Properties

Label 531.8.a.e.1.16
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(1\)
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: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(-17.7235\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+16.7235 q^{2} +151.676 q^{4} +298.819 q^{5} +200.095 q^{7} +395.941 q^{8} +O(q^{10})\) \(q+16.7235 q^{2} +151.676 q^{4} +298.819 q^{5} +200.095 q^{7} +395.941 q^{8} +4997.30 q^{10} -6819.80 q^{11} -7653.89 q^{13} +3346.30 q^{14} -12793.0 q^{16} +23868.2 q^{17} +41453.1 q^{19} +45323.6 q^{20} -114051. q^{22} -87045.3 q^{23} +11167.7 q^{25} -128000. q^{26} +30349.6 q^{28} +28849.4 q^{29} -136082. q^{31} -264624. q^{32} +399161. q^{34} +59792.3 q^{35} -238332. q^{37} +693241. q^{38} +118315. q^{40} +387202. q^{41} -560156. q^{43} -1.03440e6 q^{44} -1.45570e6 q^{46} -343766. q^{47} -783505. q^{49} +186763. q^{50} -1.16091e6 q^{52} -1.90828e6 q^{53} -2.03788e6 q^{55} +79226.0 q^{56} +482464. q^{58} -205379. q^{59} +2.52219e6 q^{61} -2.27577e6 q^{62} -2.78794e6 q^{64} -2.28713e6 q^{65} +1.23640e6 q^{67} +3.62023e6 q^{68} +999937. q^{70} +5.34185e6 q^{71} -4.51521e6 q^{73} -3.98574e6 q^{74} +6.28743e6 q^{76} -1.36461e6 q^{77} -1.40570e6 q^{79} -3.82278e6 q^{80} +6.47538e6 q^{82} -8.95946e6 q^{83} +7.13228e6 q^{85} -9.36777e6 q^{86} -2.70024e6 q^{88} +1.30962e6 q^{89} -1.53151e6 q^{91} -1.32027e7 q^{92} -5.74897e6 q^{94} +1.23870e7 q^{95} +1.27503e7 q^{97} -1.31029e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 24q^{2} + 1358q^{4} - 678q^{5} + 3081q^{7} - 4107q^{8} + O(q^{10}) \) \( 18q - 24q^{2} + 1358q^{4} - 678q^{5} + 3081q^{7} - 4107q^{8} + 3609q^{10} - 15070q^{11} + 13662q^{13} - 20861q^{14} + 60482q^{16} - 71919q^{17} + 56231q^{19} - 143053q^{20} + 274198q^{22} - 150029q^{23} + 399672q^{25} - 182846q^{26} + 434150q^{28} - 591285q^{29} + 426733q^{31} - 1205630q^{32} + 403548q^{34} - 912879q^{35} + 7703q^{37} + 417859q^{38} + 618020q^{40} - 770959q^{41} + 793050q^{43} - 2591274q^{44} - 4068019q^{46} - 1410373q^{47} + 1637427q^{49} - 1021549q^{50} - 3749190q^{52} - 1037934q^{53} + 331974q^{55} + 391748q^{56} + 653724q^{58} - 3696822q^{59} - 1374623q^{61} - 5251718q^{62} + 5077197q^{64} - 3257170q^{65} - 2436904q^{67} - 14119909q^{68} + 5185580q^{70} - 14289172q^{71} + 5482515q^{73} - 14934154q^{74} + 3822912q^{76} - 23157109q^{77} + 19786414q^{79} - 31978143q^{80} + 9749509q^{82} - 30227337q^{83} + 9946981q^{85} - 44295864q^{86} + 39970897q^{88} - 31061677q^{89} + 26377785q^{91} - 4719698q^{92} + 44488296q^{94} - 15534599q^{95} + 12084118q^{97} - 42274744q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.7235 1.47816 0.739082 0.673616i \(-0.235260\pi\)
0.739082 + 0.673616i \(0.235260\pi\)
\(3\) 0 0
\(4\) 151.676 1.18497
\(5\) 298.819 1.06909 0.534543 0.845141i \(-0.320484\pi\)
0.534543 + 0.845141i \(0.320484\pi\)
\(6\) 0 0
\(7\) 200.095 0.220493 0.110246 0.993904i \(-0.464836\pi\)
0.110246 + 0.993904i \(0.464836\pi\)
\(8\) 395.941 0.273411
\(9\) 0 0
\(10\) 4997.30 1.58028
\(11\) −6819.80 −1.54489 −0.772444 0.635082i \(-0.780966\pi\)
−0.772444 + 0.635082i \(0.780966\pi\)
\(12\) 0 0
\(13\) −7653.89 −0.966230 −0.483115 0.875557i \(-0.660495\pi\)
−0.483115 + 0.875557i \(0.660495\pi\)
\(14\) 3346.30 0.325924
\(15\) 0 0
\(16\) −12793.0 −0.780821
\(17\) 23868.2 1.17828 0.589141 0.808030i \(-0.299466\pi\)
0.589141 + 0.808030i \(0.299466\pi\)
\(18\) 0 0
\(19\) 41453.1 1.38650 0.693249 0.720698i \(-0.256179\pi\)
0.693249 + 0.720698i \(0.256179\pi\)
\(20\) 45323.6 1.26683
\(21\) 0 0
\(22\) −114051. −2.28360
\(23\) −87045.3 −1.49176 −0.745878 0.666082i \(-0.767970\pi\)
−0.745878 + 0.666082i \(0.767970\pi\)
\(24\) 0 0
\(25\) 11167.7 0.142947
\(26\) −128000. −1.42825
\(27\) 0 0
\(28\) 30349.6 0.261276
\(29\) 28849.4 0.219657 0.109828 0.993951i \(-0.464970\pi\)
0.109828 + 0.993951i \(0.464970\pi\)
\(30\) 0 0
\(31\) −136082. −0.820419 −0.410209 0.911991i \(-0.634544\pi\)
−0.410209 + 0.911991i \(0.634544\pi\)
\(32\) −264624. −1.42759
\(33\) 0 0
\(34\) 399161. 1.74169
\(35\) 59792.3 0.235726
\(36\) 0 0
\(37\) −238332. −0.773527 −0.386764 0.922179i \(-0.626407\pi\)
−0.386764 + 0.922179i \(0.626407\pi\)
\(38\) 693241. 2.04947
\(39\) 0 0
\(40\) 118315. 0.292300
\(41\) 387202. 0.877394 0.438697 0.898635i \(-0.355440\pi\)
0.438697 + 0.898635i \(0.355440\pi\)
\(42\) 0 0
\(43\) −560156. −1.07441 −0.537204 0.843452i \(-0.680519\pi\)
−0.537204 + 0.843452i \(0.680519\pi\)
\(44\) −1.03440e6 −1.83064
\(45\) 0 0
\(46\) −1.45570e6 −2.20506
\(47\) −343766. −0.482970 −0.241485 0.970405i \(-0.577635\pi\)
−0.241485 + 0.970405i \(0.577635\pi\)
\(48\) 0 0
\(49\) −783505. −0.951383
\(50\) 186763. 0.211298
\(51\) 0 0
\(52\) −1.16091e6 −1.14495
\(53\) −1.90828e6 −1.76066 −0.880332 0.474358i \(-0.842680\pi\)
−0.880332 + 0.474358i \(0.842680\pi\)
\(54\) 0 0
\(55\) −2.03788e6 −1.65162
\(56\) 79226.0 0.0602851
\(57\) 0 0
\(58\) 482464. 0.324689
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) 2.52219e6 1.42273 0.711366 0.702822i \(-0.248077\pi\)
0.711366 + 0.702822i \(0.248077\pi\)
\(62\) −2.27577e6 −1.21271
\(63\) 0 0
\(64\) −2.78794e6 −1.32939
\(65\) −2.28713e6 −1.03298
\(66\) 0 0
\(67\) 1.23640e6 0.502224 0.251112 0.967958i \(-0.419204\pi\)
0.251112 + 0.967958i \(0.419204\pi\)
\(68\) 3.62023e6 1.39622
\(69\) 0 0
\(70\) 999937. 0.348441
\(71\) 5.34185e6 1.77128 0.885640 0.464373i \(-0.153720\pi\)
0.885640 + 0.464373i \(0.153720\pi\)
\(72\) 0 0
\(73\) −4.51521e6 −1.35846 −0.679232 0.733923i \(-0.737687\pi\)
−0.679232 + 0.733923i \(0.737687\pi\)
\(74\) −3.98574e6 −1.14340
\(75\) 0 0
\(76\) 6.28743e6 1.64295
\(77\) −1.36461e6 −0.340637
\(78\) 0 0
\(79\) −1.40570e6 −0.320772 −0.160386 0.987054i \(-0.551274\pi\)
−0.160386 + 0.987054i \(0.551274\pi\)
\(80\) −3.82278e6 −0.834765
\(81\) 0 0
\(82\) 6.47538e6 1.29693
\(83\) −8.95946e6 −1.71992 −0.859961 0.510361i \(-0.829512\pi\)
−0.859961 + 0.510361i \(0.829512\pi\)
\(84\) 0 0
\(85\) 7.13228e6 1.25969
\(86\) −9.36777e6 −1.58815
\(87\) 0 0
\(88\) −2.70024e6 −0.422389
\(89\) 1.30962e6 0.196916 0.0984581 0.995141i \(-0.468609\pi\)
0.0984581 + 0.995141i \(0.468609\pi\)
\(90\) 0 0
\(91\) −1.53151e6 −0.213047
\(92\) −1.32027e7 −1.76768
\(93\) 0 0
\(94\) −5.74897e6 −0.713909
\(95\) 1.23870e7 1.48229
\(96\) 0 0
\(97\) 1.27503e7 1.41847 0.709236 0.704971i \(-0.249040\pi\)
0.709236 + 0.704971i \(0.249040\pi\)
\(98\) −1.31029e7 −1.40630
\(99\) 0 0
\(100\) 1.69387e6 0.169387
\(101\) −1.88135e7 −1.81696 −0.908478 0.417933i \(-0.862755\pi\)
−0.908478 + 0.417933i \(0.862755\pi\)
\(102\) 0 0
\(103\) 1.54829e7 1.39611 0.698057 0.716042i \(-0.254048\pi\)
0.698057 + 0.716042i \(0.254048\pi\)
\(104\) −3.03049e6 −0.264178
\(105\) 0 0
\(106\) −3.19131e7 −2.60255
\(107\) −2.71022e6 −0.213876 −0.106938 0.994266i \(-0.534105\pi\)
−0.106938 + 0.994266i \(0.534105\pi\)
\(108\) 0 0
\(109\) 4.25363e6 0.314606 0.157303 0.987550i \(-0.449720\pi\)
0.157303 + 0.987550i \(0.449720\pi\)
\(110\) −3.40806e7 −2.44136
\(111\) 0 0
\(112\) −2.55981e6 −0.172165
\(113\) −1.36740e7 −0.891501 −0.445750 0.895157i \(-0.647063\pi\)
−0.445750 + 0.895157i \(0.647063\pi\)
\(114\) 0 0
\(115\) −2.60108e7 −1.59482
\(116\) 4.37576e6 0.260286
\(117\) 0 0
\(118\) −3.43466e6 −0.192440
\(119\) 4.77593e6 0.259802
\(120\) 0 0
\(121\) 2.70225e7 1.38668
\(122\) 4.21798e7 2.10303
\(123\) 0 0
\(124\) −2.06404e7 −0.972169
\(125\) −2.00081e7 −0.916264
\(126\) 0 0
\(127\) 3.13310e7 1.35726 0.678628 0.734482i \(-0.262575\pi\)
0.678628 + 0.734482i \(0.262575\pi\)
\(128\) −1.27523e7 −0.537468
\(129\) 0 0
\(130\) −3.82488e7 −1.52692
\(131\) 2.66364e6 0.103520 0.0517602 0.998660i \(-0.483517\pi\)
0.0517602 + 0.998660i \(0.483517\pi\)
\(132\) 0 0
\(133\) 8.29457e6 0.305713
\(134\) 2.06770e7 0.742369
\(135\) 0 0
\(136\) 9.45042e6 0.322155
\(137\) −2.83551e6 −0.0942126 −0.0471063 0.998890i \(-0.515000\pi\)
−0.0471063 + 0.998890i \(0.515000\pi\)
\(138\) 0 0
\(139\) 3.75740e7 1.18669 0.593343 0.804950i \(-0.297808\pi\)
0.593343 + 0.804950i \(0.297808\pi\)
\(140\) 9.06904e6 0.279327
\(141\) 0 0
\(142\) 8.93344e7 2.61824
\(143\) 5.21980e7 1.49272
\(144\) 0 0
\(145\) 8.62076e6 0.234832
\(146\) −7.55102e7 −2.00803
\(147\) 0 0
\(148\) −3.61491e7 −0.916604
\(149\) −2.26234e7 −0.560280 −0.280140 0.959959i \(-0.590381\pi\)
−0.280140 + 0.959959i \(0.590381\pi\)
\(150\) 0 0
\(151\) −2.07436e7 −0.490303 −0.245152 0.969485i \(-0.578838\pi\)
−0.245152 + 0.969485i \(0.578838\pi\)
\(152\) 1.64130e7 0.379084
\(153\) 0 0
\(154\) −2.28211e7 −0.503516
\(155\) −4.06639e7 −0.877099
\(156\) 0 0
\(157\) −5.37534e7 −1.10855 −0.554277 0.832332i \(-0.687005\pi\)
−0.554277 + 0.832332i \(0.687005\pi\)
\(158\) −2.35082e7 −0.474154
\(159\) 0 0
\(160\) −7.90746e7 −1.52622
\(161\) −1.74174e7 −0.328921
\(162\) 0 0
\(163\) 2.39837e7 0.433769 0.216885 0.976197i \(-0.430410\pi\)
0.216885 + 0.976197i \(0.430410\pi\)
\(164\) 5.87292e7 1.03968
\(165\) 0 0
\(166\) −1.49834e8 −2.54232
\(167\) −7.25545e7 −1.20547 −0.602736 0.797941i \(-0.705923\pi\)
−0.602736 + 0.797941i \(0.705923\pi\)
\(168\) 0 0
\(169\) −4.16645e6 −0.0663992
\(170\) 1.19277e8 1.86202
\(171\) 0 0
\(172\) −8.49620e7 −1.27314
\(173\) −8.99082e7 −1.32020 −0.660098 0.751180i \(-0.729485\pi\)
−0.660098 + 0.751180i \(0.729485\pi\)
\(174\) 0 0
\(175\) 2.23461e6 0.0315187
\(176\) 8.72455e7 1.20628
\(177\) 0 0
\(178\) 2.19015e7 0.291074
\(179\) −2.66685e7 −0.347547 −0.173774 0.984786i \(-0.555596\pi\)
−0.173774 + 0.984786i \(0.555596\pi\)
\(180\) 0 0
\(181\) −1.27888e8 −1.60308 −0.801539 0.597943i \(-0.795985\pi\)
−0.801539 + 0.597943i \(0.795985\pi\)
\(182\) −2.56122e7 −0.314918
\(183\) 0 0
\(184\) −3.44648e7 −0.407862
\(185\) −7.12180e7 −0.826968
\(186\) 0 0
\(187\) −1.62777e8 −1.82031
\(188\) −5.21410e7 −0.572304
\(189\) 0 0
\(190\) 2.07153e8 2.19106
\(191\) 3.33882e7 0.346718 0.173359 0.984859i \(-0.444538\pi\)
0.173359 + 0.984859i \(0.444538\pi\)
\(192\) 0 0
\(193\) 6.61151e7 0.661988 0.330994 0.943633i \(-0.392616\pi\)
0.330994 + 0.943633i \(0.392616\pi\)
\(194\) 2.13230e8 2.09673
\(195\) 0 0
\(196\) −1.18839e8 −1.12736
\(197\) 7.50492e6 0.0699381 0.0349691 0.999388i \(-0.488867\pi\)
0.0349691 + 0.999388i \(0.488867\pi\)
\(198\) 0 0
\(199\) 2.88579e7 0.259584 0.129792 0.991541i \(-0.458569\pi\)
0.129792 + 0.991541i \(0.458569\pi\)
\(200\) 4.42175e6 0.0390832
\(201\) 0 0
\(202\) −3.14627e8 −2.68576
\(203\) 5.77264e6 0.0484327
\(204\) 0 0
\(205\) 1.15703e8 0.938010
\(206\) 2.58928e8 2.06368
\(207\) 0 0
\(208\) 9.79160e7 0.754453
\(209\) −2.82702e8 −2.14199
\(210\) 0 0
\(211\) −1.61521e8 −1.18370 −0.591849 0.806049i \(-0.701602\pi\)
−0.591849 + 0.806049i \(0.701602\pi\)
\(212\) −2.89440e8 −2.08633
\(213\) 0 0
\(214\) −4.53245e7 −0.316144
\(215\) −1.67385e8 −1.14863
\(216\) 0 0
\(217\) −2.72294e7 −0.180896
\(218\) 7.11356e7 0.465039
\(219\) 0 0
\(220\) −3.09098e8 −1.95711
\(221\) −1.82685e8 −1.13849
\(222\) 0 0
\(223\) 1.37272e8 0.828924 0.414462 0.910067i \(-0.363970\pi\)
0.414462 + 0.910067i \(0.363970\pi\)
\(224\) −5.29500e7 −0.314773
\(225\) 0 0
\(226\) −2.28678e8 −1.31778
\(227\) 5.03590e7 0.285750 0.142875 0.989741i \(-0.454365\pi\)
0.142875 + 0.989741i \(0.454365\pi\)
\(228\) 0 0
\(229\) 3.29634e8 1.81388 0.906938 0.421264i \(-0.138413\pi\)
0.906938 + 0.421264i \(0.138413\pi\)
\(230\) −4.34991e8 −2.35740
\(231\) 0 0
\(232\) 1.14227e7 0.0600565
\(233\) −4.35459e7 −0.225529 −0.112764 0.993622i \(-0.535971\pi\)
−0.112764 + 0.993622i \(0.535971\pi\)
\(234\) 0 0
\(235\) −1.02724e8 −0.516337
\(236\) −3.11510e7 −0.154270
\(237\) 0 0
\(238\) 7.98703e7 0.384030
\(239\) 1.16895e8 0.553865 0.276933 0.960889i \(-0.410682\pi\)
0.276933 + 0.960889i \(0.410682\pi\)
\(240\) 0 0
\(241\) −3.76459e8 −1.73244 −0.866219 0.499664i \(-0.833457\pi\)
−0.866219 + 0.499664i \(0.833457\pi\)
\(242\) 4.51911e8 2.04974
\(243\) 0 0
\(244\) 3.82555e8 1.68589
\(245\) −2.34126e8 −1.01711
\(246\) 0 0
\(247\) −3.17277e8 −1.33968
\(248\) −5.38806e7 −0.224311
\(249\) 0 0
\(250\) −3.34606e8 −1.35439
\(251\) −3.72328e8 −1.48617 −0.743084 0.669198i \(-0.766638\pi\)
−0.743084 + 0.669198i \(0.766638\pi\)
\(252\) 0 0
\(253\) 5.93631e8 2.30460
\(254\) 5.23965e8 2.00625
\(255\) 0 0
\(256\) 1.43594e8 0.534928
\(257\) 4.73110e8 1.73859 0.869294 0.494296i \(-0.164574\pi\)
0.869294 + 0.494296i \(0.164574\pi\)
\(258\) 0 0
\(259\) −4.76891e7 −0.170557
\(260\) −3.46902e8 −1.22405
\(261\) 0 0
\(262\) 4.45454e7 0.153020
\(263\) 4.08150e8 1.38349 0.691743 0.722144i \(-0.256843\pi\)
0.691743 + 0.722144i \(0.256843\pi\)
\(264\) 0 0
\(265\) −5.70230e8 −1.88230
\(266\) 1.38714e8 0.451893
\(267\) 0 0
\(268\) 1.87532e8 0.595119
\(269\) −5.16986e8 −1.61937 −0.809684 0.586865i \(-0.800362\pi\)
−0.809684 + 0.586865i \(0.800362\pi\)
\(270\) 0 0
\(271\) −5.66936e7 −0.173038 −0.0865189 0.996250i \(-0.527574\pi\)
−0.0865189 + 0.996250i \(0.527574\pi\)
\(272\) −3.05346e8 −0.920027
\(273\) 0 0
\(274\) −4.74196e7 −0.139262
\(275\) −7.61615e7 −0.220837
\(276\) 0 0
\(277\) 2.31128e8 0.653392 0.326696 0.945129i \(-0.394065\pi\)
0.326696 + 0.945129i \(0.394065\pi\)
\(278\) 6.28370e8 1.75412
\(279\) 0 0
\(280\) 2.36742e7 0.0644500
\(281\) −3.68378e8 −0.990426 −0.495213 0.868772i \(-0.664910\pi\)
−0.495213 + 0.868772i \(0.664910\pi\)
\(282\) 0 0
\(283\) 5.19507e8 1.36251 0.681254 0.732047i \(-0.261435\pi\)
0.681254 + 0.732047i \(0.261435\pi\)
\(284\) 8.10228e8 2.09891
\(285\) 0 0
\(286\) 8.72934e8 2.20648
\(287\) 7.74774e7 0.193459
\(288\) 0 0
\(289\) 1.59354e8 0.388349
\(290\) 1.44169e8 0.347120
\(291\) 0 0
\(292\) −6.84848e8 −1.60974
\(293\) 8.25048e8 1.91621 0.958104 0.286421i \(-0.0924657\pi\)
0.958104 + 0.286421i \(0.0924657\pi\)
\(294\) 0 0
\(295\) −6.13711e7 −0.139183
\(296\) −9.43653e7 −0.211491
\(297\) 0 0
\(298\) −3.78342e8 −0.828186
\(299\) 6.66235e8 1.44138
\(300\) 0 0
\(301\) −1.12085e8 −0.236899
\(302\) −3.46906e8 −0.724748
\(303\) 0 0
\(304\) −5.30308e8 −1.08261
\(305\) 7.53677e8 1.52102
\(306\) 0 0
\(307\) −2.01480e8 −0.397418 −0.198709 0.980059i \(-0.563675\pi\)
−0.198709 + 0.980059i \(0.563675\pi\)
\(308\) −2.06978e8 −0.403643
\(309\) 0 0
\(310\) −6.80044e8 −1.29650
\(311\) −1.01602e8 −0.191533 −0.0957663 0.995404i \(-0.530530\pi\)
−0.0957663 + 0.995404i \(0.530530\pi\)
\(312\) 0 0
\(313\) −5.09494e8 −0.939148 −0.469574 0.882893i \(-0.655592\pi\)
−0.469574 + 0.882893i \(0.655592\pi\)
\(314\) −8.98946e8 −1.63863
\(315\) 0 0
\(316\) −2.13210e8 −0.380104
\(317\) −5.25634e8 −0.926779 −0.463389 0.886155i \(-0.653367\pi\)
−0.463389 + 0.886155i \(0.653367\pi\)
\(318\) 0 0
\(319\) −1.96747e8 −0.339345
\(320\) −8.33088e8 −1.42124
\(321\) 0 0
\(322\) −2.91279e8 −0.486199
\(323\) 9.89412e8 1.63369
\(324\) 0 0
\(325\) −8.54764e7 −0.138119
\(326\) 4.01091e8 0.641182
\(327\) 0 0
\(328\) 1.53309e8 0.239889
\(329\) −6.87860e7 −0.106491
\(330\) 0 0
\(331\) 2.47062e8 0.374463 0.187231 0.982316i \(-0.440049\pi\)
0.187231 + 0.982316i \(0.440049\pi\)
\(332\) −1.35893e9 −2.03805
\(333\) 0 0
\(334\) −1.21337e9 −1.78188
\(335\) 3.69460e8 0.536921
\(336\) 0 0
\(337\) 1.28783e9 1.83296 0.916482 0.400076i \(-0.131016\pi\)
0.916482 + 0.400076i \(0.131016\pi\)
\(338\) −6.96777e7 −0.0981488
\(339\) 0 0
\(340\) 1.08179e9 1.49269
\(341\) 9.28054e8 1.26746
\(342\) 0 0
\(343\) −3.21563e8 −0.430266
\(344\) −2.21789e8 −0.293755
\(345\) 0 0
\(346\) −1.50358e9 −1.95146
\(347\) 1.16047e9 1.49101 0.745504 0.666502i \(-0.232209\pi\)
0.745504 + 0.666502i \(0.232209\pi\)
\(348\) 0 0
\(349\) 1.00669e9 1.26767 0.633836 0.773467i \(-0.281479\pi\)
0.633836 + 0.773467i \(0.281479\pi\)
\(350\) 3.73705e7 0.0465897
\(351\) 0 0
\(352\) 1.80468e9 2.20547
\(353\) −2.36378e8 −0.286019 −0.143010 0.989721i \(-0.545678\pi\)
−0.143010 + 0.989721i \(0.545678\pi\)
\(354\) 0 0
\(355\) 1.59624e9 1.89365
\(356\) 1.98638e8 0.233339
\(357\) 0 0
\(358\) −4.45992e8 −0.513731
\(359\) 4.28699e8 0.489015 0.244507 0.969647i \(-0.421374\pi\)
0.244507 + 0.969647i \(0.421374\pi\)
\(360\) 0 0
\(361\) 8.24486e8 0.922377
\(362\) −2.13873e9 −2.36961
\(363\) 0 0
\(364\) −2.32293e8 −0.252453
\(365\) −1.34923e9 −1.45232
\(366\) 0 0
\(367\) 1.21528e9 1.28335 0.641677 0.766975i \(-0.278239\pi\)
0.641677 + 0.766975i \(0.278239\pi\)
\(368\) 1.11357e9 1.16479
\(369\) 0 0
\(370\) −1.19101e9 −1.22239
\(371\) −3.81838e8 −0.388213
\(372\) 0 0
\(373\) 9.41218e8 0.939094 0.469547 0.882907i \(-0.344417\pi\)
0.469547 + 0.882907i \(0.344417\pi\)
\(374\) −2.72220e9 −2.69072
\(375\) 0 0
\(376\) −1.36111e8 −0.132049
\(377\) −2.20811e8 −0.212239
\(378\) 0 0
\(379\) 2.27326e8 0.214492 0.107246 0.994233i \(-0.465797\pi\)
0.107246 + 0.994233i \(0.465797\pi\)
\(380\) 1.87880e9 1.75646
\(381\) 0 0
\(382\) 5.58368e8 0.512506
\(383\) −1.40852e9 −1.28105 −0.640525 0.767938i \(-0.721283\pi\)
−0.640525 + 0.767938i \(0.721283\pi\)
\(384\) 0 0
\(385\) −4.07771e8 −0.364170
\(386\) 1.10568e9 0.978527
\(387\) 0 0
\(388\) 1.93392e9 1.68084
\(389\) −1.22433e9 −1.05457 −0.527285 0.849689i \(-0.676790\pi\)
−0.527285 + 0.849689i \(0.676790\pi\)
\(390\) 0 0
\(391\) −2.07762e9 −1.75771
\(392\) −3.10222e8 −0.260118
\(393\) 0 0
\(394\) 1.25509e8 0.103380
\(395\) −4.20049e8 −0.342933
\(396\) 0 0
\(397\) −3.38844e8 −0.271790 −0.135895 0.990723i \(-0.543391\pi\)
−0.135895 + 0.990723i \(0.543391\pi\)
\(398\) 4.82605e8 0.383708
\(399\) 0 0
\(400\) −1.42868e8 −0.111616
\(401\) −1.32434e8 −0.102564 −0.0512820 0.998684i \(-0.516331\pi\)
−0.0512820 + 0.998684i \(0.516331\pi\)
\(402\) 0 0
\(403\) 1.04156e9 0.792713
\(404\) −2.85355e9 −2.15303
\(405\) 0 0
\(406\) 9.65388e7 0.0715914
\(407\) 1.62537e9 1.19501
\(408\) 0 0
\(409\) 1.81103e9 1.30886 0.654431 0.756122i \(-0.272908\pi\)
0.654431 + 0.756122i \(0.272908\pi\)
\(410\) 1.93497e9 1.38653
\(411\) 0 0
\(412\) 2.34837e9 1.65435
\(413\) −4.10954e7 −0.0287057
\(414\) 0 0
\(415\) −2.67726e9 −1.83874
\(416\) 2.02540e9 1.37938
\(417\) 0 0
\(418\) −4.72776e9 −3.16620
\(419\) 2.75900e9 1.83233 0.916164 0.400805i \(-0.131269\pi\)
0.916164 + 0.400805i \(0.131269\pi\)
\(420\) 0 0
\(421\) −6.24044e8 −0.407594 −0.203797 0.979013i \(-0.565328\pi\)
−0.203797 + 0.979013i \(0.565328\pi\)
\(422\) −2.70120e9 −1.74970
\(423\) 0 0
\(424\) −7.55566e8 −0.481385
\(425\) 2.66554e8 0.168431
\(426\) 0 0
\(427\) 5.04678e8 0.313702
\(428\) −4.11075e8 −0.253436
\(429\) 0 0
\(430\) −2.79927e9 −1.69787
\(431\) −2.11891e9 −1.27480 −0.637401 0.770532i \(-0.719990\pi\)
−0.637401 + 0.770532i \(0.719990\pi\)
\(432\) 0 0
\(433\) −1.12211e9 −0.664244 −0.332122 0.943236i \(-0.607765\pi\)
−0.332122 + 0.943236i \(0.607765\pi\)
\(434\) −4.55372e8 −0.267394
\(435\) 0 0
\(436\) 6.45172e8 0.372797
\(437\) −3.60830e9 −2.06832
\(438\) 0 0
\(439\) −1.19818e9 −0.675922 −0.337961 0.941160i \(-0.609737\pi\)
−0.337961 + 0.941160i \(0.609737\pi\)
\(440\) −8.06882e8 −0.451571
\(441\) 0 0
\(442\) −3.05513e9 −1.68288
\(443\) 5.64409e8 0.308447 0.154224 0.988036i \(-0.450712\pi\)
0.154224 + 0.988036i \(0.450712\pi\)
\(444\) 0 0
\(445\) 3.91340e8 0.210521
\(446\) 2.29567e9 1.22529
\(447\) 0 0
\(448\) −5.57854e8 −0.293121
\(449\) 1.48373e9 0.773558 0.386779 0.922172i \(-0.373588\pi\)
0.386779 + 0.922172i \(0.373588\pi\)
\(450\) 0 0
\(451\) −2.64064e9 −1.35548
\(452\) −2.07402e9 −1.05640
\(453\) 0 0
\(454\) 8.42179e8 0.422386
\(455\) −4.57644e8 −0.227765
\(456\) 0 0
\(457\) −1.51906e9 −0.744505 −0.372253 0.928131i \(-0.621415\pi\)
−0.372253 + 0.928131i \(0.621415\pi\)
\(458\) 5.51264e9 2.68121
\(459\) 0 0
\(460\) −3.94520e9 −1.88980
\(461\) −8.02602e8 −0.381546 −0.190773 0.981634i \(-0.561099\pi\)
−0.190773 + 0.981634i \(0.561099\pi\)
\(462\) 0 0
\(463\) 9.47165e8 0.443499 0.221749 0.975104i \(-0.428823\pi\)
0.221749 + 0.975104i \(0.428823\pi\)
\(464\) −3.69070e8 −0.171513
\(465\) 0 0
\(466\) −7.28241e8 −0.333368
\(467\) −1.58449e9 −0.719914 −0.359957 0.932969i \(-0.617209\pi\)
−0.359957 + 0.932969i \(0.617209\pi\)
\(468\) 0 0
\(469\) 2.47398e8 0.110737
\(470\) −1.71790e9 −0.763231
\(471\) 0 0
\(472\) −8.13180e7 −0.0355951
\(473\) 3.82015e9 1.65984
\(474\) 0 0
\(475\) 4.62936e8 0.198195
\(476\) 7.24392e8 0.307857
\(477\) 0 0
\(478\) 1.95490e9 0.818703
\(479\) 2.85017e6 0.00118494 0.000592471 1.00000i \(-0.499811\pi\)
0.000592471 1.00000i \(0.499811\pi\)
\(480\) 0 0
\(481\) 1.82416e9 0.747405
\(482\) −6.29571e9 −2.56083
\(483\) 0 0
\(484\) 4.09866e9 1.64317
\(485\) 3.81004e9 1.51647
\(486\) 0 0
\(487\) 1.28087e9 0.502521 0.251260 0.967920i \(-0.419155\pi\)
0.251260 + 0.967920i \(0.419155\pi\)
\(488\) 9.98638e8 0.388990
\(489\) 0 0
\(490\) −3.91541e9 −1.50346
\(491\) 1.37484e8 0.0524164 0.0262082 0.999657i \(-0.491657\pi\)
0.0262082 + 0.999657i \(0.491657\pi\)
\(492\) 0 0
\(493\) 6.88586e8 0.258818
\(494\) −5.30599e9 −1.98026
\(495\) 0 0
\(496\) 1.74090e9 0.640600
\(497\) 1.06888e9 0.390554
\(498\) 0 0
\(499\) 1.97156e8 0.0710327 0.0355163 0.999369i \(-0.488692\pi\)
0.0355163 + 0.999369i \(0.488692\pi\)
\(500\) −3.03474e9 −1.08574
\(501\) 0 0
\(502\) −6.22663e9 −2.19680
\(503\) 2.00636e9 0.702946 0.351473 0.936198i \(-0.385681\pi\)
0.351473 + 0.936198i \(0.385681\pi\)
\(504\) 0 0
\(505\) −5.62182e9 −1.94248
\(506\) 9.92760e9 3.40657
\(507\) 0 0
\(508\) 4.75216e9 1.60830
\(509\) −4.88057e9 −1.64043 −0.820216 0.572053i \(-0.806147\pi\)
−0.820216 + 0.572053i \(0.806147\pi\)
\(510\) 0 0
\(511\) −9.03474e8 −0.299531
\(512\) 4.03368e9 1.32818
\(513\) 0 0
\(514\) 7.91206e9 2.56992
\(515\) 4.62657e9 1.49257
\(516\) 0 0
\(517\) 2.34442e9 0.746135
\(518\) −7.97528e8 −0.252111
\(519\) 0 0
\(520\) −9.05568e8 −0.282429
\(521\) −3.48111e9 −1.07842 −0.539208 0.842173i \(-0.681276\pi\)
−0.539208 + 0.842173i \(0.681276\pi\)
\(522\) 0 0
\(523\) −8.79645e8 −0.268876 −0.134438 0.990922i \(-0.542923\pi\)
−0.134438 + 0.990922i \(0.542923\pi\)
\(524\) 4.04010e8 0.122668
\(525\) 0 0
\(526\) 6.82570e9 2.04502
\(527\) −3.24804e9 −0.966685
\(528\) 0 0
\(529\) 4.17206e9 1.22534
\(530\) −9.53625e9 −2.78235
\(531\) 0 0
\(532\) 1.25809e9 0.362259
\(533\) −2.96360e9 −0.847764
\(534\) 0 0
\(535\) −8.09866e8 −0.228652
\(536\) 4.89542e8 0.137314
\(537\) 0 0
\(538\) −8.64582e9 −2.39369
\(539\) 5.34335e9 1.46978
\(540\) 0 0
\(541\) −5.31588e9 −1.44339 −0.721697 0.692209i \(-0.756638\pi\)
−0.721697 + 0.692209i \(0.756638\pi\)
\(542\) −9.48115e8 −0.255778
\(543\) 0 0
\(544\) −6.31611e9 −1.68211
\(545\) 1.27106e9 0.336341
\(546\) 0 0
\(547\) 4.68079e9 1.22282 0.611412 0.791313i \(-0.290602\pi\)
0.611412 + 0.791313i \(0.290602\pi\)
\(548\) −4.30078e8 −0.111639
\(549\) 0 0
\(550\) −1.27369e9 −0.326433
\(551\) 1.19590e9 0.304554
\(552\) 0 0
\(553\) −2.81274e8 −0.0707279
\(554\) 3.86527e9 0.965820
\(555\) 0 0
\(556\) 5.69907e9 1.40618
\(557\) −9.19162e8 −0.225371 −0.112686 0.993631i \(-0.535945\pi\)
−0.112686 + 0.993631i \(0.535945\pi\)
\(558\) 0 0
\(559\) 4.28737e9 1.03813
\(560\) −7.64921e8 −0.184060
\(561\) 0 0
\(562\) −6.16058e9 −1.46401
\(563\) −2.14197e9 −0.505865 −0.252933 0.967484i \(-0.581395\pi\)
−0.252933 + 0.967484i \(0.581395\pi\)
\(564\) 0 0
\(565\) −4.08606e9 −0.953092
\(566\) 8.68798e9 2.01401
\(567\) 0 0
\(568\) 2.11506e9 0.484287
\(569\) 6.07580e9 1.38265 0.691323 0.722546i \(-0.257028\pi\)
0.691323 + 0.722546i \(0.257028\pi\)
\(570\) 0 0
\(571\) −6.78352e9 −1.52486 −0.762428 0.647073i \(-0.775993\pi\)
−0.762428 + 0.647073i \(0.775993\pi\)
\(572\) 7.91717e9 1.76882
\(573\) 0 0
\(574\) 1.29569e9 0.285964
\(575\) −9.72096e8 −0.213242
\(576\) 0 0
\(577\) −2.20797e9 −0.478495 −0.239248 0.970959i \(-0.576901\pi\)
−0.239248 + 0.970959i \(0.576901\pi\)
\(578\) 2.66497e9 0.574043
\(579\) 0 0
\(580\) 1.30756e9 0.278268
\(581\) −1.79275e9 −0.379230
\(582\) 0 0
\(583\) 1.30141e10 2.72003
\(584\) −1.78776e9 −0.371419
\(585\) 0 0
\(586\) 1.37977e10 2.83247
\(587\) 3.43972e9 0.701924 0.350962 0.936390i \(-0.385855\pi\)
0.350962 + 0.936390i \(0.385855\pi\)
\(588\) 0 0
\(589\) −5.64103e9 −1.13751
\(590\) −1.02634e9 −0.205736
\(591\) 0 0
\(592\) 3.04897e9 0.603986
\(593\) 3.62804e9 0.714464 0.357232 0.934016i \(-0.383721\pi\)
0.357232 + 0.934016i \(0.383721\pi\)
\(594\) 0 0
\(595\) 1.42714e9 0.277751
\(596\) −3.43142e9 −0.663914
\(597\) 0 0
\(598\) 1.11418e10 2.13059
\(599\) −3.45165e9 −0.656194 −0.328097 0.944644i \(-0.606407\pi\)
−0.328097 + 0.944644i \(0.606407\pi\)
\(600\) 0 0
\(601\) 7.18642e9 1.35037 0.675184 0.737649i \(-0.264064\pi\)
0.675184 + 0.737649i \(0.264064\pi\)
\(602\) −1.87445e9 −0.350175
\(603\) 0 0
\(604\) −3.14630e9 −0.580993
\(605\) 8.07483e9 1.48248
\(606\) 0 0
\(607\) 7.76353e9 1.40896 0.704480 0.709724i \(-0.251180\pi\)
0.704480 + 0.709724i \(0.251180\pi\)
\(608\) −1.09695e10 −1.97935
\(609\) 0 0
\(610\) 1.26041e10 2.24832
\(611\) 2.63115e9 0.466660
\(612\) 0 0
\(613\) 2.87864e9 0.504750 0.252375 0.967630i \(-0.418788\pi\)
0.252375 + 0.967630i \(0.418788\pi\)
\(614\) −3.36945e9 −0.587449
\(615\) 0 0
\(616\) −5.40306e8 −0.0931337
\(617\) −9.59531e9 −1.64460 −0.822301 0.569053i \(-0.807310\pi\)
−0.822301 + 0.569053i \(0.807310\pi\)
\(618\) 0 0
\(619\) 6.93284e8 0.117488 0.0587441 0.998273i \(-0.481290\pi\)
0.0587441 + 0.998273i \(0.481290\pi\)
\(620\) −6.16773e9 −1.03933
\(621\) 0 0
\(622\) −1.69915e9 −0.283116
\(623\) 2.62050e8 0.0434186
\(624\) 0 0
\(625\) −6.85127e9 −1.12251
\(626\) −8.52053e9 −1.38821
\(627\) 0 0
\(628\) −8.15309e9 −1.31360
\(629\) −5.68856e9 −0.911433
\(630\) 0 0
\(631\) 4.90432e9 0.777099 0.388549 0.921428i \(-0.372976\pi\)
0.388549 + 0.921428i \(0.372976\pi\)
\(632\) −5.56573e8 −0.0877026
\(633\) 0 0
\(634\) −8.79044e9 −1.36993
\(635\) 9.36231e9 1.45102
\(636\) 0 0
\(637\) 5.99686e9 0.919255
\(638\) −3.29031e9 −0.501608
\(639\) 0 0
\(640\) −3.81062e9 −0.574599
\(641\) 9.16838e9 1.37496 0.687479 0.726204i \(-0.258717\pi\)
0.687479 + 0.726204i \(0.258717\pi\)
\(642\) 0 0
\(643\) 4.70285e9 0.697626 0.348813 0.937192i \(-0.386585\pi\)
0.348813 + 0.937192i \(0.386585\pi\)
\(644\) −2.64179e9 −0.389761
\(645\) 0 0
\(646\) 1.65464e10 2.41485
\(647\) −5.90195e9 −0.856703 −0.428352 0.903612i \(-0.640906\pi\)
−0.428352 + 0.903612i \(0.640906\pi\)
\(648\) 0 0
\(649\) 1.40064e9 0.201127
\(650\) −1.42947e9 −0.204163
\(651\) 0 0
\(652\) 3.63774e9 0.514002
\(653\) −7.53884e9 −1.05952 −0.529759 0.848148i \(-0.677718\pi\)
−0.529759 + 0.848148i \(0.677718\pi\)
\(654\) 0 0
\(655\) 7.95947e8 0.110672
\(656\) −4.95347e9 −0.685087
\(657\) 0 0
\(658\) −1.15034e9 −0.157412
\(659\) 8.90395e9 1.21195 0.605974 0.795484i \(-0.292784\pi\)
0.605974 + 0.795484i \(0.292784\pi\)
\(660\) 0 0
\(661\) −2.51361e9 −0.338527 −0.169264 0.985571i \(-0.554139\pi\)
−0.169264 + 0.985571i \(0.554139\pi\)
\(662\) 4.13175e9 0.553517
\(663\) 0 0
\(664\) −3.54742e9 −0.470245
\(665\) 2.47857e9 0.326833
\(666\) 0 0
\(667\) −2.51121e9 −0.327674
\(668\) −1.10048e10 −1.42844
\(669\) 0 0
\(670\) 6.17867e9 0.793657
\(671\) −1.72008e10 −2.19796
\(672\) 0 0
\(673\) 1.80021e9 0.227651 0.113826 0.993501i \(-0.463690\pi\)
0.113826 + 0.993501i \(0.463690\pi\)
\(674\) 2.15370e10 2.70942
\(675\) 0 0
\(676\) −6.31949e8 −0.0786808
\(677\) −7.89862e9 −0.978343 −0.489171 0.872188i \(-0.662701\pi\)
−0.489171 + 0.872188i \(0.662701\pi\)
\(678\) 0 0
\(679\) 2.55128e9 0.312762
\(680\) 2.82396e9 0.344412
\(681\) 0 0
\(682\) 1.55203e10 1.87351
\(683\) −3.31446e9 −0.398053 −0.199026 0.979994i \(-0.563778\pi\)
−0.199026 + 0.979994i \(0.563778\pi\)
\(684\) 0 0
\(685\) −8.47303e8 −0.100721
\(686\) −5.37766e9 −0.636003
\(687\) 0 0
\(688\) 7.16605e9 0.838920
\(689\) 1.46058e10 1.70121
\(690\) 0 0
\(691\) −5.78938e9 −0.667512 −0.333756 0.942659i \(-0.608316\pi\)
−0.333756 + 0.942659i \(0.608316\pi\)
\(692\) −1.36369e10 −1.56439
\(693\) 0 0
\(694\) 1.94071e10 2.20395
\(695\) 1.12278e10 1.26867
\(696\) 0 0
\(697\) 9.24184e9 1.03382
\(698\) 1.68354e10 1.87383
\(699\) 0 0
\(700\) 3.38936e8 0.0373486
\(701\) 1.14194e10 1.25207 0.626036 0.779794i \(-0.284676\pi\)
0.626036 + 0.779794i \(0.284676\pi\)
\(702\) 0 0
\(703\) −9.87958e9 −1.07249
\(704\) 1.90132e10 2.05376
\(705\) 0 0
\(706\) −3.95307e9 −0.422783
\(707\) −3.76449e9 −0.400625
\(708\) 0 0
\(709\) −1.23857e10 −1.30515 −0.652575 0.757725i \(-0.726311\pi\)
−0.652575 + 0.757725i \(0.726311\pi\)
\(710\) 2.66948e10 2.79913
\(711\) 0 0
\(712\) 5.18534e8 0.0538390
\(713\) 1.18453e10 1.22386
\(714\) 0 0
\(715\) 1.55978e10 1.59585
\(716\) −4.04497e9 −0.411832
\(717\) 0 0
\(718\) 7.16935e9 0.722844
\(719\) 1.16266e10 1.16655 0.583274 0.812275i \(-0.301771\pi\)
0.583274 + 0.812275i \(0.301771\pi\)
\(720\) 0 0
\(721\) 3.09805e9 0.307833
\(722\) 1.37883e10 1.36342
\(723\) 0 0
\(724\) −1.93975e10 −1.89959
\(725\) 3.22182e8 0.0313992
\(726\) 0 0
\(727\) 8.72736e9 0.842389 0.421194 0.906970i \(-0.361611\pi\)
0.421194 + 0.906970i \(0.361611\pi\)
\(728\) −6.06387e8 −0.0582492
\(729\) 0 0
\(730\) −2.25639e10 −2.14676
\(731\) −1.33699e10 −1.26596
\(732\) 0 0
\(733\) 2.02816e10 1.90212 0.951062 0.309000i \(-0.0999943\pi\)
0.951062 + 0.309000i \(0.0999943\pi\)
\(734\) 2.03238e10 1.89701
\(735\) 0 0
\(736\) 2.30343e10 2.12962
\(737\) −8.43201e9 −0.775881
\(738\) 0 0
\(739\) −3.02731e9 −0.275932 −0.137966 0.990437i \(-0.544056\pi\)
−0.137966 + 0.990437i \(0.544056\pi\)
\(740\) −1.08020e10 −0.979929
\(741\) 0 0
\(742\) −6.38567e9 −0.573843
\(743\) −1.35696e10 −1.21369 −0.606843 0.794822i \(-0.707564\pi\)
−0.606843 + 0.794822i \(0.707564\pi\)
\(744\) 0 0
\(745\) −6.76029e9 −0.598988
\(746\) 1.57405e10 1.38813
\(747\) 0 0
\(748\) −2.46893e10 −2.15701
\(749\) −5.42304e8 −0.0471581
\(750\) 0 0
\(751\) −1.38238e10 −1.19094 −0.595469 0.803378i \(-0.703034\pi\)
−0.595469 + 0.803378i \(0.703034\pi\)
\(752\) 4.39779e9 0.377113
\(753\) 0 0
\(754\) −3.69273e9 −0.313724
\(755\) −6.19858e9 −0.524177
\(756\) 0 0
\(757\) −1.53342e9 −0.128477 −0.0642386 0.997935i \(-0.520462\pi\)
−0.0642386 + 0.997935i \(0.520462\pi\)
\(758\) 3.80168e9 0.317054
\(759\) 0 0
\(760\) 4.90451e9 0.405273
\(761\) 8.31351e9 0.683814 0.341907 0.939734i \(-0.388927\pi\)
0.341907 + 0.939734i \(0.388927\pi\)
\(762\) 0 0
\(763\) 8.51131e8 0.0693682
\(764\) 5.06418e9 0.410849
\(765\) 0 0
\(766\) −2.35553e10 −1.89360
\(767\) 1.57195e9 0.125792
\(768\) 0 0
\(769\) 1.21279e10 0.961710 0.480855 0.876800i \(-0.340326\pi\)
0.480855 + 0.876800i \(0.340326\pi\)
\(770\) −6.81937e9 −0.538303
\(771\) 0 0
\(772\) 1.00281e10 0.784434
\(773\) 8.47150e9 0.659678 0.329839 0.944037i \(-0.393006\pi\)
0.329839 + 0.944037i \(0.393006\pi\)
\(774\) 0 0
\(775\) −1.51973e9 −0.117276
\(776\) 5.04838e9 0.387825
\(777\) 0 0
\(778\) −2.04751e10 −1.55883
\(779\) 1.60507e10 1.21650
\(780\) 0 0
\(781\) −3.64303e10 −2.73643
\(782\) −3.47451e10 −2.59818
\(783\) 0 0
\(784\) 1.00234e10 0.742860
\(785\) −1.60625e10 −1.18514
\(786\) 0 0
\(787\) −1.35820e10 −0.993233 −0.496617 0.867970i \(-0.665424\pi\)
−0.496617 + 0.867970i \(0.665424\pi\)
\(788\) 1.13831e9 0.0828744
\(789\) 0 0
\(790\) −7.02469e9 −0.506912
\(791\) −2.73611e9 −0.196569
\(792\) 0 0
\(793\) −1.93045e10 −1.37469
\(794\) −5.66666e9 −0.401750
\(795\) 0 0
\(796\) 4.37704e9 0.307599
\(797\) 2.11682e10 1.48108 0.740542 0.672010i \(-0.234569\pi\)
0.740542 + 0.672010i \(0.234569\pi\)
\(798\) 0 0
\(799\) −8.20509e9 −0.569075
\(800\) −2.95524e9 −0.204069
\(801\) 0 0
\(802\) −2.21477e9 −0.151606
\(803\) 3.07929e10 2.09868
\(804\) 0 0
\(805\) −5.20464e9 −0.351645
\(806\) 1.74185e10 1.17176
\(807\) 0 0
\(808\) −7.44903e9 −0.496775
\(809\) 5.28973e9 0.351248 0.175624 0.984457i \(-0.443806\pi\)
0.175624 + 0.984457i \(0.443806\pi\)
\(810\) 0 0
\(811\) 2.57240e10 1.69342 0.846710 0.532054i \(-0.178580\pi\)
0.846710 + 0.532054i \(0.178580\pi\)
\(812\) 8.75570e8 0.0573911
\(813\) 0 0
\(814\) 2.71819e10 1.76642
\(815\) 7.16677e9 0.463737
\(816\) 0 0
\(817\) −2.32202e10 −1.48966
\(818\) 3.02868e10 1.93471
\(819\) 0 0
\(820\) 1.75494e10 1.11151
\(821\) 3.97061e8 0.0250413 0.0125206 0.999922i \(-0.496014\pi\)
0.0125206 + 0.999922i \(0.496014\pi\)
\(822\) 0 0
\(823\) −2.17465e10 −1.35985 −0.679923 0.733283i \(-0.737987\pi\)
−0.679923 + 0.733283i \(0.737987\pi\)
\(824\) 6.13030e9 0.381713
\(825\) 0 0
\(826\) −6.87259e8 −0.0424317
\(827\) −1.11227e10 −0.683821 −0.341910 0.939733i \(-0.611074\pi\)
−0.341910 + 0.939733i \(0.611074\pi\)
\(828\) 0 0
\(829\) −9.06605e9 −0.552684 −0.276342 0.961059i \(-0.589122\pi\)
−0.276342 + 0.961059i \(0.589122\pi\)
\(830\) −4.47731e10 −2.71797
\(831\) 0 0
\(832\) 2.13386e10 1.28450
\(833\) −1.87009e10 −1.12100
\(834\) 0 0
\(835\) −2.16807e10 −1.28875
\(836\) −4.28790e10 −2.53818
\(837\) 0 0
\(838\) 4.61402e10 2.70848
\(839\) −5.43132e9 −0.317496 −0.158748 0.987319i \(-0.550746\pi\)
−0.158748 + 0.987319i \(0.550746\pi\)
\(840\) 0 0
\(841\) −1.64176e10 −0.951751
\(842\) −1.04362e10 −0.602491
\(843\) 0 0
\(844\) −2.44988e10 −1.40264
\(845\) −1.24501e9 −0.0709865
\(846\) 0 0
\(847\) 5.40708e9 0.305753
\(848\) 2.44126e10 1.37476
\(849\) 0 0
\(850\) 4.45771e9 0.248969
\(851\) 2.07456e10 1.15391
\(852\) 0 0
\(853\) −6.82993e9 −0.376786 −0.188393 0.982094i \(-0.560328\pi\)
−0.188393 + 0.982094i \(0.560328\pi\)
\(854\) 8.43999e9 0.463702
\(855\) 0 0
\(856\) −1.07309e9 −0.0584760
\(857\) −3.60369e10 −1.95576 −0.977878 0.209178i \(-0.932921\pi\)
−0.977878 + 0.209178i \(0.932921\pi\)
\(858\) 0 0
\(859\) 1.90837e10 1.02727 0.513637 0.858008i \(-0.328298\pi\)
0.513637 + 0.858008i \(0.328298\pi\)
\(860\) −2.53882e10 −1.36109
\(861\) 0 0
\(862\) −3.54357e10 −1.88437
\(863\) 3.33278e10 1.76510 0.882550 0.470218i \(-0.155825\pi\)
0.882550 + 0.470218i \(0.155825\pi\)
\(864\) 0 0
\(865\) −2.68663e10 −1.41140
\(866\) −1.87656e10 −0.981861
\(867\) 0 0
\(868\) −4.13004e9 −0.214356
\(869\) 9.58657e9 0.495558
\(870\) 0 0
\(871\) −9.46328e9 −0.485264
\(872\) 1.68419e9 0.0860166
\(873\) 0 0
\(874\) −6.03434e10 −3.05731
\(875\) −4.00353e9 −0.202030
\(876\) 0 0
\(877\) 2.92873e10 1.46616 0.733079 0.680143i \(-0.238082\pi\)
0.733079 + 0.680143i \(0.238082\pi\)
\(878\) −2.00378e10 −0.999124
\(879\) 0 0
\(880\) 2.60706e10 1.28962
\(881\) 7.06390e9 0.348040 0.174020 0.984742i \(-0.444324\pi\)
0.174020 + 0.984742i \(0.444324\pi\)
\(882\) 0 0
\(883\) −2.11700e10 −1.03480 −0.517401 0.855743i \(-0.673100\pi\)
−0.517401 + 0.855743i \(0.673100\pi\)
\(884\) −2.77089e10 −1.34907
\(885\) 0 0
\(886\) 9.43890e9 0.455935
\(887\) −3.92843e9 −0.189011 −0.0945053 0.995524i \(-0.530127\pi\)
−0.0945053 + 0.995524i \(0.530127\pi\)
\(888\) 0 0
\(889\) 6.26920e9 0.299265
\(890\) 6.54458e9 0.311184
\(891\) 0 0
\(892\) 2.08208e10 0.982248
\(893\) −1.42502e10 −0.669637
\(894\) 0 0
\(895\) −7.96906e9 −0.371558
\(896\) −2.55167e9 −0.118508
\(897\) 0 0
\(898\) 2.48132e10 1.14345
\(899\) −3.92590e9 −0.180211
\(900\) 0 0
\(901\) −4.55473e10 −2.07456
\(902\) −4.41608e10 −2.00361
\(903\) 0 0
\(904\) −5.41411e9 −0.243746
\(905\) −3.82153e10 −1.71383
\(906\) 0 0
\(907\) 1.73810e10 0.773482 0.386741 0.922188i \(-0.373601\pi\)
0.386741 + 0.922188i \(0.373601\pi\)
\(908\) 7.63824e9 0.338605
\(909\) 0 0
\(910\) −7.65341e9 −0.336674
\(911\) −3.94515e9 −0.172882 −0.0864409 0.996257i \(-0.527549\pi\)
−0.0864409 + 0.996257i \(0.527549\pi\)
\(912\) 0 0
\(913\) 6.11017e10 2.65709
\(914\) −2.54040e10 −1.10050
\(915\) 0 0
\(916\) 4.99975e10 2.14938
\(917\) 5.32983e8 0.0228255
\(918\) 0 0
\(919\) −1.68687e10 −0.716930 −0.358465 0.933543i \(-0.616700\pi\)
−0.358465 + 0.933543i \(0.616700\pi\)
\(920\) −1.02987e10 −0.436040
\(921\) 0 0
\(922\) −1.34223e10 −0.563987
\(923\) −4.08859e10 −1.71146
\(924\) 0 0
\(925\) −2.66162e9 −0.110573
\(926\) 1.58399e10 0.655564
\(927\) 0 0
\(928\) −7.63425e9 −0.313580
\(929\) −5.86634e9 −0.240056 −0.120028 0.992771i \(-0.538298\pi\)
−0.120028 + 0.992771i \(0.538298\pi\)
\(930\) 0 0
\(931\) −3.24787e10 −1.31909
\(932\) −6.60486e9 −0.267244
\(933\) 0 0
\(934\) −2.64983e10 −1.06415
\(935\) −4.86407e10 −1.94607
\(936\) 0 0
\(937\) 2.42022e10 0.961093 0.480546 0.876969i \(-0.340438\pi\)
0.480546 + 0.876969i \(0.340438\pi\)
\(938\) 4.13737e9 0.163687
\(939\) 0 0
\(940\) −1.55807e10 −0.611842
\(941\) −2.22320e10 −0.869792 −0.434896 0.900481i \(-0.643215\pi\)
−0.434896 + 0.900481i \(0.643215\pi\)
\(942\) 0 0
\(943\) −3.37041e10 −1.30886
\(944\) 2.62741e9 0.101654
\(945\) 0 0
\(946\) 6.38863e10 2.45351
\(947\) −2.37539e9 −0.0908889 −0.0454444 0.998967i \(-0.514470\pi\)
−0.0454444 + 0.998967i \(0.514470\pi\)
\(948\) 0 0
\(949\) 3.45590e10 1.31259
\(950\) 7.74191e9 0.292965
\(951\) 0 0
\(952\) 1.89099e9 0.0710328
\(953\) 2.11402e10 0.791197 0.395599 0.918423i \(-0.370537\pi\)
0.395599 + 0.918423i \(0.370537\pi\)
\(954\) 0 0
\(955\) 9.97702e9 0.370671
\(956\) 1.77302e10 0.656312
\(957\) 0 0
\(958\) 4.76649e7 0.00175154
\(959\) −5.67372e8 −0.0207732
\(960\) 0 0
\(961\) −8.99423e9 −0.326913
\(962\) 3.05064e10 1.10479
\(963\) 0 0
\(964\) −5.70996e10 −2.05288
\(965\) 1.97564e10 0.707723
\(966\) 0 0
\(967\) −9.04540e9 −0.321688 −0.160844 0.986980i \(-0.551422\pi\)
−0.160844 + 0.986980i \(0.551422\pi\)
\(968\) 1.06993e10 0.379134
\(969\) 0 0
\(970\) 6.37173e10 2.24159
\(971\) 4.75592e9 0.166712 0.0833560 0.996520i \(-0.473436\pi\)
0.0833560 + 0.996520i \(0.473436\pi\)
\(972\) 0 0
\(973\) 7.51839e9 0.261656
\(974\) 2.14207e10 0.742808
\(975\) 0 0
\(976\) −3.22663e10 −1.11090
\(977\) −1.97475e9 −0.0677456 −0.0338728 0.999426i \(-0.510784\pi\)
−0.0338728 + 0.999426i \(0.510784\pi\)
\(978\) 0 0
\(979\) −8.93137e9 −0.304214
\(980\) −3.55112e10 −1.20524
\(981\) 0 0
\(982\) 2.29922e9 0.0774801
\(983\) −1.84818e10 −0.620592 −0.310296 0.950640i \(-0.600428\pi\)
−0.310296 + 0.950640i \(0.600428\pi\)
\(984\) 0 0
\(985\) 2.24261e9 0.0747699
\(986\) 1.15156e10 0.382575
\(987\) 0 0
\(988\) −4.81233e10 −1.58747
\(989\) 4.87589e10 1.60275
\(990\) 0 0
\(991\) 1.09246e9 0.0356573 0.0178286 0.999841i \(-0.494325\pi\)
0.0178286 + 0.999841i \(0.494325\pi\)
\(992\) 3.60106e10 1.17122
\(993\) 0 0
\(994\) 1.78754e10 0.577303
\(995\) 8.62327e9 0.277518
\(996\) 0 0
\(997\) −7.48711e9 −0.239266 −0.119633 0.992818i \(-0.538172\pi\)
−0.119633 + 0.992818i \(0.538172\pi\)
\(998\) 3.29714e9 0.104998
\(999\) 0 0
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 531.8.a.e.1.16 18
3.2 odd 2 177.8.a.d.1.3 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.3 18 3.2 odd 2
531.8.a.e.1.16 18 1.1 even 1 trivial