Properties

Label 531.8.a.b.1.5
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-13.0039\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-13.0039 q^{2} +41.1009 q^{4} -167.303 q^{5} +887.373 q^{7} +1130.03 q^{8} +O(q^{10})\) \(q-13.0039 q^{2} +41.1009 q^{4} -167.303 q^{5} +887.373 q^{7} +1130.03 q^{8} +2175.59 q^{10} +7784.37 q^{11} +4609.99 q^{13} -11539.3 q^{14} -19955.6 q^{16} +6055.74 q^{17} +13552.7 q^{19} -6876.32 q^{20} -101227. q^{22} -35274.4 q^{23} -50134.6 q^{25} -59947.7 q^{26} +36471.8 q^{28} -78335.1 q^{29} -170575. q^{31} +114857. q^{32} -78748.2 q^{34} -148460. q^{35} -263573. q^{37} -176238. q^{38} -189057. q^{40} +198905. q^{41} -444435. q^{43} +319945. q^{44} +458704. q^{46} +531622. q^{47} -36112.2 q^{49} +651945. q^{50} +189475. q^{52} +1.00107e6 q^{53} -1.30235e6 q^{55} +1.00275e6 q^{56} +1.01866e6 q^{58} -205379. q^{59} -1.77312e6 q^{61} +2.21813e6 q^{62} +1.06073e6 q^{64} -771266. q^{65} -2.37225e6 q^{67} +248897. q^{68} +1.93056e6 q^{70} +5.87275e6 q^{71} -2.90906e6 q^{73} +3.42748e6 q^{74} +557030. q^{76} +6.90764e6 q^{77} -1.66441e6 q^{79} +3.33864e6 q^{80} -2.58653e6 q^{82} -8.18425e6 q^{83} -1.01315e6 q^{85} +5.77938e6 q^{86} +8.79653e6 q^{88} -1.13356e7 q^{89} +4.09078e6 q^{91} -1.44981e6 q^{92} -6.91315e6 q^{94} -2.26742e6 q^{95} +243947. q^{97} +469599. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −13.0039 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(3\) 0 0
\(4\) 41.1009 0.321101
\(5\) −167.303 −0.598562 −0.299281 0.954165i \(-0.596747\pi\)
−0.299281 + 0.954165i \(0.596747\pi\)
\(6\) 0 0
\(7\) 887.373 0.977829 0.488915 0.872332i \(-0.337393\pi\)
0.488915 + 0.872332i \(0.337393\pi\)
\(8\) 1130.03 0.780321
\(9\) 0 0
\(10\) 2175.59 0.687982
\(11\) 7784.37 1.76339 0.881696 0.471818i \(-0.156402\pi\)
0.881696 + 0.471818i \(0.156402\pi\)
\(12\) 0 0
\(13\) 4609.99 0.581967 0.290983 0.956728i \(-0.406018\pi\)
0.290983 + 0.956728i \(0.406018\pi\)
\(14\) −11539.3 −1.12391
\(15\) 0 0
\(16\) −19955.6 −1.21800
\(17\) 6055.74 0.298948 0.149474 0.988766i \(-0.452242\pi\)
0.149474 + 0.988766i \(0.452242\pi\)
\(18\) 0 0
\(19\) 13552.7 0.453304 0.226652 0.973976i \(-0.427222\pi\)
0.226652 + 0.973976i \(0.427222\pi\)
\(20\) −6876.32 −0.192199
\(21\) 0 0
\(22\) −101227. −2.02683
\(23\) −35274.4 −0.604523 −0.302261 0.953225i \(-0.597742\pi\)
−0.302261 + 0.953225i \(0.597742\pi\)
\(24\) 0 0
\(25\) −50134.6 −0.641723
\(26\) −59947.7 −0.668907
\(27\) 0 0
\(28\) 36471.8 0.313982
\(29\) −78335.1 −0.596435 −0.298218 0.954498i \(-0.596392\pi\)
−0.298218 + 0.954498i \(0.596392\pi\)
\(30\) 0 0
\(31\) −170575. −1.02837 −0.514184 0.857680i \(-0.671905\pi\)
−0.514184 + 0.857680i \(0.671905\pi\)
\(32\) 114857. 0.619632
\(33\) 0 0
\(34\) −78748.2 −0.343609
\(35\) −148460. −0.585292
\(36\) 0 0
\(37\) −263573. −0.855451 −0.427726 0.903909i \(-0.640685\pi\)
−0.427726 + 0.903909i \(0.640685\pi\)
\(38\) −176238. −0.521024
\(39\) 0 0
\(40\) −189057. −0.467071
\(41\) 198905. 0.450714 0.225357 0.974276i \(-0.427645\pi\)
0.225357 + 0.974276i \(0.427645\pi\)
\(42\) 0 0
\(43\) −444435. −0.852449 −0.426225 0.904617i \(-0.640157\pi\)
−0.426225 + 0.904617i \(0.640157\pi\)
\(44\) 319945. 0.566227
\(45\) 0 0
\(46\) 458704. 0.694833
\(47\) 531622. 0.746896 0.373448 0.927651i \(-0.378175\pi\)
0.373448 + 0.927651i \(0.378175\pi\)
\(48\) 0 0
\(49\) −36112.2 −0.0438499
\(50\) 651945. 0.737591
\(51\) 0 0
\(52\) 189475. 0.186870
\(53\) 1.00107e6 0.923631 0.461816 0.886976i \(-0.347198\pi\)
0.461816 + 0.886976i \(0.347198\pi\)
\(54\) 0 0
\(55\) −1.30235e6 −1.05550
\(56\) 1.00275e6 0.763021
\(57\) 0 0
\(58\) 1.01866e6 0.685538
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) −1.77312e6 −1.00019 −0.500097 0.865969i \(-0.666702\pi\)
−0.500097 + 0.865969i \(0.666702\pi\)
\(62\) 2.21813e6 1.18200
\(63\) 0 0
\(64\) 1.06073e6 0.505795
\(65\) −771266. −0.348343
\(66\) 0 0
\(67\) −2.37225e6 −0.963606 −0.481803 0.876280i \(-0.660018\pi\)
−0.481803 + 0.876280i \(0.660018\pi\)
\(68\) 248897. 0.0959926
\(69\) 0 0
\(70\) 1.93056e6 0.672729
\(71\) 5.87275e6 1.94732 0.973659 0.228008i \(-0.0732212\pi\)
0.973659 + 0.228008i \(0.0732212\pi\)
\(72\) 0 0
\(73\) −2.90906e6 −0.875232 −0.437616 0.899162i \(-0.644177\pi\)
−0.437616 + 0.899162i \(0.644177\pi\)
\(74\) 3.42748e6 0.983249
\(75\) 0 0
\(76\) 557030. 0.145556
\(77\) 6.90764e6 1.72430
\(78\) 0 0
\(79\) −1.66441e6 −0.379808 −0.189904 0.981803i \(-0.560818\pi\)
−0.189904 + 0.981803i \(0.560818\pi\)
\(80\) 3.33864e6 0.729046
\(81\) 0 0
\(82\) −2.58653e6 −0.518047
\(83\) −8.18425e6 −1.57111 −0.785553 0.618794i \(-0.787621\pi\)
−0.785553 + 0.618794i \(0.787621\pi\)
\(84\) 0 0
\(85\) −1.01315e6 −0.178939
\(86\) 5.77938e6 0.979798
\(87\) 0 0
\(88\) 8.79653e6 1.37601
\(89\) −1.13356e7 −1.70444 −0.852219 0.523185i \(-0.824744\pi\)
−0.852219 + 0.523185i \(0.824744\pi\)
\(90\) 0 0
\(91\) 4.09078e6 0.569064
\(92\) −1.44981e6 −0.194113
\(93\) 0 0
\(94\) −6.91315e6 −0.858476
\(95\) −2.26742e6 −0.271331
\(96\) 0 0
\(97\) 243947. 0.0271390 0.0135695 0.999908i \(-0.495681\pi\)
0.0135695 + 0.999908i \(0.495681\pi\)
\(98\) 469599. 0.0504006
\(99\) 0 0
\(100\) −2.06058e6 −0.206058
\(101\) −1.91745e7 −1.85183 −0.925913 0.377736i \(-0.876703\pi\)
−0.925913 + 0.377736i \(0.876703\pi\)
\(102\) 0 0
\(103\) 8.63345e6 0.778492 0.389246 0.921134i \(-0.372736\pi\)
0.389246 + 0.921134i \(0.372736\pi\)
\(104\) 5.20940e6 0.454121
\(105\) 0 0
\(106\) −1.30178e7 −1.06161
\(107\) 1.22715e7 0.968397 0.484199 0.874958i \(-0.339111\pi\)
0.484199 + 0.874958i \(0.339111\pi\)
\(108\) 0 0
\(109\) −6.90858e6 −0.510971 −0.255485 0.966813i \(-0.582235\pi\)
−0.255485 + 0.966813i \(0.582235\pi\)
\(110\) 1.69356e7 1.21318
\(111\) 0 0
\(112\) −1.77081e7 −1.19099
\(113\) −9.23590e6 −0.602150 −0.301075 0.953601i \(-0.597345\pi\)
−0.301075 + 0.953601i \(0.597345\pi\)
\(114\) 0 0
\(115\) 5.90153e6 0.361844
\(116\) −3.21964e6 −0.191516
\(117\) 0 0
\(118\) 2.67072e6 0.149638
\(119\) 5.37370e6 0.292320
\(120\) 0 0
\(121\) 4.11092e7 2.10955
\(122\) 2.30575e7 1.14961
\(123\) 0 0
\(124\) −7.01078e6 −0.330210
\(125\) 2.14583e7 0.982674
\(126\) 0 0
\(127\) 3.43987e6 0.149015 0.0745073 0.997220i \(-0.476262\pi\)
0.0745073 + 0.997220i \(0.476262\pi\)
\(128\) −2.84953e7 −1.20099
\(129\) 0 0
\(130\) 1.00295e7 0.400383
\(131\) 2.36186e6 0.0917919 0.0458960 0.998946i \(-0.485386\pi\)
0.0458960 + 0.998946i \(0.485386\pi\)
\(132\) 0 0
\(133\) 1.20263e7 0.443254
\(134\) 3.08485e7 1.10756
\(135\) 0 0
\(136\) 6.84314e6 0.233276
\(137\) −3.14805e6 −0.104597 −0.0522985 0.998631i \(-0.516655\pi\)
−0.0522985 + 0.998631i \(0.516655\pi\)
\(138\) 0 0
\(139\) 1.60628e7 0.507305 0.253652 0.967295i \(-0.418368\pi\)
0.253652 + 0.967295i \(0.418368\pi\)
\(140\) −6.10186e6 −0.187938
\(141\) 0 0
\(142\) −7.63685e7 −2.23823
\(143\) 3.58858e7 1.02624
\(144\) 0 0
\(145\) 1.31057e7 0.357004
\(146\) 3.78291e7 1.00598
\(147\) 0 0
\(148\) −1.08331e7 −0.274686
\(149\) 3.30906e7 0.819506 0.409753 0.912196i \(-0.365615\pi\)
0.409753 + 0.912196i \(0.365615\pi\)
\(150\) 0 0
\(151\) −8.23311e7 −1.94601 −0.973003 0.230791i \(-0.925869\pi\)
−0.973003 + 0.230791i \(0.925869\pi\)
\(152\) 1.53149e7 0.353722
\(153\) 0 0
\(154\) −8.98261e7 −1.98189
\(155\) 2.85377e7 0.615543
\(156\) 0 0
\(157\) 7.06660e7 1.45734 0.728671 0.684863i \(-0.240138\pi\)
0.728671 + 0.684863i \(0.240138\pi\)
\(158\) 2.16437e7 0.436548
\(159\) 0 0
\(160\) −1.92160e7 −0.370889
\(161\) −3.13016e7 −0.591120
\(162\) 0 0
\(163\) −5.37840e7 −0.972740 −0.486370 0.873753i \(-0.661679\pi\)
−0.486370 + 0.873753i \(0.661679\pi\)
\(164\) 8.17516e6 0.144725
\(165\) 0 0
\(166\) 1.06427e8 1.80582
\(167\) −4.12989e7 −0.686169 −0.343084 0.939305i \(-0.611472\pi\)
−0.343084 + 0.939305i \(0.611472\pi\)
\(168\) 0 0
\(169\) −4.14965e7 −0.661315
\(170\) 1.31748e7 0.205671
\(171\) 0 0
\(172\) −1.82667e7 −0.273722
\(173\) −5.36462e7 −0.787731 −0.393865 0.919168i \(-0.628862\pi\)
−0.393865 + 0.919168i \(0.628862\pi\)
\(174\) 0 0
\(175\) −4.44881e7 −0.627496
\(176\) −1.55342e8 −2.14780
\(177\) 0 0
\(178\) 1.47407e8 1.95907
\(179\) 3.62629e7 0.472581 0.236291 0.971682i \(-0.424068\pi\)
0.236291 + 0.971682i \(0.424068\pi\)
\(180\) 0 0
\(181\) −1.64625e7 −0.206358 −0.103179 0.994663i \(-0.532901\pi\)
−0.103179 + 0.994663i \(0.532901\pi\)
\(182\) −5.31960e7 −0.654077
\(183\) 0 0
\(184\) −3.98610e7 −0.471722
\(185\) 4.40967e7 0.512041
\(186\) 0 0
\(187\) 4.71401e7 0.527163
\(188\) 2.18501e7 0.239829
\(189\) 0 0
\(190\) 2.94852e7 0.311865
\(191\) 7.01561e7 0.728532 0.364266 0.931295i \(-0.381320\pi\)
0.364266 + 0.931295i \(0.381320\pi\)
\(192\) 0 0
\(193\) 9.06871e7 0.908019 0.454010 0.890997i \(-0.349993\pi\)
0.454010 + 0.890997i \(0.349993\pi\)
\(194\) −3.17225e6 −0.0311933
\(195\) 0 0
\(196\) −1.48425e6 −0.0140802
\(197\) 3.02563e7 0.281958 0.140979 0.990013i \(-0.454975\pi\)
0.140979 + 0.990013i \(0.454975\pi\)
\(198\) 0 0
\(199\) 1.78612e8 1.60666 0.803332 0.595532i \(-0.203059\pi\)
0.803332 + 0.595532i \(0.203059\pi\)
\(200\) −5.66534e7 −0.500750
\(201\) 0 0
\(202\) 2.49343e8 2.12847
\(203\) −6.95125e7 −0.583212
\(204\) 0 0
\(205\) −3.32774e7 −0.269781
\(206\) −1.12268e8 −0.894792
\(207\) 0 0
\(208\) −9.19952e7 −0.708832
\(209\) 1.05499e8 0.799352
\(210\) 0 0
\(211\) −1.01493e8 −0.743786 −0.371893 0.928276i \(-0.621291\pi\)
−0.371893 + 0.928276i \(0.621291\pi\)
\(212\) 4.11449e7 0.296579
\(213\) 0 0
\(214\) −1.59577e8 −1.11307
\(215\) 7.43554e7 0.510244
\(216\) 0 0
\(217\) −1.51363e8 −1.00557
\(218\) 8.98383e7 0.587305
\(219\) 0 0
\(220\) −5.35278e7 −0.338922
\(221\) 2.79169e7 0.173978
\(222\) 0 0
\(223\) 1.17677e8 0.710600 0.355300 0.934752i \(-0.384379\pi\)
0.355300 + 0.934752i \(0.384379\pi\)
\(224\) 1.01921e8 0.605895
\(225\) 0 0
\(226\) 1.20102e8 0.692106
\(227\) −2.17121e8 −1.23200 −0.616000 0.787746i \(-0.711248\pi\)
−0.616000 + 0.787746i \(0.711248\pi\)
\(228\) 0 0
\(229\) 6.89428e7 0.379372 0.189686 0.981845i \(-0.439253\pi\)
0.189686 + 0.981845i \(0.439253\pi\)
\(230\) −7.67427e7 −0.415901
\(231\) 0 0
\(232\) −8.85207e7 −0.465411
\(233\) −2.42127e8 −1.25400 −0.627000 0.779019i \(-0.715718\pi\)
−0.627000 + 0.779019i \(0.715718\pi\)
\(234\) 0 0
\(235\) −8.89421e7 −0.447064
\(236\) −8.44126e6 −0.0418038
\(237\) 0 0
\(238\) −6.98790e7 −0.335991
\(239\) −1.66165e8 −0.787312 −0.393656 0.919258i \(-0.628790\pi\)
−0.393656 + 0.919258i \(0.628790\pi\)
\(240\) 0 0
\(241\) 2.91718e7 0.134247 0.0671233 0.997745i \(-0.478618\pi\)
0.0671233 + 0.997745i \(0.478618\pi\)
\(242\) −5.34579e8 −2.42470
\(243\) 0 0
\(244\) −7.28769e7 −0.321163
\(245\) 6.04170e6 0.0262469
\(246\) 0 0
\(247\) 6.24779e7 0.263808
\(248\) −1.92754e8 −0.802458
\(249\) 0 0
\(250\) −2.79041e8 −1.12948
\(251\) −2.96094e8 −1.18188 −0.590938 0.806717i \(-0.701242\pi\)
−0.590938 + 0.806717i \(0.701242\pi\)
\(252\) 0 0
\(253\) −2.74589e8 −1.06601
\(254\) −4.47317e7 −0.171276
\(255\) 0 0
\(256\) 2.34777e8 0.874611
\(257\) −2.67505e8 −0.983027 −0.491514 0.870870i \(-0.663556\pi\)
−0.491514 + 0.870870i \(0.663556\pi\)
\(258\) 0 0
\(259\) −2.33888e8 −0.836485
\(260\) −3.16997e7 −0.111853
\(261\) 0 0
\(262\) −3.07133e7 −0.105505
\(263\) −2.10348e8 −0.713006 −0.356503 0.934294i \(-0.616031\pi\)
−0.356503 + 0.934294i \(0.616031\pi\)
\(264\) 0 0
\(265\) −1.67482e8 −0.552851
\(266\) −1.56389e8 −0.509472
\(267\) 0 0
\(268\) −9.75018e7 −0.309415
\(269\) 4.76854e8 1.49366 0.746831 0.665014i \(-0.231574\pi\)
0.746831 + 0.665014i \(0.231574\pi\)
\(270\) 0 0
\(271\) −2.49889e8 −0.762701 −0.381350 0.924431i \(-0.624541\pi\)
−0.381350 + 0.924431i \(0.624541\pi\)
\(272\) −1.20846e8 −0.364118
\(273\) 0 0
\(274\) 4.09368e7 0.120223
\(275\) −3.90266e8 −1.13161
\(276\) 0 0
\(277\) −2.02351e8 −0.572039 −0.286020 0.958224i \(-0.592332\pi\)
−0.286020 + 0.958224i \(0.592332\pi\)
\(278\) −2.08879e8 −0.583092
\(279\) 0 0
\(280\) −1.67764e8 −0.456715
\(281\) 2.77424e8 0.745885 0.372942 0.927855i \(-0.378349\pi\)
0.372942 + 0.927855i \(0.378349\pi\)
\(282\) 0 0
\(283\) −3.44360e8 −0.903151 −0.451575 0.892233i \(-0.649138\pi\)
−0.451575 + 0.892233i \(0.649138\pi\)
\(284\) 2.41375e8 0.625286
\(285\) 0 0
\(286\) −4.66655e8 −1.17955
\(287\) 1.76503e8 0.440722
\(288\) 0 0
\(289\) −3.73667e8 −0.910630
\(290\) −1.70425e8 −0.410337
\(291\) 0 0
\(292\) −1.19565e8 −0.281038
\(293\) 3.56077e8 0.827004 0.413502 0.910503i \(-0.364306\pi\)
0.413502 + 0.910503i \(0.364306\pi\)
\(294\) 0 0
\(295\) 3.43606e7 0.0779262
\(296\) −2.97845e8 −0.667527
\(297\) 0 0
\(298\) −4.30306e8 −0.941934
\(299\) −1.62615e8 −0.351812
\(300\) 0 0
\(301\) −3.94379e8 −0.833550
\(302\) 1.07062e9 2.23672
\(303\) 0 0
\(304\) −2.70453e8 −0.552122
\(305\) 2.96649e8 0.598679
\(306\) 0 0
\(307\) −1.96195e8 −0.386993 −0.193496 0.981101i \(-0.561983\pi\)
−0.193496 + 0.981101i \(0.561983\pi\)
\(308\) 2.83910e8 0.553673
\(309\) 0 0
\(310\) −3.71101e8 −0.707500
\(311\) −5.23320e7 −0.0986519 −0.0493260 0.998783i \(-0.515707\pi\)
−0.0493260 + 0.998783i \(0.515707\pi\)
\(312\) 0 0
\(313\) −3.12015e8 −0.575136 −0.287568 0.957760i \(-0.592847\pi\)
−0.287568 + 0.957760i \(0.592847\pi\)
\(314\) −9.18932e8 −1.67506
\(315\) 0 0
\(316\) −6.84086e7 −0.121957
\(317\) −5.38220e8 −0.948970 −0.474485 0.880264i \(-0.657366\pi\)
−0.474485 + 0.880264i \(0.657366\pi\)
\(318\) 0 0
\(319\) −6.09789e8 −1.05175
\(320\) −1.77463e8 −0.302750
\(321\) 0 0
\(322\) 4.07042e8 0.679428
\(323\) 8.20719e7 0.135514
\(324\) 0 0
\(325\) −2.31120e8 −0.373461
\(326\) 6.99401e8 1.11806
\(327\) 0 0
\(328\) 2.24767e8 0.351702
\(329\) 4.71747e8 0.730337
\(330\) 0 0
\(331\) −7.08848e8 −1.07437 −0.537187 0.843463i \(-0.680513\pi\)
−0.537187 + 0.843463i \(0.680513\pi\)
\(332\) −3.36380e8 −0.504483
\(333\) 0 0
\(334\) 5.37046e8 0.788676
\(335\) 3.96886e8 0.576778
\(336\) 0 0
\(337\) 1.28631e8 0.183079 0.0915397 0.995801i \(-0.470821\pi\)
0.0915397 + 0.995801i \(0.470821\pi\)
\(338\) 5.39616e8 0.760110
\(339\) 0 0
\(340\) −4.16412e7 −0.0574575
\(341\) −1.32782e9 −1.81342
\(342\) 0 0
\(343\) −7.62835e8 −1.02071
\(344\) −5.02223e8 −0.665184
\(345\) 0 0
\(346\) 6.97609e8 0.905411
\(347\) 6.00582e7 0.0771648 0.0385824 0.999255i \(-0.487716\pi\)
0.0385824 + 0.999255i \(0.487716\pi\)
\(348\) 0 0
\(349\) 1.11674e9 1.40625 0.703124 0.711067i \(-0.251788\pi\)
0.703124 + 0.711067i \(0.251788\pi\)
\(350\) 5.78518e8 0.721238
\(351\) 0 0
\(352\) 8.94092e8 1.09265
\(353\) −3.99147e8 −0.482971 −0.241486 0.970404i \(-0.577635\pi\)
−0.241486 + 0.970404i \(0.577635\pi\)
\(354\) 0 0
\(355\) −9.82530e8 −1.16559
\(356\) −4.65905e8 −0.547296
\(357\) 0 0
\(358\) −4.71558e8 −0.543181
\(359\) 6.35935e8 0.725408 0.362704 0.931904i \(-0.381854\pi\)
0.362704 + 0.931904i \(0.381854\pi\)
\(360\) 0 0
\(361\) −7.10195e8 −0.794516
\(362\) 2.14077e8 0.237186
\(363\) 0 0
\(364\) 1.68135e8 0.182727
\(365\) 4.86696e8 0.523881
\(366\) 0 0
\(367\) −2.61675e8 −0.276332 −0.138166 0.990409i \(-0.544121\pi\)
−0.138166 + 0.990409i \(0.544121\pi\)
\(368\) 7.03923e8 0.736305
\(369\) 0 0
\(370\) −5.73428e8 −0.588536
\(371\) 8.88322e8 0.903154
\(372\) 0 0
\(373\) 1.69724e9 1.69341 0.846706 0.532060i \(-0.178582\pi\)
0.846706 + 0.532060i \(0.178582\pi\)
\(374\) −6.13004e8 −0.605917
\(375\) 0 0
\(376\) 6.00746e8 0.582819
\(377\) −3.61124e8 −0.347106
\(378\) 0 0
\(379\) 1.10612e9 1.04367 0.521837 0.853045i \(-0.325247\pi\)
0.521837 + 0.853045i \(0.325247\pi\)
\(380\) −9.31929e7 −0.0871245
\(381\) 0 0
\(382\) −9.12301e8 −0.837368
\(383\) 6.71833e8 0.611034 0.305517 0.952187i \(-0.401171\pi\)
0.305517 + 0.952187i \(0.401171\pi\)
\(384\) 0 0
\(385\) −1.15567e9 −1.03210
\(386\) −1.17928e9 −1.04367
\(387\) 0 0
\(388\) 1.00264e7 0.00871435
\(389\) −1.16525e9 −1.00368 −0.501839 0.864961i \(-0.667343\pi\)
−0.501839 + 0.864961i \(0.667343\pi\)
\(390\) 0 0
\(391\) −2.13613e8 −0.180721
\(392\) −4.08077e7 −0.0342170
\(393\) 0 0
\(394\) −3.93450e8 −0.324080
\(395\) 2.78460e8 0.227339
\(396\) 0 0
\(397\) 6.93011e8 0.555870 0.277935 0.960600i \(-0.410350\pi\)
0.277935 + 0.960600i \(0.410350\pi\)
\(398\) −2.32265e9 −1.84669
\(399\) 0 0
\(400\) 1.00047e9 0.781616
\(401\) 2.25957e9 1.74993 0.874965 0.484186i \(-0.160884\pi\)
0.874965 + 0.484186i \(0.160884\pi\)
\(402\) 0 0
\(403\) −7.86348e8 −0.598476
\(404\) −7.88091e8 −0.594623
\(405\) 0 0
\(406\) 9.03932e8 0.670339
\(407\) −2.05175e9 −1.50850
\(408\) 0 0
\(409\) 8.72327e7 0.0630446 0.0315223 0.999503i \(-0.489964\pi\)
0.0315223 + 0.999503i \(0.489964\pi\)
\(410\) 4.32735e8 0.310084
\(411\) 0 0
\(412\) 3.54843e8 0.249974
\(413\) −1.82248e8 −0.127303
\(414\) 0 0
\(415\) 1.36925e9 0.940405
\(416\) 5.29491e8 0.360605
\(417\) 0 0
\(418\) −1.37190e9 −0.918769
\(419\) 8.24265e8 0.547416 0.273708 0.961813i \(-0.411750\pi\)
0.273708 + 0.961813i \(0.411750\pi\)
\(420\) 0 0
\(421\) −1.82215e9 −1.19014 −0.595068 0.803675i \(-0.702875\pi\)
−0.595068 + 0.803675i \(0.702875\pi\)
\(422\) 1.31980e9 0.854901
\(423\) 0 0
\(424\) 1.13123e9 0.720729
\(425\) −3.03602e8 −0.191842
\(426\) 0 0
\(427\) −1.57342e9 −0.978019
\(428\) 5.04369e8 0.310953
\(429\) 0 0
\(430\) −9.66909e8 −0.586470
\(431\) −3.24182e9 −1.95038 −0.975189 0.221375i \(-0.928946\pi\)
−0.975189 + 0.221375i \(0.928946\pi\)
\(432\) 0 0
\(433\) 1.71326e9 1.01418 0.507090 0.861893i \(-0.330721\pi\)
0.507090 + 0.861893i \(0.330721\pi\)
\(434\) 1.96831e9 1.15579
\(435\) 0 0
\(436\) −2.83949e8 −0.164073
\(437\) −4.78065e8 −0.274032
\(438\) 0 0
\(439\) −1.70403e9 −0.961283 −0.480642 0.876917i \(-0.659596\pi\)
−0.480642 + 0.876917i \(0.659596\pi\)
\(440\) −1.47169e9 −0.823629
\(441\) 0 0
\(442\) −3.63028e8 −0.199969
\(443\) −8.96154e7 −0.0489745 −0.0244872 0.999700i \(-0.507795\pi\)
−0.0244872 + 0.999700i \(0.507795\pi\)
\(444\) 0 0
\(445\) 1.89649e9 1.02021
\(446\) −1.53026e9 −0.816757
\(447\) 0 0
\(448\) 9.41262e8 0.494581
\(449\) 2.38975e9 1.24592 0.622959 0.782255i \(-0.285930\pi\)
0.622959 + 0.782255i \(0.285930\pi\)
\(450\) 0 0
\(451\) 1.54835e9 0.794786
\(452\) −3.79604e8 −0.193351
\(453\) 0 0
\(454\) 2.82341e9 1.41605
\(455\) −6.84401e8 −0.340620
\(456\) 0 0
\(457\) 3.10251e9 1.52057 0.760286 0.649588i \(-0.225059\pi\)
0.760286 + 0.649588i \(0.225059\pi\)
\(458\) −8.96525e8 −0.436047
\(459\) 0 0
\(460\) 2.42558e8 0.116189
\(461\) 9.42015e8 0.447821 0.223911 0.974610i \(-0.428118\pi\)
0.223911 + 0.974610i \(0.428118\pi\)
\(462\) 0 0
\(463\) −2.76277e9 −1.29363 −0.646817 0.762645i \(-0.723900\pi\)
−0.646817 + 0.762645i \(0.723900\pi\)
\(464\) 1.56323e9 0.726455
\(465\) 0 0
\(466\) 3.14859e9 1.44134
\(467\) −602426. −0.000273712 0 −0.000136856 1.00000i \(-0.500044\pi\)
−0.000136856 1.00000i \(0.500044\pi\)
\(468\) 0 0
\(469\) −2.10507e9 −0.942242
\(470\) 1.15659e9 0.513852
\(471\) 0 0
\(472\) −2.32083e8 −0.101589
\(473\) −3.45964e9 −1.50320
\(474\) 0 0
\(475\) −6.79461e8 −0.290896
\(476\) 2.20864e8 0.0938643
\(477\) 0 0
\(478\) 2.16079e9 0.904930
\(479\) 2.45402e9 1.02024 0.510121 0.860103i \(-0.329601\pi\)
0.510121 + 0.860103i \(0.329601\pi\)
\(480\) 0 0
\(481\) −1.21507e9 −0.497844
\(482\) −3.79346e8 −0.154302
\(483\) 0 0
\(484\) 1.68963e9 0.677379
\(485\) −4.08131e7 −0.0162444
\(486\) 0 0
\(487\) −1.78601e9 −0.700701 −0.350350 0.936619i \(-0.613937\pi\)
−0.350350 + 0.936619i \(0.613937\pi\)
\(488\) −2.00367e9 −0.780472
\(489\) 0 0
\(490\) −7.85655e7 −0.0301679
\(491\) 2.06965e9 0.789063 0.394532 0.918882i \(-0.370907\pi\)
0.394532 + 0.918882i \(0.370907\pi\)
\(492\) 0 0
\(493\) −4.74377e8 −0.178303
\(494\) −8.12456e8 −0.303218
\(495\) 0 0
\(496\) 3.40393e9 1.25255
\(497\) 5.21132e9 1.90415
\(498\) 0 0
\(499\) −1.63444e9 −0.588866 −0.294433 0.955672i \(-0.595131\pi\)
−0.294433 + 0.955672i \(0.595131\pi\)
\(500\) 8.81954e8 0.315537
\(501\) 0 0
\(502\) 3.85037e9 1.35844
\(503\) 1.96230e9 0.687507 0.343754 0.939060i \(-0.388301\pi\)
0.343754 + 0.939060i \(0.388301\pi\)
\(504\) 0 0
\(505\) 3.20796e9 1.10843
\(506\) 3.57072e9 1.22526
\(507\) 0 0
\(508\) 1.41382e8 0.0478487
\(509\) 1.74020e8 0.0584906 0.0292453 0.999572i \(-0.490690\pi\)
0.0292453 + 0.999572i \(0.490690\pi\)
\(510\) 0 0
\(511\) −2.58142e9 −0.855827
\(512\) 5.94396e8 0.195718
\(513\) 0 0
\(514\) 3.47860e9 1.12988
\(515\) −1.44440e9 −0.465976
\(516\) 0 0
\(517\) 4.13834e9 1.31707
\(518\) 3.04145e9 0.961449
\(519\) 0 0
\(520\) −8.71550e8 −0.271820
\(521\) 1.04646e9 0.324184 0.162092 0.986776i \(-0.448176\pi\)
0.162092 + 0.986776i \(0.448176\pi\)
\(522\) 0 0
\(523\) −3.71160e9 −1.13450 −0.567252 0.823545i \(-0.691993\pi\)
−0.567252 + 0.823545i \(0.691993\pi\)
\(524\) 9.70745e7 0.0294745
\(525\) 0 0
\(526\) 2.73534e9 0.819523
\(527\) −1.03296e9 −0.307429
\(528\) 0 0
\(529\) −2.16054e9 −0.634553
\(530\) 2.17792e9 0.635442
\(531\) 0 0
\(532\) 4.94293e8 0.142329
\(533\) 9.16948e8 0.262301
\(534\) 0 0
\(535\) −2.05306e9 −0.579646
\(536\) −2.68071e9 −0.751922
\(537\) 0 0
\(538\) −6.20095e9 −1.71680
\(539\) −2.81111e8 −0.0773245
\(540\) 0 0
\(541\) 3.26372e9 0.886180 0.443090 0.896477i \(-0.353882\pi\)
0.443090 + 0.896477i \(0.353882\pi\)
\(542\) 3.24952e9 0.876642
\(543\) 0 0
\(544\) 6.95547e8 0.185238
\(545\) 1.15583e9 0.305848
\(546\) 0 0
\(547\) 7.46387e8 0.194988 0.0974941 0.995236i \(-0.468917\pi\)
0.0974941 + 0.995236i \(0.468917\pi\)
\(548\) −1.29388e8 −0.0335862
\(549\) 0 0
\(550\) 5.07498e9 1.30066
\(551\) −1.06166e9 −0.270366
\(552\) 0 0
\(553\) −1.47695e9 −0.371388
\(554\) 2.63135e9 0.657497
\(555\) 0 0
\(556\) 6.60195e8 0.162896
\(557\) −5.33461e9 −1.30801 −0.654003 0.756492i \(-0.726912\pi\)
−0.654003 + 0.756492i \(0.726912\pi\)
\(558\) 0 0
\(559\) −2.04884e9 −0.496097
\(560\) 2.96262e9 0.712883
\(561\) 0 0
\(562\) −3.60759e9 −0.857313
\(563\) −7.90248e9 −1.86631 −0.933156 0.359472i \(-0.882957\pi\)
−0.933156 + 0.359472i \(0.882957\pi\)
\(564\) 0 0
\(565\) 1.54520e9 0.360424
\(566\) 4.47802e9 1.03807
\(567\) 0 0
\(568\) 6.63635e9 1.51953
\(569\) −5.54526e9 −1.26191 −0.630956 0.775819i \(-0.717337\pi\)
−0.630956 + 0.775819i \(0.717337\pi\)
\(570\) 0 0
\(571\) 5.09241e9 1.14471 0.572357 0.820004i \(-0.306029\pi\)
0.572357 + 0.820004i \(0.306029\pi\)
\(572\) 1.47494e9 0.329525
\(573\) 0 0
\(574\) −2.29522e9 −0.506562
\(575\) 1.76847e9 0.387936
\(576\) 0 0
\(577\) 1.69420e9 0.367154 0.183577 0.983005i \(-0.441232\pi\)
0.183577 + 0.983005i \(0.441232\pi\)
\(578\) 4.85912e9 1.04667
\(579\) 0 0
\(580\) 5.38657e8 0.114634
\(581\) −7.26248e9 −1.53627
\(582\) 0 0
\(583\) 7.79269e9 1.62872
\(584\) −3.28732e9 −0.682962
\(585\) 0 0
\(586\) −4.63039e9 −0.950551
\(587\) 8.65701e9 1.76659 0.883293 0.468821i \(-0.155321\pi\)
0.883293 + 0.468821i \(0.155321\pi\)
\(588\) 0 0
\(589\) −2.31175e9 −0.466163
\(590\) −4.46821e8 −0.0895677
\(591\) 0 0
\(592\) 5.25977e9 1.04194
\(593\) 7.39803e9 1.45688 0.728441 0.685108i \(-0.240245\pi\)
0.728441 + 0.685108i \(0.240245\pi\)
\(594\) 0 0
\(595\) −8.99038e8 −0.174972
\(596\) 1.36005e9 0.263144
\(597\) 0 0
\(598\) 2.11462e9 0.404370
\(599\) −4.39303e9 −0.835162 −0.417581 0.908640i \(-0.637122\pi\)
−0.417581 + 0.908640i \(0.637122\pi\)
\(600\) 0 0
\(601\) −1.08818e9 −0.204475 −0.102238 0.994760i \(-0.532600\pi\)
−0.102238 + 0.994760i \(0.532600\pi\)
\(602\) 5.12846e9 0.958075
\(603\) 0 0
\(604\) −3.38388e9 −0.624864
\(605\) −6.87770e9 −1.26270
\(606\) 0 0
\(607\) −5.98074e9 −1.08541 −0.542706 0.839923i \(-0.682600\pi\)
−0.542706 + 0.839923i \(0.682600\pi\)
\(608\) 1.55663e9 0.280882
\(609\) 0 0
\(610\) −3.85759e9 −0.688116
\(611\) 2.45077e9 0.434669
\(612\) 0 0
\(613\) −5.60656e9 −0.983070 −0.491535 0.870858i \(-0.663564\pi\)
−0.491535 + 0.870858i \(0.663564\pi\)
\(614\) 2.55129e9 0.444806
\(615\) 0 0
\(616\) 7.80580e9 1.34550
\(617\) 9.07967e9 1.55622 0.778112 0.628126i \(-0.216178\pi\)
0.778112 + 0.628126i \(0.216178\pi\)
\(618\) 0 0
\(619\) −3.61181e9 −0.612079 −0.306040 0.952019i \(-0.599004\pi\)
−0.306040 + 0.952019i \(0.599004\pi\)
\(620\) 1.17293e9 0.197651
\(621\) 0 0
\(622\) 6.80518e8 0.113390
\(623\) −1.00589e10 −1.66665
\(624\) 0 0
\(625\) 3.26731e8 0.0535316
\(626\) 4.05741e9 0.661056
\(627\) 0 0
\(628\) 2.90444e9 0.467954
\(629\) −1.59613e9 −0.255736
\(630\) 0 0
\(631\) 2.73746e9 0.433756 0.216878 0.976199i \(-0.430413\pi\)
0.216878 + 0.976199i \(0.430413\pi\)
\(632\) −1.88082e9 −0.296372
\(633\) 0 0
\(634\) 6.99895e9 1.09074
\(635\) −5.75502e8 −0.0891946
\(636\) 0 0
\(637\) −1.66477e8 −0.0255191
\(638\) 7.92963e9 1.20887
\(639\) 0 0
\(640\) 4.76736e9 0.718867
\(641\) 7.16076e9 1.07388 0.536940 0.843620i \(-0.319580\pi\)
0.536940 + 0.843620i \(0.319580\pi\)
\(642\) 0 0
\(643\) −1.59933e9 −0.237247 −0.118623 0.992939i \(-0.537848\pi\)
−0.118623 + 0.992939i \(0.537848\pi\)
\(644\) −1.28652e9 −0.189809
\(645\) 0 0
\(646\) −1.06725e9 −0.155759
\(647\) 1.56410e9 0.227038 0.113519 0.993536i \(-0.463788\pi\)
0.113519 + 0.993536i \(0.463788\pi\)
\(648\) 0 0
\(649\) −1.59875e9 −0.229574
\(650\) 3.00546e9 0.429253
\(651\) 0 0
\(652\) −2.21057e9 −0.312348
\(653\) −1.10948e10 −1.55928 −0.779639 0.626229i \(-0.784598\pi\)
−0.779639 + 0.626229i \(0.784598\pi\)
\(654\) 0 0
\(655\) −3.95147e8 −0.0549432
\(656\) −3.96927e9 −0.548968
\(657\) 0 0
\(658\) −6.13454e9 −0.839443
\(659\) −8.31935e9 −1.13238 −0.566188 0.824276i \(-0.691582\pi\)
−0.566188 + 0.824276i \(0.691582\pi\)
\(660\) 0 0
\(661\) 5.78095e9 0.778563 0.389282 0.921119i \(-0.372723\pi\)
0.389282 + 0.921119i \(0.372723\pi\)
\(662\) 9.21778e9 1.23488
\(663\) 0 0
\(664\) −9.24841e9 −1.22597
\(665\) −2.01204e9 −0.265315
\(666\) 0 0
\(667\) 2.76323e9 0.360559
\(668\) −1.69742e9 −0.220329
\(669\) 0 0
\(670\) −5.16106e9 −0.662944
\(671\) −1.38026e10 −1.76373
\(672\) 0 0
\(673\) −6.94500e9 −0.878252 −0.439126 0.898425i \(-0.644712\pi\)
−0.439126 + 0.898425i \(0.644712\pi\)
\(674\) −1.67270e9 −0.210430
\(675\) 0 0
\(676\) −1.70554e9 −0.212349
\(677\) −2.90331e9 −0.359611 −0.179806 0.983702i \(-0.557547\pi\)
−0.179806 + 0.983702i \(0.557547\pi\)
\(678\) 0 0
\(679\) 2.16472e8 0.0265373
\(680\) −1.14488e9 −0.139630
\(681\) 0 0
\(682\) 1.72668e10 2.08433
\(683\) 4.42751e9 0.531726 0.265863 0.964011i \(-0.414343\pi\)
0.265863 + 0.964011i \(0.414343\pi\)
\(684\) 0 0
\(685\) 5.26679e8 0.0626078
\(686\) 9.91981e9 1.17319
\(687\) 0 0
\(688\) 8.86898e9 1.03828
\(689\) 4.61492e9 0.537523
\(690\) 0 0
\(691\) 3.87979e9 0.447337 0.223669 0.974665i \(-0.428197\pi\)
0.223669 + 0.974665i \(0.428197\pi\)
\(692\) −2.20491e9 −0.252941
\(693\) 0 0
\(694\) −7.80989e8 −0.0886925
\(695\) −2.68736e9 −0.303654
\(696\) 0 0
\(697\) 1.20451e9 0.134740
\(698\) −1.45219e10 −1.61633
\(699\) 0 0
\(700\) −1.82850e9 −0.201489
\(701\) 1.56016e10 1.71063 0.855315 0.518108i \(-0.173364\pi\)
0.855315 + 0.518108i \(0.173364\pi\)
\(702\) 0 0
\(703\) −3.57214e9 −0.387779
\(704\) 8.25710e9 0.891915
\(705\) 0 0
\(706\) 5.19046e9 0.555123
\(707\) −1.70150e10 −1.81077
\(708\) 0 0
\(709\) 1.35775e10 1.43073 0.715364 0.698752i \(-0.246261\pi\)
0.715364 + 0.698752i \(0.246261\pi\)
\(710\) 1.27767e10 1.33972
\(711\) 0 0
\(712\) −1.28096e10 −1.33001
\(713\) 6.01693e9 0.621672
\(714\) 0 0
\(715\) −6.00382e9 −0.614266
\(716\) 1.49044e9 0.151746
\(717\) 0 0
\(718\) −8.26963e9 −0.833778
\(719\) −1.51786e10 −1.52293 −0.761466 0.648205i \(-0.775520\pi\)
−0.761466 + 0.648205i \(0.775520\pi\)
\(720\) 0 0
\(721\) 7.66109e9 0.761232
\(722\) 9.23529e9 0.913210
\(723\) 0 0
\(724\) −6.76625e8 −0.0662618
\(725\) 3.92730e9 0.382746
\(726\) 0 0
\(727\) −1.49034e10 −1.43852 −0.719260 0.694741i \(-0.755519\pi\)
−0.719260 + 0.694741i \(0.755519\pi\)
\(728\) 4.62268e9 0.444053
\(729\) 0 0
\(730\) −6.32893e9 −0.602144
\(731\) −2.69138e9 −0.254838
\(732\) 0 0
\(733\) 1.05230e10 0.986905 0.493453 0.869773i \(-0.335735\pi\)
0.493453 + 0.869773i \(0.335735\pi\)
\(734\) 3.40279e9 0.317614
\(735\) 0 0
\(736\) −4.05153e9 −0.374582
\(737\) −1.84665e10 −1.69921
\(738\) 0 0
\(739\) 1.82290e10 1.66153 0.830764 0.556624i \(-0.187904\pi\)
0.830764 + 0.556624i \(0.187904\pi\)
\(740\) 1.81241e9 0.164417
\(741\) 0 0
\(742\) −1.15516e10 −1.03808
\(743\) 1.41694e9 0.126733 0.0633667 0.997990i \(-0.479816\pi\)
0.0633667 + 0.997990i \(0.479816\pi\)
\(744\) 0 0
\(745\) −5.53616e9 −0.490526
\(746\) −2.20707e10 −1.94639
\(747\) 0 0
\(748\) 1.93750e9 0.169273
\(749\) 1.08894e10 0.946927
\(750\) 0 0
\(751\) −1.51027e10 −1.30111 −0.650554 0.759460i \(-0.725463\pi\)
−0.650554 + 0.759460i \(0.725463\pi\)
\(752\) −1.06088e10 −0.909716
\(753\) 0 0
\(754\) 4.69601e9 0.398960
\(755\) 1.37743e10 1.16481
\(756\) 0 0
\(757\) −8.01056e9 −0.671162 −0.335581 0.942011i \(-0.608933\pi\)
−0.335581 + 0.942011i \(0.608933\pi\)
\(758\) −1.43838e10 −1.19959
\(759\) 0 0
\(760\) −2.56224e9 −0.211725
\(761\) −9.09051e9 −0.747725 −0.373863 0.927484i \(-0.621967\pi\)
−0.373863 + 0.927484i \(0.621967\pi\)
\(762\) 0 0
\(763\) −6.13049e9 −0.499642
\(764\) 2.88348e9 0.233932
\(765\) 0 0
\(766\) −8.73643e9 −0.702317
\(767\) −9.46795e8 −0.0757656
\(768\) 0 0
\(769\) −1.56881e10 −1.24402 −0.622009 0.783010i \(-0.713683\pi\)
−0.622009 + 0.783010i \(0.713683\pi\)
\(770\) 1.50282e10 1.18629
\(771\) 0 0
\(772\) 3.72732e9 0.291566
\(773\) −2.02623e10 −1.57783 −0.788916 0.614502i \(-0.789357\pi\)
−0.788916 + 0.614502i \(0.789357\pi\)
\(774\) 0 0
\(775\) 8.55170e9 0.659928
\(776\) 2.75666e8 0.0211771
\(777\) 0 0
\(778\) 1.51527e10 1.15362
\(779\) 2.69570e9 0.204310
\(780\) 0 0
\(781\) 4.57156e10 3.43389
\(782\) 2.77780e9 0.207719
\(783\) 0 0
\(784\) 7.20643e8 0.0534089
\(785\) −1.18227e10 −0.872311
\(786\) 0 0
\(787\) 2.40184e9 0.175644 0.0878219 0.996136i \(-0.472009\pi\)
0.0878219 + 0.996136i \(0.472009\pi\)
\(788\) 1.24356e9 0.0905370
\(789\) 0 0
\(790\) −3.62107e9 −0.261301
\(791\) −8.19568e9 −0.588800
\(792\) 0 0
\(793\) −8.17407e9 −0.582080
\(794\) −9.01184e9 −0.638913
\(795\) 0 0
\(796\) 7.34111e9 0.515901
\(797\) −2.37890e9 −0.166446 −0.0832228 0.996531i \(-0.526521\pi\)
−0.0832228 + 0.996531i \(0.526521\pi\)
\(798\) 0 0
\(799\) 3.21936e9 0.223283
\(800\) −5.75833e9 −0.397632
\(801\) 0 0
\(802\) −2.93832e10 −2.01136
\(803\) −2.26452e10 −1.54338
\(804\) 0 0
\(805\) 5.23685e9 0.353822
\(806\) 1.02256e10 0.687884
\(807\) 0 0
\(808\) −2.16677e10 −1.44502
\(809\) −1.39600e10 −0.926967 −0.463483 0.886106i \(-0.653401\pi\)
−0.463483 + 0.886106i \(0.653401\pi\)
\(810\) 0 0
\(811\) −1.88244e10 −1.23922 −0.619611 0.784909i \(-0.712710\pi\)
−0.619611 + 0.784909i \(0.712710\pi\)
\(812\) −2.85703e9 −0.187270
\(813\) 0 0
\(814\) 2.66807e10 1.73385
\(815\) 8.99825e9 0.582246
\(816\) 0 0
\(817\) −6.02331e9 −0.386418
\(818\) −1.13436e9 −0.0724629
\(819\) 0 0
\(820\) −1.36773e9 −0.0866268
\(821\) −7.81215e9 −0.492685 −0.246343 0.969183i \(-0.579229\pi\)
−0.246343 + 0.969183i \(0.579229\pi\)
\(822\) 0 0
\(823\) −1.87867e10 −1.17476 −0.587381 0.809310i \(-0.699841\pi\)
−0.587381 + 0.809310i \(0.699841\pi\)
\(824\) 9.75602e9 0.607474
\(825\) 0 0
\(826\) 2.36993e9 0.146320
\(827\) 1.42506e10 0.876119 0.438059 0.898946i \(-0.355666\pi\)
0.438059 + 0.898946i \(0.355666\pi\)
\(828\) 0 0
\(829\) 7.49011e8 0.0456612 0.0228306 0.999739i \(-0.492732\pi\)
0.0228306 + 0.999739i \(0.492732\pi\)
\(830\) −1.78056e10 −1.08089
\(831\) 0 0
\(832\) 4.88995e9 0.294356
\(833\) −2.18686e8 −0.0131088
\(834\) 0 0
\(835\) 6.90944e9 0.410715
\(836\) 4.33612e9 0.256673
\(837\) 0 0
\(838\) −1.07186e10 −0.629196
\(839\) 2.65131e10 1.54987 0.774933 0.632043i \(-0.217783\pi\)
0.774933 + 0.632043i \(0.217783\pi\)
\(840\) 0 0
\(841\) −1.11135e10 −0.644265
\(842\) 2.36950e10 1.36793
\(843\) 0 0
\(844\) −4.17146e9 −0.238830
\(845\) 6.94250e9 0.395838
\(846\) 0 0
\(847\) 3.64792e10 2.06278
\(848\) −1.99770e10 −1.12498
\(849\) 0 0
\(850\) 3.94801e9 0.220502
\(851\) 9.29739e9 0.517140
\(852\) 0 0
\(853\) −1.34893e10 −0.744163 −0.372081 0.928200i \(-0.621356\pi\)
−0.372081 + 0.928200i \(0.621356\pi\)
\(854\) 2.04606e10 1.12413
\(855\) 0 0
\(856\) 1.38671e10 0.755660
\(857\) −2.26740e10 −1.23054 −0.615269 0.788317i \(-0.710953\pi\)
−0.615269 + 0.788317i \(0.710953\pi\)
\(858\) 0 0
\(859\) −1.97914e10 −1.06537 −0.532685 0.846314i \(-0.678817\pi\)
−0.532685 + 0.846314i \(0.678817\pi\)
\(860\) 3.05607e9 0.163840
\(861\) 0 0
\(862\) 4.21563e10 2.24175
\(863\) 1.08103e10 0.572534 0.286267 0.958150i \(-0.407585\pi\)
0.286267 + 0.958150i \(0.407585\pi\)
\(864\) 0 0
\(865\) 8.97519e9 0.471506
\(866\) −2.22790e10 −1.16569
\(867\) 0 0
\(868\) −6.22117e9 −0.322889
\(869\) −1.29563e10 −0.669751
\(870\) 0 0
\(871\) −1.09361e10 −0.560786
\(872\) −7.80687e9 −0.398721
\(873\) 0 0
\(874\) 6.21670e9 0.314970
\(875\) 1.90415e10 0.960887
\(876\) 0 0
\(877\) −8.46440e7 −0.00423738 −0.00211869 0.999998i \(-0.500674\pi\)
−0.00211869 + 0.999998i \(0.500674\pi\)
\(878\) 2.21590e10 1.10489
\(879\) 0 0
\(880\) 2.59892e10 1.28559
\(881\) 3.78092e10 1.86286 0.931432 0.363915i \(-0.118560\pi\)
0.931432 + 0.363915i \(0.118560\pi\)
\(882\) 0 0
\(883\) 3.77215e10 1.84385 0.921927 0.387364i \(-0.126614\pi\)
0.921927 + 0.387364i \(0.126614\pi\)
\(884\) 1.14741e9 0.0558645
\(885\) 0 0
\(886\) 1.16535e9 0.0562908
\(887\) −1.24757e10 −0.600250 −0.300125 0.953900i \(-0.597028\pi\)
−0.300125 + 0.953900i \(0.597028\pi\)
\(888\) 0 0
\(889\) 3.05245e9 0.145711
\(890\) −2.46617e10 −1.17262
\(891\) 0 0
\(892\) 4.83664e9 0.228174
\(893\) 7.20493e9 0.338571
\(894\) 0 0
\(895\) −6.06689e9 −0.282869
\(896\) −2.52860e10 −1.17436
\(897\) 0 0
\(898\) −3.10760e10 −1.43205
\(899\) 1.33620e10 0.613356
\(900\) 0 0
\(901\) 6.06222e9 0.276118
\(902\) −2.01345e10 −0.913520
\(903\) 0 0
\(904\) −1.04368e10 −0.469870
\(905\) 2.75424e9 0.123518
\(906\) 0 0
\(907\) 1.59259e8 0.00708725 0.00354363 0.999994i \(-0.498872\pi\)
0.00354363 + 0.999994i \(0.498872\pi\)
\(908\) −8.92386e9 −0.395596
\(909\) 0 0
\(910\) 8.89986e9 0.391506
\(911\) −2.28968e10 −1.00337 −0.501683 0.865051i \(-0.667286\pi\)
−0.501683 + 0.865051i \(0.667286\pi\)
\(912\) 0 0
\(913\) −6.37092e10 −2.77048
\(914\) −4.03447e10 −1.74773
\(915\) 0 0
\(916\) 2.83361e9 0.121817
\(917\) 2.09585e9 0.0897568
\(918\) 0 0
\(919\) −1.19936e10 −0.509734 −0.254867 0.966976i \(-0.582032\pi\)
−0.254867 + 0.966976i \(0.582032\pi\)
\(920\) 6.66887e9 0.282355
\(921\) 0 0
\(922\) −1.22498e10 −0.514722
\(923\) 2.70733e10 1.13327
\(924\) 0 0
\(925\) 1.32141e10 0.548963
\(926\) 3.59268e10 1.48689
\(927\) 0 0
\(928\) −8.99737e9 −0.369571
\(929\) 5.52148e9 0.225944 0.112972 0.993598i \(-0.463963\pi\)
0.112972 + 0.993598i \(0.463963\pi\)
\(930\) 0 0
\(931\) −4.89420e8 −0.0198773
\(932\) −9.95165e9 −0.402661
\(933\) 0 0
\(934\) 7.83387e6 0.000314603 0
\(935\) −7.88670e9 −0.315540
\(936\) 0 0
\(937\) 4.28153e10 1.70024 0.850120 0.526589i \(-0.176529\pi\)
0.850120 + 0.526589i \(0.176529\pi\)
\(938\) 2.73741e10 1.08300
\(939\) 0 0
\(940\) −3.65560e9 −0.143553
\(941\) −4.62683e10 −1.81017 −0.905087 0.425227i \(-0.860194\pi\)
−0.905087 + 0.425227i \(0.860194\pi\)
\(942\) 0 0
\(943\) −7.01624e9 −0.272467
\(944\) 4.09847e9 0.158569
\(945\) 0 0
\(946\) 4.49888e10 1.72777
\(947\) −1.84946e10 −0.707651 −0.353826 0.935311i \(-0.615119\pi\)
−0.353826 + 0.935311i \(0.615119\pi\)
\(948\) 0 0
\(949\) −1.34107e10 −0.509356
\(950\) 8.83563e9 0.334353
\(951\) 0 0
\(952\) 6.07242e9 0.228104
\(953\) 1.08055e10 0.404409 0.202205 0.979343i \(-0.435189\pi\)
0.202205 + 0.979343i \(0.435189\pi\)
\(954\) 0 0
\(955\) −1.17373e10 −0.436072
\(956\) −6.82953e9 −0.252807
\(957\) 0 0
\(958\) −3.19117e10 −1.17266
\(959\) −2.79349e9 −0.102278
\(960\) 0 0
\(961\) 1.58314e9 0.0575423
\(962\) 1.58006e10 0.572218
\(963\) 0 0
\(964\) 1.19899e9 0.0431067
\(965\) −1.51723e10 −0.543506
\(966\) 0 0
\(967\) 2.41210e10 0.857834 0.428917 0.903344i \(-0.358895\pi\)
0.428917 + 0.903344i \(0.358895\pi\)
\(968\) 4.64544e10 1.64613
\(969\) 0 0
\(970\) 5.30728e8 0.0186712
\(971\) 3.65187e10 1.28011 0.640057 0.768328i \(-0.278911\pi\)
0.640057 + 0.768328i \(0.278911\pi\)
\(972\) 0 0
\(973\) 1.42537e10 0.496058
\(974\) 2.32251e10 0.805379
\(975\) 0 0
\(976\) 3.53838e10 1.21823
\(977\) −1.82046e10 −0.624524 −0.312262 0.949996i \(-0.601087\pi\)
−0.312262 + 0.949996i \(0.601087\pi\)
\(978\) 0 0
\(979\) −8.82408e10 −3.00559
\(980\) 2.48319e8 0.00842789
\(981\) 0 0
\(982\) −2.69135e10 −0.906942
\(983\) 4.38690e10 1.47306 0.736531 0.676404i \(-0.236463\pi\)
0.736531 + 0.676404i \(0.236463\pi\)
\(984\) 0 0
\(985\) −5.06199e9 −0.168770
\(986\) 6.16875e9 0.204940
\(987\) 0 0
\(988\) 2.56790e9 0.0847089
\(989\) 1.56772e10 0.515325
\(990\) 0 0
\(991\) −2.39803e10 −0.782702 −0.391351 0.920242i \(-0.627992\pi\)
−0.391351 + 0.920242i \(0.627992\pi\)
\(992\) −1.95918e10 −0.637210
\(993\) 0 0
\(994\) −6.77673e10 −2.18861
\(995\) −2.98824e10 −0.961688
\(996\) 0 0
\(997\) 5.54581e10 1.77228 0.886139 0.463419i \(-0.153378\pi\)
0.886139 + 0.463419i \(0.153378\pi\)
\(998\) 2.12540e10 0.676837
\(999\) 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.b.1.5 16
3.2 odd 2 177.8.a.a.1.12 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.12 16 3.2 odd 2
531.8.a.b.1.5 16 1.1 even 1 trivial