Properties

Label 177.8.a.b.1.5
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 1.5
Root \(-11.3335\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-13.3335 q^{2} +27.0000 q^{3} +49.7814 q^{4} +152.093 q^{5} -360.004 q^{6} -1328.47 q^{7} +1042.93 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-13.3335 q^{2} +27.0000 q^{3} +49.7814 q^{4} +152.093 q^{5} -360.004 q^{6} -1328.47 q^{7} +1042.93 q^{8} +729.000 q^{9} -2027.93 q^{10} +5132.02 q^{11} +1344.10 q^{12} -12251.2 q^{13} +17713.2 q^{14} +4106.51 q^{15} -20277.8 q^{16} +26601.9 q^{17} -9720.10 q^{18} +33197.2 q^{19} +7571.39 q^{20} -35868.8 q^{21} -68427.7 q^{22} -97057.0 q^{23} +28159.0 q^{24} -54992.8 q^{25} +163351. q^{26} +19683.0 q^{27} -66133.3 q^{28} +209269. q^{29} -54754.0 q^{30} -308874. q^{31} +136879. q^{32} +138565. q^{33} -354695. q^{34} -202052. q^{35} +36290.6 q^{36} +287842. q^{37} -442634. q^{38} -330783. q^{39} +158622. q^{40} +581611. q^{41} +478256. q^{42} +540792. q^{43} +255479. q^{44} +110876. q^{45} +1.29411e6 q^{46} -650609. q^{47} -547501. q^{48} +941302. q^{49} +733244. q^{50} +718250. q^{51} -609883. q^{52} -49948.0 q^{53} -262443. q^{54} +780544. q^{55} -1.38550e6 q^{56} +896326. q^{57} -2.79028e6 q^{58} -205379. q^{59} +204428. q^{60} -20025.0 q^{61} +4.11836e6 q^{62} -968458. q^{63} +770486. q^{64} -1.86332e6 q^{65} -1.84755e6 q^{66} -1.38680e6 q^{67} +1.32428e6 q^{68} -2.62054e6 q^{69} +2.69405e6 q^{70} +1.24373e6 q^{71} +760293. q^{72} -4.83751e6 q^{73} -3.83794e6 q^{74} -1.48480e6 q^{75} +1.65260e6 q^{76} -6.81777e6 q^{77} +4.41048e6 q^{78} -2.94089e6 q^{79} -3.08411e6 q^{80} +531441. q^{81} -7.75490e6 q^{82} -8.62802e6 q^{83} -1.78560e6 q^{84} +4.04595e6 q^{85} -7.21064e6 q^{86} +5.65025e6 q^{87} +5.35232e6 q^{88} -879309. q^{89} -1.47836e6 q^{90} +1.62754e7 q^{91} -4.83163e6 q^{92} -8.33960e6 q^{93} +8.67488e6 q^{94} +5.04907e6 q^{95} +3.69574e6 q^{96} -1.62017e7 q^{97} -1.25508e7 q^{98} +3.74125e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −13.3335 −1.17852 −0.589262 0.807942i \(-0.700581\pi\)
−0.589262 + 0.807942i \(0.700581\pi\)
\(3\) 27.0000 0.577350
\(4\) 49.7814 0.388917
\(5\) 152.093 0.544144 0.272072 0.962277i \(-0.412291\pi\)
0.272072 + 0.962277i \(0.412291\pi\)
\(6\) −360.004 −0.680421
\(7\) −1328.47 −1.46390 −0.731948 0.681361i \(-0.761389\pi\)
−0.731948 + 0.681361i \(0.761389\pi\)
\(8\) 1042.93 0.720176
\(9\) 729.000 0.333333
\(10\) −2027.93 −0.641286
\(11\) 5132.02 1.16256 0.581279 0.813705i \(-0.302553\pi\)
0.581279 + 0.813705i \(0.302553\pi\)
\(12\) 1344.10 0.224541
\(13\) −12251.2 −1.54660 −0.773299 0.634041i \(-0.781395\pi\)
−0.773299 + 0.634041i \(0.781395\pi\)
\(14\) 17713.2 1.72524
\(15\) 4106.51 0.314162
\(16\) −20277.8 −1.23766
\(17\) 26601.9 1.31323 0.656615 0.754226i \(-0.271988\pi\)
0.656615 + 0.754226i \(0.271988\pi\)
\(18\) −9720.10 −0.392841
\(19\) 33197.2 1.11036 0.555181 0.831730i \(-0.312649\pi\)
0.555181 + 0.831730i \(0.312649\pi\)
\(20\) 7571.39 0.211627
\(21\) −35868.8 −0.845181
\(22\) −68427.7 −1.37010
\(23\) −97057.0 −1.66333 −0.831667 0.555274i \(-0.812613\pi\)
−0.831667 + 0.555274i \(0.812613\pi\)
\(24\) 28159.0 0.415794
\(25\) −54992.8 −0.703907
\(26\) 163351. 1.82270
\(27\) 19683.0 0.192450
\(28\) −66133.3 −0.569334
\(29\) 209269. 1.59335 0.796675 0.604408i \(-0.206590\pi\)
0.796675 + 0.604408i \(0.206590\pi\)
\(30\) −54754.0 −0.370247
\(31\) −308874. −1.86215 −0.931077 0.364823i \(-0.881129\pi\)
−0.931077 + 0.364823i \(0.881129\pi\)
\(32\) 136879. 0.738436
\(33\) 138565. 0.671203
\(34\) −354695. −1.54767
\(35\) −202052. −0.796570
\(36\) 36290.6 0.129639
\(37\) 287842. 0.934219 0.467110 0.884199i \(-0.345295\pi\)
0.467110 + 0.884199i \(0.345295\pi\)
\(38\) −442634. −1.30859
\(39\) −330783. −0.892929
\(40\) 158622. 0.391879
\(41\) 581611. 1.31792 0.658960 0.752178i \(-0.270996\pi\)
0.658960 + 0.752178i \(0.270996\pi\)
\(42\) 478256. 0.996065
\(43\) 540792. 1.03727 0.518634 0.854996i \(-0.326441\pi\)
0.518634 + 0.854996i \(0.326441\pi\)
\(44\) 255479. 0.452138
\(45\) 110876. 0.181381
\(46\) 1.29411e6 1.96028
\(47\) −650609. −0.914066 −0.457033 0.889450i \(-0.651088\pi\)
−0.457033 + 0.889450i \(0.651088\pi\)
\(48\) −547501. −0.714564
\(49\) 941302. 1.14299
\(50\) 733244. 0.829571
\(51\) 718250. 0.758194
\(52\) −609883. −0.601499
\(53\) −49948.0 −0.0460843 −0.0230421 0.999734i \(-0.507335\pi\)
−0.0230421 + 0.999734i \(0.507335\pi\)
\(54\) −262443. −0.226807
\(55\) 780544. 0.632599
\(56\) −1.38550e6 −1.05426
\(57\) 896326. 0.641068
\(58\) −2.79028e6 −1.87780
\(59\) −205379. −0.130189
\(60\) 204428. 0.122183
\(61\) −20025.0 −0.0112958 −0.00564791 0.999984i \(-0.501798\pi\)
−0.00564791 + 0.999984i \(0.501798\pi\)
\(62\) 4.11836e6 2.19459
\(63\) −968458. −0.487965
\(64\) 770486. 0.367396
\(65\) −1.86332e6 −0.841573
\(66\) −1.84755e6 −0.791028
\(67\) −1.38680e6 −0.563314 −0.281657 0.959515i \(-0.590884\pi\)
−0.281657 + 0.959515i \(0.590884\pi\)
\(68\) 1.32428e6 0.510738
\(69\) −2.62054e6 −0.960327
\(70\) 2.69405e6 0.938776
\(71\) 1.24373e6 0.412403 0.206201 0.978510i \(-0.433890\pi\)
0.206201 + 0.978510i \(0.433890\pi\)
\(72\) 760293. 0.240059
\(73\) −4.83751e6 −1.45543 −0.727716 0.685879i \(-0.759418\pi\)
−0.727716 + 0.685879i \(0.759418\pi\)
\(74\) −3.83794e6 −1.10100
\(75\) −1.48480e6 −0.406401
\(76\) 1.65260e6 0.431839
\(77\) −6.81777e6 −1.70186
\(78\) 4.41048e6 1.05234
\(79\) −2.94089e6 −0.671095 −0.335547 0.942023i \(-0.608921\pi\)
−0.335547 + 0.942023i \(0.608921\pi\)
\(80\) −3.08411e6 −0.673466
\(81\) 531441. 0.111111
\(82\) −7.75490e6 −1.55320
\(83\) −8.62802e6 −1.65630 −0.828148 0.560510i \(-0.810605\pi\)
−0.828148 + 0.560510i \(0.810605\pi\)
\(84\) −1.78560e6 −0.328705
\(85\) 4.04595e6 0.714586
\(86\) −7.21064e6 −1.22244
\(87\) 5.65025e6 0.919921
\(88\) 5.35232e6 0.837245
\(89\) −879309. −0.132214 −0.0661069 0.997813i \(-0.521058\pi\)
−0.0661069 + 0.997813i \(0.521058\pi\)
\(90\) −1.47836e6 −0.213762
\(91\) 1.62754e7 2.26406
\(92\) −4.83163e6 −0.646899
\(93\) −8.33960e6 −1.07511
\(94\) 8.67488e6 1.07725
\(95\) 5.04907e6 0.604197
\(96\) 3.69574e6 0.426336
\(97\) −1.62017e7 −1.80244 −0.901220 0.433362i \(-0.857327\pi\)
−0.901220 + 0.433362i \(0.857327\pi\)
\(98\) −1.25508e7 −1.34704
\(99\) 3.74125e6 0.387519
\(100\) −2.73761e6 −0.273761
\(101\) 617795. 0.0596650 0.0298325 0.999555i \(-0.490503\pi\)
0.0298325 + 0.999555i \(0.490503\pi\)
\(102\) −9.57677e6 −0.893549
\(103\) −6.98654e6 −0.629988 −0.314994 0.949094i \(-0.602002\pi\)
−0.314994 + 0.949094i \(0.602002\pi\)
\(104\) −1.27771e7 −1.11382
\(105\) −5.45539e6 −0.459900
\(106\) 665980. 0.0543114
\(107\) −7.47019e6 −0.589506 −0.294753 0.955573i \(-0.595237\pi\)
−0.294753 + 0.955573i \(0.595237\pi\)
\(108\) 979847. 0.0748471
\(109\) −2.25015e7 −1.66425 −0.832125 0.554588i \(-0.812876\pi\)
−0.832125 + 0.554588i \(0.812876\pi\)
\(110\) −1.04074e7 −0.745532
\(111\) 7.77175e6 0.539372
\(112\) 2.69386e7 1.81181
\(113\) 7.33737e6 0.478372 0.239186 0.970974i \(-0.423119\pi\)
0.239186 + 0.970974i \(0.423119\pi\)
\(114\) −1.19511e7 −0.755513
\(115\) −1.47617e7 −0.905094
\(116\) 1.04177e7 0.619681
\(117\) −8.93114e6 −0.515533
\(118\) 2.73841e6 0.153431
\(119\) −3.53399e7 −1.92243
\(120\) 4.28278e6 0.226252
\(121\) 6.85051e6 0.351539
\(122\) 267002. 0.0133124
\(123\) 1.57035e7 0.760902
\(124\) −1.53762e7 −0.724223
\(125\) −2.02463e7 −0.927171
\(126\) 1.29129e7 0.575078
\(127\) −1.12559e7 −0.487605 −0.243803 0.969825i \(-0.578395\pi\)
−0.243803 + 0.969825i \(0.578395\pi\)
\(128\) −2.77938e7 −1.17142
\(129\) 1.46014e7 0.598867
\(130\) 2.48446e7 0.991813
\(131\) 4.30156e6 0.167177 0.0835885 0.996500i \(-0.473362\pi\)
0.0835885 + 0.996500i \(0.473362\pi\)
\(132\) 6.89794e6 0.261042
\(133\) −4.41017e7 −1.62545
\(134\) 1.84908e7 0.663879
\(135\) 2.99364e6 0.104721
\(136\) 2.77438e7 0.945756
\(137\) −1.85481e7 −0.616281 −0.308140 0.951341i \(-0.599707\pi\)
−0.308140 + 0.951341i \(0.599707\pi\)
\(138\) 3.49409e7 1.13177
\(139\) −4.12956e7 −1.30422 −0.652112 0.758123i \(-0.726117\pi\)
−0.652112 + 0.758123i \(0.726117\pi\)
\(140\) −1.00584e7 −0.309800
\(141\) −1.75664e7 −0.527736
\(142\) −1.65832e7 −0.486026
\(143\) −6.28736e7 −1.79801
\(144\) −1.47825e7 −0.412554
\(145\) 3.18283e7 0.867012
\(146\) 6.45008e7 1.71526
\(147\) 2.54152e7 0.659906
\(148\) 1.43292e7 0.363334
\(149\) 2.77964e7 0.688394 0.344197 0.938897i \(-0.388151\pi\)
0.344197 + 0.938897i \(0.388151\pi\)
\(150\) 1.97976e7 0.478953
\(151\) 5.05159e7 1.19401 0.597006 0.802236i \(-0.296357\pi\)
0.597006 + 0.802236i \(0.296357\pi\)
\(152\) 3.46223e7 0.799655
\(153\) 1.93928e7 0.437743
\(154\) 9.09045e7 2.00568
\(155\) −4.69775e7 −1.01328
\(156\) −1.64668e7 −0.347275
\(157\) −3.27653e7 −0.675717 −0.337858 0.941197i \(-0.609703\pi\)
−0.337858 + 0.941197i \(0.609703\pi\)
\(158\) 3.92122e7 0.790901
\(159\) −1.34860e6 −0.0266068
\(160\) 2.08184e7 0.401816
\(161\) 1.28938e8 2.43495
\(162\) −7.08595e6 −0.130947
\(163\) 3.27745e7 0.592761 0.296381 0.955070i \(-0.404220\pi\)
0.296381 + 0.955070i \(0.404220\pi\)
\(164\) 2.89534e7 0.512562
\(165\) 2.10747e7 0.365231
\(166\) 1.15041e8 1.95198
\(167\) −4.91242e7 −0.816183 −0.408092 0.912941i \(-0.633806\pi\)
−0.408092 + 0.912941i \(0.633806\pi\)
\(168\) −3.74085e7 −0.608678
\(169\) 8.73439e7 1.39197
\(170\) −5.39466e7 −0.842157
\(171\) 2.42008e7 0.370121
\(172\) 2.69214e7 0.403411
\(173\) 2.08181e7 0.305688 0.152844 0.988250i \(-0.451157\pi\)
0.152844 + 0.988250i \(0.451157\pi\)
\(174\) −7.53375e7 −1.08415
\(175\) 7.30565e7 1.03045
\(176\) −1.04066e8 −1.43885
\(177\) −5.54523e6 −0.0751646
\(178\) 1.17242e7 0.155817
\(179\) 8.64093e7 1.12609 0.563047 0.826425i \(-0.309629\pi\)
0.563047 + 0.826425i \(0.309629\pi\)
\(180\) 5.51954e6 0.0705423
\(181\) −6.54723e7 −0.820696 −0.410348 0.911929i \(-0.634593\pi\)
−0.410348 + 0.911929i \(0.634593\pi\)
\(182\) −2.17008e8 −2.66825
\(183\) −540674. −0.00652164
\(184\) −1.01223e8 −1.19789
\(185\) 4.37788e7 0.508350
\(186\) 1.11196e8 1.26705
\(187\) 1.36521e8 1.52671
\(188\) −3.23882e7 −0.355496
\(189\) −2.61484e7 −0.281727
\(190\) −6.73215e7 −0.712060
\(191\) −8.45503e7 −0.878007 −0.439004 0.898485i \(-0.644669\pi\)
−0.439004 + 0.898485i \(0.644669\pi\)
\(192\) 2.08031e7 0.212116
\(193\) −2.04209e7 −0.204467 −0.102234 0.994760i \(-0.532599\pi\)
−0.102234 + 0.994760i \(0.532599\pi\)
\(194\) 2.16026e8 2.12422
\(195\) −5.03097e7 −0.485882
\(196\) 4.68593e7 0.444529
\(197\) 3.23191e7 0.301181 0.150591 0.988596i \(-0.451882\pi\)
0.150591 + 0.988596i \(0.451882\pi\)
\(198\) −4.98838e7 −0.456700
\(199\) −7.22071e7 −0.649522 −0.324761 0.945796i \(-0.605284\pi\)
−0.324761 + 0.945796i \(0.605284\pi\)
\(200\) −5.73534e7 −0.506937
\(201\) −3.74435e7 −0.325229
\(202\) −8.23736e6 −0.0703166
\(203\) −2.78008e8 −2.33250
\(204\) 3.57555e7 0.294874
\(205\) 8.84589e7 0.717139
\(206\) 9.31549e7 0.742455
\(207\) −7.07546e7 −0.554445
\(208\) 2.48428e8 1.91416
\(209\) 1.70369e8 1.29086
\(210\) 7.27393e7 0.542003
\(211\) −1.90681e8 −1.39739 −0.698696 0.715419i \(-0.746236\pi\)
−0.698696 + 0.715419i \(0.746236\pi\)
\(212\) −2.48648e6 −0.0179230
\(213\) 3.35807e7 0.238101
\(214\) 9.96036e7 0.694747
\(215\) 8.22507e7 0.564423
\(216\) 2.05279e7 0.138598
\(217\) 4.10331e8 2.72600
\(218\) 3.00023e8 1.96136
\(219\) −1.30613e8 −0.840294
\(220\) 3.88566e7 0.246028
\(221\) −3.25905e8 −2.03104
\(222\) −1.03624e8 −0.635662
\(223\) 2.03359e8 1.22800 0.613998 0.789307i \(-0.289560\pi\)
0.613998 + 0.789307i \(0.289560\pi\)
\(224\) −1.81841e8 −1.08099
\(225\) −4.00897e7 −0.234636
\(226\) −9.78326e7 −0.563773
\(227\) 3.52165e7 0.199828 0.0999139 0.994996i \(-0.468143\pi\)
0.0999139 + 0.994996i \(0.468143\pi\)
\(228\) 4.46203e7 0.249322
\(229\) −3.50737e8 −1.93000 −0.965000 0.262249i \(-0.915536\pi\)
−0.965000 + 0.262249i \(0.915536\pi\)
\(230\) 1.96824e8 1.06667
\(231\) −1.84080e8 −0.982571
\(232\) 2.18252e8 1.14749
\(233\) −2.71870e8 −1.40804 −0.704020 0.710180i \(-0.748613\pi\)
−0.704020 + 0.710180i \(0.748613\pi\)
\(234\) 1.19083e8 0.607568
\(235\) −9.89530e7 −0.497384
\(236\) −1.02240e7 −0.0506327
\(237\) −7.94040e7 −0.387457
\(238\) 4.71203e8 2.26563
\(239\) −7.12076e7 −0.337391 −0.168696 0.985668i \(-0.553955\pi\)
−0.168696 + 0.985668i \(0.553955\pi\)
\(240\) −8.32711e7 −0.388826
\(241\) −2.36356e8 −1.08770 −0.543848 0.839184i \(-0.683033\pi\)
−0.543848 + 0.839184i \(0.683033\pi\)
\(242\) −9.13410e7 −0.414297
\(243\) 1.43489e7 0.0641500
\(244\) −996870. −0.00439313
\(245\) 1.43165e8 0.621952
\(246\) −2.09382e8 −0.896741
\(247\) −4.06707e8 −1.71728
\(248\) −3.22133e8 −1.34108
\(249\) −2.32957e8 −0.956263
\(250\) 2.69953e8 1.09269
\(251\) 3.04691e7 0.121619 0.0608094 0.998149i \(-0.480632\pi\)
0.0608094 + 0.998149i \(0.480632\pi\)
\(252\) −4.82112e7 −0.189778
\(253\) −4.98099e8 −1.93372
\(254\) 1.50081e8 0.574654
\(255\) 1.09241e8 0.412567
\(256\) 2.71966e8 1.01315
\(257\) 2.69817e8 0.991524 0.495762 0.868458i \(-0.334889\pi\)
0.495762 + 0.868458i \(0.334889\pi\)
\(258\) −1.94687e8 −0.705779
\(259\) −3.82391e8 −1.36760
\(260\) −9.27588e7 −0.327302
\(261\) 1.52557e8 0.531117
\(262\) −5.73548e7 −0.197022
\(263\) −3.30750e8 −1.12113 −0.560564 0.828111i \(-0.689415\pi\)
−0.560564 + 0.828111i \(0.689415\pi\)
\(264\) 1.44513e8 0.483384
\(265\) −7.59674e6 −0.0250765
\(266\) 5.88029e8 1.91564
\(267\) −2.37413e7 −0.0763336
\(268\) −6.90366e7 −0.219082
\(269\) 3.87843e8 1.21485 0.607426 0.794377i \(-0.292202\pi\)
0.607426 + 0.794377i \(0.292202\pi\)
\(270\) −3.99157e7 −0.123416
\(271\) 3.62357e8 1.10597 0.552986 0.833191i \(-0.313488\pi\)
0.552986 + 0.833191i \(0.313488\pi\)
\(272\) −5.39428e8 −1.62533
\(273\) 4.39437e8 1.30716
\(274\) 2.47311e8 0.726301
\(275\) −2.82224e8 −0.818333
\(276\) −1.30454e8 −0.373487
\(277\) 3.92785e8 1.11039 0.555195 0.831721i \(-0.312644\pi\)
0.555195 + 0.831721i \(0.312644\pi\)
\(278\) 5.50614e8 1.53706
\(279\) −2.25169e8 −0.620718
\(280\) −2.10725e8 −0.573670
\(281\) 2.62473e7 0.0705686 0.0352843 0.999377i \(-0.488766\pi\)
0.0352843 + 0.999377i \(0.488766\pi\)
\(282\) 2.34222e8 0.621950
\(283\) 3.02721e8 0.793945 0.396972 0.917831i \(-0.370061\pi\)
0.396972 + 0.917831i \(0.370061\pi\)
\(284\) 6.19146e7 0.160390
\(285\) 1.36325e8 0.348833
\(286\) 8.38323e8 2.11900
\(287\) −7.72656e8 −1.92930
\(288\) 9.97850e7 0.246145
\(289\) 2.97321e8 0.724574
\(290\) −4.24381e8 −1.02179
\(291\) −4.37447e8 −1.04064
\(292\) −2.40818e8 −0.566042
\(293\) 4.44159e8 1.03158 0.515789 0.856716i \(-0.327499\pi\)
0.515789 + 0.856716i \(0.327499\pi\)
\(294\) −3.38872e8 −0.777715
\(295\) −3.12367e7 −0.0708415
\(296\) 3.00198e8 0.672802
\(297\) 1.01014e8 0.223734
\(298\) −3.70623e8 −0.811288
\(299\) 1.18907e9 2.57251
\(300\) −7.39156e7 −0.158056
\(301\) −7.18429e8 −1.51845
\(302\) −6.73552e8 −1.40717
\(303\) 1.66805e7 0.0344476
\(304\) −6.73168e8 −1.37425
\(305\) −3.04566e6 −0.00614655
\(306\) −2.58573e8 −0.515891
\(307\) −7.17875e8 −1.41600 −0.708002 0.706211i \(-0.750403\pi\)
−0.708002 + 0.706211i \(0.750403\pi\)
\(308\) −3.39398e8 −0.661883
\(309\) −1.88637e8 −0.363724
\(310\) 6.26374e8 1.19417
\(311\) −9.91134e8 −1.86841 −0.934203 0.356743i \(-0.883887\pi\)
−0.934203 + 0.356743i \(0.883887\pi\)
\(312\) −3.44982e8 −0.643066
\(313\) −1.77188e7 −0.0326610 −0.0163305 0.999867i \(-0.505198\pi\)
−0.0163305 + 0.999867i \(0.505198\pi\)
\(314\) 4.36875e8 0.796348
\(315\) −1.47296e8 −0.265523
\(316\) −1.46401e8 −0.261000
\(317\) 6.67310e8 1.17658 0.588289 0.808651i \(-0.299802\pi\)
0.588289 + 0.808651i \(0.299802\pi\)
\(318\) 1.79815e7 0.0313567
\(319\) 1.07397e9 1.85236
\(320\) 1.17185e8 0.199917
\(321\) −2.01695e8 −0.340352
\(322\) −1.71919e9 −2.86964
\(323\) 8.83109e8 1.45816
\(324\) 2.64559e7 0.0432130
\(325\) 6.73728e8 1.08866
\(326\) −4.36998e8 −0.698583
\(327\) −6.07540e8 −0.960855
\(328\) 6.06577e8 0.949134
\(329\) 8.64318e8 1.33810
\(330\) −2.80999e8 −0.430433
\(331\) 1.21309e9 1.83863 0.919313 0.393526i \(-0.128745\pi\)
0.919313 + 0.393526i \(0.128745\pi\)
\(332\) −4.29515e8 −0.644161
\(333\) 2.09837e8 0.311406
\(334\) 6.54996e8 0.961891
\(335\) −2.10922e8 −0.306524
\(336\) 7.27342e8 1.04605
\(337\) 4.03850e8 0.574799 0.287399 0.957811i \(-0.407209\pi\)
0.287399 + 0.957811i \(0.407209\pi\)
\(338\) −1.16460e9 −1.64047
\(339\) 1.98109e8 0.276188
\(340\) 2.01413e8 0.277915
\(341\) −1.58515e9 −2.16486
\(342\) −3.22681e8 −0.436196
\(343\) −1.56440e8 −0.209324
\(344\) 5.64006e8 0.747015
\(345\) −3.98565e8 −0.522556
\(346\) −2.77577e8 −0.360261
\(347\) −5.73452e8 −0.736791 −0.368395 0.929669i \(-0.620093\pi\)
−0.368395 + 0.929669i \(0.620093\pi\)
\(348\) 2.81277e8 0.357773
\(349\) −3.21244e8 −0.404525 −0.202263 0.979331i \(-0.564829\pi\)
−0.202263 + 0.979331i \(0.564829\pi\)
\(350\) −9.74096e8 −1.21441
\(351\) −2.41141e8 −0.297643
\(352\) 7.02468e8 0.858474
\(353\) 3.98442e8 0.482118 0.241059 0.970510i \(-0.422505\pi\)
0.241059 + 0.970510i \(0.422505\pi\)
\(354\) 7.39372e7 0.0885832
\(355\) 1.89162e8 0.224407
\(356\) −4.37732e7 −0.0514202
\(357\) −9.54177e8 −1.10992
\(358\) −1.15214e9 −1.32713
\(359\) 1.40704e9 1.60500 0.802501 0.596651i \(-0.203502\pi\)
0.802501 + 0.596651i \(0.203502\pi\)
\(360\) 1.15635e8 0.130626
\(361\) 2.08186e8 0.232903
\(362\) 8.72973e8 0.967210
\(363\) 1.84964e8 0.202961
\(364\) 8.10214e8 0.880531
\(365\) −7.35751e8 −0.791965
\(366\) 7.20906e6 0.00768590
\(367\) −9.65705e8 −1.01980 −0.509898 0.860235i \(-0.670317\pi\)
−0.509898 + 0.860235i \(0.670317\pi\)
\(368\) 1.96811e9 2.05864
\(369\) 4.23995e8 0.439307
\(370\) −5.83723e8 −0.599102
\(371\) 6.63547e7 0.0674626
\(372\) −4.15157e8 −0.418130
\(373\) −6.50998e8 −0.649529 −0.324765 0.945795i \(-0.605285\pi\)
−0.324765 + 0.945795i \(0.605285\pi\)
\(374\) −1.82030e9 −1.79926
\(375\) −5.46649e8 −0.535302
\(376\) −6.78537e8 −0.658288
\(377\) −2.56380e9 −2.46427
\(378\) 3.48648e8 0.332022
\(379\) −3.97991e7 −0.0375522 −0.0187761 0.999824i \(-0.505977\pi\)
−0.0187761 + 0.999824i \(0.505977\pi\)
\(380\) 2.51349e8 0.234982
\(381\) −3.03910e8 −0.281519
\(382\) 1.12735e9 1.03475
\(383\) 6.27745e7 0.0570936 0.0285468 0.999592i \(-0.490912\pi\)
0.0285468 + 0.999592i \(0.490912\pi\)
\(384\) −7.50433e8 −0.676320
\(385\) −1.03693e9 −0.926058
\(386\) 2.72281e8 0.240970
\(387\) 3.94238e8 0.345756
\(388\) −8.06545e8 −0.701000
\(389\) −4.14596e8 −0.357109 −0.178555 0.983930i \(-0.557142\pi\)
−0.178555 + 0.983930i \(0.557142\pi\)
\(390\) 6.70803e8 0.572623
\(391\) −2.58190e9 −2.18434
\(392\) 9.81708e8 0.823154
\(393\) 1.16142e8 0.0965197
\(394\) −4.30926e8 −0.354949
\(395\) −4.47288e8 −0.365172
\(396\) 1.86244e8 0.150713
\(397\) −5.40817e8 −0.433794 −0.216897 0.976194i \(-0.569594\pi\)
−0.216897 + 0.976194i \(0.569594\pi\)
\(398\) 9.62771e8 0.765477
\(399\) −1.19075e9 −0.938456
\(400\) 1.11513e9 0.871198
\(401\) 1.82362e9 1.41231 0.706153 0.708059i \(-0.250429\pi\)
0.706153 + 0.708059i \(0.250429\pi\)
\(402\) 4.99251e8 0.383290
\(403\) 3.78408e9 2.88000
\(404\) 3.07547e7 0.0232047
\(405\) 8.08284e7 0.0604604
\(406\) 3.70681e9 2.74890
\(407\) 1.47721e9 1.08608
\(408\) 7.49082e8 0.546033
\(409\) 1.97200e9 1.42520 0.712599 0.701571i \(-0.247518\pi\)
0.712599 + 0.701571i \(0.247518\pi\)
\(410\) −1.17946e9 −0.845165
\(411\) −5.00800e8 −0.355810
\(412\) −3.47800e8 −0.245013
\(413\) 2.72841e8 0.190583
\(414\) 9.43404e8 0.653426
\(415\) −1.31226e9 −0.901263
\(416\) −1.67694e9 −1.14206
\(417\) −1.11498e9 −0.752994
\(418\) −2.27161e9 −1.52131
\(419\) 4.90370e8 0.325668 0.162834 0.986653i \(-0.447936\pi\)
0.162834 + 0.986653i \(0.447936\pi\)
\(420\) −2.71577e8 −0.178863
\(421\) −1.72536e8 −0.112692 −0.0563459 0.998411i \(-0.517945\pi\)
−0.0563459 + 0.998411i \(0.517945\pi\)
\(422\) 2.54244e9 1.64686
\(423\) −4.74294e8 −0.304689
\(424\) −5.20921e7 −0.0331888
\(425\) −1.46291e9 −0.924392
\(426\) −4.47747e8 −0.280607
\(427\) 2.66027e7 0.0165359
\(428\) −3.71876e8 −0.229269
\(429\) −1.69759e9 −1.03808
\(430\) −1.09669e9 −0.665186
\(431\) 1.99685e9 1.20136 0.600681 0.799489i \(-0.294896\pi\)
0.600681 + 0.799489i \(0.294896\pi\)
\(432\) −3.99129e8 −0.238188
\(433\) −6.53171e8 −0.386651 −0.193326 0.981135i \(-0.561927\pi\)
−0.193326 + 0.981135i \(0.561927\pi\)
\(434\) −5.47114e9 −3.21265
\(435\) 8.59364e8 0.500570
\(436\) −1.12015e9 −0.647255
\(437\) −3.22203e9 −1.84690
\(438\) 1.74152e9 0.990306
\(439\) −2.18962e9 −1.23521 −0.617607 0.786487i \(-0.711898\pi\)
−0.617607 + 0.786487i \(0.711898\pi\)
\(440\) 8.14050e8 0.455582
\(441\) 6.86209e8 0.380997
\(442\) 4.34545e9 2.39363
\(443\) 2.98274e9 1.63006 0.815028 0.579422i \(-0.196722\pi\)
0.815028 + 0.579422i \(0.196722\pi\)
\(444\) 3.86888e8 0.209771
\(445\) −1.33737e8 −0.0719433
\(446\) −2.71149e9 −1.44722
\(447\) 7.50504e8 0.397444
\(448\) −1.02357e9 −0.537830
\(449\) −1.54617e9 −0.806111 −0.403055 0.915176i \(-0.632052\pi\)
−0.403055 + 0.915176i \(0.632052\pi\)
\(450\) 5.34535e8 0.276524
\(451\) 2.98484e9 1.53216
\(452\) 3.65264e8 0.186047
\(453\) 1.36393e9 0.689364
\(454\) −4.69558e8 −0.235502
\(455\) 2.47538e9 1.23197
\(456\) 9.34801e8 0.461681
\(457\) −7.00396e8 −0.343271 −0.171635 0.985161i \(-0.554905\pi\)
−0.171635 + 0.985161i \(0.554905\pi\)
\(458\) 4.67654e9 2.27455
\(459\) 5.23605e8 0.252731
\(460\) −7.34857e8 −0.352006
\(461\) −2.39080e9 −1.13655 −0.568277 0.822838i \(-0.692390\pi\)
−0.568277 + 0.822838i \(0.692390\pi\)
\(462\) 2.45442e9 1.15798
\(463\) 2.71887e9 1.27308 0.636539 0.771244i \(-0.280365\pi\)
0.636539 + 0.771244i \(0.280365\pi\)
\(464\) −4.24352e9 −1.97203
\(465\) −1.26839e9 −0.585017
\(466\) 3.62496e9 1.65941
\(467\) 3.63245e9 1.65041 0.825203 0.564836i \(-0.191061\pi\)
0.825203 + 0.564836i \(0.191061\pi\)
\(468\) −4.44604e8 −0.200500
\(469\) 1.84232e9 0.824633
\(470\) 1.31939e9 0.586178
\(471\) −8.84662e8 −0.390125
\(472\) −2.14195e8 −0.0937589
\(473\) 2.77536e9 1.20588
\(474\) 1.05873e9 0.456627
\(475\) −1.82561e9 −0.781592
\(476\) −1.75927e9 −0.747666
\(477\) −3.64121e7 −0.0153614
\(478\) 9.49444e8 0.397623
\(479\) −7.60336e8 −0.316105 −0.158053 0.987431i \(-0.550522\pi\)
−0.158053 + 0.987431i \(0.550522\pi\)
\(480\) 5.62096e8 0.231988
\(481\) −3.52642e9 −1.44486
\(482\) 3.15145e9 1.28188
\(483\) 3.48132e9 1.40582
\(484\) 3.41028e8 0.136720
\(485\) −2.46417e9 −0.980787
\(486\) −1.91321e8 −0.0756023
\(487\) −3.11666e9 −1.22275 −0.611375 0.791341i \(-0.709383\pi\)
−0.611375 + 0.791341i \(0.709383\pi\)
\(488\) −2.08846e7 −0.00813497
\(489\) 8.84912e8 0.342231
\(490\) −1.90889e9 −0.732984
\(491\) −2.02161e9 −0.770749 −0.385374 0.922760i \(-0.625928\pi\)
−0.385374 + 0.922760i \(0.625928\pi\)
\(492\) 7.81742e8 0.295928
\(493\) 5.56694e9 2.09244
\(494\) 5.42281e9 2.02386
\(495\) 5.69017e8 0.210866
\(496\) 6.26330e9 2.30471
\(497\) −1.65226e9 −0.603715
\(498\) 3.10612e9 1.12698
\(499\) −3.90803e9 −1.40801 −0.704006 0.710194i \(-0.748607\pi\)
−0.704006 + 0.710194i \(0.748607\pi\)
\(500\) −1.00789e9 −0.360593
\(501\) −1.32635e9 −0.471224
\(502\) −4.06258e8 −0.143331
\(503\) −5.07994e9 −1.77980 −0.889899 0.456158i \(-0.849225\pi\)
−0.889899 + 0.456158i \(0.849225\pi\)
\(504\) −1.01003e9 −0.351421
\(505\) 9.39623e7 0.0324664
\(506\) 6.64139e9 2.27894
\(507\) 2.35829e9 0.803653
\(508\) −5.60336e8 −0.189638
\(509\) −2.67138e9 −0.897892 −0.448946 0.893559i \(-0.648200\pi\)
−0.448946 + 0.893559i \(0.648200\pi\)
\(510\) −1.45656e9 −0.486219
\(511\) 6.42651e9 2.13060
\(512\) −6.86375e7 −0.0226004
\(513\) 6.53421e8 0.213689
\(514\) −3.59760e9 −1.16853
\(515\) −1.06260e9 −0.342804
\(516\) 7.26878e8 0.232910
\(517\) −3.33894e9 −1.06265
\(518\) 5.09860e9 1.61175
\(519\) 5.62088e8 0.176489
\(520\) −1.94331e9 −0.606080
\(521\) 1.52466e9 0.472326 0.236163 0.971713i \(-0.424110\pi\)
0.236163 + 0.971713i \(0.424110\pi\)
\(522\) −2.03411e9 −0.625933
\(523\) 9.55727e8 0.292131 0.146066 0.989275i \(-0.453339\pi\)
0.146066 + 0.989275i \(0.453339\pi\)
\(524\) 2.14138e8 0.0650180
\(525\) 1.97253e9 0.594929
\(526\) 4.41005e9 1.32127
\(527\) −8.21663e9 −2.44544
\(528\) −2.80979e9 −0.830721
\(529\) 6.01524e9 1.76668
\(530\) 1.01291e8 0.0295532
\(531\) −1.49721e8 −0.0433963
\(532\) −2.19544e9 −0.632167
\(533\) −7.12545e9 −2.03829
\(534\) 3.16554e8 0.0899610
\(535\) −1.13616e9 −0.320776
\(536\) −1.44632e9 −0.405685
\(537\) 2.33305e9 0.650151
\(538\) −5.17129e9 −1.43173
\(539\) 4.83079e9 1.32879
\(540\) 1.49028e8 0.0407276
\(541\) 1.12167e9 0.304561 0.152280 0.988337i \(-0.451338\pi\)
0.152280 + 0.988337i \(0.451338\pi\)
\(542\) −4.83147e9 −1.30341
\(543\) −1.76775e9 −0.473829
\(544\) 3.64125e9 0.969737
\(545\) −3.42232e9 −0.905591
\(546\) −5.85922e9 −1.54051
\(547\) 5.80841e8 0.151740 0.0758702 0.997118i \(-0.475827\pi\)
0.0758702 + 0.997118i \(0.475827\pi\)
\(548\) −9.23352e8 −0.239682
\(549\) −1.45982e7 −0.00376527
\(550\) 3.76303e9 0.964424
\(551\) 6.94715e9 1.76920
\(552\) −2.73303e9 −0.691604
\(553\) 3.90690e9 0.982413
\(554\) −5.23718e9 −1.30862
\(555\) 1.18203e9 0.293496
\(556\) −2.05575e9 −0.507235
\(557\) 3.50741e9 0.859989 0.429994 0.902832i \(-0.358516\pi\)
0.429994 + 0.902832i \(0.358516\pi\)
\(558\) 3.00229e9 0.731530
\(559\) −6.62537e9 −1.60424
\(560\) 4.09717e9 0.985883
\(561\) 3.68608e9 0.881444
\(562\) −3.49967e8 −0.0831668
\(563\) 1.57440e9 0.371822 0.185911 0.982567i \(-0.440476\pi\)
0.185911 + 0.982567i \(0.440476\pi\)
\(564\) −8.74482e8 −0.205246
\(565\) 1.11596e9 0.260303
\(566\) −4.03632e9 −0.935682
\(567\) −7.06006e8 −0.162655
\(568\) 1.29712e9 0.297003
\(569\) −2.71007e7 −0.00616719 −0.00308360 0.999995i \(-0.500982\pi\)
−0.00308360 + 0.999995i \(0.500982\pi\)
\(570\) −1.81768e9 −0.411108
\(571\) −8.12943e9 −1.82740 −0.913700 0.406390i \(-0.866787\pi\)
−0.913700 + 0.406390i \(0.866787\pi\)
\(572\) −3.12993e9 −0.699276
\(573\) −2.28286e9 −0.506918
\(574\) 1.03022e10 2.27372
\(575\) 5.33743e9 1.17083
\(576\) 5.61684e8 0.122465
\(577\) −8.89512e8 −0.192769 −0.0963844 0.995344i \(-0.530728\pi\)
−0.0963844 + 0.995344i \(0.530728\pi\)
\(578\) −3.96431e9 −0.853927
\(579\) −5.51364e8 −0.118049
\(580\) 1.58446e9 0.337196
\(581\) 1.14621e10 2.42464
\(582\) 5.83269e9 1.22642
\(583\) −2.56334e8 −0.0535756
\(584\) −5.04516e9 −1.04817
\(585\) −1.35836e9 −0.280524
\(586\) −5.92218e9 −1.21574
\(587\) 8.09762e8 0.165243 0.0826217 0.996581i \(-0.473671\pi\)
0.0826217 + 0.996581i \(0.473671\pi\)
\(588\) 1.26520e9 0.256649
\(589\) −1.02538e10 −2.06766
\(590\) 4.16493e8 0.0834884
\(591\) 8.72617e8 0.173887
\(592\) −5.83682e9 −1.15625
\(593\) −6.48925e9 −1.27792 −0.638959 0.769241i \(-0.720635\pi\)
−0.638959 + 0.769241i \(0.720635\pi\)
\(594\) −1.34686e9 −0.263676
\(595\) −5.37495e9 −1.04608
\(596\) 1.38374e9 0.267728
\(597\) −1.94959e9 −0.375002
\(598\) −1.58544e10 −3.03176
\(599\) 6.51857e8 0.123925 0.0619624 0.998078i \(-0.480264\pi\)
0.0619624 + 0.998078i \(0.480264\pi\)
\(600\) −1.54854e9 −0.292680
\(601\) −2.97476e9 −0.558974 −0.279487 0.960150i \(-0.590164\pi\)
−0.279487 + 0.960150i \(0.590164\pi\)
\(602\) 9.57915e9 1.78953
\(603\) −1.01097e9 −0.187771
\(604\) 2.51475e9 0.464372
\(605\) 1.04191e9 0.191288
\(606\) −2.22409e8 −0.0405973
\(607\) −4.76572e9 −0.864905 −0.432452 0.901657i \(-0.642352\pi\)
−0.432452 + 0.901657i \(0.642352\pi\)
\(608\) 4.54402e9 0.819931
\(609\) −7.50622e9 −1.34667
\(610\) 4.06091e7 0.00724385
\(611\) 7.97076e9 1.41369
\(612\) 9.65398e8 0.170246
\(613\) 4.10844e9 0.720385 0.360193 0.932878i \(-0.382711\pi\)
0.360193 + 0.932878i \(0.382711\pi\)
\(614\) 9.57176e9 1.66879
\(615\) 2.38839e9 0.414040
\(616\) −7.11042e9 −1.22564
\(617\) 8.36433e9 1.43362 0.716809 0.697270i \(-0.245602\pi\)
0.716809 + 0.697270i \(0.245602\pi\)
\(618\) 2.51518e9 0.428657
\(619\) −1.37677e8 −0.0233315 −0.0116658 0.999932i \(-0.503713\pi\)
−0.0116658 + 0.999932i \(0.503713\pi\)
\(620\) −2.33861e9 −0.394082
\(621\) −1.91037e9 −0.320109
\(622\) 1.32153e10 2.20196
\(623\) 1.16814e9 0.193547
\(624\) 6.70756e9 1.10514
\(625\) 1.21700e9 0.199393
\(626\) 2.36253e8 0.0384917
\(627\) 4.59997e9 0.745278
\(628\) −1.63110e9 −0.262798
\(629\) 7.65715e9 1.22685
\(630\) 1.96396e9 0.312925
\(631\) −1.86229e9 −0.295083 −0.147542 0.989056i \(-0.547136\pi\)
−0.147542 + 0.989056i \(0.547136\pi\)
\(632\) −3.06713e9 −0.483306
\(633\) −5.14838e9 −0.806784
\(634\) −8.89756e9 −1.38662
\(635\) −1.71195e9 −0.265328
\(636\) −6.71350e7 −0.0103478
\(637\) −1.15321e10 −1.76775
\(638\) −1.43198e10 −2.18305
\(639\) 9.06679e8 0.137468
\(640\) −4.22724e9 −0.637422
\(641\) 1.04508e10 1.56727 0.783637 0.621219i \(-0.213362\pi\)
0.783637 + 0.621219i \(0.213362\pi\)
\(642\) 2.68930e9 0.401112
\(643\) −1.17114e10 −1.73728 −0.868642 0.495440i \(-0.835007\pi\)
−0.868642 + 0.495440i \(0.835007\pi\)
\(644\) 6.41870e9 0.946993
\(645\) 2.22077e9 0.325870
\(646\) −1.17749e10 −1.71848
\(647\) −2.82698e7 −0.00410353 −0.00205176 0.999998i \(-0.500653\pi\)
−0.00205176 + 0.999998i \(0.500653\pi\)
\(648\) 5.54253e8 0.0800195
\(649\) −1.05401e9 −0.151352
\(650\) −8.98314e9 −1.28301
\(651\) 1.10789e10 1.57386
\(652\) 1.63156e9 0.230535
\(653\) 1.03288e10 1.45163 0.725813 0.687892i \(-0.241464\pi\)
0.725813 + 0.687892i \(0.241464\pi\)
\(654\) 8.10062e9 1.13239
\(655\) 6.54237e8 0.0909684
\(656\) −1.17938e10 −1.63114
\(657\) −3.52654e9 −0.485144
\(658\) −1.15244e10 −1.57698
\(659\) −8.85377e9 −1.20512 −0.602558 0.798075i \(-0.705852\pi\)
−0.602558 + 0.798075i \(0.705852\pi\)
\(660\) 1.04913e9 0.142045
\(661\) 3.94438e8 0.0531219 0.0265610 0.999647i \(-0.491544\pi\)
0.0265610 + 0.999647i \(0.491544\pi\)
\(662\) −1.61746e10 −2.16686
\(663\) −8.79944e9 −1.17262
\(664\) −8.99838e9 −1.19282
\(665\) −6.70756e9 −0.884481
\(666\) −2.79786e9 −0.367000
\(667\) −2.03110e10 −2.65027
\(668\) −2.44547e9 −0.317427
\(669\) 5.49070e9 0.708984
\(670\) 2.81232e9 0.361246
\(671\) −1.02769e8 −0.0131320
\(672\) −4.90970e9 −0.624112
\(673\) −3.32239e8 −0.0420144 −0.0210072 0.999779i \(-0.506687\pi\)
−0.0210072 + 0.999779i \(0.506687\pi\)
\(674\) −5.38472e9 −0.677413
\(675\) −1.08242e9 −0.135467
\(676\) 4.34810e9 0.541360
\(677\) −7.35957e9 −0.911574 −0.455787 0.890089i \(-0.650642\pi\)
−0.455787 + 0.890089i \(0.650642\pi\)
\(678\) −2.64148e9 −0.325494
\(679\) 2.15236e10 2.63858
\(680\) 4.21963e9 0.514628
\(681\) 9.50846e8 0.115371
\(682\) 2.11355e10 2.55134
\(683\) −1.53063e10 −1.83822 −0.919111 0.393999i \(-0.871091\pi\)
−0.919111 + 0.393999i \(0.871091\pi\)
\(684\) 1.20475e9 0.143946
\(685\) −2.82104e9 −0.335345
\(686\) 2.08589e9 0.246693
\(687\) −9.46990e9 −1.11429
\(688\) −1.09661e10 −1.28379
\(689\) 6.11924e8 0.0712739
\(690\) 5.31426e9 0.615844
\(691\) −1.49200e10 −1.72026 −0.860132 0.510072i \(-0.829619\pi\)
−0.860132 + 0.510072i \(0.829619\pi\)
\(692\) 1.03635e9 0.118887
\(693\) −4.97015e9 −0.567288
\(694\) 7.64611e9 0.868325
\(695\) −6.28077e9 −0.709686
\(696\) 5.89280e9 0.662505
\(697\) 1.54719e10 1.73073
\(698\) 4.28329e9 0.476742
\(699\) −7.34048e9 −0.812932
\(700\) 3.63685e9 0.400758
\(701\) −6.52424e9 −0.715347 −0.357673 0.933847i \(-0.616430\pi\)
−0.357673 + 0.933847i \(0.616430\pi\)
\(702\) 3.21524e9 0.350779
\(703\) 9.55558e9 1.03732
\(704\) 3.95415e9 0.427119
\(705\) −2.67173e9 −0.287165
\(706\) −5.31262e9 −0.568188
\(707\) −8.20726e8 −0.0873434
\(708\) −2.76049e8 −0.0292328
\(709\) 4.55527e9 0.480012 0.240006 0.970771i \(-0.422851\pi\)
0.240006 + 0.970771i \(0.422851\pi\)
\(710\) −2.52219e9 −0.264468
\(711\) −2.14391e9 −0.223698
\(712\) −9.17054e8 −0.0952171
\(713\) 2.99784e10 3.09738
\(714\) 1.27225e10 1.30806
\(715\) −9.56262e9 −0.978376
\(716\) 4.30157e9 0.437957
\(717\) −1.92260e9 −0.194793
\(718\) −1.87607e10 −1.89153
\(719\) 6.09158e9 0.611194 0.305597 0.952161i \(-0.401144\pi\)
0.305597 + 0.952161i \(0.401144\pi\)
\(720\) −2.24832e9 −0.224489
\(721\) 9.28145e9 0.922236
\(722\) −2.77584e9 −0.274482
\(723\) −6.38162e9 −0.627982
\(724\) −3.25930e9 −0.319183
\(725\) −1.15083e10 −1.12157
\(726\) −2.46621e9 −0.239195
\(727\) 1.68599e10 1.62737 0.813684 0.581308i \(-0.197459\pi\)
0.813684 + 0.581308i \(0.197459\pi\)
\(728\) 1.69741e10 1.63052
\(729\) 3.87420e8 0.0370370
\(730\) 9.81011e9 0.933349
\(731\) 1.43861e10 1.36217
\(732\) −2.69155e7 −0.00253638
\(733\) −1.44578e9 −0.135593 −0.0677966 0.997699i \(-0.521597\pi\)
−0.0677966 + 0.997699i \(0.521597\pi\)
\(734\) 1.28762e10 1.20185
\(735\) 3.86546e9 0.359084
\(736\) −1.32851e10 −1.22827
\(737\) −7.11707e9 −0.654885
\(738\) −5.65332e9 −0.517733
\(739\) 8.56684e8 0.0780845 0.0390423 0.999238i \(-0.487569\pi\)
0.0390423 + 0.999238i \(0.487569\pi\)
\(740\) 2.17937e9 0.197706
\(741\) −1.09811e10 −0.991475
\(742\) −8.84738e8 −0.0795062
\(743\) 1.09080e10 0.975630 0.487815 0.872947i \(-0.337794\pi\)
0.487815 + 0.872947i \(0.337794\pi\)
\(744\) −8.69758e9 −0.774271
\(745\) 4.22764e9 0.374585
\(746\) 8.68006e9 0.765485
\(747\) −6.28983e9 −0.552098
\(748\) 6.79622e9 0.593762
\(749\) 9.92396e9 0.862976
\(750\) 7.28873e9 0.630866
\(751\) −5.27485e9 −0.454433 −0.227217 0.973844i \(-0.572963\pi\)
−0.227217 + 0.973844i \(0.572963\pi\)
\(752\) 1.31929e10 1.13130
\(753\) 8.22665e8 0.0702167
\(754\) 3.41843e10 2.90420
\(755\) 7.68311e9 0.649715
\(756\) −1.30170e9 −0.109568
\(757\) −2.07054e10 −1.73479 −0.867395 0.497620i \(-0.834208\pi\)
−0.867395 + 0.497620i \(0.834208\pi\)
\(758\) 5.30659e8 0.0442562
\(759\) −1.34487e10 −1.11643
\(760\) 5.26580e9 0.435128
\(761\) 1.35780e10 1.11683 0.558417 0.829561i \(-0.311409\pi\)
0.558417 + 0.829561i \(0.311409\pi\)
\(762\) 4.05218e9 0.331777
\(763\) 2.98927e10 2.43629
\(764\) −4.20903e9 −0.341472
\(765\) 2.94950e9 0.238195
\(766\) −8.37001e8 −0.0672861
\(767\) 2.51614e9 0.201350
\(768\) 7.34307e9 0.584943
\(769\) 1.62978e10 1.29237 0.646184 0.763182i \(-0.276364\pi\)
0.646184 + 0.763182i \(0.276364\pi\)
\(770\) 1.38259e10 1.09138
\(771\) 7.28506e9 0.572457
\(772\) −1.01658e9 −0.0795208
\(773\) −2.64697e9 −0.206120 −0.103060 0.994675i \(-0.532863\pi\)
−0.103060 + 0.994675i \(0.532863\pi\)
\(774\) −5.25656e9 −0.407482
\(775\) 1.69858e10 1.31078
\(776\) −1.68972e10 −1.29807
\(777\) −1.03246e10 −0.789584
\(778\) 5.52800e9 0.420862
\(779\) 1.93079e10 1.46337
\(780\) −2.50449e9 −0.188968
\(781\) 6.38285e9 0.479442
\(782\) 3.44257e10 2.57430
\(783\) 4.11904e9 0.306640
\(784\) −1.90876e10 −1.41463
\(785\) −4.98336e9 −0.367687
\(786\) −1.54858e9 −0.113751
\(787\) −7.25919e9 −0.530856 −0.265428 0.964131i \(-0.585513\pi\)
−0.265428 + 0.964131i \(0.585513\pi\)
\(788\) 1.60889e9 0.117134
\(789\) −8.93025e9 −0.647283
\(790\) 5.96390e9 0.430364
\(791\) −9.74751e9 −0.700287
\(792\) 3.90184e9 0.279082
\(793\) 2.45330e8 0.0174701
\(794\) 7.21097e9 0.511237
\(795\) −2.05112e8 −0.0144779
\(796\) −3.59457e9 −0.252610
\(797\) −3.67130e9 −0.256871 −0.128436 0.991718i \(-0.540996\pi\)
−0.128436 + 0.991718i \(0.540996\pi\)
\(798\) 1.58768e10 1.10599
\(799\) −1.73074e10 −1.20038
\(800\) −7.52737e9 −0.519791
\(801\) −6.41016e8 −0.0440712
\(802\) −2.43152e10 −1.66444
\(803\) −2.48262e10 −1.69202
\(804\) −1.86399e9 −0.126487
\(805\) 1.96105e10 1.32496
\(806\) −5.04550e10 −3.39415
\(807\) 1.04718e10 0.701395
\(808\) 6.44315e8 0.0429693
\(809\) −5.97080e9 −0.396472 −0.198236 0.980154i \(-0.563521\pi\)
−0.198236 + 0.980154i \(0.563521\pi\)
\(810\) −1.07772e9 −0.0712540
\(811\) 2.51680e10 1.65682 0.828409 0.560123i \(-0.189246\pi\)
0.828409 + 0.560123i \(0.189246\pi\)
\(812\) −1.38396e10 −0.907148
\(813\) 9.78363e9 0.638533
\(814\) −1.96964e10 −1.27997
\(815\) 4.98477e9 0.322548
\(816\) −1.45646e10 −0.938387
\(817\) 1.79528e10 1.15174
\(818\) −2.62936e10 −1.67963
\(819\) 1.18648e10 0.754686
\(820\) 4.40361e9 0.278907
\(821\) 1.74411e10 1.09995 0.549975 0.835181i \(-0.314637\pi\)
0.549975 + 0.835181i \(0.314637\pi\)
\(822\) 6.67740e9 0.419330
\(823\) 1.10753e10 0.692557 0.346278 0.938132i \(-0.387445\pi\)
0.346278 + 0.938132i \(0.387445\pi\)
\(824\) −7.28644e9 −0.453702
\(825\) −7.62005e9 −0.472464
\(826\) −3.63791e9 −0.224606
\(827\) −7.16696e9 −0.440622 −0.220311 0.975430i \(-0.570707\pi\)
−0.220311 + 0.975430i \(0.570707\pi\)
\(828\) −3.52226e9 −0.215633
\(829\) 1.09285e9 0.0666224 0.0333112 0.999445i \(-0.489395\pi\)
0.0333112 + 0.999445i \(0.489395\pi\)
\(830\) 1.74970e10 1.06216
\(831\) 1.06052e10 0.641083
\(832\) −9.43940e9 −0.568215
\(833\) 2.50404e10 1.50101
\(834\) 1.48666e10 0.887421
\(835\) −7.47144e9 −0.444121
\(836\) 8.48121e9 0.502037
\(837\) −6.07957e9 −0.358372
\(838\) −6.53834e9 −0.383807
\(839\) −2.38628e10 −1.39494 −0.697469 0.716615i \(-0.745691\pi\)
−0.697469 + 0.716615i \(0.745691\pi\)
\(840\) −5.68957e9 −0.331209
\(841\) 2.65435e10 1.53877
\(842\) 2.30050e9 0.132810
\(843\) 7.08676e8 0.0407428
\(844\) −9.49235e9 −0.543469
\(845\) 1.32844e10 0.757431
\(846\) 6.32398e9 0.359083
\(847\) −9.10073e9 −0.514617
\(848\) 1.01284e9 0.0570367
\(849\) 8.17347e9 0.458384
\(850\) 1.95057e10 1.08942
\(851\) −2.79371e10 −1.55392
\(852\) 1.67169e9 0.0926015
\(853\) 1.71276e10 0.944878 0.472439 0.881363i \(-0.343374\pi\)
0.472439 + 0.881363i \(0.343374\pi\)
\(854\) −3.54706e8 −0.0194879
\(855\) 3.68077e9 0.201399
\(856\) −7.79086e9 −0.424548
\(857\) 3.37860e8 0.0183360 0.00916798 0.999958i \(-0.497082\pi\)
0.00916798 + 0.999958i \(0.497082\pi\)
\(858\) 2.26347e10 1.22340
\(859\) 3.39086e9 0.182530 0.0912649 0.995827i \(-0.470909\pi\)
0.0912649 + 0.995827i \(0.470909\pi\)
\(860\) 4.09455e9 0.219514
\(861\) −2.08617e10 −1.11388
\(862\) −2.66249e10 −1.41583
\(863\) 9.19359e9 0.486909 0.243454 0.969912i \(-0.421719\pi\)
0.243454 + 0.969912i \(0.421719\pi\)
\(864\) 2.69420e9 0.142112
\(865\) 3.16628e9 0.166339
\(866\) 8.70904e9 0.455678
\(867\) 8.02766e9 0.418333
\(868\) 2.04269e10 1.06019
\(869\) −1.50927e10 −0.780186
\(870\) −1.14583e10 −0.589933
\(871\) 1.69899e10 0.871221
\(872\) −2.34674e10 −1.19855
\(873\) −1.18111e10 −0.600813
\(874\) 4.29608e10 2.17662
\(875\) 2.68966e10 1.35728
\(876\) −6.50208e9 −0.326805
\(877\) −1.78531e10 −0.893747 −0.446874 0.894597i \(-0.647463\pi\)
−0.446874 + 0.894597i \(0.647463\pi\)
\(878\) 2.91952e10 1.45573
\(879\) 1.19923e10 0.595582
\(880\) −1.58277e10 −0.782942
\(881\) −1.00212e10 −0.493746 −0.246873 0.969048i \(-0.579403\pi\)
−0.246873 + 0.969048i \(0.579403\pi\)
\(882\) −9.14955e9 −0.449014
\(883\) 4.80376e8 0.0234811 0.0117406 0.999931i \(-0.496263\pi\)
0.0117406 + 0.999931i \(0.496263\pi\)
\(884\) −1.62240e10 −0.789906
\(885\) −8.43390e8 −0.0409004
\(886\) −3.97703e10 −1.92106
\(887\) −2.86366e9 −0.137781 −0.0688903 0.997624i \(-0.521946\pi\)
−0.0688903 + 0.997624i \(0.521946\pi\)
\(888\) 8.10535e9 0.388442
\(889\) 1.49532e10 0.713804
\(890\) 1.78317e9 0.0847869
\(891\) 2.72737e9 0.129173
\(892\) 1.01235e10 0.477589
\(893\) −2.15984e10 −1.01494
\(894\) −1.00068e10 −0.468398
\(895\) 1.31422e10 0.612758
\(896\) 3.69234e10 1.71484
\(897\) 3.21048e10 1.48524
\(898\) 2.06158e10 0.950020
\(899\) −6.46377e10 −2.96706
\(900\) −1.99572e9 −0.0912538
\(901\) −1.32871e9 −0.0605193
\(902\) −3.97983e10 −1.80568
\(903\) −1.93976e10 −0.876679
\(904\) 7.65233e9 0.344512
\(905\) −9.95787e9 −0.446577
\(906\) −1.81859e10 −0.812431
\(907\) −5.67695e9 −0.252633 −0.126316 0.991990i \(-0.540315\pi\)
−0.126316 + 0.991990i \(0.540315\pi\)
\(908\) 1.75313e9 0.0777164
\(909\) 4.50373e8 0.0198883
\(910\) −3.30054e10 −1.45191
\(911\) 2.67566e10 1.17251 0.586256 0.810126i \(-0.300601\pi\)
0.586256 + 0.810126i \(0.300601\pi\)
\(912\) −1.81755e10 −0.793424
\(913\) −4.42792e10 −1.92554
\(914\) 9.33871e9 0.404553
\(915\) −8.22327e7 −0.00354871
\(916\) −1.74602e10 −0.750610
\(917\) −5.71452e9 −0.244730
\(918\) −6.98146e9 −0.297850
\(919\) 2.00200e10 0.850862 0.425431 0.904991i \(-0.360122\pi\)
0.425431 + 0.904991i \(0.360122\pi\)
\(920\) −1.53953e10 −0.651826
\(921\) −1.93826e10 −0.817530
\(922\) 3.18776e10 1.33945
\(923\) −1.52372e10 −0.637822
\(924\) −9.16374e9 −0.382138
\(925\) −1.58293e10 −0.657604
\(926\) −3.62520e10 −1.50035
\(927\) −5.09319e9 −0.209996
\(928\) 2.86446e10 1.17659
\(929\) 2.42743e10 0.993326 0.496663 0.867943i \(-0.334559\pi\)
0.496663 + 0.867943i \(0.334559\pi\)
\(930\) 1.69121e10 0.689457
\(931\) 3.12486e10 1.26913
\(932\) −1.35340e10 −0.547611
\(933\) −2.67606e10 −1.07872
\(934\) −4.84332e10 −1.94504
\(935\) 2.07639e10 0.830748
\(936\) −9.31452e9 −0.371274
\(937\) 2.31846e10 0.920683 0.460342 0.887742i \(-0.347727\pi\)
0.460342 + 0.887742i \(0.347727\pi\)
\(938\) −2.45645e10 −0.971849
\(939\) −4.78408e8 −0.0188568
\(940\) −4.92602e9 −0.193441
\(941\) −4.82845e10 −1.88905 −0.944527 0.328434i \(-0.893479\pi\)
−0.944527 + 0.328434i \(0.893479\pi\)
\(942\) 1.17956e10 0.459772
\(943\) −5.64495e10 −2.19214
\(944\) 4.16464e9 0.161130
\(945\) −3.97698e9 −0.153300
\(946\) −3.70052e10 −1.42116
\(947\) −9.48483e9 −0.362915 −0.181457 0.983399i \(-0.558081\pi\)
−0.181457 + 0.983399i \(0.558081\pi\)
\(948\) −3.95284e9 −0.150688
\(949\) 5.92654e10 2.25097
\(950\) 2.43417e10 0.921124
\(951\) 1.80174e10 0.679297
\(952\) −3.68569e10 −1.38449
\(953\) 3.13620e10 1.17376 0.586880 0.809674i \(-0.300356\pi\)
0.586880 + 0.809674i \(0.300356\pi\)
\(954\) 4.85500e8 0.0181038
\(955\) −1.28595e10 −0.477762
\(956\) −3.54481e9 −0.131217
\(957\) 2.89972e10 1.06946
\(958\) 1.01379e10 0.372537
\(959\) 2.46407e10 0.902171
\(960\) 3.16401e9 0.115422
\(961\) 6.78906e10 2.46762
\(962\) 4.70194e10 1.70280
\(963\) −5.44577e9 −0.196502
\(964\) −1.17661e10 −0.423024
\(965\) −3.10587e9 −0.111260
\(966\) −4.64181e10 −1.65679
\(967\) −4.56505e10 −1.62350 −0.811752 0.584003i \(-0.801486\pi\)
−0.811752 + 0.584003i \(0.801486\pi\)
\(968\) 7.14457e9 0.253170
\(969\) 2.38439e10 0.841869
\(970\) 3.28559e10 1.15588
\(971\) −4.75876e10 −1.66812 −0.834059 0.551675i \(-0.813989\pi\)
−0.834059 + 0.551675i \(0.813989\pi\)
\(972\) 7.14308e8 0.0249490
\(973\) 5.48602e10 1.90925
\(974\) 4.15558e10 1.44104
\(975\) 1.81907e10 0.628539
\(976\) 4.06063e8 0.0139804
\(977\) −3.64883e10 −1.25176 −0.625882 0.779918i \(-0.715261\pi\)
−0.625882 + 0.779918i \(0.715261\pi\)
\(978\) −1.17990e10 −0.403327
\(979\) −4.51264e9 −0.153706
\(980\) 7.12697e9 0.241888
\(981\) −1.64036e10 −0.554750
\(982\) 2.69551e10 0.908346
\(983\) −2.58715e10 −0.868729 −0.434365 0.900737i \(-0.643027\pi\)
−0.434365 + 0.900737i \(0.643027\pi\)
\(984\) 1.63776e10 0.547983
\(985\) 4.91551e9 0.163886
\(986\) −7.42266e10 −2.46598
\(987\) 2.33366e10 0.772551
\(988\) −2.02464e10 −0.667881
\(989\) −5.24877e10 −1.72532
\(990\) −7.58697e9 −0.248511
\(991\) 5.08417e10 1.65944 0.829722 0.558177i \(-0.188499\pi\)
0.829722 + 0.558177i \(0.188499\pi\)
\(992\) −4.22785e10 −1.37508
\(993\) 3.27533e10 1.06153
\(994\) 2.20304e10 0.711492
\(995\) −1.09822e10 −0.353434
\(996\) −1.15969e10 −0.371907
\(997\) −7.59411e9 −0.242686 −0.121343 0.992611i \(-0.538720\pi\)
−0.121343 + 0.992611i \(0.538720\pi\)
\(998\) 5.21076e10 1.65937
\(999\) 5.66560e9 0.179791
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.1.5 17
3.2 odd 2 531.8.a.d.1.13 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.5 17 1.1 even 1 trivial
531.8.a.d.1.13 17 3.2 odd 2