Properties

Label 177.8.a.d.1.16
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $18$
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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + 3375594921 x^{11} + 115310342333 x^{10} - 774932111214 x^{9} - 14600047830166 x^{8} + 102185027148481 x^{7} + 988557475638619 x^{6} - 7379206238519716 x^{5} - 30152342836849520 x^{4} + 260578770749067175 x^{3} + 182609347488069978 x^{2} - 3481290425710753600 x + 5164646074739714048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(17.9457\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+18.9457 q^{2} +27.0000 q^{3} +230.941 q^{4} +260.561 q^{5} +511.535 q^{6} +254.948 q^{7} +1950.29 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+18.9457 q^{2} +27.0000 q^{3} +230.941 q^{4} +260.561 q^{5} +511.535 q^{6} +254.948 q^{7} +1950.29 q^{8} +729.000 q^{9} +4936.51 q^{10} -3600.84 q^{11} +6235.40 q^{12} +10076.4 q^{13} +4830.18 q^{14} +7035.14 q^{15} +7389.22 q^{16} +4423.02 q^{17} +13811.4 q^{18} +16970.9 q^{19} +60174.1 q^{20} +6883.60 q^{21} -68220.5 q^{22} +63177.4 q^{23} +52657.8 q^{24} -10233.2 q^{25} +190906. q^{26} +19683.0 q^{27} +58877.9 q^{28} -27436.7 q^{29} +133286. q^{30} -22223.7 q^{31} -109643. q^{32} -97222.6 q^{33} +83797.4 q^{34} +66429.5 q^{35} +168356. q^{36} +111541. q^{37} +321525. q^{38} +272064. q^{39} +508168. q^{40} +87545.8 q^{41} +130415. q^{42} -454839. q^{43} -831580. q^{44} +189949. q^{45} +1.19694e6 q^{46} -747408. q^{47} +199509. q^{48} -758544. q^{49} -193875. q^{50} +119422. q^{51} +2.32706e6 q^{52} +1.10581e6 q^{53} +372909. q^{54} -938236. q^{55} +497223. q^{56} +458213. q^{57} -519807. q^{58} +205379. q^{59} +1.62470e6 q^{60} -227215. q^{61} -421044. q^{62} +185857. q^{63} -3.02308e6 q^{64} +2.62552e6 q^{65} -1.84195e6 q^{66} +1.66699e6 q^{67} +1.02146e6 q^{68} +1.70579e6 q^{69} +1.25856e6 q^{70} +1.63889e6 q^{71} +1.42176e6 q^{72} -379956. q^{73} +2.11322e6 q^{74} -276295. q^{75} +3.91926e6 q^{76} -918027. q^{77} +5.15445e6 q^{78} -1.69232e6 q^{79} +1.92534e6 q^{80} +531441. q^{81} +1.65862e6 q^{82} +3.64673e6 q^{83} +1.58970e6 q^{84} +1.15247e6 q^{85} -8.61726e6 q^{86} -740790. q^{87} -7.02267e6 q^{88} -7.90873e6 q^{89} +3.59872e6 q^{90} +2.56897e6 q^{91} +1.45902e7 q^{92} -600040. q^{93} -1.41602e7 q^{94} +4.42194e6 q^{95} -2.96035e6 q^{96} -905108. q^{97} -1.43712e7 q^{98} -2.62501e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 18.9457 1.67458 0.837291 0.546758i \(-0.184138\pi\)
0.837291 + 0.546758i \(0.184138\pi\)
\(3\) 27.0000 0.577350
\(4\) 230.941 1.80422
\(5\) 260.561 0.932210 0.466105 0.884729i \(-0.345657\pi\)
0.466105 + 0.884729i \(0.345657\pi\)
\(6\) 511.535 0.966820
\(7\) 254.948 0.280937 0.140468 0.990085i \(-0.455139\pi\)
0.140468 + 0.990085i \(0.455139\pi\)
\(8\) 1950.29 1.34674
\(9\) 729.000 0.333333
\(10\) 4936.51 1.56106
\(11\) −3600.84 −0.815697 −0.407849 0.913050i \(-0.633721\pi\)
−0.407849 + 0.913050i \(0.633721\pi\)
\(12\) 6235.40 1.04167
\(13\) 10076.4 1.27205 0.636027 0.771667i \(-0.280577\pi\)
0.636027 + 0.771667i \(0.280577\pi\)
\(14\) 4830.18 0.470452
\(15\) 7035.14 0.538212
\(16\) 7389.22 0.451002
\(17\) 4423.02 0.218347 0.109174 0.994023i \(-0.465180\pi\)
0.109174 + 0.994023i \(0.465180\pi\)
\(18\) 13811.4 0.558194
\(19\) 16970.9 0.567631 0.283815 0.958879i \(-0.408400\pi\)
0.283815 + 0.958879i \(0.408400\pi\)
\(20\) 60174.1 1.68192
\(21\) 6883.60 0.162199
\(22\) −68220.5 −1.36595
\(23\) 63177.4 1.08272 0.541358 0.840792i \(-0.317910\pi\)
0.541358 + 0.840792i \(0.317910\pi\)
\(24\) 52657.8 0.777541
\(25\) −10233.2 −0.130984
\(26\) 190906. 2.13016
\(27\) 19683.0 0.192450
\(28\) 58877.9 0.506873
\(29\) −27436.7 −0.208900 −0.104450 0.994530i \(-0.533308\pi\)
−0.104450 + 0.994530i \(0.533308\pi\)
\(30\) 133286. 0.901280
\(31\) −22223.7 −0.133983 −0.0669916 0.997754i \(-0.521340\pi\)
−0.0669916 + 0.997754i \(0.521340\pi\)
\(32\) −109643. −0.591500
\(33\) −97222.6 −0.470943
\(34\) 83797.4 0.365641
\(35\) 66429.5 0.261892
\(36\) 168356. 0.601408
\(37\) 111541. 0.362016 0.181008 0.983482i \(-0.442064\pi\)
0.181008 + 0.983482i \(0.442064\pi\)
\(38\) 321525. 0.950544
\(39\) 272064. 0.734421
\(40\) 508168. 1.25544
\(41\) 87545.8 0.198377 0.0991886 0.995069i \(-0.468375\pi\)
0.0991886 + 0.995069i \(0.468375\pi\)
\(42\) 130415. 0.271616
\(43\) −454839. −0.872405 −0.436202 0.899849i \(-0.643677\pi\)
−0.436202 + 0.899849i \(0.643677\pi\)
\(44\) −831580. −1.47170
\(45\) 189949. 0.310737
\(46\) 1.19694e6 1.81310
\(47\) −747408. −1.05006 −0.525031 0.851083i \(-0.675946\pi\)
−0.525031 + 0.851083i \(0.675946\pi\)
\(48\) 199509. 0.260386
\(49\) −758544. −0.921074
\(50\) −193875. −0.219344
\(51\) 119422. 0.126063
\(52\) 2.32706e6 2.29507
\(53\) 1.10581e6 1.02027 0.510136 0.860094i \(-0.329595\pi\)
0.510136 + 0.860094i \(0.329595\pi\)
\(54\) 372909. 0.322273
\(55\) −938236. −0.760401
\(56\) 497223. 0.378349
\(57\) 458213. 0.327722
\(58\) −519807. −0.349820
\(59\) 205379. 0.130189
\(60\) 1.62470e6 0.971055
\(61\) −227215. −0.128169 −0.0640844 0.997944i \(-0.520413\pi\)
−0.0640844 + 0.997944i \(0.520413\pi\)
\(62\) −421044. −0.224366
\(63\) 185857. 0.0936457
\(64\) −3.02308e6 −1.44152
\(65\) 2.62552e6 1.18582
\(66\) −1.84195e6 −0.788633
\(67\) 1.66699e6 0.677128 0.338564 0.940943i \(-0.390059\pi\)
0.338564 + 0.940943i \(0.390059\pi\)
\(68\) 1.02146e6 0.393948
\(69\) 1.70579e6 0.625106
\(70\) 1.25856e6 0.438560
\(71\) 1.63889e6 0.543432 0.271716 0.962377i \(-0.412409\pi\)
0.271716 + 0.962377i \(0.412409\pi\)
\(72\) 1.42176e6 0.448913
\(73\) −379956. −0.114315 −0.0571575 0.998365i \(-0.518204\pi\)
−0.0571575 + 0.998365i \(0.518204\pi\)
\(74\) 2.11322e6 0.606225
\(75\) −276295. −0.0756239
\(76\) 3.91926e6 1.02413
\(77\) −918027. −0.229159
\(78\) 5.15445e6 1.22985
\(79\) −1.69232e6 −0.386179 −0.193090 0.981181i \(-0.561851\pi\)
−0.193090 + 0.981181i \(0.561851\pi\)
\(80\) 1.92534e6 0.420429
\(81\) 531441. 0.111111
\(82\) 1.65862e6 0.332199
\(83\) 3.64673e6 0.700052 0.350026 0.936740i \(-0.386173\pi\)
0.350026 + 0.936740i \(0.386173\pi\)
\(84\) 1.58970e6 0.292644
\(85\) 1.15247e6 0.203546
\(86\) −8.61726e6 −1.46091
\(87\) −740790. −0.120608
\(88\) −7.02267e6 −1.09853
\(89\) −7.90873e6 −1.18916 −0.594582 0.804035i \(-0.702682\pi\)
−0.594582 + 0.804035i \(0.702682\pi\)
\(90\) 3.59872e6 0.520354
\(91\) 2.56897e6 0.357367
\(92\) 1.45902e7 1.95346
\(93\) −600040. −0.0773552
\(94\) −1.41602e7 −1.75842
\(95\) 4.42194e6 0.529151
\(96\) −2.96035e6 −0.341503
\(97\) −905108. −0.100693 −0.0503465 0.998732i \(-0.516033\pi\)
−0.0503465 + 0.998732i \(0.516033\pi\)
\(98\) −1.43712e7 −1.54241
\(99\) −2.62501e6 −0.271899
\(100\) −2.36325e6 −0.236325
\(101\) −5.51628e6 −0.532748 −0.266374 0.963870i \(-0.585826\pi\)
−0.266374 + 0.963870i \(0.585826\pi\)
\(102\) 2.26253e6 0.211103
\(103\) −8.52758e6 −0.768945 −0.384473 0.923136i \(-0.625617\pi\)
−0.384473 + 0.923136i \(0.625617\pi\)
\(104\) 1.96520e7 1.71313
\(105\) 1.79360e6 0.151204
\(106\) 2.09504e7 1.70853
\(107\) 6.00654e6 0.474003 0.237001 0.971509i \(-0.423835\pi\)
0.237001 + 0.971509i \(0.423835\pi\)
\(108\) 4.54561e6 0.347223
\(109\) −4.33355e6 −0.320517 −0.160258 0.987075i \(-0.551233\pi\)
−0.160258 + 0.987075i \(0.551233\pi\)
\(110\) −1.77756e7 −1.27335
\(111\) 3.01160e6 0.209010
\(112\) 1.88387e6 0.126703
\(113\) −2.30765e7 −1.50451 −0.752255 0.658872i \(-0.771034\pi\)
−0.752255 + 0.658872i \(0.771034\pi\)
\(114\) 8.68118e6 0.548797
\(115\) 1.64615e7 1.00932
\(116\) −6.33624e6 −0.376902
\(117\) 7.34573e6 0.424018
\(118\) 3.89106e6 0.218012
\(119\) 1.12764e6 0.0613419
\(120\) 1.37205e7 0.724831
\(121\) −6.52115e6 −0.334638
\(122\) −4.30475e6 −0.214629
\(123\) 2.36374e6 0.114533
\(124\) −5.13236e6 −0.241736
\(125\) −2.30227e7 −1.05432
\(126\) 3.52120e6 0.156817
\(127\) 8.51811e6 0.369003 0.184502 0.982832i \(-0.440933\pi\)
0.184502 + 0.982832i \(0.440933\pi\)
\(128\) −4.32402e7 −1.82244
\(129\) −1.22807e7 −0.503683
\(130\) 4.97425e7 1.98576
\(131\) −2.97734e7 −1.15712 −0.578560 0.815640i \(-0.696385\pi\)
−0.578560 + 0.815640i \(0.696385\pi\)
\(132\) −2.24527e7 −0.849687
\(133\) 4.32669e6 0.159469
\(134\) 3.15823e7 1.13391
\(135\) 5.12861e6 0.179404
\(136\) 8.62617e6 0.294057
\(137\) 1.89565e7 0.629849 0.314924 0.949117i \(-0.398021\pi\)
0.314924 + 0.949117i \(0.398021\pi\)
\(138\) 3.23174e7 1.04679
\(139\) 2.62226e7 0.828177 0.414088 0.910237i \(-0.364100\pi\)
0.414088 + 0.910237i \(0.364100\pi\)
\(140\) 1.53413e7 0.472512
\(141\) −2.01800e7 −0.606254
\(142\) 3.10500e7 0.910022
\(143\) −3.62836e7 −1.03761
\(144\) 5.38674e6 0.150334
\(145\) −7.14891e6 −0.194739
\(146\) −7.19854e6 −0.191430
\(147\) −2.04807e7 −0.531783
\(148\) 2.57593e7 0.653158
\(149\) −2.30589e7 −0.571066 −0.285533 0.958369i \(-0.592171\pi\)
−0.285533 + 0.958369i \(0.592171\pi\)
\(150\) −5.23462e6 −0.126638
\(151\) 4.51544e7 1.06728 0.533642 0.845710i \(-0.320823\pi\)
0.533642 + 0.845710i \(0.320823\pi\)
\(152\) 3.30981e7 0.764451
\(153\) 3.22438e6 0.0727825
\(154\) −1.73927e7 −0.383746
\(155\) −5.79062e6 −0.124900
\(156\) 6.28307e7 1.32506
\(157\) −3.09695e7 −0.638684 −0.319342 0.947640i \(-0.603462\pi\)
−0.319342 + 0.947640i \(0.603462\pi\)
\(158\) −3.20623e7 −0.646689
\(159\) 2.98569e7 0.589054
\(160\) −2.85686e7 −0.551402
\(161\) 1.61070e7 0.304175
\(162\) 1.00685e7 0.186065
\(163\) 8.52913e7 1.54258 0.771290 0.636483i \(-0.219612\pi\)
0.771290 + 0.636483i \(0.219612\pi\)
\(164\) 2.02179e7 0.357917
\(165\) −2.53324e7 −0.439018
\(166\) 6.90900e7 1.17229
\(167\) −4.10631e7 −0.682250 −0.341125 0.940018i \(-0.610808\pi\)
−0.341125 + 0.940018i \(0.610808\pi\)
\(168\) 1.34250e7 0.218440
\(169\) 3.87862e7 0.618122
\(170\) 2.18343e7 0.340854
\(171\) 1.23718e7 0.189210
\(172\) −1.05041e8 −1.57401
\(173\) 3.80863e7 0.559252 0.279626 0.960109i \(-0.409790\pi\)
0.279626 + 0.960109i \(0.409790\pi\)
\(174\) −1.40348e7 −0.201969
\(175\) −2.60893e6 −0.0367984
\(176\) −2.66074e7 −0.367881
\(177\) 5.54523e6 0.0751646
\(178\) −1.49837e8 −1.99135
\(179\) 2.59663e7 0.338395 0.169198 0.985582i \(-0.445882\pi\)
0.169198 + 0.985582i \(0.445882\pi\)
\(180\) 4.38669e7 0.560639
\(181\) −4.28031e6 −0.0536537 −0.0268269 0.999640i \(-0.508540\pi\)
−0.0268269 + 0.999640i \(0.508540\pi\)
\(182\) 4.86711e7 0.598440
\(183\) −6.13480e6 −0.0739983
\(184\) 1.23214e8 1.45814
\(185\) 2.90631e7 0.337475
\(186\) −1.13682e7 −0.129538
\(187\) −1.59266e7 −0.178105
\(188\) −1.72607e8 −1.89455
\(189\) 5.01815e6 0.0540663
\(190\) 8.37768e7 0.886107
\(191\) 6.15146e6 0.0638795 0.0319398 0.999490i \(-0.489832\pi\)
0.0319398 + 0.999490i \(0.489832\pi\)
\(192\) −8.16232e7 −0.832261
\(193\) −1.33350e8 −1.33519 −0.667594 0.744525i \(-0.732676\pi\)
−0.667594 + 0.744525i \(0.732676\pi\)
\(194\) −1.71479e7 −0.168619
\(195\) 7.08892e7 0.684634
\(196\) −1.75179e8 −1.66183
\(197\) −1.28756e8 −1.19987 −0.599936 0.800048i \(-0.704807\pi\)
−0.599936 + 0.800048i \(0.704807\pi\)
\(198\) −4.97327e7 −0.455317
\(199\) −1.66821e8 −1.50060 −0.750301 0.661096i \(-0.770092\pi\)
−0.750301 + 0.661096i \(0.770092\pi\)
\(200\) −1.99576e7 −0.176402
\(201\) 4.50087e7 0.390940
\(202\) −1.04510e8 −0.892130
\(203\) −6.99493e6 −0.0586877
\(204\) 2.75793e7 0.227446
\(205\) 2.28110e7 0.184929
\(206\) −1.61561e8 −1.28766
\(207\) 4.60563e7 0.360905
\(208\) 7.44570e7 0.573699
\(209\) −6.11093e7 −0.463015
\(210\) 3.39810e7 0.253203
\(211\) 2.09952e7 0.153862 0.0769310 0.997036i \(-0.475488\pi\)
0.0769310 + 0.997036i \(0.475488\pi\)
\(212\) 2.55377e8 1.84080
\(213\) 4.42500e7 0.313751
\(214\) 1.13798e8 0.793757
\(215\) −1.18513e8 −0.813265
\(216\) 3.83875e7 0.259180
\(217\) −5.66589e6 −0.0376408
\(218\) −8.21022e7 −0.536732
\(219\) −1.02588e7 −0.0659998
\(220\) −2.16677e8 −1.37193
\(221\) 4.45684e7 0.277750
\(222\) 5.70570e7 0.350004
\(223\) −1.74632e8 −1.05453 −0.527263 0.849702i \(-0.676782\pi\)
−0.527263 + 0.849702i \(0.676782\pi\)
\(224\) −2.79532e7 −0.166174
\(225\) −7.45997e6 −0.0436615
\(226\) −4.37201e8 −2.51942
\(227\) −3.02436e8 −1.71610 −0.858051 0.513565i \(-0.828324\pi\)
−0.858051 + 0.513565i \(0.828324\pi\)
\(228\) 1.05820e8 0.591284
\(229\) −1.49255e8 −0.821307 −0.410653 0.911792i \(-0.634699\pi\)
−0.410653 + 0.911792i \(0.634699\pi\)
\(230\) 3.11876e8 1.69019
\(231\) −2.47867e7 −0.132305
\(232\) −5.35094e7 −0.281334
\(233\) −2.80934e8 −1.45499 −0.727493 0.686115i \(-0.759314\pi\)
−0.727493 + 0.686115i \(0.759314\pi\)
\(234\) 1.39170e8 0.710053
\(235\) −1.94745e8 −0.978879
\(236\) 4.74304e7 0.234890
\(237\) −4.56928e7 −0.222961
\(238\) 2.13640e7 0.102722
\(239\) 9.57324e6 0.0453593 0.0226796 0.999743i \(-0.492780\pi\)
0.0226796 + 0.999743i \(0.492780\pi\)
\(240\) 5.19842e7 0.242735
\(241\) 7.63167e7 0.351204 0.175602 0.984461i \(-0.443813\pi\)
0.175602 + 0.984461i \(0.443813\pi\)
\(242\) −1.23548e8 −0.560379
\(243\) 1.43489e7 0.0641500
\(244\) −5.24732e7 −0.231245
\(245\) −1.97647e8 −0.858635
\(246\) 4.47827e7 0.191795
\(247\) 1.71006e8 0.722057
\(248\) −4.33426e7 −0.180441
\(249\) 9.84617e7 0.404175
\(250\) −4.36181e8 −1.76554
\(251\) 3.07061e8 1.22565 0.612825 0.790218i \(-0.290033\pi\)
0.612825 + 0.790218i \(0.290033\pi\)
\(252\) 4.29220e7 0.168958
\(253\) −2.27492e8 −0.883168
\(254\) 1.61382e8 0.617926
\(255\) 3.11166e7 0.117517
\(256\) −4.32263e8 −1.61031
\(257\) 3.52469e8 1.29525 0.647626 0.761958i \(-0.275762\pi\)
0.647626 + 0.761958i \(0.275762\pi\)
\(258\) −2.32666e8 −0.843459
\(259\) 2.84371e7 0.101704
\(260\) 6.06341e8 2.13949
\(261\) −2.00013e7 −0.0696333
\(262\) −5.64078e8 −1.93769
\(263\) 3.39887e8 1.15210 0.576048 0.817416i \(-0.304594\pi\)
0.576048 + 0.817416i \(0.304594\pi\)
\(264\) −1.89612e8 −0.634238
\(265\) 2.88131e8 0.951107
\(266\) 8.19723e7 0.267043
\(267\) −2.13536e8 −0.686564
\(268\) 3.84975e8 1.22169
\(269\) 4.31047e8 1.35018 0.675090 0.737735i \(-0.264105\pi\)
0.675090 + 0.737735i \(0.264105\pi\)
\(270\) 9.71654e7 0.300427
\(271\) −1.49089e8 −0.455044 −0.227522 0.973773i \(-0.573062\pi\)
−0.227522 + 0.973773i \(0.573062\pi\)
\(272\) 3.26827e7 0.0984752
\(273\) 6.93622e7 0.206326
\(274\) 3.59145e8 1.05473
\(275\) 3.68479e7 0.106844
\(276\) 3.93936e8 1.12783
\(277\) 3.21914e8 0.910041 0.455020 0.890481i \(-0.349632\pi\)
0.455020 + 0.890481i \(0.349632\pi\)
\(278\) 4.96805e8 1.38685
\(279\) −1.62011e7 −0.0446611
\(280\) 1.29557e8 0.352701
\(281\) 6.09518e8 1.63876 0.819378 0.573253i \(-0.194319\pi\)
0.819378 + 0.573253i \(0.194319\pi\)
\(282\) −3.82325e8 −1.01522
\(283\) −4.38418e8 −1.14984 −0.574918 0.818211i \(-0.694966\pi\)
−0.574918 + 0.818211i \(0.694966\pi\)
\(284\) 3.78486e8 0.980474
\(285\) 1.19392e8 0.305506
\(286\) −6.87420e8 −1.73756
\(287\) 2.23196e7 0.0557315
\(288\) −7.99295e7 −0.197167
\(289\) −3.90776e8 −0.952324
\(290\) −1.35441e8 −0.326106
\(291\) −2.44379e7 −0.0581351
\(292\) −8.77473e7 −0.206250
\(293\) −4.48586e7 −0.104186 −0.0520930 0.998642i \(-0.516589\pi\)
−0.0520930 + 0.998642i \(0.516589\pi\)
\(294\) −3.88022e8 −0.890513
\(295\) 5.35137e7 0.121363
\(296\) 2.17537e8 0.487541
\(297\) −7.08753e7 −0.156981
\(298\) −4.36868e8 −0.956297
\(299\) 6.36604e8 1.37727
\(300\) −6.38078e7 −0.136442
\(301\) −1.15960e8 −0.245091
\(302\) 8.55482e8 1.78726
\(303\) −1.48940e8 −0.307582
\(304\) 1.25401e8 0.256003
\(305\) −5.92032e7 −0.119480
\(306\) 6.10883e7 0.121880
\(307\) 7.78794e7 0.153617 0.0768083 0.997046i \(-0.475527\pi\)
0.0768083 + 0.997046i \(0.475527\pi\)
\(308\) −2.12010e8 −0.413455
\(309\) −2.30245e8 −0.443951
\(310\) −1.09708e8 −0.209156
\(311\) 4.98982e8 0.940640 0.470320 0.882496i \(-0.344139\pi\)
0.470320 + 0.882496i \(0.344139\pi\)
\(312\) 5.30603e8 0.989074
\(313\) 3.57809e8 0.659547 0.329774 0.944060i \(-0.393028\pi\)
0.329774 + 0.944060i \(0.393028\pi\)
\(314\) −5.86741e8 −1.06953
\(315\) 4.84271e7 0.0872974
\(316\) −3.90827e8 −0.696754
\(317\) 1.30653e8 0.230363 0.115181 0.993344i \(-0.463255\pi\)
0.115181 + 0.993344i \(0.463255\pi\)
\(318\) 5.65661e8 0.986419
\(319\) 9.87949e7 0.170399
\(320\) −7.87696e8 −1.34380
\(321\) 1.62176e8 0.273666
\(322\) 3.05158e8 0.509366
\(323\) 7.50625e7 0.123941
\(324\) 1.22731e8 0.200469
\(325\) −1.03114e8 −0.166619
\(326\) 1.61591e9 2.58318
\(327\) −1.17006e8 −0.185050
\(328\) 1.70739e8 0.267162
\(329\) −1.90550e8 −0.295001
\(330\) −4.79940e8 −0.735171
\(331\) 9.92186e8 1.50382 0.751909 0.659267i \(-0.229133\pi\)
0.751909 + 0.659267i \(0.229133\pi\)
\(332\) 8.42179e8 1.26305
\(333\) 8.13132e7 0.120672
\(334\) −7.77970e8 −1.14248
\(335\) 4.34351e8 0.631225
\(336\) 5.08644e7 0.0731521
\(337\) −2.68966e7 −0.0382818 −0.0191409 0.999817i \(-0.506093\pi\)
−0.0191409 + 0.999817i \(0.506093\pi\)
\(338\) 7.34833e8 1.03510
\(339\) −6.23065e8 −0.868629
\(340\) 2.66151e8 0.367242
\(341\) 8.00239e7 0.109290
\(342\) 2.34392e8 0.316848
\(343\) −4.03350e8 −0.539701
\(344\) −8.87067e8 −1.17490
\(345\) 4.44462e8 0.582730
\(346\) 7.21573e8 0.936513
\(347\) 1.17341e9 1.50764 0.753821 0.657080i \(-0.228209\pi\)
0.753821 + 0.657080i \(0.228209\pi\)
\(348\) −1.71079e8 −0.217605
\(349\) 1.13126e9 1.42454 0.712269 0.701907i \(-0.247668\pi\)
0.712269 + 0.701907i \(0.247668\pi\)
\(350\) −4.94280e7 −0.0616219
\(351\) 1.98335e8 0.244807
\(352\) 3.94805e8 0.482485
\(353\) 1.22377e9 1.48078 0.740389 0.672179i \(-0.234641\pi\)
0.740389 + 0.672179i \(0.234641\pi\)
\(354\) 1.05058e8 0.125869
\(355\) 4.27030e8 0.506593
\(356\) −1.82645e9 −2.14552
\(357\) 3.04463e7 0.0354157
\(358\) 4.91950e8 0.566670
\(359\) −6.37571e7 −0.0727274 −0.0363637 0.999339i \(-0.511577\pi\)
−0.0363637 + 0.999339i \(0.511577\pi\)
\(360\) 3.70455e8 0.418482
\(361\) −6.05862e8 −0.677795
\(362\) −8.10936e7 −0.0898476
\(363\) −1.76071e8 −0.193203
\(364\) 5.93280e8 0.644770
\(365\) −9.90015e7 −0.106566
\(366\) −1.16228e8 −0.123916
\(367\) 1.18246e9 1.24870 0.624348 0.781147i \(-0.285365\pi\)
0.624348 + 0.781147i \(0.285365\pi\)
\(368\) 4.66832e8 0.488307
\(369\) 6.38209e7 0.0661257
\(370\) 5.50622e8 0.565129
\(371\) 2.81925e8 0.286632
\(372\) −1.38574e8 −0.139566
\(373\) 5.09652e8 0.508502 0.254251 0.967138i \(-0.418171\pi\)
0.254251 + 0.967138i \(0.418171\pi\)
\(374\) −3.01741e8 −0.298252
\(375\) −6.21612e8 −0.608709
\(376\) −1.45766e9 −1.41416
\(377\) −2.76464e8 −0.265732
\(378\) 9.50725e7 0.0905385
\(379\) 5.82373e8 0.549496 0.274748 0.961516i \(-0.411406\pi\)
0.274748 + 0.961516i \(0.411406\pi\)
\(380\) 1.02121e9 0.954708
\(381\) 2.29989e8 0.213044
\(382\) 1.16544e8 0.106971
\(383\) −2.95800e8 −0.269031 −0.134515 0.990912i \(-0.542948\pi\)
−0.134515 + 0.990912i \(0.542948\pi\)
\(384\) −1.16749e9 −1.05219
\(385\) −2.39202e8 −0.213625
\(386\) −2.52641e9 −2.23588
\(387\) −3.31578e8 −0.290802
\(388\) −2.09026e8 −0.181673
\(389\) −1.67193e9 −1.44011 −0.720055 0.693917i \(-0.755884\pi\)
−0.720055 + 0.693917i \(0.755884\pi\)
\(390\) 1.34305e9 1.14648
\(391\) 2.79435e8 0.236408
\(392\) −1.47938e9 −1.24045
\(393\) −8.03881e8 −0.668064
\(394\) −2.43937e9 −2.00928
\(395\) −4.40953e8 −0.360000
\(396\) −6.06222e8 −0.490567
\(397\) 3.26565e7 0.0261940 0.0130970 0.999914i \(-0.495831\pi\)
0.0130970 + 0.999914i \(0.495831\pi\)
\(398\) −3.16055e9 −2.51288
\(399\) 1.16821e8 0.0920692
\(400\) −7.56150e7 −0.0590742
\(401\) 2.13725e7 0.0165520 0.00827600 0.999966i \(-0.497366\pi\)
0.00827600 + 0.999966i \(0.497366\pi\)
\(402\) 8.52722e8 0.654661
\(403\) −2.23936e8 −0.170434
\(404\) −1.27393e9 −0.961197
\(405\) 1.38473e8 0.103579
\(406\) −1.32524e8 −0.0982773
\(407\) −4.01640e8 −0.295295
\(408\) 2.32907e8 0.169774
\(409\) 7.70641e8 0.556955 0.278478 0.960443i \(-0.410170\pi\)
0.278478 + 0.960443i \(0.410170\pi\)
\(410\) 4.32171e8 0.309679
\(411\) 5.11826e8 0.363643
\(412\) −1.96937e9 −1.38735
\(413\) 5.23610e7 0.0365749
\(414\) 8.72571e8 0.604365
\(415\) 9.50195e8 0.652596
\(416\) −1.10481e9 −0.752420
\(417\) 7.08009e8 0.478148
\(418\) −1.15776e9 −0.775356
\(419\) 1.32286e9 0.878547 0.439274 0.898353i \(-0.355236\pi\)
0.439274 + 0.898353i \(0.355236\pi\)
\(420\) 4.14214e8 0.272805
\(421\) −5.11479e8 −0.334073 −0.167036 0.985951i \(-0.553420\pi\)
−0.167036 + 0.985951i \(0.553420\pi\)
\(422\) 3.97769e8 0.257655
\(423\) −5.44860e8 −0.350021
\(424\) 2.15665e9 1.37404
\(425\) −4.52615e7 −0.0286001
\(426\) 8.38349e8 0.525402
\(427\) −5.79280e7 −0.0360073
\(428\) 1.38715e9 0.855208
\(429\) −9.79658e8 −0.599065
\(430\) −2.24532e9 −1.36188
\(431\) −1.38021e8 −0.0830374 −0.0415187 0.999138i \(-0.513220\pi\)
−0.0415187 + 0.999138i \(0.513220\pi\)
\(432\) 1.45442e8 0.0867954
\(433\) −1.42354e9 −0.842680 −0.421340 0.906903i \(-0.638440\pi\)
−0.421340 + 0.906903i \(0.638440\pi\)
\(434\) −1.07344e8 −0.0630326
\(435\) −1.93021e8 −0.112432
\(436\) −1.00079e9 −0.578284
\(437\) 1.07217e9 0.614583
\(438\) −1.94361e8 −0.110522
\(439\) −6.95019e8 −0.392077 −0.196038 0.980596i \(-0.562808\pi\)
−0.196038 + 0.980596i \(0.562808\pi\)
\(440\) −1.82983e9 −1.02406
\(441\) −5.52979e8 −0.307025
\(442\) 8.44380e8 0.465115
\(443\) 2.52605e9 1.38047 0.690237 0.723583i \(-0.257506\pi\)
0.690237 + 0.723583i \(0.257506\pi\)
\(444\) 6.95501e8 0.377101
\(445\) −2.06070e9 −1.10855
\(446\) −3.30853e9 −1.76589
\(447\) −6.22590e8 −0.329705
\(448\) −7.70729e8 −0.404976
\(449\) 1.92004e9 1.00103 0.500517 0.865727i \(-0.333143\pi\)
0.500517 + 0.865727i \(0.333143\pi\)
\(450\) −1.41335e8 −0.0731147
\(451\) −3.15238e8 −0.161816
\(452\) −5.32930e9 −2.71447
\(453\) 1.21917e9 0.616197
\(454\) −5.72987e9 −2.87375
\(455\) 6.69373e8 0.333141
\(456\) 8.93647e8 0.441356
\(457\) 2.28317e9 1.11900 0.559502 0.828829i \(-0.310992\pi\)
0.559502 + 0.828829i \(0.310992\pi\)
\(458\) −2.82775e9 −1.37535
\(459\) 8.70584e7 0.0420210
\(460\) 3.80164e9 1.82104
\(461\) 8.39344e8 0.399013 0.199506 0.979897i \(-0.436066\pi\)
0.199506 + 0.979897i \(0.436066\pi\)
\(462\) −4.69603e8 −0.221556
\(463\) −1.31128e9 −0.613990 −0.306995 0.951711i \(-0.599323\pi\)
−0.306995 + 0.951711i \(0.599323\pi\)
\(464\) −2.02735e8 −0.0942143
\(465\) −1.56347e8 −0.0721113
\(466\) −5.32250e9 −2.43649
\(467\) 2.02299e9 0.919145 0.459572 0.888140i \(-0.348003\pi\)
0.459572 + 0.888140i \(0.348003\pi\)
\(468\) 1.69643e9 0.765024
\(469\) 4.24996e8 0.190230
\(470\) −3.68959e9 −1.63921
\(471\) −8.36178e8 −0.368744
\(472\) 4.00548e8 0.175331
\(473\) 1.63780e9 0.711618
\(474\) −8.65683e8 −0.373366
\(475\) −1.73665e8 −0.0743508
\(476\) 2.60419e8 0.110675
\(477\) 8.06137e8 0.340091
\(478\) 1.81372e8 0.0759578
\(479\) 1.74780e9 0.726635 0.363318 0.931665i \(-0.381644\pi\)
0.363318 + 0.931665i \(0.381644\pi\)
\(480\) −7.71352e8 −0.318352
\(481\) 1.12393e9 0.460504
\(482\) 1.44588e9 0.588120
\(483\) 4.34888e8 0.175615
\(484\) −1.50600e9 −0.603762
\(485\) −2.35836e8 −0.0938671
\(486\) 2.71851e8 0.107424
\(487\) −2.38890e8 −0.0937231 −0.0468616 0.998901i \(-0.514922\pi\)
−0.0468616 + 0.998901i \(0.514922\pi\)
\(488\) −4.43134e8 −0.172610
\(489\) 2.30286e9 0.890609
\(490\) −3.74456e9 −1.43785
\(491\) 2.15282e9 0.820772 0.410386 0.911912i \(-0.365394\pi\)
0.410386 + 0.911912i \(0.365394\pi\)
\(492\) 5.45883e8 0.206643
\(493\) −1.21353e8 −0.0456128
\(494\) 3.23983e9 1.20914
\(495\) −6.83974e8 −0.253467
\(496\) −1.64216e8 −0.0604267
\(497\) 4.17832e8 0.152670
\(498\) 1.86543e9 0.676825
\(499\) 9.90364e8 0.356815 0.178407 0.983957i \(-0.442906\pi\)
0.178407 + 0.983957i \(0.442906\pi\)
\(500\) −5.31687e9 −1.90222
\(501\) −1.10870e9 −0.393897
\(502\) 5.81750e9 2.05245
\(503\) 1.94133e9 0.680161 0.340080 0.940396i \(-0.389546\pi\)
0.340080 + 0.940396i \(0.389546\pi\)
\(504\) 3.62475e8 0.126116
\(505\) −1.43733e9 −0.496633
\(506\) −4.30999e9 −1.47894
\(507\) 1.04723e9 0.356873
\(508\) 1.96718e9 0.665764
\(509\) 1.30601e9 0.438970 0.219485 0.975616i \(-0.429562\pi\)
0.219485 + 0.975616i \(0.429562\pi\)
\(510\) 5.89526e8 0.196792
\(511\) −9.68690e7 −0.0321153
\(512\) −2.65480e9 −0.874150
\(513\) 3.34037e8 0.109241
\(514\) 6.67778e9 2.16901
\(515\) −2.22195e9 −0.716818
\(516\) −2.83610e9 −0.908758
\(517\) 2.69129e9 0.856533
\(518\) 5.38762e8 0.170311
\(519\) 1.02833e9 0.322884
\(520\) 5.12053e9 1.59699
\(521\) 4.39027e9 1.36006 0.680031 0.733183i \(-0.261966\pi\)
0.680031 + 0.733183i \(0.261966\pi\)
\(522\) −3.78940e8 −0.116607
\(523\) 3.06969e9 0.938293 0.469146 0.883120i \(-0.344562\pi\)
0.469146 + 0.883120i \(0.344562\pi\)
\(524\) −6.87589e9 −2.08771
\(525\) −7.04410e7 −0.0212455
\(526\) 6.43940e9 1.92928
\(527\) −9.82959e7 −0.0292549
\(528\) −7.18399e8 −0.212396
\(529\) 5.86560e8 0.172273
\(530\) 5.45885e9 1.59271
\(531\) 1.49721e8 0.0433963
\(532\) 9.99209e8 0.287717
\(533\) 8.82150e8 0.252346
\(534\) −4.04559e9 −1.14971
\(535\) 1.56507e9 0.441870
\(536\) 3.25111e9 0.911915
\(537\) 7.01089e8 0.195372
\(538\) 8.16650e9 2.26099
\(539\) 2.73139e9 0.751318
\(540\) 1.18441e9 0.323685
\(541\) −7.85502e7 −0.0213283 −0.0106642 0.999943i \(-0.503395\pi\)
−0.0106642 + 0.999943i \(0.503395\pi\)
\(542\) −2.82460e9 −0.762009
\(543\) −1.15568e8 −0.0309770
\(544\) −4.84952e8 −0.129153
\(545\) −1.12915e9 −0.298789
\(546\) 1.31412e9 0.345510
\(547\) −6.56046e9 −1.71387 −0.856937 0.515422i \(-0.827635\pi\)
−0.856937 + 0.515422i \(0.827635\pi\)
\(548\) 4.37783e9 1.13639
\(549\) −1.65640e8 −0.0427229
\(550\) 6.98111e8 0.178918
\(551\) −4.65623e8 −0.118578
\(552\) 3.32678e9 0.841856
\(553\) −4.31455e8 −0.108492
\(554\) 6.09890e9 1.52394
\(555\) 7.84704e8 0.194841
\(556\) 6.05586e9 1.49422
\(557\) −5.79150e9 −1.42003 −0.710015 0.704186i \(-0.751312\pi\)
−0.710015 + 0.704186i \(0.751312\pi\)
\(558\) −3.06941e8 −0.0747886
\(559\) −4.58316e9 −1.10975
\(560\) 4.90862e8 0.118114
\(561\) −4.30018e8 −0.102829
\(562\) 1.15478e10 2.74423
\(563\) 1.04570e9 0.246961 0.123480 0.992347i \(-0.460594\pi\)
0.123480 + 0.992347i \(0.460594\pi\)
\(564\) −4.66039e9 −1.09382
\(565\) −6.01282e9 −1.40252
\(566\) −8.30614e9 −1.92549
\(567\) 1.35490e8 0.0312152
\(568\) 3.19631e9 0.731862
\(569\) 7.96871e8 0.181341 0.0906704 0.995881i \(-0.471099\pi\)
0.0906704 + 0.995881i \(0.471099\pi\)
\(570\) 2.26197e9 0.511594
\(571\) 1.01872e9 0.228996 0.114498 0.993423i \(-0.463474\pi\)
0.114498 + 0.993423i \(0.463474\pi\)
\(572\) −8.37937e9 −1.87208
\(573\) 1.66089e8 0.0368808
\(574\) 4.22862e8 0.0933269
\(575\) −6.46504e8 −0.141819
\(576\) −2.20383e9 −0.480506
\(577\) −1.71883e9 −0.372492 −0.186246 0.982503i \(-0.559632\pi\)
−0.186246 + 0.982503i \(0.559632\pi\)
\(578\) −7.40353e9 −1.59475
\(579\) −3.60045e9 −0.770872
\(580\) −1.65098e9 −0.351352
\(581\) 9.29728e8 0.196671
\(582\) −4.62994e8 −0.0973521
\(583\) −3.98185e9 −0.832233
\(584\) −7.41023e8 −0.153953
\(585\) 1.91401e9 0.395274
\(586\) −8.49879e8 −0.174468
\(587\) −4.84773e9 −0.989249 −0.494624 0.869107i \(-0.664694\pi\)
−0.494624 + 0.869107i \(0.664694\pi\)
\(588\) −4.72983e9 −0.959455
\(589\) −3.77155e8 −0.0760530
\(590\) 1.01386e9 0.203233
\(591\) −3.47641e9 −0.692747
\(592\) 8.24199e8 0.163270
\(593\) −6.50386e9 −1.28080 −0.640398 0.768043i \(-0.721230\pi\)
−0.640398 + 0.768043i \(0.721230\pi\)
\(594\) −1.34278e9 −0.262878
\(595\) 2.93819e8 0.0571835
\(596\) −5.32524e9 −1.03033
\(597\) −4.50418e9 −0.866373
\(598\) 1.20609e10 2.30636
\(599\) 3.67574e9 0.698797 0.349398 0.936974i \(-0.386386\pi\)
0.349398 + 0.936974i \(0.386386\pi\)
\(600\) −5.38855e8 −0.101846
\(601\) 2.39919e9 0.450821 0.225410 0.974264i \(-0.427628\pi\)
0.225410 + 0.974264i \(0.427628\pi\)
\(602\) −2.19695e9 −0.410425
\(603\) 1.21523e9 0.225709
\(604\) 1.04280e10 1.92562
\(605\) −1.69916e9 −0.311953
\(606\) −2.82177e9 −0.515072
\(607\) −6.08714e9 −1.10472 −0.552361 0.833605i \(-0.686273\pi\)
−0.552361 + 0.833605i \(0.686273\pi\)
\(608\) −1.86073e9 −0.335754
\(609\) −1.88863e8 −0.0338834
\(610\) −1.12165e9 −0.200079
\(611\) −7.53121e9 −1.33574
\(612\) 7.44642e8 0.131316
\(613\) 9.32738e9 1.63549 0.817745 0.575581i \(-0.195224\pi\)
0.817745 + 0.575581i \(0.195224\pi\)
\(614\) 1.47548e9 0.257244
\(615\) 6.15896e8 0.106769
\(616\) −1.79042e9 −0.308618
\(617\) 6.37210e9 1.09216 0.546078 0.837734i \(-0.316120\pi\)
0.546078 + 0.837734i \(0.316120\pi\)
\(618\) −4.36215e9 −0.743432
\(619\) −4.05521e9 −0.687220 −0.343610 0.939112i \(-0.611650\pi\)
−0.343610 + 0.939112i \(0.611650\pi\)
\(620\) −1.33729e9 −0.225348
\(621\) 1.24352e9 0.208369
\(622\) 9.45358e9 1.57518
\(623\) −2.01632e9 −0.334080
\(624\) 2.01034e9 0.331225
\(625\) −5.19933e9 −0.851859
\(626\) 6.77895e9 1.10447
\(627\) −1.64995e9 −0.267322
\(628\) −7.15213e9 −1.15233
\(629\) 4.93347e8 0.0790452
\(630\) 9.17487e8 0.146187
\(631\) −8.06865e9 −1.27849 −0.639246 0.769002i \(-0.720754\pi\)
−0.639246 + 0.769002i \(0.720754\pi\)
\(632\) −3.30052e9 −0.520083
\(633\) 5.66870e8 0.0888323
\(634\) 2.47532e9 0.385762
\(635\) 2.21948e9 0.343988
\(636\) 6.89518e9 1.06279
\(637\) −7.64343e9 −1.17166
\(638\) 1.87174e9 0.285347
\(639\) 1.19475e9 0.181144
\(640\) −1.12667e10 −1.69890
\(641\) −3.59478e9 −0.539100 −0.269550 0.962986i \(-0.586875\pi\)
−0.269550 + 0.962986i \(0.586875\pi\)
\(642\) 3.07255e9 0.458276
\(643\) −7.51354e9 −1.11457 −0.557283 0.830322i \(-0.688156\pi\)
−0.557283 + 0.830322i \(0.688156\pi\)
\(644\) 3.71976e9 0.548800
\(645\) −3.19985e9 −0.469539
\(646\) 1.42211e9 0.207549
\(647\) 1.51619e9 0.220084 0.110042 0.993927i \(-0.464901\pi\)
0.110042 + 0.993927i \(0.464901\pi\)
\(648\) 1.03646e9 0.149638
\(649\) −7.39536e8 −0.106195
\(650\) −1.95357e9 −0.279018
\(651\) −1.52979e8 −0.0217319
\(652\) 1.96972e10 2.78316
\(653\) 1.30271e10 1.83084 0.915421 0.402497i \(-0.131858\pi\)
0.915421 + 0.402497i \(0.131858\pi\)
\(654\) −2.21676e9 −0.309882
\(655\) −7.75777e9 −1.07868
\(656\) 6.46895e8 0.0894685
\(657\) −2.76988e8 −0.0381050
\(658\) −3.61011e9 −0.494004
\(659\) 8.76958e8 0.119366 0.0596829 0.998217i \(-0.480991\pi\)
0.0596829 + 0.998217i \(0.480991\pi\)
\(660\) −5.85028e9 −0.792087
\(661\) −9.86047e9 −1.32798 −0.663991 0.747740i \(-0.731139\pi\)
−0.663991 + 0.747740i \(0.731139\pi\)
\(662\) 1.87977e10 2.51827
\(663\) 1.20335e9 0.160359
\(664\) 7.11218e9 0.942788
\(665\) 1.12736e9 0.148658
\(666\) 1.54054e9 0.202075
\(667\) −1.73338e9 −0.226179
\(668\) −9.48314e9 −1.23093
\(669\) −4.71507e9 −0.608831
\(670\) 8.22910e9 1.05704
\(671\) 8.18163e8 0.104547
\(672\) −7.54737e8 −0.0959408
\(673\) 5.88788e9 0.744571 0.372286 0.928118i \(-0.378574\pi\)
0.372286 + 0.928118i \(0.378574\pi\)
\(674\) −5.09576e8 −0.0641061
\(675\) −2.01419e8 −0.0252080
\(676\) 8.95732e9 1.11523
\(677\) 2.79927e9 0.346724 0.173362 0.984858i \(-0.444537\pi\)
0.173362 + 0.984858i \(0.444537\pi\)
\(678\) −1.18044e10 −1.45459
\(679\) −2.30756e8 −0.0282884
\(680\) 2.24764e9 0.274123
\(681\) −8.16577e9 −0.990792
\(682\) 1.51611e9 0.183015
\(683\) 2.69530e9 0.323694 0.161847 0.986816i \(-0.448255\pi\)
0.161847 + 0.986816i \(0.448255\pi\)
\(684\) 2.85714e9 0.341378
\(685\) 4.93932e9 0.587152
\(686\) −7.64177e9 −0.903773
\(687\) −4.02989e9 −0.474182
\(688\) −3.36090e9 −0.393456
\(689\) 1.11427e10 1.29784
\(690\) 8.42065e9 0.975830
\(691\) −6.92452e9 −0.798393 −0.399196 0.916865i \(-0.630711\pi\)
−0.399196 + 0.916865i \(0.630711\pi\)
\(692\) 8.79568e9 1.00902
\(693\) −6.69242e8 −0.0763865
\(694\) 2.22312e10 2.52467
\(695\) 6.83256e9 0.772035
\(696\) −1.44475e9 −0.162428
\(697\) 3.87217e8 0.0433151
\(698\) 2.14326e10 2.38551
\(699\) −7.58522e9 −0.840036
\(700\) −6.02507e8 −0.0663925
\(701\) −1.39420e10 −1.52867 −0.764333 0.644822i \(-0.776932\pi\)
−0.764333 + 0.644822i \(0.776932\pi\)
\(702\) 3.75760e9 0.409949
\(703\) 1.89294e9 0.205491
\(704\) 1.08856e10 1.17584
\(705\) −5.25811e9 −0.565156
\(706\) 2.31853e10 2.47968
\(707\) −1.40637e9 −0.149669
\(708\) 1.28062e9 0.135614
\(709\) 1.10083e10 1.16000 0.579998 0.814618i \(-0.303053\pi\)
0.579998 + 0.814618i \(0.303053\pi\)
\(710\) 8.09040e9 0.848332
\(711\) −1.23370e9 −0.128726
\(712\) −1.54243e10 −1.60149
\(713\) −1.40404e9 −0.145066
\(714\) 5.76828e8 0.0593066
\(715\) −9.45408e9 −0.967271
\(716\) 5.99667e9 0.610541
\(717\) 2.58477e8 0.0261882
\(718\) −1.20793e9 −0.121788
\(719\) 5.92255e9 0.594234 0.297117 0.954841i \(-0.403975\pi\)
0.297117 + 0.954841i \(0.403975\pi\)
\(720\) 1.40357e9 0.140143
\(721\) −2.17409e9 −0.216025
\(722\) −1.14785e10 −1.13502
\(723\) 2.06055e9 0.202768
\(724\) −9.88498e8 −0.0968034
\(725\) 2.80764e8 0.0273626
\(726\) −3.33580e9 −0.323535
\(727\) 2.95171e9 0.284907 0.142453 0.989802i \(-0.454501\pi\)
0.142453 + 0.989802i \(0.454501\pi\)
\(728\) 5.01024e9 0.481281
\(729\) 3.87420e8 0.0370370
\(730\) −1.87566e9 −0.178453
\(731\) −2.01176e9 −0.190487
\(732\) −1.41678e9 −0.133510
\(733\) 1.40788e9 0.132038 0.0660192 0.997818i \(-0.478970\pi\)
0.0660192 + 0.997818i \(0.478970\pi\)
\(734\) 2.24026e10 2.09104
\(735\) −5.33646e9 −0.495733
\(736\) −6.92694e9 −0.640427
\(737\) −6.00255e9 −0.552331
\(738\) 1.20913e9 0.110733
\(739\) 1.57441e10 1.43504 0.717518 0.696540i \(-0.245278\pi\)
0.717518 + 0.696540i \(0.245278\pi\)
\(740\) 6.71186e9 0.608880
\(741\) 4.61716e9 0.416880
\(742\) 5.34127e9 0.479989
\(743\) −1.31118e10 −1.17274 −0.586369 0.810044i \(-0.699443\pi\)
−0.586369 + 0.810044i \(0.699443\pi\)
\(744\) −1.17025e9 −0.104177
\(745\) −6.00824e9 −0.532354
\(746\) 9.65572e9 0.851528
\(747\) 2.65847e9 0.233351
\(748\) −3.67810e9 −0.321342
\(749\) 1.53136e9 0.133165
\(750\) −1.17769e10 −1.01933
\(751\) −1.05574e10 −0.909532 −0.454766 0.890611i \(-0.650277\pi\)
−0.454766 + 0.890611i \(0.650277\pi\)
\(752\) −5.52276e9 −0.473580
\(753\) 8.29065e9 0.707630
\(754\) −5.23781e9 −0.444990
\(755\) 1.17654e10 0.994934
\(756\) 1.15889e9 0.0975478
\(757\) 7.96333e8 0.0667205 0.0333602 0.999443i \(-0.489379\pi\)
0.0333602 + 0.999443i \(0.489379\pi\)
\(758\) 1.10335e10 0.920175
\(759\) −6.14227e9 −0.509897
\(760\) 8.62405e9 0.712629
\(761\) 2.66159e9 0.218925 0.109462 0.993991i \(-0.465087\pi\)
0.109462 + 0.993991i \(0.465087\pi\)
\(762\) 4.35731e9 0.356760
\(763\) −1.10483e9 −0.0900450
\(764\) 1.42062e9 0.115253
\(765\) 8.40148e8 0.0678486
\(766\) −5.60414e9 −0.450514
\(767\) 2.06949e9 0.165607
\(768\) −1.16711e10 −0.929711
\(769\) 3.74953e9 0.297327 0.148664 0.988888i \(-0.452503\pi\)
0.148664 + 0.988888i \(0.452503\pi\)
\(770\) −4.53185e9 −0.357732
\(771\) 9.51665e9 0.747815
\(772\) −3.07960e10 −2.40898
\(773\) −2.17332e10 −1.69237 −0.846187 0.532886i \(-0.821107\pi\)
−0.846187 + 0.532886i \(0.821107\pi\)
\(774\) −6.28198e9 −0.486971
\(775\) 2.27419e8 0.0175497
\(776\) −1.76522e9 −0.135607
\(777\) 7.67802e8 0.0587186
\(778\) −3.16760e10 −2.41158
\(779\) 1.48573e9 0.112605
\(780\) 1.63712e10 1.23523
\(781\) −5.90137e9 −0.443276
\(782\) 5.29411e9 0.395885
\(783\) −5.40036e8 −0.0402028
\(784\) −5.60505e9 −0.415406
\(785\) −8.06944e9 −0.595388
\(786\) −1.52301e10 −1.11873
\(787\) −5.76067e9 −0.421271 −0.210636 0.977565i \(-0.567553\pi\)
−0.210636 + 0.977565i \(0.567553\pi\)
\(788\) −2.97350e10 −2.16484
\(789\) 9.17694e9 0.665163
\(790\) −8.35418e9 −0.602850
\(791\) −5.88331e9 −0.422672
\(792\) −5.11952e9 −0.366177
\(793\) −2.28952e9 −0.163038
\(794\) 6.18701e8 0.0438641
\(795\) 7.77954e9 0.549122
\(796\) −3.85258e10 −2.70742
\(797\) −1.81058e9 −0.126682 −0.0633409 0.997992i \(-0.520176\pi\)
−0.0633409 + 0.997992i \(0.520176\pi\)
\(798\) 2.21325e9 0.154177
\(799\) −3.30580e9 −0.229278
\(800\) 1.12199e9 0.0774773
\(801\) −5.76546e9 −0.396388
\(802\) 4.04918e8 0.0277177
\(803\) 1.36816e9 0.0932464
\(804\) 1.03943e10 0.705343
\(805\) 4.19684e9 0.283555
\(806\) −4.24263e9 −0.285405
\(807\) 1.16383e10 0.779527
\(808\) −1.07583e10 −0.717473
\(809\) −1.38975e10 −0.922823 −0.461411 0.887186i \(-0.652657\pi\)
−0.461411 + 0.887186i \(0.652657\pi\)
\(810\) 2.62346e9 0.173451
\(811\) 2.46372e10 1.62188 0.810940 0.585129i \(-0.198956\pi\)
0.810940 + 0.585129i \(0.198956\pi\)
\(812\) −1.61541e9 −0.105886
\(813\) −4.02541e9 −0.262720
\(814\) −7.60936e9 −0.494496
\(815\) 2.22235e10 1.43801
\(816\) 8.82433e8 0.0568547
\(817\) −7.71900e9 −0.495204
\(818\) 1.46004e10 0.932667
\(819\) 1.87278e9 0.119122
\(820\) 5.26798e9 0.333654
\(821\) −6.82682e8 −0.0430544 −0.0215272 0.999768i \(-0.506853\pi\)
−0.0215272 + 0.999768i \(0.506853\pi\)
\(822\) 9.69691e9 0.608951
\(823\) −2.59117e10 −1.62030 −0.810151 0.586222i \(-0.800615\pi\)
−0.810151 + 0.586222i \(0.800615\pi\)
\(824\) −1.66312e10 −1.03557
\(825\) 9.94894e8 0.0616862
\(826\) 9.92018e8 0.0612476
\(827\) −1.70204e10 −1.04641 −0.523203 0.852208i \(-0.675263\pi\)
−0.523203 + 0.852208i \(0.675263\pi\)
\(828\) 1.06363e10 0.651154
\(829\) −1.88349e10 −1.14821 −0.574107 0.818780i \(-0.694651\pi\)
−0.574107 + 0.818780i \(0.694651\pi\)
\(830\) 1.80021e10 1.09283
\(831\) 8.69168e9 0.525412
\(832\) −3.04619e10 −1.83369
\(833\) −3.35506e9 −0.201114
\(834\) 1.34137e10 0.800698
\(835\) −1.06994e10 −0.636001
\(836\) −1.41126e10 −0.835383
\(837\) −4.37429e8 −0.0257851
\(838\) 2.50626e10 1.47120
\(839\) 8.65181e9 0.505755 0.252877 0.967498i \(-0.418623\pi\)
0.252877 + 0.967498i \(0.418623\pi\)
\(840\) 3.49803e9 0.203632
\(841\) −1.64971e10 −0.956361
\(842\) −9.69035e9 −0.559432
\(843\) 1.64570e10 0.946136
\(844\) 4.84865e9 0.277602
\(845\) 1.01062e10 0.576219
\(846\) −1.03228e10 −0.586138
\(847\) −1.66256e9 −0.0940122
\(848\) 8.17109e9 0.460145
\(849\) −1.18373e10 −0.663858
\(850\) −8.57512e8 −0.0478932
\(851\) 7.04685e9 0.391960
\(852\) 1.02191e10 0.566077
\(853\) 1.84081e10 1.01552 0.507758 0.861500i \(-0.330474\pi\)
0.507758 + 0.861500i \(0.330474\pi\)
\(854\) −1.09749e9 −0.0602973
\(855\) 3.22359e9 0.176384
\(856\) 1.17145e10 0.638359
\(857\) 2.23324e10 1.21200 0.605999 0.795466i \(-0.292774\pi\)
0.605999 + 0.795466i \(0.292774\pi\)
\(858\) −1.85603e10 −1.00318
\(859\) 1.81305e10 0.975966 0.487983 0.872853i \(-0.337733\pi\)
0.487983 + 0.872853i \(0.337733\pi\)
\(860\) −2.73695e10 −1.46731
\(861\) 6.02630e8 0.0321766
\(862\) −2.61490e9 −0.139053
\(863\) −9.25181e8 −0.0489992 −0.0244996 0.999700i \(-0.507799\pi\)
−0.0244996 + 0.999700i \(0.507799\pi\)
\(864\) −2.15810e9 −0.113834
\(865\) 9.92379e9 0.521340
\(866\) −2.69700e10 −1.41114
\(867\) −1.05509e10 −0.549825
\(868\) −1.30849e9 −0.0679125
\(869\) 6.09378e9 0.315005
\(870\) −3.65692e9 −0.188277
\(871\) 1.67973e10 0.861343
\(872\) −8.45167e9 −0.431653
\(873\) −6.59824e8 −0.0335643
\(874\) 2.03131e10 1.02917
\(875\) −5.86959e9 −0.296196
\(876\) −2.36918e9 −0.119078
\(877\) −9.58624e9 −0.479899 −0.239949 0.970785i \(-0.577131\pi\)
−0.239949 + 0.970785i \(0.577131\pi\)
\(878\) −1.31676e10 −0.656565
\(879\) −1.21118e9 −0.0601518
\(880\) −6.93283e9 −0.342942
\(881\) 1.89685e9 0.0934583 0.0467291 0.998908i \(-0.485120\pi\)
0.0467291 + 0.998908i \(0.485120\pi\)
\(882\) −1.04766e10 −0.514138
\(883\) 3.26755e10 1.59720 0.798600 0.601862i \(-0.205574\pi\)
0.798600 + 0.601862i \(0.205574\pi\)
\(884\) 1.02927e10 0.501123
\(885\) 1.44487e9 0.0700692
\(886\) 4.78578e10 2.31172
\(887\) 2.36089e10 1.13591 0.567955 0.823060i \(-0.307735\pi\)
0.567955 + 0.823060i \(0.307735\pi\)
\(888\) 5.87349e9 0.281482
\(889\) 2.17168e9 0.103667
\(890\) −3.90415e10 −1.85636
\(891\) −1.91363e9 −0.0906330
\(892\) −4.03297e10 −1.90260
\(893\) −1.26841e10 −0.596048
\(894\) −1.17954e10 −0.552119
\(895\) 6.76579e9 0.315455
\(896\) −1.10240e10 −0.511991
\(897\) 1.71883e10 0.795169
\(898\) 3.63766e10 1.67631
\(899\) 6.09744e8 0.0279891
\(900\) −1.72281e9 −0.0787751
\(901\) 4.89103e9 0.222774
\(902\) −5.97241e9 −0.270974
\(903\) −3.13093e9 −0.141503
\(904\) −4.50058e10 −2.02618
\(905\) −1.11528e9 −0.0500166
\(906\) 2.30980e10 1.03187
\(907\) −1.15848e10 −0.515540 −0.257770 0.966206i \(-0.582988\pi\)
−0.257770 + 0.966206i \(0.582988\pi\)
\(908\) −6.98448e10 −3.09623
\(909\) −4.02137e9 −0.177583
\(910\) 1.26818e10 0.557872
\(911\) 4.24123e10 1.85856 0.929281 0.369373i \(-0.120427\pi\)
0.929281 + 0.369373i \(0.120427\pi\)
\(912\) 3.38584e9 0.147803
\(913\) −1.31313e10 −0.571031
\(914\) 4.32564e10 1.87386
\(915\) −1.59849e9 −0.0689819
\(916\) −3.44691e10 −1.48182
\(917\) −7.59067e9 −0.325078
\(918\) 1.64939e9 0.0703676
\(919\) −1.45194e10 −0.617085 −0.308543 0.951211i \(-0.599841\pi\)
−0.308543 + 0.951211i \(0.599841\pi\)
\(920\) 3.21048e10 1.35929
\(921\) 2.10274e9 0.0886906
\(922\) 1.59020e10 0.668180
\(923\) 1.65142e10 0.691275
\(924\) −5.72426e9 −0.238708
\(925\) −1.14141e9 −0.0474184
\(926\) −2.48431e10 −1.02818
\(927\) −6.21660e9 −0.256315
\(928\) 3.00823e9 0.123564
\(929\) 1.76962e9 0.0724145 0.0362072 0.999344i \(-0.488472\pi\)
0.0362072 + 0.999344i \(0.488472\pi\)
\(930\) −2.96210e9 −0.120756
\(931\) −1.28731e10 −0.522830
\(932\) −6.48791e10 −2.62512
\(933\) 1.34725e10 0.543079
\(934\) 3.83270e10 1.53918
\(935\) −4.14984e9 −0.166032
\(936\) 1.43263e10 0.571042
\(937\) 2.29343e10 0.910747 0.455373 0.890301i \(-0.349506\pi\)
0.455373 + 0.890301i \(0.349506\pi\)
\(938\) 8.05185e9 0.318556
\(939\) 9.66084e9 0.380790
\(940\) −4.49746e10 −1.76612
\(941\) −1.81317e10 −0.709375 −0.354687 0.934985i \(-0.615413\pi\)
−0.354687 + 0.934985i \(0.615413\pi\)
\(942\) −1.58420e10 −0.617493
\(943\) 5.53091e9 0.214786
\(944\) 1.51759e9 0.0587155
\(945\) 1.30753e9 0.0504012
\(946\) 3.10293e10 1.19166
\(947\) −4.46859e10 −1.70980 −0.854901 0.518791i \(-0.826382\pi\)
−0.854901 + 0.518791i \(0.826382\pi\)
\(948\) −1.05523e10 −0.402271
\(949\) −3.82860e9 −0.145415
\(950\) −3.29022e9 −0.124507
\(951\) 3.52764e9 0.133000
\(952\) 2.19923e9 0.0826116
\(953\) −4.37413e10 −1.63707 −0.818534 0.574458i \(-0.805213\pi\)
−0.818534 + 0.574458i \(0.805213\pi\)
\(954\) 1.52729e10 0.569509
\(955\) 1.60283e9 0.0595491
\(956\) 2.21085e9 0.0818383
\(957\) 2.66746e9 0.0983799
\(958\) 3.31133e10 1.21681
\(959\) 4.83293e9 0.176948
\(960\) −2.12678e10 −0.775842
\(961\) −2.70187e10 −0.982049
\(962\) 2.12938e10 0.771151
\(963\) 4.37877e9 0.158001
\(964\) 1.76246e10 0.633652
\(965\) −3.47458e10 −1.24468
\(966\) 8.23928e9 0.294082
\(967\) −1.87281e10 −0.666042 −0.333021 0.942919i \(-0.608068\pi\)
−0.333021 + 0.942919i \(0.608068\pi\)
\(968\) −1.27181e10 −0.450671
\(969\) 2.02669e9 0.0715572
\(970\) −4.46808e9 −0.157188
\(971\) 4.41287e10 1.54687 0.773434 0.633877i \(-0.218537\pi\)
0.773434 + 0.633877i \(0.218537\pi\)
\(972\) 3.31375e9 0.115741
\(973\) 6.68539e9 0.232666
\(974\) −4.52595e9 −0.156947
\(975\) −2.78407e9 −0.0961977
\(976\) −1.67894e9 −0.0578044
\(977\) 2.33709e10 0.801762 0.400881 0.916130i \(-0.368704\pi\)
0.400881 + 0.916130i \(0.368704\pi\)
\(978\) 4.36294e10 1.49140
\(979\) 2.84780e10 0.969997
\(980\) −4.56447e10 −1.54917
\(981\) −3.15916e9 −0.106839
\(982\) 4.07868e10 1.37445
\(983\) 1.07329e10 0.360396 0.180198 0.983630i \(-0.442326\pi\)
0.180198 + 0.983630i \(0.442326\pi\)
\(984\) 4.60996e9 0.154246
\(985\) −3.35487e10 −1.11853
\(986\) −2.29912e9 −0.0763823
\(987\) −5.14486e9 −0.170319
\(988\) 3.94922e10 1.30275
\(989\) −2.87355e10 −0.944566
\(990\) −1.29584e10 −0.424451
\(991\) 2.64397e10 0.862975 0.431488 0.902119i \(-0.357989\pi\)
0.431488 + 0.902119i \(0.357989\pi\)
\(992\) 2.43667e9 0.0792511
\(993\) 2.67890e10 0.868230
\(994\) 7.91614e9 0.255659
\(995\) −4.34671e10 −1.39888
\(996\) 2.27388e10 0.729223
\(997\) −3.85050e10 −1.23051 −0.615253 0.788330i \(-0.710946\pi\)
−0.615253 + 0.788330i \(0.710946\pi\)
\(998\) 1.87632e10 0.597516
\(999\) 2.19546e9 0.0696700
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.d.1.16 18
3.2 odd 2 531.8.a.e.1.3 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.16 18 1.1 even 1 trivial
531.8.a.e.1.3 18 3.2 odd 2