\(837\) 8.66761e8 0.0510928
\(838\) −3.26981e9 −0.191942
\(839\) −2.04480e10 −1.19532 −0.597660 0.801750i \(-0.703903\pi\)
−0.597660 + 0.801750i \(0.703903\pi\)
\(840\) 5.15569e7 0.00300130
\(841\) −1.45413e10 −0.842981
\(842\) 2.36618e10 1.36602
\(843\) −1.08517e10 −0.623878
\(844\) 5.04446e9 0.288813
\(845\) 6.78233e9 0.386706
\(846\) 1.28308e10 0.728546
\(847\) −2.74095e8 −0.0154992
\(848\) −5.61932e9 −0.316445
\(849\) 3.82152e8 0.0214318
\(850\) 3.51987e9 0.196590
\(851\) −6.33673e8 −0.0352461
\(852\) 3.06022e9 0.169517
\(853\) 9.28799e9 0.512389 0.256195 0.966625i \(-0.417531\pi\)
0.256195 + 0.966625i \(0.417531\pi\)
\(854\) 3.01547e7 0.00165673
\(855\) −3.46936e9 −0.189831
\(856\) 6.84490e9 0.373000
\(857\) −7.57637e9 −0.411177 −0.205588 0.978639i \(-0.565911\pi\)
−0.205588 + 0.978639i \(0.565911\pi\)
\(858\) 8.27644e8 0.0447340
\(859\) 5.27186e9 0.283784 0.141892 0.989882i \(-0.454681\pi\)
0.141892 + 0.989882i \(0.454681\pi\)
\(860\) −1.67579e9 −0.0898412
\(861\) 1.75547e8 0.00937308
\(862\) −8.93517e9 −0.475146
\(863\) 2.11332e10 1.11925 0.559625 0.828746i \(-0.310945\pi\)
0.559625 + 0.828746i \(0.310945\pi\)
\(864\) −3.31987e9 −0.175115
\(865\) 1.79880e10 0.944988
\(866\) −3.49012e10 −1.82611
\(867\) 1.05172e10 0.548064
\(868\) −3.97719e7 −0.00206423
\(869\) 3.42276e9 0.176932
\(870\) −2.90482e9 −0.149555
\(871\) 8.60885e9 0.441450
\(872\) 7.83360e9 0.400086
\(873\) 4.61843e9 0.234933
\(874\) 2.46734e10 1.25008
\(875\) 2.85668e8 0.0144156
\(876\) −5.68928e9 −0.285952
\(877\) 1.67780e10 0.839928 0.419964 0.907541i \(-0.362043\pi\)
0.419964 + 0.907541i \(0.362043\pi\)
\(878\) 2.45453e9 0.122387
\(879\) 1.60237e9 0.0795796
\(880\) 1.62774e9 0.0805186
\(881\) 4.60252e9 0.226767 0.113384 0.993551i \(-0.463831\pi\)
0.113384 + 0.993551i \(0.463831\pi\)
\(882\) 8.30121e9 0.407382
\(883\) −1.60291e10 −0.783515 −0.391758 0.920068i \(-0.628133\pi\)
−0.391758 + 0.920068i \(0.628133\pi\)
\(884\) 1.20332e9 0.0585864
\(885\) −8.28840e8 −0.0401948
\(886\) −2.54177e10 −1.22777
\(887\) −3.53761e10 −1.70207 −0.851034 0.525111i \(-0.824024\pi\)
−0.851034 + 0.525111i \(0.824024\pi\)
\(888\) −2.73324e8 −0.0130988
\(889\) −4.50477e8 −0.0215038
\(890\) 7.63057e9 0.362820
\(891\) 2.82599e8 0.0133844
\(892\) 8.56109e9 0.403880
\(893\) −4.05192e10 −1.90406
\(894\) 1.62362e9 0.0759983
\(895\) 1.99043e9 0.0928040
\(896\) −3.65110e8 −0.0169569
\(897\) 6.30546e9 0.291704
\(898\) 1.23584e10 0.569504
\(899\) −2.29180e9 −0.105200
\(900\) −2.57339e9 −0.117668
\(901\) 1.25183e9 0.0570177
\(902\) 3.35027e9 0.152005
\(903\) −6.82754e7 −0.00308572
\(904\) −1.57873e10 −0.710753
\(905\) −1.79754e9 −0.0806136
\(906\) 2.39971e8 0.0107204
\(907\) −6.61497e9 −0.294376 −0.147188 0.989109i \(-0.547022\pi\)
−0.147188 + 0.989109i \(0.547022\pi\)
\(908\) −1.24818e10 −0.553320
\(909\) 6.04130e9 0.266782
\(910\) 1.22975e8 0.00540967
\(911\) 1.46424e10 0.641651 0.320826 0.947138i \(-0.396040\pi\)
0.320826 + 0.947138i \(0.396040\pi\)
\(912\) 1.76057e10 0.768549
\(913\) −6.14187e8 −0.0267087
\(914\) 8.13359e9 0.352347
\(915\) −6.16503e8 −0.0266049
\(916\) 8.54297e9 0.367261
\(917\) −2.20155e8 −0.00942834
\(918\) 1.24196e9 0.0529857
\(919\) 4.03688e10 1.71570 0.857850 0.513901i \(-0.171800\pi\)
0.857850 + 0.513901i \(0.171800\pi\)
\(920\) 7.49629e9 0.317387
\(921\) −7.15371e9 −0.301733
\(922\) 3.66942e10 1.54184
\(923\) −7.46534e9 −0.312495
\(924\) −1.29672e7 −0.000540749 0
\(925\) −6.30887e8 −0.0262093
\(926\) −1.52880e10 −0.632723
\(927\) 9.76630e9 0.402672
\(928\) 8.77806e9 0.360563
\(929\) −2.36038e10 −0.965889 −0.482945 0.875651i \(-0.660433\pi\)
−0.482945 + 0.875651i \(0.660433\pi\)
\(930\) 2.45786e9 0.100200
\(931\) −2.62150e10 −1.06470
\(932\) 8.62604e9 0.349024
\(933\) −6.79766e9 −0.274015
\(934\) −9.26811e9 −0.372200
\(935\) −3.62617e8 −0.0145080
\(936\) 2.71975e9 0.108408
\(937\) 2.58838e10 1.02787 0.513937 0.857828i \(-0.328186\pi\)
0.513937 + 0.857828i \(0.328186\pi\)
\(938\) −4.07709e8 −0.0161302
\(939\) −4.82419e9 −0.190149
\(940\) 1.20368e10 0.472675
\(941\) 1.53080e10 0.598902 0.299451 0.954112i \(-0.403196\pi\)
0.299451 + 0.954112i \(0.403196\pi\)
\(942\) 1.69727e10 0.661566
\(943\) 2.55242e10 0.991201
\(944\) 4.20606e9 0.162732
\(945\) 4.19897e7 0.00161857
\(946\) −1.30302e9 −0.0500417
\(947\) 9.43008e9 0.360820 0.180410 0.983592i \(-0.442258\pi\)
0.180410 + 0.983592i \(0.442258\pi\)
\(948\) −1.09975e10 −0.419241
\(949\) 1.38789e10 0.527136
\(950\) 2.45649e10 0.929571
\(951\) −2.05890e10 −0.776253
\(952\) 5.82846e7 0.00218940
\(953\) −2.53714e10 −0.949555 −0.474777 0.880106i \(-0.657471\pi\)
−0.474777 + 0.880106i \(0.657471\pi\)
\(954\) −2.76648e9 −0.103159
\(955\) 4.42687e8 0.0164469
\(956\) 8.32100e9 0.308016
\(957\) −7.47219e8 −0.0275585
\(958\) 1.29853e9 0.0477168
\(959\) 3.90442e7 0.00142953
\(960\) 1.16486e9 0.0424937
\(961\) −2.55734e10 −0.929517
\(962\) −6.51938e8 −0.0236099
\(963\) 5.57472e9 0.201155
\(964\) 1.57978e10 0.567973
\(965\) 1.32850e9 0.0475900
\(966\) −2.98622e8 −0.0106587
\(967\) −2.23509e10 −0.794883 −0.397442 0.917627i \(-0.630102\pi\)
−0.397442 + 0.917627i \(0.630102\pi\)
\(968\) −1.71899e10 −0.609128
\(969\) −3.92208e9 −0.138479
\(970\) 1.30964e10 0.460736
\(971\) −2.51106e10 −0.880217 −0.440108 0.897945i \(-0.645060\pi\)
−0.440108 + 0.897945i \(0.645060\pi\)
\(972\) −9.08003e8 −0.0317143
\(973\) 3.15622e8 0.0109843
\(974\) −1.89758e10 −0.658028
\(975\) 6.27773e9 0.216914
\(976\) 3.12852e9 0.107712
\(977\) −1.40783e10 −0.482970 −0.241485 0.970405i \(-0.577634\pi\)
−0.241485 + 0.970405i \(0.577634\pi\)
\(978\) 3.40638e10 1.16441
\(979\) 1.96284e9 0.0668569
\(980\) 7.78751e9 0.264306
\(981\) 6.37995e9 0.215762
\(982\) −4.20155e10 −1.41586
\(983\) −4.03081e10 −1.35349 −0.676744 0.736218i \(-0.736610\pi\)
−0.676744 + 0.736218i \(0.736610\pi\)
\(984\) 1.10094e10 0.368368
\(985\) −7.59316e9 −0.253160
\(986\) −3.28387e9 −0.109098
\(987\) 4.90404e8 0.0162347
\(988\) 8.39785e9 0.277025
\(989\) −9.92713e9 −0.326315
\(990\) 8.01363e8 0.0262486
\(991\) 5.94799e10 1.94139 0.970694 0.240318i \(-0.0772517\pi\)
0.970694 + 0.240318i \(0.0772517\pi\)
\(992\) −7.42741e9 −0.241572
\(993\) −1.06411e10 −0.344876
\(994\) 3.53554e8 0.0114183
\(995\) −5.96163e9 −0.191860
\(996\) 1.97341e9 0.0632863
\(997\) 1.24113e10 0.396630 0.198315 0.980138i \(-0.436453\pi\)
0.198315 + 0.980138i \(0.436453\pi\)
\(998\) 4.76099e10 1.51614
\(999\) −2.22604e8 −0.00706405
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.8.a.c.1.4 17
3.2 odd 2 531.8.a.c.1.14 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.4 17 1.1 even 1 trivial
531.8.a.c.1.14 17 3.2 odd 2