Properties

Label 16.50.a.b.1.2
Level $16$
Weight $50$
Character 16.1
Self dual yes
Analytic conductor $243.306$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 50 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(243.305928158\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 104434803447206332 x + 4289992005756109702361620\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.42020e8\) of defining polynomial
Character \(\chi\) \(=\) 16.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.33407e11 q^{3} +1.87739e17 q^{5} -8.28497e20 q^{7} -2.21502e23 q^{9} +O(q^{10})\) \(q+1.33407e11 q^{3} +1.87739e17 q^{5} -8.28497e20 q^{7} -2.21502e23 q^{9} +3.05508e25 q^{11} -4.92824e25 q^{13} +2.50456e28 q^{15} +2.50818e30 q^{17} -5.26388e30 q^{19} -1.10527e32 q^{21} +7.99827e32 q^{23} +1.74823e34 q^{25} -6.14740e34 q^{27} -2.91459e35 q^{29} -5.98168e36 q^{31} +4.07569e36 q^{33} -1.55541e38 q^{35} +1.31540e38 q^{37} -6.57461e36 q^{39} -2.57335e39 q^{41} -3.01162e39 q^{43} -4.15845e40 q^{45} +1.35707e41 q^{47} +4.29484e41 q^{49} +3.34609e41 q^{51} +3.38284e42 q^{53} +5.73558e42 q^{55} -7.02238e41 q^{57} +1.39260e42 q^{59} -8.43699e43 q^{61} +1.83514e44 q^{63} -9.25222e42 q^{65} -4.28654e44 q^{67} +1.06702e44 q^{69} +1.65460e45 q^{71} -5.25815e45 q^{73} +2.33226e45 q^{75} -2.53113e46 q^{77} -1.45663e46 q^{79} +4.48042e46 q^{81} +9.24216e46 q^{83} +4.70884e47 q^{85} -3.88826e46 q^{87} +8.22598e46 q^{89} +4.08303e46 q^{91} -7.97996e47 q^{93} -9.88236e47 q^{95} -4.76751e48 q^{97} -6.76707e48 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 16203614388q^{3} - 101813016401840430q^{5} + 125479179230203797096q^{7} + 389432931519696922052199q^{9} + O(q^{10}) \) \( 3q + 16203614388q^{3} - 101813016401840430q^{5} + \)\(12\!\cdots\!96\)\(q^{7} + \)\(38\!\cdots\!99\)\(q^{9} + \)\(25\!\cdots\!04\)\(q^{11} + \)\(29\!\cdots\!42\)\(q^{13} - \)\(87\!\cdots\!80\)\(q^{15} + \)\(39\!\cdots\!54\)\(q^{17} - \)\(14\!\cdots\!80\)\(q^{19} - \)\(10\!\cdots\!84\)\(q^{21} + \)\(49\!\cdots\!08\)\(q^{23} + \)\(39\!\cdots\!25\)\(q^{25} - \)\(19\!\cdots\!00\)\(q^{27} + \)\(93\!\cdots\!30\)\(q^{29} - \)\(24\!\cdots\!56\)\(q^{31} + \)\(84\!\cdots\!84\)\(q^{33} - \)\(30\!\cdots\!60\)\(q^{35} + \)\(30\!\cdots\!74\)\(q^{37} - \)\(36\!\cdots\!68\)\(q^{39} - \)\(11\!\cdots\!54\)\(q^{41} + \)\(17\!\cdots\!48\)\(q^{43} - \)\(11\!\cdots\!90\)\(q^{45} + \)\(21\!\cdots\!56\)\(q^{47} + \)\(37\!\cdots\!51\)\(q^{49} - \)\(22\!\cdots\!16\)\(q^{51} + \)\(74\!\cdots\!02\)\(q^{53} - \)\(32\!\cdots\!40\)\(q^{55} - \)\(20\!\cdots\!80\)\(q^{57} + \)\(53\!\cdots\!40\)\(q^{59} - \)\(12\!\cdots\!94\)\(q^{61} + \)\(46\!\cdots\!68\)\(q^{63} - \)\(42\!\cdots\!20\)\(q^{65} - \)\(11\!\cdots\!24\)\(q^{67} + \)\(77\!\cdots\!68\)\(q^{69} + \)\(40\!\cdots\!24\)\(q^{71} - \)\(10\!\cdots\!18\)\(q^{73} + \)\(38\!\cdots\!00\)\(q^{75} - \)\(23\!\cdots\!72\)\(q^{77} - \)\(36\!\cdots\!60\)\(q^{79} - \)\(14\!\cdots\!17\)\(q^{81} - \)\(15\!\cdots\!92\)\(q^{83} + \)\(56\!\cdots\!60\)\(q^{85} - \)\(10\!\cdots\!20\)\(q^{87} + \)\(54\!\cdots\!70\)\(q^{89} + \)\(14\!\cdots\!44\)\(q^{91} - \)\(23\!\cdots\!76\)\(q^{93} + \)\(28\!\cdots\!00\)\(q^{95} + \)\(43\!\cdots\!14\)\(q^{97} - \)\(17\!\cdots\!68\)\(q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.33407e11 0.272714 0.136357 0.990660i \(-0.456461\pi\)
0.136357 + 0.990660i \(0.456461\pi\)
\(4\) 0 0
\(5\) 1.87739e17 1.40860 0.704302 0.709900i \(-0.251260\pi\)
0.704302 + 0.709900i \(0.251260\pi\)
\(6\) 0 0
\(7\) −8.28497e20 −1.63452 −0.817258 0.576272i \(-0.804507\pi\)
−0.817258 + 0.576272i \(0.804507\pi\)
\(8\) 0 0
\(9\) −2.21502e23 −0.925627
\(10\) 0 0
\(11\) 3.05508e25 0.935195 0.467598 0.883941i \(-0.345120\pi\)
0.467598 + 0.883941i \(0.345120\pi\)
\(12\) 0 0
\(13\) −4.92824e25 −0.0251814 −0.0125907 0.999921i \(-0.504008\pi\)
−0.0125907 + 0.999921i \(0.504008\pi\)
\(14\) 0 0
\(15\) 2.50456e28 0.384146
\(16\) 0 0
\(17\) 2.50818e30 1.79209 0.896047 0.443959i \(-0.146427\pi\)
0.896047 + 0.443959i \(0.146427\pi\)
\(18\) 0 0
\(19\) −5.26388e30 −0.246515 −0.123257 0.992375i \(-0.539334\pi\)
−0.123257 + 0.992375i \(0.539334\pi\)
\(20\) 0 0
\(21\) −1.10527e32 −0.445755
\(22\) 0 0
\(23\) 7.99827e32 0.347268 0.173634 0.984810i \(-0.444449\pi\)
0.173634 + 0.984810i \(0.444449\pi\)
\(24\) 0 0
\(25\) 1.74823e34 0.984168
\(26\) 0 0
\(27\) −6.14740e34 −0.525145
\(28\) 0 0
\(29\) −2.91459e35 −0.432341 −0.216171 0.976356i \(-0.569357\pi\)
−0.216171 + 0.976356i \(0.569357\pi\)
\(30\) 0 0
\(31\) −5.98168e36 −1.73166 −0.865828 0.500342i \(-0.833208\pi\)
−0.865828 + 0.500342i \(0.833208\pi\)
\(32\) 0 0
\(33\) 4.07569e36 0.255041
\(34\) 0 0
\(35\) −1.55541e38 −2.30239
\(36\) 0 0
\(37\) 1.31540e38 0.499018 0.249509 0.968372i \(-0.419731\pi\)
0.249509 + 0.968372i \(0.419731\pi\)
\(38\) 0 0
\(39\) −6.57461e36 −0.00686731
\(40\) 0 0
\(41\) −2.57335e39 −0.789394 −0.394697 0.918811i \(-0.629150\pi\)
−0.394697 + 0.918811i \(0.629150\pi\)
\(42\) 0 0
\(43\) −3.01162e39 −0.287622 −0.143811 0.989605i \(-0.545936\pi\)
−0.143811 + 0.989605i \(0.545936\pi\)
\(44\) 0 0
\(45\) −4.15845e40 −1.30384
\(46\) 0 0
\(47\) 1.35707e41 1.46624 0.733118 0.680102i \(-0.238064\pi\)
0.733118 + 0.680102i \(0.238064\pi\)
\(48\) 0 0
\(49\) 4.29484e41 1.67164
\(50\) 0 0
\(51\) 3.34609e41 0.488729
\(52\) 0 0
\(53\) 3.38284e42 1.92541 0.962704 0.270557i \(-0.0872079\pi\)
0.962704 + 0.270557i \(0.0872079\pi\)
\(54\) 0 0
\(55\) 5.73558e42 1.31732
\(56\) 0 0
\(57\) −7.02238e41 −0.0672280
\(58\) 0 0
\(59\) 1.39260e42 0.0572732 0.0286366 0.999590i \(-0.490883\pi\)
0.0286366 + 0.999590i \(0.490883\pi\)
\(60\) 0 0
\(61\) −8.43699e43 −1.53323 −0.766614 0.642108i \(-0.778060\pi\)
−0.766614 + 0.642108i \(0.778060\pi\)
\(62\) 0 0
\(63\) 1.83514e44 1.51295
\(64\) 0 0
\(65\) −9.25222e42 −0.0354706
\(66\) 0 0
\(67\) −4.28654e44 −0.782121 −0.391060 0.920365i \(-0.627892\pi\)
−0.391060 + 0.920365i \(0.627892\pi\)
\(68\) 0 0
\(69\) 1.06702e44 0.0947046
\(70\) 0 0
\(71\) 1.65460e45 0.729228 0.364614 0.931159i \(-0.381201\pi\)
0.364614 + 0.931159i \(0.381201\pi\)
\(72\) 0 0
\(73\) −5.25815e45 −1.17334 −0.586669 0.809827i \(-0.699561\pi\)
−0.586669 + 0.809827i \(0.699561\pi\)
\(74\) 0 0
\(75\) 2.33226e45 0.268396
\(76\) 0 0
\(77\) −2.53113e46 −1.52859
\(78\) 0 0
\(79\) −1.45663e46 −0.469338 −0.234669 0.972075i \(-0.575401\pi\)
−0.234669 + 0.972075i \(0.575401\pi\)
\(80\) 0 0
\(81\) 4.48042e46 0.782413
\(82\) 0 0
\(83\) 9.24216e46 0.887892 0.443946 0.896054i \(-0.353578\pi\)
0.443946 + 0.896054i \(0.353578\pi\)
\(84\) 0 0
\(85\) 4.70884e47 2.52435
\(86\) 0 0
\(87\) −3.88826e46 −0.117905
\(88\) 0 0
\(89\) 8.22598e46 0.142933 0.0714665 0.997443i \(-0.477232\pi\)
0.0714665 + 0.997443i \(0.477232\pi\)
\(90\) 0 0
\(91\) 4.08303e46 0.0411594
\(92\) 0 0
\(93\) −7.97996e47 −0.472246
\(94\) 0 0
\(95\) −9.88236e47 −0.347242
\(96\) 0 0
\(97\) −4.76751e48 −1.00551 −0.502753 0.864430i \(-0.667679\pi\)
−0.502753 + 0.864430i \(0.667679\pi\)
\(98\) 0 0
\(99\) −6.76707e48 −0.865642
\(100\) 0 0
\(101\) 1.44113e49 1.12935 0.564674 0.825314i \(-0.309002\pi\)
0.564674 + 0.825314i \(0.309002\pi\)
\(102\) 0 0
\(103\) 1.10349e49 0.534879 0.267440 0.963575i \(-0.413822\pi\)
0.267440 + 0.963575i \(0.413822\pi\)
\(104\) 0 0
\(105\) −2.07502e49 −0.627893
\(106\) 0 0
\(107\) 7.92403e49 1.51023 0.755116 0.655591i \(-0.227581\pi\)
0.755116 + 0.655591i \(0.227581\pi\)
\(108\) 0 0
\(109\) −1.47552e48 −0.0178647 −0.00893236 0.999960i \(-0.502843\pi\)
−0.00893236 + 0.999960i \(0.502843\pi\)
\(110\) 0 0
\(111\) 1.75484e49 0.136089
\(112\) 0 0
\(113\) 2.29547e49 0.114935 0.0574673 0.998347i \(-0.481698\pi\)
0.0574673 + 0.998347i \(0.481698\pi\)
\(114\) 0 0
\(115\) 1.50159e50 0.489163
\(116\) 0 0
\(117\) 1.09161e49 0.0233086
\(118\) 0 0
\(119\) −2.07802e51 −2.92921
\(120\) 0 0
\(121\) −1.33836e50 −0.125410
\(122\) 0 0
\(123\) −3.43302e50 −0.215279
\(124\) 0 0
\(125\) −5.27986e49 −0.0223011
\(126\) 0 0
\(127\) 5.51820e51 1.57981 0.789907 0.613227i \(-0.210129\pi\)
0.789907 + 0.613227i \(0.210129\pi\)
\(128\) 0 0
\(129\) −4.01770e50 −0.0784385
\(130\) 0 0
\(131\) 1.30117e51 0.174256 0.0871282 0.996197i \(-0.472231\pi\)
0.0871282 + 0.996197i \(0.472231\pi\)
\(132\) 0 0
\(133\) 4.36111e51 0.402932
\(134\) 0 0
\(135\) −1.15411e52 −0.739722
\(136\) 0 0
\(137\) −1.00417e52 −0.448902 −0.224451 0.974485i \(-0.572059\pi\)
−0.224451 + 0.974485i \(0.572059\pi\)
\(138\) 0 0
\(139\) 3.07477e52 0.963719 0.481860 0.876248i \(-0.339962\pi\)
0.481860 + 0.876248i \(0.339962\pi\)
\(140\) 0 0
\(141\) 1.81042e52 0.399863
\(142\) 0 0
\(143\) −1.50562e51 −0.0235495
\(144\) 0 0
\(145\) −5.47182e52 −0.608998
\(146\) 0 0
\(147\) 5.72961e52 0.455880
\(148\) 0 0
\(149\) 1.89224e53 1.08122 0.540608 0.841275i \(-0.318194\pi\)
0.540608 + 0.841275i \(0.318194\pi\)
\(150\) 0 0
\(151\) −2.34497e52 −0.0966500 −0.0483250 0.998832i \(-0.515388\pi\)
−0.0483250 + 0.998832i \(0.515388\pi\)
\(152\) 0 0
\(153\) −5.55568e53 −1.65881
\(154\) 0 0
\(155\) −1.12299e54 −2.43922
\(156\) 0 0
\(157\) 1.13167e54 1.79546 0.897732 0.440542i \(-0.145214\pi\)
0.897732 + 0.440542i \(0.145214\pi\)
\(158\) 0 0
\(159\) 4.51294e53 0.525085
\(160\) 0 0
\(161\) −6.62655e53 −0.567614
\(162\) 0 0
\(163\) 2.02699e54 1.28308 0.641542 0.767088i \(-0.278295\pi\)
0.641542 + 0.767088i \(0.278295\pi\)
\(164\) 0 0
\(165\) 7.65165e53 0.359252
\(166\) 0 0
\(167\) 2.35877e54 0.824389 0.412194 0.911096i \(-0.364762\pi\)
0.412194 + 0.911096i \(0.364762\pi\)
\(168\) 0 0
\(169\) −3.82780e54 −0.999366
\(170\) 0 0
\(171\) 1.16596e54 0.228181
\(172\) 0 0
\(173\) 6.16168e53 0.0906921 0.0453461 0.998971i \(-0.485561\pi\)
0.0453461 + 0.998971i \(0.485561\pi\)
\(174\) 0 0
\(175\) −1.44841e55 −1.60864
\(176\) 0 0
\(177\) 1.85782e53 0.0156192
\(178\) 0 0
\(179\) −3.10481e54 −0.198215 −0.0991073 0.995077i \(-0.531599\pi\)
−0.0991073 + 0.995077i \(0.531599\pi\)
\(180\) 0 0
\(181\) −8.59693e54 −0.418041 −0.209020 0.977911i \(-0.567027\pi\)
−0.209020 + 0.977911i \(0.567027\pi\)
\(182\) 0 0
\(183\) −1.12555e55 −0.418132
\(184\) 0 0
\(185\) 2.46952e55 0.702920
\(186\) 0 0
\(187\) 7.66271e55 1.67596
\(188\) 0 0
\(189\) 5.09310e55 0.858358
\(190\) 0 0
\(191\) 2.27680e55 0.296488 0.148244 0.988951i \(-0.452638\pi\)
0.148244 + 0.988951i \(0.452638\pi\)
\(192\) 0 0
\(193\) 6.99559e55 0.705782 0.352891 0.935664i \(-0.385199\pi\)
0.352891 + 0.935664i \(0.385199\pi\)
\(194\) 0 0
\(195\) −1.23431e54 −0.00967333
\(196\) 0 0
\(197\) −1.40903e56 −0.860000 −0.430000 0.902829i \(-0.641486\pi\)
−0.430000 + 0.902829i \(0.641486\pi\)
\(198\) 0 0
\(199\) −3.14321e56 −1.49786 −0.748932 0.662647i \(-0.769433\pi\)
−0.748932 + 0.662647i \(0.769433\pi\)
\(200\) 0 0
\(201\) −5.71853e55 −0.213295
\(202\) 0 0
\(203\) 2.41473e56 0.706668
\(204\) 0 0
\(205\) −4.83117e56 −1.11194
\(206\) 0 0
\(207\) −1.77163e56 −0.321440
\(208\) 0 0
\(209\) −1.60816e56 −0.230539
\(210\) 0 0
\(211\) 1.36710e57 1.55196 0.775978 0.630760i \(-0.217257\pi\)
0.775978 + 0.630760i \(0.217257\pi\)
\(212\) 0 0
\(213\) 2.20735e56 0.198870
\(214\) 0 0
\(215\) −5.65398e56 −0.405146
\(216\) 0 0
\(217\) 4.95580e57 2.83042
\(218\) 0 0
\(219\) −7.01473e56 −0.319985
\(220\) 0 0
\(221\) −1.23609e56 −0.0451274
\(222\) 0 0
\(223\) −1.75293e56 −0.0513210 −0.0256605 0.999671i \(-0.508169\pi\)
−0.0256605 + 0.999671i \(0.508169\pi\)
\(224\) 0 0
\(225\) −3.87237e57 −0.910973
\(226\) 0 0
\(227\) 4.13704e57 0.783531 0.391765 0.920065i \(-0.371865\pi\)
0.391765 + 0.920065i \(0.371865\pi\)
\(228\) 0 0
\(229\) 6.25222e57 0.955136 0.477568 0.878595i \(-0.341518\pi\)
0.477568 + 0.878595i \(0.341518\pi\)
\(230\) 0 0
\(231\) −3.37670e57 −0.416868
\(232\) 0 0
\(233\) −5.79416e57 −0.579123 −0.289561 0.957159i \(-0.593509\pi\)
−0.289561 + 0.957159i \(0.593509\pi\)
\(234\) 0 0
\(235\) 2.54774e58 2.06535
\(236\) 0 0
\(237\) −1.94324e57 −0.127995
\(238\) 0 0
\(239\) −1.99860e58 −1.07146 −0.535732 0.844388i \(-0.679964\pi\)
−0.535732 + 0.844388i \(0.679964\pi\)
\(240\) 0 0
\(241\) −8.71374e57 −0.380879 −0.190440 0.981699i \(-0.560991\pi\)
−0.190440 + 0.981699i \(0.560991\pi\)
\(242\) 0 0
\(243\) 2.06879e58 0.738520
\(244\) 0 0
\(245\) 8.06309e58 2.35468
\(246\) 0 0
\(247\) 2.59417e56 0.00620759
\(248\) 0 0
\(249\) 1.23297e58 0.242140
\(250\) 0 0
\(251\) 7.09795e58 1.14585 0.572924 0.819609i \(-0.305809\pi\)
0.572924 + 0.819609i \(0.305809\pi\)
\(252\) 0 0
\(253\) 2.44354e58 0.324763
\(254\) 0 0
\(255\) 6.28191e58 0.688426
\(256\) 0 0
\(257\) −3.39451e58 −0.307196 −0.153598 0.988133i \(-0.549086\pi\)
−0.153598 + 0.988133i \(0.549086\pi\)
\(258\) 0 0
\(259\) −1.08981e59 −0.815653
\(260\) 0 0
\(261\) 6.45587e58 0.400187
\(262\) 0 0
\(263\) 2.34522e59 1.20578 0.602890 0.797825i \(-0.294016\pi\)
0.602890 + 0.797825i \(0.294016\pi\)
\(264\) 0 0
\(265\) 6.35091e59 2.71214
\(266\) 0 0
\(267\) 1.09740e58 0.0389798
\(268\) 0 0
\(269\) −2.15592e59 −0.637826 −0.318913 0.947784i \(-0.603318\pi\)
−0.318913 + 0.947784i \(0.603318\pi\)
\(270\) 0 0
\(271\) 3.52922e59 0.870825 0.435413 0.900231i \(-0.356602\pi\)
0.435413 + 0.900231i \(0.356602\pi\)
\(272\) 0 0
\(273\) 5.44704e57 0.0112247
\(274\) 0 0
\(275\) 5.34100e59 0.920389
\(276\) 0 0
\(277\) 2.94410e59 0.424813 0.212406 0.977181i \(-0.431870\pi\)
0.212406 + 0.977181i \(0.431870\pi\)
\(278\) 0 0
\(279\) 1.32495e60 1.60287
\(280\) 0 0
\(281\) 4.56150e59 0.463237 0.231618 0.972807i \(-0.425598\pi\)
0.231618 + 0.972807i \(0.425598\pi\)
\(282\) 0 0
\(283\) −5.39564e59 −0.460548 −0.230274 0.973126i \(-0.573962\pi\)
−0.230274 + 0.973126i \(0.573962\pi\)
\(284\) 0 0
\(285\) −1.31837e59 −0.0946977
\(286\) 0 0
\(287\) 2.13201e60 1.29028
\(288\) 0 0
\(289\) 4.33215e60 2.21160
\(290\) 0 0
\(291\) −6.36018e59 −0.274215
\(292\) 0 0
\(293\) 3.90821e60 1.42469 0.712346 0.701828i \(-0.247632\pi\)
0.712346 + 0.701828i \(0.247632\pi\)
\(294\) 0 0
\(295\) 2.61445e59 0.0806753
\(296\) 0 0
\(297\) −1.87808e60 −0.491113
\(298\) 0 0
\(299\) −3.94174e58 −0.00874468
\(300\) 0 0
\(301\) 2.49512e60 0.470123
\(302\) 0 0
\(303\) 1.92256e60 0.307989
\(304\) 0 0
\(305\) −1.58395e61 −2.15971
\(306\) 0 0
\(307\) −2.34251e59 −0.0272140 −0.0136070 0.999907i \(-0.504331\pi\)
−0.0136070 + 0.999907i \(0.504331\pi\)
\(308\) 0 0
\(309\) 1.47213e60 0.145869
\(310\) 0 0
\(311\) 4.35273e60 0.368241 0.184120 0.982904i \(-0.441056\pi\)
0.184120 + 0.982904i \(0.441056\pi\)
\(312\) 0 0
\(313\) 1.93545e61 1.39941 0.699706 0.714431i \(-0.253314\pi\)
0.699706 + 0.714431i \(0.253314\pi\)
\(314\) 0 0
\(315\) 3.44527e61 2.13115
\(316\) 0 0
\(317\) 2.36551e61 1.25306 0.626530 0.779397i \(-0.284475\pi\)
0.626530 + 0.779397i \(0.284475\pi\)
\(318\) 0 0
\(319\) −8.90431e60 −0.404323
\(320\) 0 0
\(321\) 1.05712e61 0.411861
\(322\) 0 0
\(323\) −1.32028e61 −0.441778
\(324\) 0 0
\(325\) −8.61571e59 −0.0247827
\(326\) 0 0
\(327\) −1.96844e59 −0.00487196
\(328\) 0 0
\(329\) −1.12433e62 −2.39658
\(330\) 0 0
\(331\) −6.39438e61 −1.17493 −0.587466 0.809249i \(-0.699874\pi\)
−0.587466 + 0.809249i \(0.699874\pi\)
\(332\) 0 0
\(333\) −2.91364e61 −0.461905
\(334\) 0 0
\(335\) −8.04750e61 −1.10170
\(336\) 0 0
\(337\) 9.36305e61 1.10786 0.553930 0.832563i \(-0.313128\pi\)
0.553930 + 0.832563i \(0.313128\pi\)
\(338\) 0 0
\(339\) 3.06232e60 0.0313442
\(340\) 0 0
\(341\) −1.82745e62 −1.61944
\(342\) 0 0
\(343\) −1.42966e62 −1.09781
\(344\) 0 0
\(345\) 2.00322e61 0.133401
\(346\) 0 0
\(347\) 9.31722e61 0.538535 0.269268 0.963065i \(-0.413218\pi\)
0.269268 + 0.963065i \(0.413218\pi\)
\(348\) 0 0
\(349\) 2.03454e62 1.02151 0.510755 0.859726i \(-0.329366\pi\)
0.510755 + 0.859726i \(0.329366\pi\)
\(350\) 0 0
\(351\) 3.02959e60 0.0132239
\(352\) 0 0
\(353\) −2.78134e62 −1.05627 −0.528133 0.849161i \(-0.677108\pi\)
−0.528133 + 0.849161i \(0.677108\pi\)
\(354\) 0 0
\(355\) 3.10633e62 1.02719
\(356\) 0 0
\(357\) −2.77222e62 −0.798835
\(358\) 0 0
\(359\) −7.25567e62 −1.82332 −0.911662 0.410942i \(-0.865200\pi\)
−0.911662 + 0.410942i \(0.865200\pi\)
\(360\) 0 0
\(361\) −4.28251e62 −0.939230
\(362\) 0 0
\(363\) −1.78546e61 −0.0342010
\(364\) 0 0
\(365\) −9.87160e62 −1.65277
\(366\) 0 0
\(367\) 3.66239e62 0.536347 0.268174 0.963371i \(-0.413580\pi\)
0.268174 + 0.963371i \(0.413580\pi\)
\(368\) 0 0
\(369\) 5.70001e62 0.730685
\(370\) 0 0
\(371\) −2.80267e63 −3.14711
\(372\) 0 0
\(373\) 1.01303e63 0.997140 0.498570 0.866850i \(-0.333859\pi\)
0.498570 + 0.866850i \(0.333859\pi\)
\(374\) 0 0
\(375\) −7.04369e60 −0.00608183
\(376\) 0 0
\(377\) 1.43638e61 0.0108869
\(378\) 0 0
\(379\) 4.10768e62 0.273486 0.136743 0.990607i \(-0.456336\pi\)
0.136743 + 0.990607i \(0.456336\pi\)
\(380\) 0 0
\(381\) 7.36165e62 0.430837
\(382\) 0 0
\(383\) 1.90644e63 0.981415 0.490708 0.871324i \(-0.336738\pi\)
0.490708 + 0.871324i \(0.336738\pi\)
\(384\) 0 0
\(385\) −4.75191e63 −2.15318
\(386\) 0 0
\(387\) 6.67079e62 0.266231
\(388\) 0 0
\(389\) 3.21862e63 1.13215 0.566076 0.824353i \(-0.308461\pi\)
0.566076 + 0.824353i \(0.308461\pi\)
\(390\) 0 0
\(391\) 2.00611e63 0.622336
\(392\) 0 0
\(393\) 1.73586e62 0.0475221
\(394\) 0 0
\(395\) −2.73466e63 −0.661111
\(396\) 0 0
\(397\) −1.66988e63 −0.356713 −0.178356 0.983966i \(-0.557078\pi\)
−0.178356 + 0.983966i \(0.557078\pi\)
\(398\) 0 0
\(399\) 5.81802e62 0.109885
\(400\) 0 0
\(401\) −3.21505e63 −0.537218 −0.268609 0.963249i \(-0.586564\pi\)
−0.268609 + 0.963249i \(0.586564\pi\)
\(402\) 0 0
\(403\) 2.94791e62 0.0436055
\(404\) 0 0
\(405\) 8.41150e63 1.10211
\(406\) 0 0
\(407\) 4.01867e63 0.466680
\(408\) 0 0
\(409\) 1.21477e64 1.25104 0.625522 0.780207i \(-0.284886\pi\)
0.625522 + 0.780207i \(0.284886\pi\)
\(410\) 0 0
\(411\) −1.33962e63 −0.122422
\(412\) 0 0
\(413\) −1.15376e63 −0.0936139
\(414\) 0 0
\(415\) 1.73511e64 1.25069
\(416\) 0 0
\(417\) 4.10196e63 0.262820
\(418\) 0 0
\(419\) 9.43742e63 0.537786 0.268893 0.963170i \(-0.413342\pi\)
0.268893 + 0.963170i \(0.413342\pi\)
\(420\) 0 0
\(421\) −7.07535e63 −0.358787 −0.179394 0.983777i \(-0.557414\pi\)
−0.179394 + 0.983777i \(0.557414\pi\)
\(422\) 0 0
\(423\) −3.00593e64 −1.35719
\(424\) 0 0
\(425\) 4.38489e64 1.76372
\(426\) 0 0
\(427\) 6.99002e64 2.50608
\(428\) 0 0
\(429\) −2.00860e62 −0.00642228
\(430\) 0 0
\(431\) −1.47866e64 −0.421867 −0.210933 0.977500i \(-0.567650\pi\)
−0.210933 + 0.977500i \(0.567650\pi\)
\(432\) 0 0
\(433\) 5.40840e64 1.37757 0.688787 0.724963i \(-0.258143\pi\)
0.688787 + 0.724963i \(0.258143\pi\)
\(434\) 0 0
\(435\) −7.29978e63 −0.166082
\(436\) 0 0
\(437\) −4.21020e63 −0.0856066
\(438\) 0 0
\(439\) 3.69191e63 0.0671229 0.0335614 0.999437i \(-0.489315\pi\)
0.0335614 + 0.999437i \(0.489315\pi\)
\(440\) 0 0
\(441\) −9.51316e64 −1.54732
\(442\) 0 0
\(443\) 7.49563e64 1.09123 0.545616 0.838036i \(-0.316296\pi\)
0.545616 + 0.838036i \(0.316296\pi\)
\(444\) 0 0
\(445\) 1.54434e64 0.201336
\(446\) 0 0
\(447\) 2.52437e64 0.294862
\(448\) 0 0
\(449\) −5.65052e64 −0.591634 −0.295817 0.955245i \(-0.595592\pi\)
−0.295817 + 0.955245i \(0.595592\pi\)
\(450\) 0 0
\(451\) −7.86179e64 −0.738238
\(452\) 0 0
\(453\) −3.12835e63 −0.0263578
\(454\) 0 0
\(455\) 7.66544e63 0.0579773
\(456\) 0 0
\(457\) 5.93488e64 0.403148 0.201574 0.979473i \(-0.435394\pi\)
0.201574 + 0.979473i \(0.435394\pi\)
\(458\) 0 0
\(459\) −1.54188e65 −0.941109
\(460\) 0 0
\(461\) 1.06042e65 0.581840 0.290920 0.956747i \(-0.406039\pi\)
0.290920 + 0.956747i \(0.406039\pi\)
\(462\) 0 0
\(463\) −1.86764e65 −0.921637 −0.460818 0.887495i \(-0.652444\pi\)
−0.460818 + 0.887495i \(0.652444\pi\)
\(464\) 0 0
\(465\) −1.49815e65 −0.665209
\(466\) 0 0
\(467\) 2.28278e64 0.0912432 0.0456216 0.998959i \(-0.485473\pi\)
0.0456216 + 0.998959i \(0.485473\pi\)
\(468\) 0 0
\(469\) 3.55138e65 1.27839
\(470\) 0 0
\(471\) 1.50972e65 0.489648
\(472\) 0 0
\(473\) −9.20074e64 −0.268983
\(474\) 0 0
\(475\) −9.20250e64 −0.242612
\(476\) 0 0
\(477\) −7.49306e65 −1.78221
\(478\) 0 0
\(479\) 1.07480e65 0.230732 0.115366 0.993323i \(-0.463196\pi\)
0.115366 + 0.993323i \(0.463196\pi\)
\(480\) 0 0
\(481\) −6.48262e63 −0.0125660
\(482\) 0 0
\(483\) −8.84026e64 −0.154796
\(484\) 0 0
\(485\) −8.95047e65 −1.41636
\(486\) 0 0
\(487\) −9.52950e64 −0.136336 −0.0681680 0.997674i \(-0.521715\pi\)
−0.0681680 + 0.997674i \(0.521715\pi\)
\(488\) 0 0
\(489\) 2.70414e65 0.349915
\(490\) 0 0
\(491\) −7.75386e65 −0.907865 −0.453933 0.891036i \(-0.649979\pi\)
−0.453933 + 0.891036i \(0.649979\pi\)
\(492\) 0 0
\(493\) −7.31033e65 −0.774796
\(494\) 0 0
\(495\) −1.27044e66 −1.21935
\(496\) 0 0
\(497\) −1.37083e66 −1.19193
\(498\) 0 0
\(499\) −1.56740e66 −1.23514 −0.617569 0.786517i \(-0.711882\pi\)
−0.617569 + 0.786517i \(0.711882\pi\)
\(500\) 0 0
\(501\) 3.14676e65 0.224822
\(502\) 0 0
\(503\) 5.11802e65 0.331655 0.165827 0.986155i \(-0.446971\pi\)
0.165827 + 0.986155i \(0.446971\pi\)
\(504\) 0 0
\(505\) 2.70555e66 1.59081
\(506\) 0 0
\(507\) −5.10654e65 −0.272541
\(508\) 0 0
\(509\) −1.63079e66 −0.790335 −0.395167 0.918609i \(-0.629313\pi\)
−0.395167 + 0.918609i \(0.629313\pi\)
\(510\) 0 0
\(511\) 4.35636e66 1.91784
\(512\) 0 0
\(513\) 3.23592e65 0.129456
\(514\) 0 0
\(515\) 2.07168e66 0.753434
\(516\) 0 0
\(517\) 4.14595e66 1.37122
\(518\) 0 0
\(519\) 8.22010e64 0.0247330
\(520\) 0 0
\(521\) 3.56843e65 0.0977133 0.0488566 0.998806i \(-0.484442\pi\)
0.0488566 + 0.998806i \(0.484442\pi\)
\(522\) 0 0
\(523\) 7.86554e65 0.196082 0.0980408 0.995182i \(-0.468742\pi\)
0.0980408 + 0.995182i \(0.468742\pi\)
\(524\) 0 0
\(525\) −1.93227e66 −0.438698
\(526\) 0 0
\(527\) −1.50031e67 −3.10329
\(528\) 0 0
\(529\) −4.66501e66 −0.879405
\(530\) 0 0
\(531\) −3.08463e65 −0.0530136
\(532\) 0 0
\(533\) 1.26821e65 0.0198780
\(534\) 0 0
\(535\) 1.48765e67 2.12732
\(536\) 0 0
\(537\) −4.14202e65 −0.0540558
\(538\) 0 0
\(539\) 1.31211e67 1.56331
\(540\) 0 0
\(541\) 9.49189e66 1.03281 0.516404 0.856345i \(-0.327270\pi\)
0.516404 + 0.856345i \(0.327270\pi\)
\(542\) 0 0
\(543\) −1.14689e66 −0.114005
\(544\) 0 0
\(545\) −2.77013e65 −0.0251643
\(546\) 0 0
\(547\) 1.44217e67 1.19764 0.598819 0.800884i \(-0.295637\pi\)
0.598819 + 0.800884i \(0.295637\pi\)
\(548\) 0 0
\(549\) 1.86881e67 1.41920
\(550\) 0 0
\(551\) 1.53421e66 0.106578
\(552\) 0 0
\(553\) 1.20681e67 0.767140
\(554\) 0 0
\(555\) 3.29451e66 0.191696
\(556\) 0 0
\(557\) 7.89683e65 0.0420727 0.0210364 0.999779i \(-0.493303\pi\)
0.0210364 + 0.999779i \(0.493303\pi\)
\(558\) 0 0
\(559\) 1.48420e65 0.00724273
\(560\) 0 0
\(561\) 1.02226e67 0.457057
\(562\) 0 0
\(563\) −2.05636e67 −0.842643 −0.421321 0.906911i \(-0.638434\pi\)
−0.421321 + 0.906911i \(0.638434\pi\)
\(564\) 0 0
\(565\) 4.30950e66 0.161897
\(566\) 0 0
\(567\) −3.71202e67 −1.27887
\(568\) 0 0
\(569\) −2.99189e67 −0.945573 −0.472787 0.881177i \(-0.656752\pi\)
−0.472787 + 0.881177i \(0.656752\pi\)
\(570\) 0 0
\(571\) 1.50438e67 0.436289 0.218144 0.975917i \(-0.430000\pi\)
0.218144 + 0.975917i \(0.430000\pi\)
\(572\) 0 0
\(573\) 3.03741e66 0.0808564
\(574\) 0 0
\(575\) 1.39828e67 0.341770
\(576\) 0 0
\(577\) 1.42499e67 0.319893 0.159947 0.987126i \(-0.448868\pi\)
0.159947 + 0.987126i \(0.448868\pi\)
\(578\) 0 0
\(579\) 9.33259e66 0.192476
\(580\) 0 0
\(581\) −7.65710e67 −1.45127
\(582\) 0 0
\(583\) 1.03349e68 1.80063
\(584\) 0 0
\(585\) 2.04939e66 0.0328326
\(586\) 0 0
\(587\) 4.53396e67 0.668104 0.334052 0.942555i \(-0.391584\pi\)
0.334052 + 0.942555i \(0.391584\pi\)
\(588\) 0 0
\(589\) 3.14869e67 0.426879
\(590\) 0 0
\(591\) −1.87974e67 −0.234534
\(592\) 0 0
\(593\) 5.39408e67 0.619552 0.309776 0.950810i \(-0.399746\pi\)
0.309776 + 0.950810i \(0.399746\pi\)
\(594\) 0 0
\(595\) −3.90126e68 −4.12609
\(596\) 0 0
\(597\) −4.19325e67 −0.408488
\(598\) 0 0
\(599\) 1.84268e68 1.65383 0.826917 0.562323i \(-0.190092\pi\)
0.826917 + 0.562323i \(0.190092\pi\)
\(600\) 0 0
\(601\) −6.15406e67 −0.509023 −0.254511 0.967070i \(-0.581915\pi\)
−0.254511 + 0.967070i \(0.581915\pi\)
\(602\) 0 0
\(603\) 9.49476e67 0.723952
\(604\) 0 0
\(605\) −2.51262e67 −0.176653
\(606\) 0 0
\(607\) −2.91666e68 −1.89131 −0.945654 0.325175i \(-0.894577\pi\)
−0.945654 + 0.325175i \(0.894577\pi\)
\(608\) 0 0
\(609\) 3.22141e67 0.192718
\(610\) 0 0
\(611\) −6.68795e66 −0.0369218
\(612\) 0 0
\(613\) −1.34057e68 −0.683139 −0.341569 0.939857i \(-0.610958\pi\)
−0.341569 + 0.939857i \(0.610958\pi\)
\(614\) 0 0
\(615\) −6.44511e67 −0.303243
\(616\) 0 0
\(617\) −2.63476e68 −1.14487 −0.572433 0.819951i \(-0.694000\pi\)
−0.572433 + 0.819951i \(0.694000\pi\)
\(618\) 0 0
\(619\) −1.18534e68 −0.475799 −0.237899 0.971290i \(-0.576459\pi\)
−0.237899 + 0.971290i \(0.576459\pi\)
\(620\) 0 0
\(621\) −4.91686e67 −0.182366
\(622\) 0 0
\(623\) −6.81520e67 −0.233626
\(624\) 0 0
\(625\) −3.20461e68 −1.01558
\(626\) 0 0
\(627\) −2.14539e67 −0.0628713
\(628\) 0 0
\(629\) 3.29927e68 0.894288
\(630\) 0 0
\(631\) −4.68843e68 −1.17573 −0.587866 0.808959i \(-0.700032\pi\)
−0.587866 + 0.808959i \(0.700032\pi\)
\(632\) 0 0
\(633\) 1.82380e68 0.423240
\(634\) 0 0
\(635\) 1.03598e69 2.22533
\(636\) 0 0
\(637\) −2.11660e67 −0.0420943
\(638\) 0 0
\(639\) −3.66497e68 −0.674993
\(640\) 0 0
\(641\) −2.45988e68 −0.419654 −0.209827 0.977738i \(-0.567290\pi\)
−0.209827 + 0.977738i \(0.567290\pi\)
\(642\) 0 0
\(643\) −6.14694e68 −0.971605 −0.485803 0.874069i \(-0.661473\pi\)
−0.485803 + 0.874069i \(0.661473\pi\)
\(644\) 0 0
\(645\) −7.54279e67 −0.110489
\(646\) 0 0
\(647\) 9.77317e68 1.32703 0.663515 0.748163i \(-0.269064\pi\)
0.663515 + 0.748163i \(0.269064\pi\)
\(648\) 0 0
\(649\) 4.25450e67 0.0535616
\(650\) 0 0
\(651\) 6.61138e68 0.771894
\(652\) 0 0
\(653\) 8.35256e67 0.0904580 0.0452290 0.998977i \(-0.485598\pi\)
0.0452290 + 0.998977i \(0.485598\pi\)
\(654\) 0 0
\(655\) 2.44281e68 0.245459
\(656\) 0 0
\(657\) 1.16469e69 1.08607
\(658\) 0 0
\(659\) 3.35474e68 0.290380 0.145190 0.989404i \(-0.453621\pi\)
0.145190 + 0.989404i \(0.453621\pi\)
\(660\) 0 0
\(661\) 2.63520e68 0.211776 0.105888 0.994378i \(-0.466231\pi\)
0.105888 + 0.994378i \(0.466231\pi\)
\(662\) 0 0
\(663\) −1.64903e67 −0.0123069
\(664\) 0 0
\(665\) 8.18751e68 0.567572
\(666\) 0 0
\(667\) −2.33117e68 −0.150138
\(668\) 0 0
\(669\) −2.33852e67 −0.0139959
\(670\) 0 0
\(671\) −2.57757e69 −1.43387
\(672\) 0 0
\(673\) 3.25131e69 1.68147 0.840735 0.541447i \(-0.182123\pi\)
0.840735 + 0.541447i \(0.182123\pi\)
\(674\) 0 0
\(675\) −1.07471e69 −0.516831
\(676\) 0 0
\(677\) −2.82926e69 −1.26547 −0.632733 0.774370i \(-0.718067\pi\)
−0.632733 + 0.774370i \(0.718067\pi\)
\(678\) 0 0
\(679\) 3.94987e69 1.64351
\(680\) 0 0
\(681\) 5.51909e68 0.213680
\(682\) 0 0
\(683\) 2.71804e69 0.979374 0.489687 0.871898i \(-0.337111\pi\)
0.489687 + 0.871898i \(0.337111\pi\)
\(684\) 0 0
\(685\) −1.88521e69 −0.632325
\(686\) 0 0
\(687\) 8.34089e68 0.260479
\(688\) 0 0
\(689\) −1.66715e68 −0.0484845
\(690\) 0 0
\(691\) 2.43872e69 0.660618 0.330309 0.943873i \(-0.392847\pi\)
0.330309 + 0.943873i \(0.392847\pi\)
\(692\) 0 0
\(693\) 5.60650e69 1.41491
\(694\) 0 0
\(695\) 5.77255e69 1.35750
\(696\) 0 0
\(697\) −6.45443e69 −1.41467
\(698\) 0 0
\(699\) −7.72980e68 −0.157935
\(700\) 0 0
\(701\) 7.63989e69 1.45544 0.727722 0.685872i \(-0.240579\pi\)
0.727722 + 0.685872i \(0.240579\pi\)
\(702\) 0 0
\(703\) −6.92413e68 −0.123015
\(704\) 0 0
\(705\) 3.39886e69 0.563248
\(706\) 0 0
\(707\) −1.19397e70 −1.84594
\(708\) 0 0
\(709\) 2.36736e69 0.341533 0.170766 0.985312i \(-0.445376\pi\)
0.170766 + 0.985312i \(0.445376\pi\)
\(710\) 0 0
\(711\) 3.22646e69 0.434432
\(712\) 0 0
\(713\) −4.78431e69 −0.601348
\(714\) 0 0
\(715\) −2.82663e68 −0.0331720
\(716\) 0 0
\(717\) −2.66627e69 −0.292203
\(718\) 0 0
\(719\) −9.01007e69 −0.922296 −0.461148 0.887323i \(-0.652562\pi\)
−0.461148 + 0.887323i \(0.652562\pi\)
\(720\) 0 0
\(721\) −9.14237e69 −0.874269
\(722\) 0 0
\(723\) −1.16247e69 −0.103871
\(724\) 0 0
\(725\) −5.09538e69 −0.425496
\(726\) 0 0
\(727\) −1.12644e70 −0.879257 −0.439628 0.898180i \(-0.644890\pi\)
−0.439628 + 0.898180i \(0.644890\pi\)
\(728\) 0 0
\(729\) −7.96172e69 −0.581008
\(730\) 0 0
\(731\) −7.55369e69 −0.515446
\(732\) 0 0
\(733\) 4.32996e69 0.276335 0.138168 0.990409i \(-0.455879\pi\)
0.138168 + 0.990409i \(0.455879\pi\)
\(734\) 0 0
\(735\) 1.07567e70 0.642154
\(736\) 0 0
\(737\) −1.30957e70 −0.731436
\(738\) 0 0
\(739\) 2.10376e70 1.09953 0.549766 0.835319i \(-0.314717\pi\)
0.549766 + 0.835319i \(0.314717\pi\)
\(740\) 0 0
\(741\) 3.46080e67 0.00169289
\(742\) 0 0
\(743\) −3.09494e69 −0.141718 −0.0708592 0.997486i \(-0.522574\pi\)
−0.0708592 + 0.997486i \(0.522574\pi\)
\(744\) 0 0
\(745\) 3.55247e70 1.52301
\(746\) 0 0
\(747\) −2.04716e70 −0.821857
\(748\) 0 0
\(749\) −6.56504e70 −2.46850
\(750\) 0 0
\(751\) −2.36292e70 −0.832281 −0.416140 0.909300i \(-0.636617\pi\)
−0.416140 + 0.909300i \(0.636617\pi\)
\(752\) 0 0
\(753\) 9.46915e69 0.312488
\(754\) 0 0
\(755\) −4.40242e69 −0.136142
\(756\) 0 0
\(757\) 4.84445e70 1.40409 0.702045 0.712132i \(-0.252270\pi\)
0.702045 + 0.712132i \(0.252270\pi\)
\(758\) 0 0
\(759\) 3.25985e69 0.0885673
\(760\) 0 0
\(761\) −1.48623e70 −0.378584 −0.189292 0.981921i \(-0.560619\pi\)
−0.189292 + 0.981921i \(0.560619\pi\)
\(762\) 0 0
\(763\) 1.22247e69 0.0292002
\(764\) 0 0
\(765\) −1.04302e71 −2.33661
\(766\) 0 0
\(767\) −6.86305e67 −0.00144222
\(768\) 0 0
\(769\) −6.67570e70 −1.31614 −0.658069 0.752957i \(-0.728627\pi\)
−0.658069 + 0.752957i \(0.728627\pi\)
\(770\) 0 0
\(771\) −4.52850e69 −0.0837765
\(772\) 0 0
\(773\) 8.33587e70 1.44728 0.723642 0.690176i \(-0.242467\pi\)
0.723642 + 0.690176i \(0.242467\pi\)
\(774\) 0 0
\(775\) −1.04574e71 −1.70424
\(776\) 0 0
\(777\) −1.45388e70 −0.222440
\(778\) 0 0
\(779\) 1.35458e70 0.194597
\(780\) 0 0
\(781\) 5.05494e70 0.681970
\(782\) 0 0
\(783\) 1.79172e70 0.227042
\(784\) 0 0
\(785\) 2.12458e71 2.52910
\(786\) 0 0
\(787\) 1.52479e71 1.70541 0.852706 0.522390i \(-0.174960\pi\)
0.852706 + 0.522390i \(0.174960\pi\)
\(788\) 0 0
\(789\) 3.12868e70 0.328833
\(790\) 0 0
\(791\) −1.90179e70 −0.187862
\(792\) 0 0
\(793\) 4.15795e69 0.0386088
\(794\) 0 0
\(795\) 8.47254e70 0.739638
\(796\) 0 0
\(797\) 8.15352e70 0.669293 0.334647 0.942344i \(-0.391383\pi\)
0.334647 + 0.942344i \(0.391383\pi\)
\(798\) 0 0
\(799\) 3.40377e71 2.62763
\(800\) 0 0
\(801\) −1.82207e70 −0.132303
\(802\) 0 0
\(803\) −1.60641e71 −1.09730
\(804\) 0 0
\(805\) −1.24406e71 −0.799544
\(806\) 0 0
\(807\) −2.87615e70 −0.173944
\(808\) 0 0
\(809\) −2.92840e71 −1.66683 −0.833416 0.552646i \(-0.813618\pi\)
−0.833416 + 0.552646i \(0.813618\pi\)
\(810\) 0 0
\(811\) −2.95951e71 −1.58565 −0.792827 0.609446i \(-0.791392\pi\)
−0.792827 + 0.609446i \(0.791392\pi\)
\(812\) 0 0
\(813\) 4.70822e70 0.237486
\(814\) 0 0
\(815\) 3.80545e71 1.80736
\(816\) 0 0
\(817\) 1.58528e70 0.0709031
\(818\) 0 0
\(819\) −9.04400e69 −0.0380982
\(820\) 0 0
\(821\) 2.31367e71 0.918107 0.459053 0.888409i \(-0.348189\pi\)
0.459053 + 0.888409i \(0.348189\pi\)
\(822\) 0 0
\(823\) −1.40957e70 −0.0526976 −0.0263488 0.999653i \(-0.508388\pi\)
−0.0263488 + 0.999653i \(0.508388\pi\)
\(824\) 0 0
\(825\) 7.12525e70 0.251003
\(826\) 0 0
\(827\) −3.13935e71 −1.04221 −0.521105 0.853493i \(-0.674480\pi\)
−0.521105 + 0.853493i \(0.674480\pi\)
\(828\) 0 0
\(829\) −1.00068e71 −0.313118 −0.156559 0.987669i \(-0.550040\pi\)
−0.156559 + 0.987669i \(0.550040\pi\)
\(830\) 0 0
\(831\) 3.92762e70 0.115852
\(832\) 0 0
\(833\) 1.07722e72 2.99574
\(834\) 0 0
\(835\) 4.42833e71 1.16124
\(836\) 0 0
\(837\) 3.67718e71 0.909370
\(838\) 0 0
\(839\) −7.28357e70 −0.169893 −0.0849466 0.996386i \(-0.527072\pi\)
−0.0849466 + 0.996386i \(0.527072\pi\)
\(840\) 0 0
\(841\) −3.69518e71 −0.813081
\(842\) 0 0
\(843\) 6.08535e70 0.126331
\(844\) 0 0
\(845\) −7.18626e71 −1.40771
\(846\) 0 0
\(847\) 1.10883e71 0.204984
\(848\) 0 0
\(849\) −7.19814e70 −0.125598
\(850\) 0 0
\(851\) 1.05209e71 0.173293
\(852\) 0 0
\(853\) −3.12687e71 −0.486249 −0.243125 0.969995i \(-0.578172\pi\)
−0.243125 + 0.969995i \(0.578172\pi\)
\(854\) 0 0
\(855\) 2.18896e71 0.321417
\(856\) 0 0
\(857\) −1.07117e72 −1.48535 −0.742675 0.669652i \(-0.766443\pi\)
−0.742675 + 0.669652i \(0.766443\pi\)
\(858\) 0 0
\(859\) −6.94017e71 −0.908945 −0.454473 0.890761i \(-0.650172\pi\)
−0.454473 + 0.890761i \(0.650172\pi\)
\(860\) 0 0
\(861\) 2.84425e71 0.351876
\(862\) 0 0
\(863\) −1.51616e72 −1.77207 −0.886037 0.463614i \(-0.846552\pi\)
−0.886037 + 0.463614i \(0.846552\pi\)
\(864\) 0 0
\(865\) 1.15679e71 0.127749
\(866\) 0 0
\(867\) 5.77938e71 0.603134
\(868\) 0 0
\(869\) −4.45012e71 −0.438922
\(870\) 0 0
\(871\) 2.11251e70 0.0196949
\(872\) 0 0
\(873\) 1.05601e72 0.930723
\(874\) 0 0
\(875\) 4.37435e70 0.0364516
\(876\) 0 0
\(877\) −1.82968e72 −1.44173 −0.720867 0.693073i \(-0.756256\pi\)
−0.720867 + 0.693073i \(0.756256\pi\)
\(878\) 0 0
\(879\) 5.21382e71 0.388533
\(880\) 0 0
\(881\) 8.23192e71 0.580218 0.290109 0.956994i \(-0.406308\pi\)
0.290109 + 0.956994i \(0.406308\pi\)
\(882\) 0 0
\(883\) −2.32646e72 −1.55116 −0.775582 0.631247i \(-0.782544\pi\)
−0.775582 + 0.631247i \(0.782544\pi\)
\(884\) 0 0
\(885\) 3.48785e70 0.0220013
\(886\) 0 0
\(887\) −2.02329e72 −1.20762 −0.603809 0.797129i \(-0.706351\pi\)
−0.603809 + 0.797129i \(0.706351\pi\)
\(888\) 0 0
\(889\) −4.57181e72 −2.58223
\(890\) 0 0
\(891\) 1.36881e72 0.731709
\(892\) 0 0
\(893\) −7.14344e71 −0.361449
\(894\) 0 0
\(895\) −5.82893e71 −0.279206
\(896\) 0 0
\(897\) −5.25855e69 −0.00238479
\(898\) 0 0
\(899\) 1.74341e72 0.748666
\(900\) 0 0
\(901\) 8.48479e72 3.45051
\(902\) 0 0
\(903\) 3.32865e71 0.128209
\(904\) 0 0
\(905\) −1.61398e72 −0.588854
\(906\) 0 0
\(907\) −5.38576e72 −1.86152 −0.930762 0.365627i \(-0.880855\pi\)
−0.930762 + 0.365627i \(0.880855\pi\)
\(908\) 0 0
\(909\) −3.19212e72 −1.04536
\(910\) 0 0
\(911\) 4.00433e72 1.24260 0.621298 0.783574i \(-0.286606\pi\)
0.621298 + 0.783574i \(0.286606\pi\)
\(912\) 0 0
\(913\) 2.82356e72 0.830352
\(914\) 0 0
\(915\) −2.11310e72 −0.588983
\(916\) 0 0
\(917\) −1.07802e72 −0.284825
\(918\) 0 0
\(919\) 6.26307e72 1.56876 0.784381 0.620279i \(-0.212981\pi\)
0.784381 + 0.620279i \(0.212981\pi\)
\(920\) 0 0
\(921\) −3.12507e70 −0.00742162
\(922\) 0 0
\(923\) −8.15426e70 −0.0183630
\(924\) 0 0
\(925\) 2.29963e72 0.491118
\(926\) 0 0
\(927\) −2.44425e72 −0.495099
\(928\) 0 0
\(929\) −3.00047e72 −0.576508 −0.288254 0.957554i \(-0.593075\pi\)
−0.288254 + 0.957554i \(0.593075\pi\)
\(930\) 0 0
\(931\) −2.26075e72 −0.412084
\(932\) 0 0
\(933\) 5.80684e71 0.100424
\(934\) 0 0
\(935\) 1.43859e73 2.36076
\(936\) 0 0
\(937\) −3.72864e72 −0.580671 −0.290336 0.956925i \(-0.593767\pi\)
−0.290336 + 0.956925i \(0.593767\pi\)
\(938\) 0 0
\(939\) 2.58202e72 0.381639
\(940\) 0 0
\(941\) −2.09160e72 −0.293449 −0.146724 0.989177i \(-0.546873\pi\)
−0.146724 + 0.989177i \(0.546873\pi\)
\(942\) 0 0
\(943\) −2.05823e72 −0.274131
\(944\) 0 0
\(945\) 9.56174e72 1.20909
\(946\) 0 0
\(947\) −8.20140e72 −0.984721 −0.492361 0.870391i \(-0.663866\pi\)
−0.492361 + 0.870391i \(0.663866\pi\)
\(948\) 0 0
\(949\) 2.59134e71 0.0295463
\(950\) 0 0
\(951\) 3.15574e72 0.341727
\(952\) 0 0
\(953\) 1.17749e73 1.21110 0.605549 0.795808i \(-0.292953\pi\)
0.605549 + 0.795808i \(0.292953\pi\)
\(954\) 0 0
\(955\) 4.27444e72 0.417635
\(956\) 0 0
\(957\) −1.18790e72 −0.110265
\(958\) 0 0
\(959\) 8.31948e72 0.733737
\(960\) 0 0
\(961\) 2.38482e73 1.99863
\(962\) 0 0
\(963\) −1.75519e73 −1.39791
\(964\) 0 0
\(965\) 1.31334e73 0.994168
\(966\) 0 0
\(967\) 3.48982e72 0.251105 0.125552 0.992087i \(-0.459930\pi\)
0.125552 + 0.992087i \(0.459930\pi\)
\(968\) 0 0
\(969\) −1.76134e72 −0.120479
\(970\) 0 0
\(971\) 2.02593e73 1.31751 0.658753 0.752359i \(-0.271084\pi\)
0.658753 + 0.752359i \(0.271084\pi\)
\(972\) 0 0
\(973\) −2.54744e73 −1.57521
\(974\) 0 0
\(975\) −1.14939e71 −0.00675859
\(976\) 0 0
\(977\) −1.10677e73 −0.618926 −0.309463 0.950911i \(-0.600149\pi\)
−0.309463 + 0.950911i \(0.600149\pi\)
\(978\) 0 0
\(979\) 2.51311e72 0.133670
\(980\) 0 0
\(981\) 3.26831e71 0.0165361
\(982\) 0 0
\(983\) 6.73348e72 0.324100 0.162050 0.986783i \(-0.448189\pi\)
0.162050 + 0.986783i \(0.448189\pi\)
\(984\) 0 0
\(985\) −2.64530e73 −1.21140
\(986\) 0 0
\(987\) −1.49993e73 −0.653582
\(988\) 0 0
\(989\) −2.40877e72 −0.0998819
\(990\) 0 0
\(991\) 2.62295e73 1.03511 0.517554 0.855650i \(-0.326843\pi\)
0.517554 + 0.855650i \(0.326843\pi\)
\(992\) 0 0
\(993\) −8.53053e72 −0.320420
\(994\) 0 0
\(995\) −5.90102e73 −2.10990
\(996\) 0 0
\(997\) 2.54966e73 0.867864 0.433932 0.900946i \(-0.357126\pi\)
0.433932 + 0.900946i \(0.357126\pi\)
\(998\) 0 0
\(999\) −8.08631e72 −0.262057
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 16.50.a.b.1.2 3
4.3 odd 2 2.50.a.b.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.50.a.b.1.2 3 4.3 odd 2
16.50.a.b.1.2 3 1.1 even 1 trivial