Properties

Label 177.8.a.b.1.13
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
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: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 1.13
Root \(12.7630\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+10.7630 q^{2} +27.0000 q^{3} -12.1577 q^{4} +451.863 q^{5} +290.601 q^{6} -504.672 q^{7} -1508.52 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+10.7630 q^{2} +27.0000 q^{3} -12.1577 q^{4} +451.863 q^{5} +290.601 q^{6} -504.672 q^{7} -1508.52 q^{8} +729.000 q^{9} +4863.40 q^{10} -8070.72 q^{11} -328.258 q^{12} -4674.69 q^{13} -5431.79 q^{14} +12200.3 q^{15} -14680.0 q^{16} -13499.0 q^{17} +7846.23 q^{18} -11364.6 q^{19} -5493.61 q^{20} -13626.1 q^{21} -86865.2 q^{22} -66475.4 q^{23} -40730.0 q^{24} +126055. q^{25} -50313.7 q^{26} +19683.0 q^{27} +6135.65 q^{28} -174542. q^{29} +131312. q^{30} +62482.4 q^{31} +35089.3 q^{32} -217909. q^{33} -145289. q^{34} -228043. q^{35} -8862.96 q^{36} +277403. q^{37} -122318. q^{38} -126217. q^{39} -681643. q^{40} +529842. q^{41} -146658. q^{42} +675111. q^{43} +98121.3 q^{44} +329408. q^{45} -715475. q^{46} +175110. q^{47} -396360. q^{48} -568849. q^{49} +1.35673e6 q^{50} -364472. q^{51} +56833.4 q^{52} -313480. q^{53} +211848. q^{54} -3.64686e6 q^{55} +761307. q^{56} -306845. q^{57} -1.87860e6 q^{58} -205379. q^{59} -148328. q^{60} -3.01895e6 q^{61} +672499. q^{62} -367906. q^{63} +2.25671e6 q^{64} -2.11232e6 q^{65} -2.34536e6 q^{66} -885587. q^{67} +164116. q^{68} -1.79484e6 q^{69} -2.45442e6 q^{70} -1.15026e6 q^{71} -1.09971e6 q^{72} +33498.4 q^{73} +2.98569e6 q^{74} +3.40349e6 q^{75} +138168. q^{76} +4.07307e6 q^{77} -1.35847e6 q^{78} -694337. q^{79} -6.63335e6 q^{80} +531441. q^{81} +5.70270e6 q^{82} -672731. q^{83} +165663. q^{84} -6.09968e6 q^{85} +7.26622e6 q^{86} -4.71263e6 q^{87} +1.21748e7 q^{88} +7.67580e6 q^{89} +3.54542e6 q^{90} +2.35918e6 q^{91} +808188. q^{92} +1.68703e6 q^{93} +1.88471e6 q^{94} -5.13525e6 q^{95} +947412. q^{96} +4.89911e6 q^{97} -6.12253e6 q^{98} -5.88355e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.7630 0.951324 0.475662 0.879628i \(-0.342208\pi\)
0.475662 + 0.879628i \(0.342208\pi\)
\(3\) 27.0000 0.577350
\(4\) −12.1577 −0.0949820
\(5\) 451.863 1.61663 0.808317 0.588748i \(-0.200379\pi\)
0.808317 + 0.588748i \(0.200379\pi\)
\(6\) 290.601 0.549247
\(7\) −504.672 −0.556117 −0.278058 0.960564i \(-0.589691\pi\)
−0.278058 + 0.960564i \(0.589691\pi\)
\(8\) −1508.52 −1.04168
\(9\) 729.000 0.333333
\(10\) 4863.40 1.53794
\(11\) −8070.72 −1.82826 −0.914129 0.405422i \(-0.867124\pi\)
−0.914129 + 0.405422i \(0.867124\pi\)
\(12\) −328.258 −0.0548379
\(13\) −4674.69 −0.590134 −0.295067 0.955477i \(-0.595342\pi\)
−0.295067 + 0.955477i \(0.595342\pi\)
\(14\) −5431.79 −0.529048
\(15\) 12200.3 0.933364
\(16\) −14680.0 −0.895996
\(17\) −13499.0 −0.666391 −0.333196 0.942858i \(-0.608127\pi\)
−0.333196 + 0.942858i \(0.608127\pi\)
\(18\) 7846.23 0.317108
\(19\) −11364.6 −0.380117 −0.190059 0.981773i \(-0.560868\pi\)
−0.190059 + 0.981773i \(0.560868\pi\)
\(20\) −5493.61 −0.153551
\(21\) −13626.1 −0.321074
\(22\) −86865.2 −1.73927
\(23\) −66475.4 −1.13924 −0.569618 0.821910i \(-0.692909\pi\)
−0.569618 + 0.821910i \(0.692909\pi\)
\(24\) −40730.0 −0.601416
\(25\) 126055. 1.61350
\(26\) −50313.7 −0.561409
\(27\) 19683.0 0.192450
\(28\) 6135.65 0.0528211
\(29\) −174542. −1.32894 −0.664472 0.747313i \(-0.731343\pi\)
−0.664472 + 0.747313i \(0.731343\pi\)
\(30\) 131312. 0.887932
\(31\) 62482.4 0.376697 0.188348 0.982102i \(-0.439687\pi\)
0.188348 + 0.982102i \(0.439687\pi\)
\(32\) 35089.3 0.189300
\(33\) −217909. −1.05555
\(34\) −145289. −0.633954
\(35\) −228043. −0.899038
\(36\) −8862.96 −0.0316607
\(37\) 277403. 0.900336 0.450168 0.892944i \(-0.351364\pi\)
0.450168 + 0.892944i \(0.351364\pi\)
\(38\) −122318. −0.361615
\(39\) −126217. −0.340714
\(40\) −681643. −1.68402
\(41\) 529842. 1.20061 0.600307 0.799770i \(-0.295045\pi\)
0.600307 + 0.799770i \(0.295045\pi\)
\(42\) −146658. −0.305446
\(43\) 675111. 1.29490 0.647449 0.762109i \(-0.275836\pi\)
0.647449 + 0.762109i \(0.275836\pi\)
\(44\) 98121.3 0.173652
\(45\) 329408. 0.538878
\(46\) −715475. −1.08378
\(47\) 175110. 0.246019 0.123010 0.992405i \(-0.460745\pi\)
0.123010 + 0.992405i \(0.460745\pi\)
\(48\) −396360. −0.517304
\(49\) −568849. −0.690734
\(50\) 1.35673e6 1.53497
\(51\) −364472. −0.384741
\(52\) 56833.4 0.0560521
\(53\) −313480. −0.289231 −0.144615 0.989488i \(-0.546195\pi\)
−0.144615 + 0.989488i \(0.546195\pi\)
\(54\) 211848. 0.183082
\(55\) −3.64686e6 −2.95563
\(56\) 761307. 0.579298
\(57\) −306845. −0.219461
\(58\) −1.87860e6 −1.26426
\(59\) −205379. −0.130189
\(60\) −148328. −0.0886528
\(61\) −3.01895e6 −1.70295 −0.851474 0.524396i \(-0.824291\pi\)
−0.851474 + 0.524396i \(0.824291\pi\)
\(62\) 672499. 0.358361
\(63\) −367906. −0.185372
\(64\) 2.25671e6 1.07608
\(65\) −2.11232e6 −0.954031
\(66\) −2.34536e6 −1.00417
\(67\) −885587. −0.359724 −0.179862 0.983692i \(-0.557565\pi\)
−0.179862 + 0.983692i \(0.557565\pi\)
\(68\) 164116. 0.0632952
\(69\) −1.79484e6 −0.657738
\(70\) −2.45442e6 −0.855276
\(71\) −1.15026e6 −0.381411 −0.190706 0.981647i \(-0.561078\pi\)
−0.190706 + 0.981647i \(0.561078\pi\)
\(72\) −1.09971e6 −0.347228
\(73\) 33498.4 0.0100785 0.00503923 0.999987i \(-0.498396\pi\)
0.00503923 + 0.999987i \(0.498396\pi\)
\(74\) 2.98569e6 0.856511
\(75\) 3.40349e6 0.931558
\(76\) 138168. 0.0361043
\(77\) 4.07307e6 1.01673
\(78\) −1.35847e6 −0.324130
\(79\) −694337. −0.158444 −0.0792219 0.996857i \(-0.525244\pi\)
−0.0792219 + 0.996857i \(0.525244\pi\)
\(80\) −6.63335e6 −1.44850
\(81\) 531441. 0.111111
\(82\) 5.70270e6 1.14217
\(83\) −672731. −0.129142 −0.0645711 0.997913i \(-0.520568\pi\)
−0.0645711 + 0.997913i \(0.520568\pi\)
\(84\) 165663. 0.0304963
\(85\) −6.09968e6 −1.07731
\(86\) 7.26622e6 1.23187
\(87\) −4.71263e6 −0.767266
\(88\) 1.21748e7 1.90447
\(89\) 7.67580e6 1.15414 0.577070 0.816695i \(-0.304196\pi\)
0.577070 + 0.816695i \(0.304196\pi\)
\(90\) 3.54542e6 0.512648
\(91\) 2.35918e6 0.328184
\(92\) 808188. 0.108207
\(93\) 1.68703e6 0.217486
\(94\) 1.88471e6 0.234044
\(95\) −5.13525e6 −0.614510
\(96\) 947412. 0.109292
\(97\) 4.89911e6 0.545025 0.272512 0.962152i \(-0.412145\pi\)
0.272512 + 0.962152i \(0.412145\pi\)
\(98\) −6.12253e6 −0.657112
\(99\) −5.88355e6 −0.609420
\(100\) −1.53254e6 −0.153254
\(101\) −1.84644e7 −1.78324 −0.891621 0.452782i \(-0.850432\pi\)
−0.891621 + 0.452782i \(0.850432\pi\)
\(102\) −3.92281e6 −0.366014
\(103\) −5.21469e6 −0.470217 −0.235109 0.971969i \(-0.575545\pi\)
−0.235109 + 0.971969i \(0.575545\pi\)
\(104\) 7.05185e6 0.614733
\(105\) −6.15715e6 −0.519060
\(106\) −3.37399e6 −0.275152
\(107\) −2.16312e6 −0.170701 −0.0853506 0.996351i \(-0.527201\pi\)
−0.0853506 + 0.996351i \(0.527201\pi\)
\(108\) −239300. −0.0182793
\(109\) −1.59770e7 −1.18169 −0.590845 0.806785i \(-0.701206\pi\)
−0.590845 + 0.806785i \(0.701206\pi\)
\(110\) −3.92511e7 −2.81176
\(111\) 7.48987e6 0.519809
\(112\) 7.40859e6 0.498279
\(113\) 9.29263e6 0.605849 0.302924 0.953015i \(-0.402037\pi\)
0.302924 + 0.953015i \(0.402037\pi\)
\(114\) −3.30257e6 −0.208778
\(115\) −3.00378e7 −1.84173
\(116\) 2.12203e6 0.126226
\(117\) −3.40785e6 −0.196711
\(118\) −2.21050e6 −0.123852
\(119\) 6.81255e6 0.370591
\(120\) −1.84044e7 −0.972269
\(121\) 4.56493e7 2.34253
\(122\) −3.24930e7 −1.62006
\(123\) 1.43057e7 0.693175
\(124\) −759642. −0.0357794
\(125\) 2.16578e7 0.991813
\(126\) −3.95977e6 −0.176349
\(127\) 8.17874e6 0.354302 0.177151 0.984184i \(-0.443312\pi\)
0.177151 + 0.984184i \(0.443312\pi\)
\(128\) 1.97975e7 0.834403
\(129\) 1.82280e7 0.747610
\(130\) −2.27349e7 −0.907593
\(131\) 4.40912e6 0.171357 0.0856787 0.996323i \(-0.472694\pi\)
0.0856787 + 0.996323i \(0.472694\pi\)
\(132\) 2.64928e6 0.100258
\(133\) 5.73541e6 0.211390
\(134\) −9.53158e6 −0.342214
\(135\) 8.89402e6 0.311121
\(136\) 2.03634e7 0.694168
\(137\) −8.21867e6 −0.273073 −0.136537 0.990635i \(-0.543597\pi\)
−0.136537 + 0.990635i \(0.543597\pi\)
\(138\) −1.93178e7 −0.625722
\(139\) −5.56805e7 −1.75854 −0.879268 0.476328i \(-0.841968\pi\)
−0.879268 + 0.476328i \(0.841968\pi\)
\(140\) 2.77247e6 0.0853924
\(141\) 4.72798e6 0.142039
\(142\) −1.23803e7 −0.362846
\(143\) 3.77281e7 1.07892
\(144\) −1.07017e7 −0.298665
\(145\) −7.88690e7 −2.14842
\(146\) 360544. 0.00958789
\(147\) −1.53589e7 −0.398795
\(148\) −3.37258e6 −0.0855157
\(149\) 2.85493e7 0.707039 0.353520 0.935427i \(-0.384985\pi\)
0.353520 + 0.935427i \(0.384985\pi\)
\(150\) 3.66318e7 0.886213
\(151\) −1.41537e7 −0.334541 −0.167271 0.985911i \(-0.553495\pi\)
−0.167271 + 0.985911i \(0.553495\pi\)
\(152\) 1.71437e7 0.395962
\(153\) −9.84074e6 −0.222130
\(154\) 4.38384e7 0.967236
\(155\) 2.82335e7 0.608981
\(156\) 1.53450e6 0.0323617
\(157\) 6.04002e7 1.24563 0.622816 0.782368i \(-0.285989\pi\)
0.622816 + 0.782368i \(0.285989\pi\)
\(158\) −7.47315e6 −0.150732
\(159\) −8.46397e6 −0.166988
\(160\) 1.58556e7 0.306029
\(161\) 3.35483e7 0.633548
\(162\) 5.71990e6 0.105703
\(163\) 7.29006e7 1.31848 0.659241 0.751931i \(-0.270878\pi\)
0.659241 + 0.751931i \(0.270878\pi\)
\(164\) −6.44166e6 −0.114037
\(165\) −9.84652e7 −1.70643
\(166\) −7.24061e6 −0.122856
\(167\) −9.50243e7 −1.57880 −0.789400 0.613879i \(-0.789608\pi\)
−0.789400 + 0.613879i \(0.789608\pi\)
\(168\) 2.05553e7 0.334458
\(169\) −4.08958e7 −0.651742
\(170\) −6.56509e7 −1.02487
\(171\) −8.28481e6 −0.126706
\(172\) −8.20779e6 −0.122992
\(173\) 5.86358e7 0.860996 0.430498 0.902592i \(-0.358338\pi\)
0.430498 + 0.902592i \(0.358338\pi\)
\(174\) −5.07221e7 −0.729919
\(175\) −6.36165e7 −0.897298
\(176\) 1.18478e8 1.63811
\(177\) −5.54523e6 −0.0751646
\(178\) 8.26147e7 1.09796
\(179\) −3.55324e7 −0.463061 −0.231531 0.972828i \(-0.574373\pi\)
−0.231531 + 0.972828i \(0.574373\pi\)
\(180\) −4.00484e6 −0.0511837
\(181\) 1.28354e8 1.60892 0.804459 0.594008i \(-0.202455\pi\)
0.804459 + 0.594008i \(0.202455\pi\)
\(182\) 2.53919e7 0.312209
\(183\) −8.15117e7 −0.983198
\(184\) 1.00279e8 1.18672
\(185\) 1.25348e8 1.45551
\(186\) 1.81575e7 0.206900
\(187\) 1.08946e8 1.21834
\(188\) −2.12894e6 −0.0233674
\(189\) −9.93346e6 −0.107025
\(190\) −5.52708e7 −0.584599
\(191\) −8.71206e7 −0.904699 −0.452350 0.891841i \(-0.649414\pi\)
−0.452350 + 0.891841i \(0.649414\pi\)
\(192\) 6.09311e7 0.621276
\(193\) 3.34478e7 0.334902 0.167451 0.985880i \(-0.446446\pi\)
0.167451 + 0.985880i \(0.446446\pi\)
\(194\) 5.27292e7 0.518495
\(195\) −5.70326e7 −0.550810
\(196\) 6.91589e6 0.0656073
\(197\) 1.43873e8 1.34074 0.670372 0.742025i \(-0.266134\pi\)
0.670372 + 0.742025i \(0.266134\pi\)
\(198\) −6.33247e7 −0.579756
\(199\) −3.39107e7 −0.305036 −0.152518 0.988301i \(-0.548738\pi\)
−0.152518 + 0.988301i \(0.548738\pi\)
\(200\) −1.90156e8 −1.68076
\(201\) −2.39109e7 −0.207687
\(202\) −1.98732e8 −1.69644
\(203\) 8.80864e7 0.739048
\(204\) 4.43114e6 0.0365435
\(205\) 2.39416e8 1.94095
\(206\) −5.61258e7 −0.447329
\(207\) −4.84606e7 −0.379745
\(208\) 6.86244e7 0.528758
\(209\) 9.17207e7 0.694953
\(210\) −6.62694e7 −0.493794
\(211\) −1.13428e8 −0.831249 −0.415624 0.909536i \(-0.636437\pi\)
−0.415624 + 0.909536i \(0.636437\pi\)
\(212\) 3.81120e6 0.0274717
\(213\) −3.10572e7 −0.220208
\(214\) −2.32816e7 −0.162392
\(215\) 3.05058e8 2.09338
\(216\) −2.96922e7 −0.200472
\(217\) −3.15331e7 −0.209487
\(218\) −1.71961e8 −1.12417
\(219\) 904458. 0.00581881
\(220\) 4.43374e7 0.280731
\(221\) 6.31034e7 0.393260
\(222\) 8.06135e7 0.494507
\(223\) −1.58906e8 −0.959565 −0.479782 0.877387i \(-0.659284\pi\)
−0.479782 + 0.877387i \(0.659284\pi\)
\(224\) −1.77086e7 −0.105273
\(225\) 9.18942e7 0.537835
\(226\) 1.00017e8 0.576358
\(227\) −2.46698e7 −0.139983 −0.0699915 0.997548i \(-0.522297\pi\)
−0.0699915 + 0.997548i \(0.522297\pi\)
\(228\) 3.73053e6 0.0208448
\(229\) 2.79166e6 0.0153617 0.00768084 0.999971i \(-0.497555\pi\)
0.00768084 + 0.999971i \(0.497555\pi\)
\(230\) −3.23297e8 −1.75208
\(231\) 1.09973e8 0.587007
\(232\) 2.63300e8 1.38434
\(233\) 1.72078e8 0.891207 0.445604 0.895230i \(-0.352989\pi\)
0.445604 + 0.895230i \(0.352989\pi\)
\(234\) −3.66787e7 −0.187136
\(235\) 7.91259e7 0.397723
\(236\) 2.49694e6 0.0123656
\(237\) −1.87471e7 −0.0914776
\(238\) 7.33235e7 0.352553
\(239\) −3.26143e8 −1.54531 −0.772656 0.634825i \(-0.781072\pi\)
−0.772656 + 0.634825i \(0.781072\pi\)
\(240\) −1.79100e8 −0.836291
\(241\) 4.12780e8 1.89958 0.949792 0.312882i \(-0.101294\pi\)
0.949792 + 0.312882i \(0.101294\pi\)
\(242\) 4.91324e8 2.22851
\(243\) 1.43489e7 0.0641500
\(244\) 3.67035e7 0.161750
\(245\) −2.57042e8 −1.11666
\(246\) 1.53973e8 0.659434
\(247\) 5.31261e7 0.224320
\(248\) −9.42559e7 −0.392399
\(249\) −1.81637e7 −0.0745603
\(250\) 2.33103e8 0.943536
\(251\) 4.66684e8 1.86279 0.931397 0.364005i \(-0.118591\pi\)
0.931397 + 0.364005i \(0.118591\pi\)
\(252\) 4.47289e6 0.0176070
\(253\) 5.36504e8 2.08282
\(254\) 8.80278e7 0.337056
\(255\) −1.64691e8 −0.621985
\(256\) −7.57777e7 −0.282294
\(257\) −9.36406e7 −0.344111 −0.172055 0.985087i \(-0.555041\pi\)
−0.172055 + 0.985087i \(0.555041\pi\)
\(258\) 1.96188e8 0.711219
\(259\) −1.39997e8 −0.500692
\(260\) 2.56809e7 0.0906158
\(261\) −1.27241e8 −0.442981
\(262\) 4.74554e7 0.163016
\(263\) −5.00867e8 −1.69776 −0.848882 0.528582i \(-0.822724\pi\)
−0.848882 + 0.528582i \(0.822724\pi\)
\(264\) 3.28720e8 1.09954
\(265\) −1.41650e8 −0.467581
\(266\) 6.17303e7 0.201100
\(267\) 2.07247e8 0.666343
\(268\) 1.07667e7 0.0341673
\(269\) 7.84305e7 0.245670 0.122835 0.992427i \(-0.460801\pi\)
0.122835 + 0.992427i \(0.460801\pi\)
\(270\) 9.57264e7 0.295977
\(271\) 3.97558e8 1.21341 0.606705 0.794927i \(-0.292491\pi\)
0.606705 + 0.794927i \(0.292491\pi\)
\(272\) 1.98165e8 0.597084
\(273\) 6.36980e7 0.189477
\(274\) −8.84576e7 −0.259781
\(275\) −1.01735e9 −2.94990
\(276\) 2.18211e7 0.0624733
\(277\) −9.64793e7 −0.272744 −0.136372 0.990658i \(-0.543544\pi\)
−0.136372 + 0.990658i \(0.543544\pi\)
\(278\) −5.99289e8 −1.67294
\(279\) 4.55497e7 0.125566
\(280\) 3.44006e8 0.936512
\(281\) −5.37086e8 −1.44402 −0.722008 0.691885i \(-0.756781\pi\)
−0.722008 + 0.691885i \(0.756781\pi\)
\(282\) 5.08873e7 0.135125
\(283\) −6.18129e8 −1.62116 −0.810582 0.585626i \(-0.800849\pi\)
−0.810582 + 0.585626i \(0.800849\pi\)
\(284\) 1.39846e7 0.0362272
\(285\) −1.38652e8 −0.354788
\(286\) 4.06067e8 1.02640
\(287\) −2.67397e8 −0.667682
\(288\) 2.55801e7 0.0630999
\(289\) −2.28117e8 −0.555923
\(290\) −8.48867e8 −2.04384
\(291\) 1.32276e8 0.314670
\(292\) −407264. −0.000957273 0
\(293\) 3.39223e8 0.787858 0.393929 0.919141i \(-0.371116\pi\)
0.393929 + 0.919141i \(0.371116\pi\)
\(294\) −1.65308e8 −0.379384
\(295\) −9.28031e7 −0.210468
\(296\) −4.18467e8 −0.937864
\(297\) −1.58856e8 −0.351849
\(298\) 3.07276e8 0.672623
\(299\) 3.10752e8 0.672302
\(300\) −4.13786e7 −0.0884812
\(301\) −3.40710e8 −0.720115
\(302\) −1.52336e8 −0.318257
\(303\) −4.98539e8 −1.02956
\(304\) 1.66833e8 0.340584
\(305\) −1.36415e9 −2.75304
\(306\) −1.05916e8 −0.211318
\(307\) −9.62816e7 −0.189915 −0.0949575 0.995481i \(-0.530272\pi\)
−0.0949575 + 0.995481i \(0.530272\pi\)
\(308\) −4.95191e7 −0.0965707
\(309\) −1.40797e8 −0.271480
\(310\) 3.03877e8 0.579338
\(311\) −7.91196e8 −1.49150 −0.745749 0.666227i \(-0.767908\pi\)
−0.745749 + 0.666227i \(0.767908\pi\)
\(312\) 1.90400e8 0.354916
\(313\) −8.83580e7 −0.162870 −0.0814349 0.996679i \(-0.525950\pi\)
−0.0814349 + 0.996679i \(0.525950\pi\)
\(314\) 6.50088e8 1.18500
\(315\) −1.66243e8 −0.299679
\(316\) 8.44154e6 0.0150493
\(317\) −3.65056e8 −0.643654 −0.321827 0.946799i \(-0.604297\pi\)
−0.321827 + 0.946799i \(0.604297\pi\)
\(318\) −9.10978e7 −0.158859
\(319\) 1.40868e9 2.42965
\(320\) 1.01972e9 1.73963
\(321\) −5.84041e7 −0.0985544
\(322\) 3.61080e8 0.602710
\(323\) 1.53411e8 0.253307
\(324\) −6.46110e6 −0.0105536
\(325\) −5.89268e8 −0.952184
\(326\) 7.84630e8 1.25430
\(327\) −4.31380e8 −0.682249
\(328\) −7.99277e8 −1.25066
\(329\) −8.83733e7 −0.136816
\(330\) −1.05978e9 −1.62337
\(331\) 8.84610e8 1.34077 0.670384 0.742014i \(-0.266129\pi\)
0.670384 + 0.742014i \(0.266129\pi\)
\(332\) 8.17886e6 0.0122662
\(333\) 2.02226e8 0.300112
\(334\) −1.02275e9 −1.50195
\(335\) −4.00164e8 −0.581542
\(336\) 2.00032e8 0.287681
\(337\) −1.01312e9 −1.44197 −0.720983 0.692953i \(-0.756309\pi\)
−0.720983 + 0.692953i \(0.756309\pi\)
\(338\) −4.40162e8 −0.620018
\(339\) 2.50901e8 0.349787
\(340\) 7.41581e7 0.102325
\(341\) −5.04278e8 −0.688699
\(342\) −8.91695e7 −0.120538
\(343\) 7.02701e8 0.940246
\(344\) −1.01842e9 −1.34887
\(345\) −8.11019e8 −1.06332
\(346\) 6.31097e8 0.819086
\(347\) −4.59045e8 −0.589796 −0.294898 0.955529i \(-0.595286\pi\)
−0.294898 + 0.955529i \(0.595286\pi\)
\(348\) 5.72947e7 0.0728765
\(349\) 6.65833e8 0.838448 0.419224 0.907883i \(-0.362302\pi\)
0.419224 + 0.907883i \(0.362302\pi\)
\(350\) −6.84705e8 −0.853621
\(351\) −9.20118e7 −0.113571
\(352\) −2.83196e8 −0.346089
\(353\) −7.46486e7 −0.0903255 −0.0451627 0.998980i \(-0.514381\pi\)
−0.0451627 + 0.998980i \(0.514381\pi\)
\(354\) −5.96834e7 −0.0715059
\(355\) −5.19762e8 −0.616603
\(356\) −9.33201e7 −0.109623
\(357\) 1.83939e8 0.213961
\(358\) −3.82435e8 −0.440521
\(359\) −1.55696e9 −1.77601 −0.888005 0.459833i \(-0.847909\pi\)
−0.888005 + 0.459833i \(0.847909\pi\)
\(360\) −4.96918e8 −0.561340
\(361\) −7.64717e8 −0.855511
\(362\) 1.38147e9 1.53060
\(363\) 1.23253e9 1.35246
\(364\) −2.86822e7 −0.0311715
\(365\) 1.51367e7 0.0162932
\(366\) −8.77311e8 −0.935340
\(367\) −4.78302e7 −0.0505092 −0.0252546 0.999681i \(-0.508040\pi\)
−0.0252546 + 0.999681i \(0.508040\pi\)
\(368\) 9.75859e8 1.02075
\(369\) 3.86255e8 0.400205
\(370\) 1.34912e9 1.38467
\(371\) 1.58205e8 0.160846
\(372\) −2.05103e7 −0.0206573
\(373\) −1.46088e9 −1.45759 −0.728793 0.684735i \(-0.759918\pi\)
−0.728793 + 0.684735i \(0.759918\pi\)
\(374\) 1.17259e9 1.15903
\(375\) 5.84761e8 0.572623
\(376\) −2.64157e8 −0.256274
\(377\) 8.15928e8 0.784255
\(378\) −1.06914e8 −0.101815
\(379\) 8.99699e8 0.848907 0.424453 0.905450i \(-0.360466\pi\)
0.424453 + 0.905450i \(0.360466\pi\)
\(380\) 6.24329e7 0.0583674
\(381\) 2.20826e8 0.204556
\(382\) −9.37680e8 −0.860662
\(383\) 7.51806e8 0.683770 0.341885 0.939742i \(-0.388935\pi\)
0.341885 + 0.939742i \(0.388935\pi\)
\(384\) 5.34533e8 0.481743
\(385\) 1.84047e9 1.64367
\(386\) 3.59999e8 0.318600
\(387\) 4.92156e8 0.431633
\(388\) −5.95619e7 −0.0517675
\(389\) 1.98944e9 1.71359 0.856794 0.515660i \(-0.172453\pi\)
0.856794 + 0.515660i \(0.172453\pi\)
\(390\) −6.13842e8 −0.523999
\(391\) 8.97349e8 0.759176
\(392\) 8.58119e8 0.719526
\(393\) 1.19046e8 0.0989332
\(394\) 1.54850e9 1.27548
\(395\) −3.13745e8 −0.256146
\(396\) 7.15305e7 0.0578839
\(397\) −2.39171e9 −1.91841 −0.959207 0.282704i \(-0.908769\pi\)
−0.959207 + 0.282704i \(0.908769\pi\)
\(398\) −3.64981e8 −0.290188
\(399\) 1.54856e8 0.122046
\(400\) −1.85049e9 −1.44569
\(401\) 1.23670e9 0.957767 0.478883 0.877879i \(-0.341042\pi\)
0.478883 + 0.877879i \(0.341042\pi\)
\(402\) −2.57353e8 −0.197577
\(403\) −2.92086e8 −0.222302
\(404\) 2.24485e8 0.169376
\(405\) 2.40138e8 0.179626
\(406\) 9.48075e8 0.703075
\(407\) −2.23884e9 −1.64605
\(408\) 5.49813e8 0.400778
\(409\) 1.08001e9 0.780545 0.390272 0.920699i \(-0.372381\pi\)
0.390272 + 0.920699i \(0.372381\pi\)
\(410\) 2.57684e9 1.84648
\(411\) −2.21904e8 −0.157659
\(412\) 6.33986e7 0.0446622
\(413\) 1.03649e8 0.0724003
\(414\) −5.21581e8 −0.361261
\(415\) −3.03982e8 −0.208776
\(416\) −1.64032e8 −0.111712
\(417\) −1.50337e9 −1.01529
\(418\) 9.87190e8 0.661125
\(419\) −1.95040e9 −1.29531 −0.647656 0.761933i \(-0.724251\pi\)
−0.647656 + 0.761933i \(0.724251\pi\)
\(420\) 7.48568e7 0.0493013
\(421\) 4.11047e8 0.268475 0.134238 0.990949i \(-0.457141\pi\)
0.134238 + 0.990949i \(0.457141\pi\)
\(422\) −1.22082e9 −0.790787
\(423\) 1.27655e8 0.0820065
\(424\) 4.72891e8 0.301287
\(425\) −1.70161e9 −1.07523
\(426\) −3.34268e8 −0.209489
\(427\) 1.52358e9 0.947039
\(428\) 2.62985e7 0.0162135
\(429\) 1.01866e9 0.622914
\(430\) 3.28334e9 1.99148
\(431\) −2.36621e9 −1.42358 −0.711792 0.702390i \(-0.752116\pi\)
−0.711792 + 0.702390i \(0.752116\pi\)
\(432\) −2.88947e8 −0.172435
\(433\) −2.22967e8 −0.131987 −0.0659937 0.997820i \(-0.521022\pi\)
−0.0659937 + 0.997820i \(0.521022\pi\)
\(434\) −3.39391e8 −0.199291
\(435\) −2.12946e9 −1.24039
\(436\) 1.94244e8 0.112239
\(437\) 7.55468e8 0.433043
\(438\) 9.73469e6 0.00553557
\(439\) −6.34602e8 −0.357994 −0.178997 0.983850i \(-0.557285\pi\)
−0.178997 + 0.983850i \(0.557285\pi\)
\(440\) 5.50135e9 3.07882
\(441\) −4.14691e8 −0.230245
\(442\) 6.79182e8 0.374118
\(443\) −1.44884e9 −0.791788 −0.395894 0.918296i \(-0.629565\pi\)
−0.395894 + 0.918296i \(0.629565\pi\)
\(444\) −9.10596e7 −0.0493725
\(445\) 3.46841e9 1.86582
\(446\) −1.71031e9 −0.912857
\(447\) 7.70831e8 0.408209
\(448\) −1.13890e9 −0.598427
\(449\) 3.21927e9 1.67840 0.839199 0.543825i \(-0.183024\pi\)
0.839199 + 0.543825i \(0.183024\pi\)
\(450\) 9.89057e8 0.511656
\(451\) −4.27621e9 −2.19503
\(452\) −1.12977e8 −0.0575447
\(453\) −3.82149e8 −0.193147
\(454\) −2.65521e8 −0.133169
\(455\) 1.06603e9 0.530553
\(456\) 4.62881e8 0.228609
\(457\) −1.03131e9 −0.505454 −0.252727 0.967538i \(-0.581328\pi\)
−0.252727 + 0.967538i \(0.581328\pi\)
\(458\) 3.00467e7 0.0146139
\(459\) −2.65700e8 −0.128247
\(460\) 3.65190e8 0.174931
\(461\) 3.21305e9 1.52744 0.763720 0.645547i \(-0.223371\pi\)
0.763720 + 0.645547i \(0.223371\pi\)
\(462\) 1.18364e9 0.558434
\(463\) −1.66945e9 −0.781698 −0.390849 0.920455i \(-0.627819\pi\)
−0.390849 + 0.920455i \(0.627819\pi\)
\(464\) 2.56228e9 1.19073
\(465\) 7.62304e8 0.351595
\(466\) 1.85207e9 0.847827
\(467\) 3.79498e8 0.172425 0.0862125 0.996277i \(-0.472524\pi\)
0.0862125 + 0.996277i \(0.472524\pi\)
\(468\) 4.14316e7 0.0186840
\(469\) 4.46931e8 0.200049
\(470\) 8.51632e8 0.378364
\(471\) 1.63081e9 0.719166
\(472\) 3.09818e8 0.135616
\(473\) −5.44863e9 −2.36741
\(474\) −2.01775e8 −0.0870249
\(475\) −1.43257e9 −0.613321
\(476\) −8.28249e7 −0.0351995
\(477\) −2.28527e8 −0.0964103
\(478\) −3.51028e9 −1.47009
\(479\) 1.60692e9 0.668068 0.334034 0.942561i \(-0.391590\pi\)
0.334034 + 0.942561i \(0.391590\pi\)
\(480\) 4.28100e8 0.176686
\(481\) −1.29677e9 −0.531319
\(482\) 4.44275e9 1.80712
\(483\) 9.05803e8 0.365779
\(484\) −5.54990e8 −0.222498
\(485\) 2.21373e9 0.881105
\(486\) 1.54437e8 0.0610275
\(487\) 4.45492e9 1.74779 0.873893 0.486118i \(-0.161587\pi\)
0.873893 + 0.486118i \(0.161587\pi\)
\(488\) 4.55414e9 1.77393
\(489\) 1.96832e9 0.761226
\(490\) −2.76654e9 −1.06231
\(491\) −3.60524e9 −1.37451 −0.687256 0.726415i \(-0.741185\pi\)
−0.687256 + 0.726415i \(0.741185\pi\)
\(492\) −1.73925e8 −0.0658391
\(493\) 2.35613e9 0.885596
\(494\) 5.71796e8 0.213401
\(495\) −2.65856e9 −0.985208
\(496\) −9.17242e8 −0.337519
\(497\) 5.80507e8 0.212109
\(498\) −1.95496e8 −0.0709310
\(499\) −2.51714e9 −0.906893 −0.453446 0.891284i \(-0.649806\pi\)
−0.453446 + 0.891284i \(0.649806\pi\)
\(500\) −2.63309e8 −0.0942044
\(501\) −2.56566e9 −0.911520
\(502\) 5.02292e9 1.77212
\(503\) 3.07735e9 1.07817 0.539087 0.842250i \(-0.318770\pi\)
0.539087 + 0.842250i \(0.318770\pi\)
\(504\) 5.54993e8 0.193099
\(505\) −8.34338e9 −2.88285
\(506\) 5.77440e9 1.98143
\(507\) −1.10419e9 −0.376283
\(508\) −9.94346e7 −0.0336523
\(509\) −5.06424e9 −1.70217 −0.851083 0.525031i \(-0.824054\pi\)
−0.851083 + 0.525031i \(0.824054\pi\)
\(510\) −1.77257e9 −0.591710
\(511\) −1.69057e7 −0.00560481
\(512\) −3.34968e9 −1.10296
\(513\) −2.23690e8 −0.0731536
\(514\) −1.00785e9 −0.327361
\(515\) −2.35633e9 −0.760169
\(516\) −2.21610e8 −0.0710095
\(517\) −1.41327e9 −0.449787
\(518\) −1.50679e9 −0.476320
\(519\) 1.58317e9 0.497096
\(520\) 3.18647e9 0.993798
\(521\) −3.99436e9 −1.23741 −0.618707 0.785622i \(-0.712343\pi\)
−0.618707 + 0.785622i \(0.712343\pi\)
\(522\) −1.36950e9 −0.421419
\(523\) 6.31987e9 1.93175 0.965877 0.259001i \(-0.0833931\pi\)
0.965877 + 0.259001i \(0.0833931\pi\)
\(524\) −5.36048e7 −0.0162759
\(525\) −1.71764e9 −0.518055
\(526\) −5.39084e9 −1.61513
\(527\) −8.43448e8 −0.251027
\(528\) 3.19891e9 0.945765
\(529\) 1.01415e9 0.297857
\(530\) −1.52458e9 −0.444821
\(531\) −1.49721e8 −0.0433963
\(532\) −6.97294e7 −0.0200782
\(533\) −2.47685e9 −0.708523
\(534\) 2.23060e9 0.633909
\(535\) −9.77432e8 −0.275961
\(536\) 1.33592e9 0.374718
\(537\) −9.59374e8 −0.267349
\(538\) 8.44148e8 0.233712
\(539\) 4.59102e9 1.26284
\(540\) −1.08131e8 −0.0295509
\(541\) −4.36436e9 −1.18503 −0.592516 0.805559i \(-0.701865\pi\)
−0.592516 + 0.805559i \(0.701865\pi\)
\(542\) 4.27892e9 1.15435
\(543\) 3.46556e9 0.928910
\(544\) −4.73670e8 −0.126148
\(545\) −7.21943e9 −1.91036
\(546\) 6.85581e8 0.180254
\(547\) −2.77236e9 −0.724258 −0.362129 0.932128i \(-0.617950\pi\)
−0.362129 + 0.932128i \(0.617950\pi\)
\(548\) 9.99201e7 0.0259371
\(549\) −2.20082e9 −0.567650
\(550\) −1.09498e10 −2.80632
\(551\) 1.98360e9 0.505154
\(552\) 2.70754e9 0.685154
\(553\) 3.50412e8 0.0881133
\(554\) −1.03841e9 −0.259468
\(555\) 3.38439e9 0.840341
\(556\) 6.76946e8 0.167029
\(557\) 6.99835e9 1.71594 0.857970 0.513699i \(-0.171725\pi\)
0.857970 + 0.513699i \(0.171725\pi\)
\(558\) 4.90252e8 0.119454
\(559\) −3.15593e9 −0.764163
\(560\) 3.34767e9 0.805534
\(561\) 2.94155e9 0.703406
\(562\) −5.78066e9 −1.37373
\(563\) 1.13060e9 0.267012 0.133506 0.991048i \(-0.457376\pi\)
0.133506 + 0.991048i \(0.457376\pi\)
\(564\) −5.74813e7 −0.0134912
\(565\) 4.19899e9 0.979435
\(566\) −6.65293e9 −1.54225
\(567\) −2.68203e8 −0.0617908
\(568\) 1.73520e9 0.397310
\(569\) −4.34076e9 −0.987810 −0.493905 0.869516i \(-0.664431\pi\)
−0.493905 + 0.869516i \(0.664431\pi\)
\(570\) −1.49231e9 −0.337518
\(571\) 1.60830e9 0.361526 0.180763 0.983527i \(-0.442143\pi\)
0.180763 + 0.983527i \(0.442143\pi\)
\(572\) −4.58686e8 −0.102478
\(573\) −2.35226e9 −0.522328
\(574\) −2.87799e9 −0.635182
\(575\) −8.37956e9 −1.83816
\(576\) 1.64514e9 0.358694
\(577\) 4.44207e9 0.962654 0.481327 0.876541i \(-0.340155\pi\)
0.481327 + 0.876541i \(0.340155\pi\)
\(578\) −2.45522e9 −0.528863
\(579\) 9.03091e8 0.193355
\(580\) 9.58865e8 0.204061
\(581\) 3.39509e8 0.0718181
\(582\) 1.42369e9 0.299353
\(583\) 2.53001e9 0.528789
\(584\) −5.05330e7 −0.0104986
\(585\) −1.53988e9 −0.318010
\(586\) 3.65106e9 0.749509
\(587\) 5.12698e8 0.104623 0.0523116 0.998631i \(-0.483341\pi\)
0.0523116 + 0.998631i \(0.483341\pi\)
\(588\) 1.86729e8 0.0378784
\(589\) −7.10089e8 −0.143189
\(590\) −9.98841e8 −0.200223
\(591\) 3.88456e9 0.774079
\(592\) −4.07227e9 −0.806698
\(593\) 4.79247e9 0.943774 0.471887 0.881659i \(-0.343573\pi\)
0.471887 + 0.881659i \(0.343573\pi\)
\(594\) −1.70977e9 −0.334722
\(595\) 3.07834e9 0.599111
\(596\) −3.47094e8 −0.0671560
\(597\) −9.15589e8 −0.176113
\(598\) 3.34462e9 0.639577
\(599\) 6.08958e9 1.15769 0.578846 0.815437i \(-0.303503\pi\)
0.578846 + 0.815437i \(0.303503\pi\)
\(600\) −5.13422e9 −0.970388
\(601\) −5.35978e9 −1.00713 −0.503566 0.863957i \(-0.667979\pi\)
−0.503566 + 0.863957i \(0.667979\pi\)
\(602\) −3.66706e9 −0.685063
\(603\) −6.45593e8 −0.119908
\(604\) 1.72076e8 0.0317754
\(605\) 2.06272e10 3.78701
\(606\) −5.36578e9 −0.979441
\(607\) 9.02137e9 1.63724 0.818619 0.574337i \(-0.194740\pi\)
0.818619 + 0.574337i \(0.194740\pi\)
\(608\) −3.98777e8 −0.0719561
\(609\) 2.37833e9 0.426690
\(610\) −1.46824e10 −2.61904
\(611\) −8.18586e8 −0.145184
\(612\) 1.19641e8 0.0210984
\(613\) 7.72450e9 1.35444 0.677219 0.735782i \(-0.263185\pi\)
0.677219 + 0.735782i \(0.263185\pi\)
\(614\) −1.03628e9 −0.180671
\(615\) 6.46424e9 1.12061
\(616\) −6.14429e9 −1.05911
\(617\) −4.39185e9 −0.752748 −0.376374 0.926468i \(-0.622829\pi\)
−0.376374 + 0.926468i \(0.622829\pi\)
\(618\) −1.51540e9 −0.258265
\(619\) 3.33323e9 0.564870 0.282435 0.959286i \(-0.408858\pi\)
0.282435 + 0.959286i \(0.408858\pi\)
\(620\) −3.43254e8 −0.0578422
\(621\) −1.30843e9 −0.219246
\(622\) −8.51565e9 −1.41890
\(623\) −3.87376e9 −0.641837
\(624\) 1.85286e9 0.305279
\(625\) −6.16859e7 −0.0101066
\(626\) −9.50998e8 −0.154942
\(627\) 2.47646e9 0.401231
\(628\) −7.34327e8 −0.118313
\(629\) −3.74465e9 −0.599976
\(630\) −1.78928e9 −0.285092
\(631\) −7.87439e9 −1.24771 −0.623856 0.781540i \(-0.714435\pi\)
−0.623856 + 0.781540i \(0.714435\pi\)
\(632\) 1.04742e9 0.165048
\(633\) −3.06255e9 −0.479922
\(634\) −3.92910e9 −0.612323
\(635\) 3.69567e9 0.572776
\(636\) 1.02902e8 0.0158608
\(637\) 2.65919e9 0.407626
\(638\) 1.51616e10 2.31139
\(639\) −8.38543e8 −0.127137
\(640\) 8.94577e9 1.34892
\(641\) 2.66936e8 0.0400316 0.0200158 0.999800i \(-0.493628\pi\)
0.0200158 + 0.999800i \(0.493628\pi\)
\(642\) −6.28604e8 −0.0937572
\(643\) −9.18242e9 −1.36213 −0.681065 0.732223i \(-0.738483\pi\)
−0.681065 + 0.732223i \(0.738483\pi\)
\(644\) −4.07870e8 −0.0601757
\(645\) 8.23655e9 1.20861
\(646\) 1.65116e9 0.240977
\(647\) 7.79548e9 1.13156 0.565781 0.824556i \(-0.308575\pi\)
0.565781 + 0.824556i \(0.308575\pi\)
\(648\) −8.01688e8 −0.115743
\(649\) 1.65756e9 0.238019
\(650\) −6.34229e9 −0.905836
\(651\) −8.51395e8 −0.120948
\(652\) −8.86303e8 −0.125232
\(653\) −7.64228e8 −0.107406 −0.0537028 0.998557i \(-0.517102\pi\)
−0.0537028 + 0.998557i \(0.517102\pi\)
\(654\) −4.64295e9 −0.649040
\(655\) 1.99232e9 0.277022
\(656\) −7.77809e9 −1.07575
\(657\) 2.44204e7 0.00335949
\(658\) −9.51162e8 −0.130156
\(659\) −4.42497e9 −0.602299 −0.301149 0.953577i \(-0.597370\pi\)
−0.301149 + 0.953577i \(0.597370\pi\)
\(660\) 1.19711e9 0.162080
\(661\) −1.21787e9 −0.164019 −0.0820097 0.996632i \(-0.526134\pi\)
−0.0820097 + 0.996632i \(0.526134\pi\)
\(662\) 9.52106e9 1.27551
\(663\) 1.70379e9 0.227049
\(664\) 1.01483e9 0.134525
\(665\) 2.59162e9 0.341740
\(666\) 2.17657e9 0.285504
\(667\) 1.16027e10 1.51398
\(668\) 1.15528e9 0.149958
\(669\) −4.29047e9 −0.554005
\(670\) −4.30697e9 −0.553235
\(671\) 2.43651e10 3.11343
\(672\) −4.78132e8 −0.0607793
\(673\) −6.63758e9 −0.839377 −0.419689 0.907668i \(-0.637861\pi\)
−0.419689 + 0.907668i \(0.637861\pi\)
\(674\) −1.09042e10 −1.37178
\(675\) 2.48114e9 0.310519
\(676\) 4.97199e8 0.0619037
\(677\) 1.51749e9 0.187961 0.0939803 0.995574i \(-0.470041\pi\)
0.0939803 + 0.995574i \(0.470041\pi\)
\(678\) 2.70045e9 0.332761
\(679\) −2.47244e9 −0.303098
\(680\) 9.20148e9 1.12222
\(681\) −6.66085e8 −0.0808193
\(682\) −5.42755e9 −0.655176
\(683\) −1.25832e10 −1.51119 −0.755595 0.655039i \(-0.772652\pi\)
−0.755595 + 0.655039i \(0.772652\pi\)
\(684\) 1.00724e8 0.0120348
\(685\) −3.71371e9 −0.441460
\(686\) 7.56318e9 0.894479
\(687\) 7.53749e7 0.00886907
\(688\) −9.91063e9 −1.16022
\(689\) 1.46542e9 0.170685
\(690\) −8.72901e9 −1.01156
\(691\) −5.99235e9 −0.690914 −0.345457 0.938435i \(-0.612276\pi\)
−0.345457 + 0.938435i \(0.612276\pi\)
\(692\) −7.12876e8 −0.0817791
\(693\) 2.96926e9 0.338909
\(694\) −4.94070e9 −0.561088
\(695\) −2.51599e10 −2.84291
\(696\) 7.10909e9 0.799248
\(697\) −7.15232e9 −0.800078
\(698\) 7.16636e9 0.797636
\(699\) 4.64610e9 0.514539
\(700\) 7.73430e8 0.0852271
\(701\) −1.16633e10 −1.27881 −0.639406 0.768869i \(-0.720820\pi\)
−0.639406 + 0.768869i \(0.720820\pi\)
\(702\) −9.90324e8 −0.108043
\(703\) −3.15258e9 −0.342233
\(704\) −1.82132e10 −1.96736
\(705\) 2.13640e9 0.229626
\(706\) −8.03443e8 −0.0859288
\(707\) 9.31847e9 0.991691
\(708\) 6.74173e7 0.00713928
\(709\) −5.32106e9 −0.560707 −0.280353 0.959897i \(-0.590452\pi\)
−0.280353 + 0.959897i \(0.590452\pi\)
\(710\) −5.59420e9 −0.586589
\(711\) −5.06172e8 −0.0528146
\(712\) −1.15791e10 −1.20225
\(713\) −4.15354e9 −0.429146
\(714\) 1.97974e9 0.203546
\(715\) 1.70479e10 1.74422
\(716\) 4.31992e8 0.0439825
\(717\) −8.80587e9 −0.892186
\(718\) −1.67575e10 −1.68956
\(719\) −1.68037e9 −0.168599 −0.0842993 0.996440i \(-0.526865\pi\)
−0.0842993 + 0.996440i \(0.526865\pi\)
\(720\) −4.83571e9 −0.482833
\(721\) 2.63171e9 0.261496
\(722\) −8.23065e9 −0.813868
\(723\) 1.11450e10 1.09673
\(724\) −1.56049e9 −0.152818
\(725\) −2.20019e10 −2.14426
\(726\) 1.32657e10 1.28663
\(727\) 6.43210e9 0.620844 0.310422 0.950599i \(-0.399530\pi\)
0.310422 + 0.950599i \(0.399530\pi\)
\(728\) −3.55887e9 −0.341863
\(729\) 3.87420e8 0.0370370
\(730\) 1.62916e8 0.0155001
\(731\) −9.11330e9 −0.862908
\(732\) 9.90994e8 0.0933861
\(733\) 9.99078e9 0.936991 0.468495 0.883466i \(-0.344796\pi\)
0.468495 + 0.883466i \(0.344796\pi\)
\(734\) −5.14796e8 −0.0480507
\(735\) −6.94013e9 −0.644706
\(736\) −2.33258e9 −0.215657
\(737\) 7.14732e9 0.657669
\(738\) 4.15727e9 0.380724
\(739\) 5.40356e9 0.492521 0.246260 0.969204i \(-0.420798\pi\)
0.246260 + 0.969204i \(0.420798\pi\)
\(740\) −1.52394e9 −0.138248
\(741\) 1.43440e9 0.129511
\(742\) 1.70276e9 0.153017
\(743\) −1.37163e10 −1.22680 −0.613402 0.789771i \(-0.710199\pi\)
−0.613402 + 0.789771i \(0.710199\pi\)
\(744\) −2.54491e9 −0.226551
\(745\) 1.29004e10 1.14302
\(746\) −1.57235e10 −1.38664
\(747\) −4.90421e8 −0.0430474
\(748\) −1.32454e9 −0.115720
\(749\) 1.09166e9 0.0949299
\(750\) 6.29379e9 0.544751
\(751\) 1.22637e9 0.105653 0.0528264 0.998604i \(-0.483177\pi\)
0.0528264 + 0.998604i \(0.483177\pi\)
\(752\) −2.57062e9 −0.220432
\(753\) 1.26005e10 1.07548
\(754\) 8.78184e9 0.746081
\(755\) −6.39552e9 −0.540831
\(756\) 1.20768e8 0.0101654
\(757\) −1.17911e10 −0.987912 −0.493956 0.869487i \(-0.664450\pi\)
−0.493956 + 0.869487i \(0.664450\pi\)
\(758\) 9.68347e9 0.807586
\(759\) 1.44856e10 1.20251
\(760\) 7.74662e9 0.640125
\(761\) −1.03626e10 −0.852359 −0.426179 0.904639i \(-0.640141\pi\)
−0.426179 + 0.904639i \(0.640141\pi\)
\(762\) 2.37675e9 0.194599
\(763\) 8.06317e9 0.657158
\(764\) 1.05919e9 0.0859301
\(765\) −4.44667e9 −0.359103
\(766\) 8.09169e9 0.650487
\(767\) 9.60082e8 0.0768289
\(768\) −2.04600e9 −0.162983
\(769\) 7.77986e9 0.616921 0.308460 0.951237i \(-0.400186\pi\)
0.308460 + 0.951237i \(0.400186\pi\)
\(770\) 1.98090e10 1.56367
\(771\) −2.52830e9 −0.198672
\(772\) −4.06648e8 −0.0318096
\(773\) 8.05208e9 0.627018 0.313509 0.949585i \(-0.398495\pi\)
0.313509 + 0.949585i \(0.398495\pi\)
\(774\) 5.29708e9 0.410623
\(775\) 7.87623e9 0.607802
\(776\) −7.39040e9 −0.567743
\(777\) −3.77993e9 −0.289075
\(778\) 2.14123e10 1.63018
\(779\) −6.02146e9 −0.456374
\(780\) 6.93385e8 0.0523170
\(781\) 9.28346e9 0.697319
\(782\) 9.65817e9 0.722223
\(783\) −3.43551e9 −0.255755
\(784\) 8.35071e9 0.618895
\(785\) 2.72926e10 2.01373
\(786\) 1.28130e9 0.0941176
\(787\) 2.48655e10 1.81838 0.909192 0.416377i \(-0.136700\pi\)
0.909192 + 0.416377i \(0.136700\pi\)
\(788\) −1.74916e9 −0.127347
\(789\) −1.35234e10 −0.980205
\(790\) −3.37684e9 −0.243678
\(791\) −4.68973e9 −0.336923
\(792\) 8.87545e9 0.634822
\(793\) 1.41126e10 1.00497
\(794\) −2.57420e10 −1.82503
\(795\) −3.82455e9 −0.269958
\(796\) 4.12276e8 0.0289729
\(797\) −1.33668e9 −0.0935241 −0.0467621 0.998906i \(-0.514890\pi\)
−0.0467621 + 0.998906i \(0.514890\pi\)
\(798\) 1.66672e9 0.116105
\(799\) −2.36381e9 −0.163945
\(800\) 4.42319e9 0.305436
\(801\) 5.59566e9 0.384713
\(802\) 1.33106e10 0.911147
\(803\) −2.70356e8 −0.0184261
\(804\) 2.90701e8 0.0197265
\(805\) 1.51592e10 1.02422
\(806\) −3.14372e9 −0.211481
\(807\) 2.11762e9 0.141838
\(808\) 2.78539e10 1.85757
\(809\) −5.48115e9 −0.363959 −0.181979 0.983302i \(-0.558250\pi\)
−0.181979 + 0.983302i \(0.558250\pi\)
\(810\) 2.58461e9 0.170883
\(811\) 1.52888e10 1.00647 0.503233 0.864151i \(-0.332144\pi\)
0.503233 + 0.864151i \(0.332144\pi\)
\(812\) −1.07093e9 −0.0701963
\(813\) 1.07341e10 0.700563
\(814\) −2.40966e10 −1.56592
\(815\) 3.29411e10 2.13150
\(816\) 5.35045e9 0.344727
\(817\) −7.67238e9 −0.492213
\(818\) 1.16242e10 0.742551
\(819\) 1.71984e9 0.109395
\(820\) −2.91075e9 −0.184356
\(821\) 5.00980e9 0.315951 0.157975 0.987443i \(-0.449503\pi\)
0.157975 + 0.987443i \(0.449503\pi\)
\(822\) −2.38835e9 −0.149985
\(823\) −1.43300e10 −0.896077 −0.448038 0.894014i \(-0.647877\pi\)
−0.448038 + 0.894014i \(0.647877\pi\)
\(824\) 7.86646e9 0.489817
\(825\) −2.74686e10 −1.70313
\(826\) 1.11558e9 0.0688761
\(827\) 2.07445e10 1.27536 0.637682 0.770300i \(-0.279893\pi\)
0.637682 + 0.770300i \(0.279893\pi\)
\(828\) 5.89169e8 0.0360689
\(829\) −3.94434e9 −0.240455 −0.120227 0.992746i \(-0.538362\pi\)
−0.120227 + 0.992746i \(0.538362\pi\)
\(830\) −3.27176e9 −0.198613
\(831\) −2.60494e9 −0.157469
\(832\) −1.05494e10 −0.635033
\(833\) 7.67887e9 0.460299
\(834\) −1.61808e10 −0.965871
\(835\) −4.29380e10 −2.55234
\(836\) −1.11511e9 −0.0660080
\(837\) 1.22984e9 0.0724953
\(838\) −2.09922e10 −1.23226
\(839\) −1.52467e10 −0.891268 −0.445634 0.895215i \(-0.647022\pi\)
−0.445634 + 0.895215i \(0.647022\pi\)
\(840\) 9.28817e9 0.540696
\(841\) 1.32150e10 0.766091
\(842\) 4.42410e9 0.255407
\(843\) −1.45013e10 −0.833703
\(844\) 1.37902e9 0.0789537
\(845\) −1.84793e10 −1.05363
\(846\) 1.37396e9 0.0780147
\(847\) −2.30379e10 −1.30272
\(848\) 4.60189e9 0.259150
\(849\) −1.66895e10 −0.935979
\(850\) −1.83145e10 −1.02289
\(851\) −1.84404e10 −1.02569
\(852\) 3.77583e8 0.0209158
\(853\) −2.14258e10 −1.18200 −0.590998 0.806673i \(-0.701266\pi\)
−0.590998 + 0.806673i \(0.701266\pi\)
\(854\) 1.63983e10 0.900941
\(855\) −3.74360e9 −0.204837
\(856\) 3.26310e9 0.177817
\(857\) 2.49225e10 1.35257 0.676283 0.736642i \(-0.263590\pi\)
0.676283 + 0.736642i \(0.263590\pi\)
\(858\) 1.09638e10 0.592593
\(859\) −8.80692e9 −0.474076 −0.237038 0.971500i \(-0.576177\pi\)
−0.237038 + 0.971500i \(0.576177\pi\)
\(860\) −3.70880e9 −0.198833
\(861\) −7.21971e9 −0.385486
\(862\) −2.54675e10 −1.35429
\(863\) 5.86301e9 0.310515 0.155257 0.987874i \(-0.450379\pi\)
0.155257 + 0.987874i \(0.450379\pi\)
\(864\) 6.90663e8 0.0364308
\(865\) 2.64953e10 1.39192
\(866\) −2.39979e9 −0.125563
\(867\) −6.15915e9 −0.320962
\(868\) 3.83370e8 0.0198975
\(869\) 5.60380e9 0.289676
\(870\) −2.29194e10 −1.18001
\(871\) 4.13984e9 0.212285
\(872\) 2.41017e10 1.23095
\(873\) 3.57145e9 0.181675
\(874\) 8.13111e9 0.411964
\(875\) −1.09301e10 −0.551564
\(876\) −1.09961e7 −0.000552682 0
\(877\) 3.56534e10 1.78485 0.892427 0.451193i \(-0.149001\pi\)
0.892427 + 0.451193i \(0.149001\pi\)
\(878\) −6.83022e9 −0.340568
\(879\) 9.15901e9 0.454870
\(880\) 5.35359e10 2.64823
\(881\) 3.92292e10 1.93283 0.966414 0.256989i \(-0.0827306\pi\)
0.966414 + 0.256989i \(0.0827306\pi\)
\(882\) −4.46332e9 −0.219037
\(883\) 2.18077e10 1.06598 0.532988 0.846123i \(-0.321069\pi\)
0.532988 + 0.846123i \(0.321069\pi\)
\(884\) −7.67192e8 −0.0373526
\(885\) −2.50569e9 −0.121514
\(886\) −1.55939e10 −0.753247
\(887\) 4.76501e9 0.229262 0.114631 0.993408i \(-0.463431\pi\)
0.114631 + 0.993408i \(0.463431\pi\)
\(888\) −1.12986e10 −0.541476
\(889\) −4.12758e9 −0.197033
\(890\) 3.73305e10 1.77500
\(891\) −4.28911e9 −0.203140
\(892\) 1.93194e9 0.0911414
\(893\) −1.99006e9 −0.0935162
\(894\) 8.29646e9 0.388339
\(895\) −1.60558e10 −0.748601
\(896\) −9.99126e9 −0.464026
\(897\) 8.39029e9 0.388153
\(898\) 3.46490e10 1.59670
\(899\) −1.09058e10 −0.500609
\(900\) −1.11722e9 −0.0510846
\(901\) 4.23166e9 0.192741
\(902\) −4.60249e10 −2.08819
\(903\) −9.19916e9 −0.415758
\(904\) −1.40181e10 −0.631102
\(905\) 5.79984e10 2.60103
\(906\) −4.11307e9 −0.183746
\(907\) −8.85912e9 −0.394244 −0.197122 0.980379i \(-0.563159\pi\)
−0.197122 + 0.980379i \(0.563159\pi\)
\(908\) 2.99928e8 0.0132959
\(909\) −1.34605e10 −0.594414
\(910\) 1.14737e10 0.504728
\(911\) −5.66060e9 −0.248055 −0.124028 0.992279i \(-0.539581\pi\)
−0.124028 + 0.992279i \(0.539581\pi\)
\(912\) 4.50449e9 0.196636
\(913\) 5.42942e9 0.236105
\(914\) −1.11000e10 −0.480851
\(915\) −3.68321e10 −1.58947
\(916\) −3.39402e7 −0.00145908
\(917\) −2.22516e9 −0.0952947
\(918\) −2.85973e9 −0.122005
\(919\) 3.15687e10 1.34169 0.670846 0.741597i \(-0.265931\pi\)
0.670846 + 0.741597i \(0.265931\pi\)
\(920\) 4.53125e10 1.91849
\(921\) −2.59960e9 −0.109647
\(922\) 3.45821e10 1.45309
\(923\) 5.37713e9 0.225084
\(924\) −1.33702e9 −0.0557551
\(925\) 3.49680e10 1.45270
\(926\) −1.79683e10 −0.743649
\(927\) −3.80151e9 −0.156739
\(928\) −6.12456e9 −0.251569
\(929\) 2.88328e10 1.17986 0.589932 0.807453i \(-0.299155\pi\)
0.589932 + 0.807453i \(0.299155\pi\)
\(930\) 8.20468e9 0.334481
\(931\) 6.46476e9 0.262560
\(932\) −2.09207e9 −0.0846487
\(933\) −2.13623e10 −0.861117
\(934\) 4.08454e9 0.164032
\(935\) 4.92288e10 1.96960
\(936\) 5.14080e9 0.204911
\(937\) −4.19782e10 −1.66700 −0.833500 0.552520i \(-0.813666\pi\)
−0.833500 + 0.552520i \(0.813666\pi\)
\(938\) 4.81032e9 0.190311
\(939\) −2.38567e9 −0.0940330
\(940\) −9.61988e8 −0.0377766
\(941\) −4.83827e10 −1.89289 −0.946447 0.322860i \(-0.895356\pi\)
−0.946447 + 0.322860i \(0.895356\pi\)
\(942\) 1.75524e10 0.684160
\(943\) −3.52215e10 −1.36778
\(944\) 3.01496e9 0.116649
\(945\) −4.48856e9 −0.173020
\(946\) −5.86436e10 −2.25217
\(947\) −7.76682e9 −0.297179 −0.148590 0.988899i \(-0.547473\pi\)
−0.148590 + 0.988899i \(0.547473\pi\)
\(948\) 2.27922e8 0.00868873
\(949\) −1.56595e8 −0.00594765
\(950\) −1.54187e10 −0.583467
\(951\) −9.85651e9 −0.371614
\(952\) −1.02769e10 −0.386039
\(953\) −4.85606e9 −0.181743 −0.0908717 0.995863i \(-0.528965\pi\)
−0.0908717 + 0.995863i \(0.528965\pi\)
\(954\) −2.45964e9 −0.0917175
\(955\) −3.93666e10 −1.46257
\(956\) 3.96515e9 0.146777
\(957\) 3.80343e10 1.40276
\(958\) 1.72953e10 0.635549
\(959\) 4.14773e9 0.151861
\(960\) 2.75325e10 1.00438
\(961\) −2.36086e10 −0.858100
\(962\) −1.39571e10 −0.505457
\(963\) −1.57691e9 −0.0569004
\(964\) −5.01845e9 −0.180426
\(965\) 1.51138e10 0.541413
\(966\) 9.74917e9 0.347975
\(967\) −2.84618e10 −1.01221 −0.506103 0.862473i \(-0.668915\pi\)
−0.506103 + 0.862473i \(0.668915\pi\)
\(968\) −6.88628e10 −2.44017
\(969\) 4.14209e9 0.146247
\(970\) 2.38264e10 0.838217
\(971\) −3.54110e10 −1.24128 −0.620642 0.784094i \(-0.713128\pi\)
−0.620642 + 0.784094i \(0.713128\pi\)
\(972\) −1.74450e8 −0.00609310
\(973\) 2.81004e10 0.977952
\(974\) 4.79483e10 1.66271
\(975\) −1.59102e10 −0.549744
\(976\) 4.43182e10 1.52584
\(977\) 4.07835e10 1.39911 0.699557 0.714577i \(-0.253381\pi\)
0.699557 + 0.714577i \(0.253381\pi\)
\(978\) 2.11850e10 0.724173
\(979\) −6.19492e10 −2.11007
\(980\) 3.12504e9 0.106063
\(981\) −1.16473e10 −0.393897
\(982\) −3.88032e10 −1.30761
\(983\) −4.57225e10 −1.53530 −0.767649 0.640871i \(-0.778573\pi\)
−0.767649 + 0.640871i \(0.778573\pi\)
\(984\) −2.15805e10 −0.722068
\(985\) 6.50107e10 2.16749
\(986\) 2.53591e10 0.842489
\(987\) −2.38608e9 −0.0789905
\(988\) −6.45891e8 −0.0213064
\(989\) −4.48782e10 −1.47519
\(990\) −2.86141e10 −0.937253
\(991\) 7.18635e9 0.234558 0.117279 0.993099i \(-0.462583\pi\)
0.117279 + 0.993099i \(0.462583\pi\)
\(992\) 2.19247e9 0.0713086
\(993\) 2.38845e10 0.774093
\(994\) 6.24800e9 0.201785
\(995\) −1.53230e10 −0.493131
\(996\) 2.20829e8 0.00708188
\(997\) 5.76916e10 1.84365 0.921827 0.387602i \(-0.126696\pi\)
0.921827 + 0.387602i \(0.126696\pi\)
\(998\) −2.70920e10 −0.862749
\(999\) 5.46012e9 0.173270
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.b.1.13 17
3.2 odd 2 531.8.a.d.1.5 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.13 17 1.1 even 1 trivial
531.8.a.d.1.5 17 3.2 odd 2