Properties

Label 4.34.a.a.1.3
Level $4$
Weight $34$
Character 4.1
Self dual yes
Analytic conductor $27.593$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 4.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5931315524\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 65185566 x - 173679864984\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{3}\cdot 7\cdot 11\cdot 29 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3139.05\) of defining polynomial
Character \(\chi\) \(=\) 4.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.34728e8 q^{3} -4.39893e11 q^{5} -5.84388e13 q^{7} +1.25925e16 q^{9} +O(q^{10})\) \(q+1.34728e8 q^{3} -4.39893e11 q^{5} -5.84388e13 q^{7} +1.25925e16 q^{9} +2.38684e17 q^{11} +1.71699e18 q^{13} -5.92657e19 q^{15} +1.90455e20 q^{17} +1.58873e21 q^{19} -7.87333e21 q^{21} +6.96105e21 q^{23} +7.70902e22 q^{25} +9.47601e23 q^{27} -1.30633e24 q^{29} +5.31348e24 q^{31} +3.21574e25 q^{33} +2.57068e25 q^{35} -8.25335e25 q^{37} +2.31327e26 q^{39} -4.09102e26 q^{41} +5.28980e26 q^{43} -5.53935e27 q^{45} -8.51099e26 q^{47} -4.31590e27 q^{49} +2.56596e28 q^{51} +2.58983e28 q^{53} -1.04995e29 q^{55} +2.14047e29 q^{57} +2.29414e29 q^{59} -8.85755e28 q^{61} -7.35891e29 q^{63} -7.55293e29 q^{65} -1.09971e30 q^{67} +9.37847e29 q^{69} +1.24605e30 q^{71} -2.58039e30 q^{73} +1.03862e31 q^{75} -1.39484e31 q^{77} +3.55401e31 q^{79} +5.76657e31 q^{81} -5.14661e31 q^{83} -8.37798e31 q^{85} -1.75999e32 q^{87} +5.16507e31 q^{89} -1.00339e32 q^{91} +7.15874e32 q^{93} -6.98872e32 q^{95} -1.10473e33 q^{97} +3.00564e33 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 92491788q^{3} - 53880683886q^{5} + 4541009914392q^{7} + 6032364433690023q^{9} + O(q^{10}) \) \( 3q + 92491788q^{3} - 53880683886q^{5} + 4541009914392q^{7} + 6032364433690023q^{9} + 227617657302449700q^{11} + 272970442217358762q^{13} - 32142473481241105848q^{15} + 93037188311816716854q^{17} + \)\(13\!\cdots\!28\)\(q^{19} - \)\(22\!\cdots\!52\)\(q^{21} + \)\(11\!\cdots\!96\)\(q^{23} + \)\(25\!\cdots\!21\)\(q^{25} + \)\(11\!\cdots\!24\)\(q^{27} + \)\(28\!\cdots\!78\)\(q^{29} + \)\(15\!\cdots\!28\)\(q^{31} + \)\(51\!\cdots\!80\)\(q^{33} + \)\(10\!\cdots\!48\)\(q^{35} + \)\(34\!\cdots\!42\)\(q^{37} + \)\(13\!\cdots\!68\)\(q^{39} - \)\(67\!\cdots\!62\)\(q^{41} - \)\(15\!\cdots\!20\)\(q^{43} - \)\(82\!\cdots\!86\)\(q^{45} - \)\(72\!\cdots\!76\)\(q^{47} - \)\(47\!\cdots\!09\)\(q^{49} + \)\(24\!\cdots\!36\)\(q^{51} + \)\(54\!\cdots\!34\)\(q^{53} + \)\(72\!\cdots\!80\)\(q^{55} + \)\(27\!\cdots\!48\)\(q^{57} + \)\(20\!\cdots\!76\)\(q^{59} + \)\(59\!\cdots\!66\)\(q^{61} - \)\(12\!\cdots\!68\)\(q^{63} - \)\(22\!\cdots\!12\)\(q^{65} - \)\(29\!\cdots\!28\)\(q^{67} - \)\(15\!\cdots\!16\)\(q^{69} + \)\(16\!\cdots\!44\)\(q^{71} + \)\(13\!\cdots\!02\)\(q^{73} + \)\(20\!\cdots\!28\)\(q^{75} + \)\(20\!\cdots\!20\)\(q^{77} + \)\(43\!\cdots\!04\)\(q^{79} + \)\(60\!\cdots\!27\)\(q^{81} - \)\(71\!\cdots\!64\)\(q^{83} - \)\(13\!\cdots\!84\)\(q^{85} - \)\(25\!\cdots\!92\)\(q^{87} - \)\(19\!\cdots\!98\)\(q^{89} - \)\(38\!\cdots\!08\)\(q^{91} + \)\(55\!\cdots\!88\)\(q^{93} - \)\(20\!\cdots\!76\)\(q^{95} + \)\(56\!\cdots\!22\)\(q^{97} + \)\(22\!\cdots\!00\)\(q^{99} + O(q^{100}) \)

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
\(3\) 1.34728e8 1.80699 0.903496 0.428596i \(-0.140991\pi\)
0.903496 + 0.428596i \(0.140991\pi\)
\(4\) 0 0
\(5\) −4.39893e11 −1.28926 −0.644632 0.764493i \(-0.722989\pi\)
−0.644632 + 0.764493i \(0.722989\pi\)
\(6\) 0 0
\(7\) −5.84388e13 −0.664635 −0.332318 0.943168i \(-0.607831\pi\)
−0.332318 + 0.943168i \(0.607831\pi\)
\(8\) 0 0
\(9\) 1.25925e16 2.26522
\(10\) 0 0
\(11\) 2.38684e17 1.56619 0.783095 0.621902i \(-0.213640\pi\)
0.783095 + 0.621902i \(0.213640\pi\)
\(12\) 0 0
\(13\) 1.71699e18 0.715654 0.357827 0.933788i \(-0.383518\pi\)
0.357827 + 0.933788i \(0.383518\pi\)
\(14\) 0 0
\(15\) −5.92657e19 −2.32969
\(16\) 0 0
\(17\) 1.90455e20 0.949261 0.474631 0.880185i \(-0.342582\pi\)
0.474631 + 0.880185i \(0.342582\pi\)
\(18\) 0 0
\(19\) 1.58873e21 1.26362 0.631810 0.775123i \(-0.282312\pi\)
0.631810 + 0.775123i \(0.282312\pi\)
\(20\) 0 0
\(21\) −7.87333e21 −1.20099
\(22\) 0 0
\(23\) 6.96105e21 0.236682 0.118341 0.992973i \(-0.462242\pi\)
0.118341 + 0.992973i \(0.462242\pi\)
\(24\) 0 0
\(25\) 7.70902e22 0.662199
\(26\) 0 0
\(27\) 9.47601e23 2.28625
\(28\) 0 0
\(29\) −1.30633e24 −0.969362 −0.484681 0.874691i \(-0.661064\pi\)
−0.484681 + 0.874691i \(0.661064\pi\)
\(30\) 0 0
\(31\) 5.31348e24 1.31193 0.655966 0.754790i \(-0.272261\pi\)
0.655966 + 0.754790i \(0.272261\pi\)
\(32\) 0 0
\(33\) 3.21574e25 2.83009
\(34\) 0 0
\(35\) 2.57068e25 0.856890
\(36\) 0 0
\(37\) −8.25335e25 −1.09977 −0.549885 0.835240i \(-0.685328\pi\)
−0.549885 + 0.835240i \(0.685328\pi\)
\(38\) 0 0
\(39\) 2.31327e26 1.29318
\(40\) 0 0
\(41\) −4.09102e26 −1.00207 −0.501035 0.865427i \(-0.667047\pi\)
−0.501035 + 0.865427i \(0.667047\pi\)
\(42\) 0 0
\(43\) 5.28980e26 0.590486 0.295243 0.955422i \(-0.404599\pi\)
0.295243 + 0.955422i \(0.404599\pi\)
\(44\) 0 0
\(45\) −5.53935e27 −2.92047
\(46\) 0 0
\(47\) −8.51099e26 −0.218960 −0.109480 0.993989i \(-0.534919\pi\)
−0.109480 + 0.993989i \(0.534919\pi\)
\(48\) 0 0
\(49\) −4.31590e27 −0.558260
\(50\) 0 0
\(51\) 2.56596e28 1.71531
\(52\) 0 0
\(53\) 2.58983e28 0.917738 0.458869 0.888504i \(-0.348255\pi\)
0.458869 + 0.888504i \(0.348255\pi\)
\(54\) 0 0
\(55\) −1.04995e29 −2.01923
\(56\) 0 0
\(57\) 2.14047e29 2.28335
\(58\) 0 0
\(59\) 2.29414e29 1.38536 0.692679 0.721246i \(-0.256430\pi\)
0.692679 + 0.721246i \(0.256430\pi\)
\(60\) 0 0
\(61\) −8.85755e28 −0.308583 −0.154291 0.988025i \(-0.549309\pi\)
−0.154291 + 0.988025i \(0.549309\pi\)
\(62\) 0 0
\(63\) −7.35891e29 −1.50555
\(64\) 0 0
\(65\) −7.55293e29 −0.922667
\(66\) 0 0
\(67\) −1.09971e30 −0.814789 −0.407394 0.913252i \(-0.633563\pi\)
−0.407394 + 0.913252i \(0.633563\pi\)
\(68\) 0 0
\(69\) 9.37847e29 0.427683
\(70\) 0 0
\(71\) 1.24605e30 0.354627 0.177314 0.984154i \(-0.443259\pi\)
0.177314 + 0.984154i \(0.443259\pi\)
\(72\) 0 0
\(73\) −2.58039e30 −0.464363 −0.232182 0.972672i \(-0.574586\pi\)
−0.232182 + 0.972672i \(0.574586\pi\)
\(74\) 0 0
\(75\) 1.03862e31 1.19659
\(76\) 0 0
\(77\) −1.39484e31 −1.04095
\(78\) 0 0
\(79\) 3.55401e31 1.73729 0.868643 0.495439i \(-0.164993\pi\)
0.868643 + 0.495439i \(0.164993\pi\)
\(80\) 0 0
\(81\) 5.76657e31 1.86601
\(82\) 0 0
\(83\) −5.14661e31 −1.11361 −0.556804 0.830644i \(-0.687973\pi\)
−0.556804 + 0.830644i \(0.687973\pi\)
\(84\) 0 0
\(85\) −8.37798e31 −1.22385
\(86\) 0 0
\(87\) −1.75999e32 −1.75163
\(88\) 0 0
\(89\) 5.16507e31 0.353298 0.176649 0.984274i \(-0.443474\pi\)
0.176649 + 0.984274i \(0.443474\pi\)
\(90\) 0 0
\(91\) −1.00339e32 −0.475649
\(92\) 0 0
\(93\) 7.15874e32 2.37065
\(94\) 0 0
\(95\) −6.98872e32 −1.62914
\(96\) 0 0
\(97\) −1.10473e33 −1.82609 −0.913043 0.407863i \(-0.866274\pi\)
−0.913043 + 0.407863i \(0.866274\pi\)
\(98\) 0 0
\(99\) 3.00564e33 3.54777
\(100\) 0 0
\(101\) −4.29066e31 −0.0364100 −0.0182050 0.999834i \(-0.505795\pi\)
−0.0182050 + 0.999834i \(0.505795\pi\)
\(102\) 0 0
\(103\) −5.10178e32 −0.313262 −0.156631 0.987657i \(-0.550063\pi\)
−0.156631 + 0.987657i \(0.550063\pi\)
\(104\) 0 0
\(105\) 3.46342e33 1.54839
\(106\) 0 0
\(107\) −2.05277e33 −0.672215 −0.336107 0.941824i \(-0.609110\pi\)
−0.336107 + 0.941824i \(0.609110\pi\)
\(108\) 0 0
\(109\) −4.05659e33 −0.978642 −0.489321 0.872104i \(-0.662755\pi\)
−0.489321 + 0.872104i \(0.662755\pi\)
\(110\) 0 0
\(111\) −1.11196e34 −1.98728
\(112\) 0 0
\(113\) 8.02059e32 0.106761 0.0533805 0.998574i \(-0.483000\pi\)
0.0533805 + 0.998574i \(0.483000\pi\)
\(114\) 0 0
\(115\) −3.06212e33 −0.305146
\(116\) 0 0
\(117\) 2.16213e34 1.62112
\(118\) 0 0
\(119\) −1.11300e34 −0.630912
\(120\) 0 0
\(121\) 3.37450e34 1.45295
\(122\) 0 0
\(123\) −5.51175e34 −1.81073
\(124\) 0 0
\(125\) 1.72988e34 0.435514
\(126\) 0 0
\(127\) 7.39847e34 1.43344 0.716721 0.697360i \(-0.245642\pi\)
0.716721 + 0.697360i \(0.245642\pi\)
\(128\) 0 0
\(129\) 7.12683e34 1.06700
\(130\) 0 0
\(131\) 4.87568e34 0.566316 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(132\) 0 0
\(133\) −9.28436e34 −0.839847
\(134\) 0 0
\(135\) −4.16843e35 −2.94758
\(136\) 0 0
\(137\) 9.18664e34 0.509645 0.254822 0.966988i \(-0.417983\pi\)
0.254822 + 0.966988i \(0.417983\pi\)
\(138\) 0 0
\(139\) −1.56271e35 −0.682552 −0.341276 0.939963i \(-0.610859\pi\)
−0.341276 + 0.939963i \(0.610859\pi\)
\(140\) 0 0
\(141\) −1.14667e35 −0.395660
\(142\) 0 0
\(143\) 4.09819e35 1.12085
\(144\) 0 0
\(145\) 5.74645e35 1.24976
\(146\) 0 0
\(147\) −5.81472e35 −1.00877
\(148\) 0 0
\(149\) 3.61831e35 0.502264 0.251132 0.967953i \(-0.419197\pi\)
0.251132 + 0.967953i \(0.419197\pi\)
\(150\) 0 0
\(151\) −1.33622e35 −0.148854 −0.0744269 0.997226i \(-0.523713\pi\)
−0.0744269 + 0.997226i \(0.523713\pi\)
\(152\) 0 0
\(153\) 2.39831e36 2.15029
\(154\) 0 0
\(155\) −2.33736e36 −1.69143
\(156\) 0 0
\(157\) −3.98805e35 −0.233570 −0.116785 0.993157i \(-0.537259\pi\)
−0.116785 + 0.993157i \(0.537259\pi\)
\(158\) 0 0
\(159\) 3.48922e36 1.65835
\(160\) 0 0
\(161\) −4.06795e35 −0.157307
\(162\) 0 0
\(163\) 6.70557e35 0.211515 0.105757 0.994392i \(-0.466273\pi\)
0.105757 + 0.994392i \(0.466273\pi\)
\(164\) 0 0
\(165\) −1.41458e37 −3.64874
\(166\) 0 0
\(167\) −3.15724e36 −0.667553 −0.333777 0.942652i \(-0.608323\pi\)
−0.333777 + 0.942652i \(0.608323\pi\)
\(168\) 0 0
\(169\) −2.80806e36 −0.487839
\(170\) 0 0
\(171\) 2.00061e37 2.86238
\(172\) 0 0
\(173\) −1.21773e36 −0.143810 −0.0719051 0.997411i \(-0.522908\pi\)
−0.0719051 + 0.997411i \(0.522908\pi\)
\(174\) 0 0
\(175\) −4.50505e36 −0.440121
\(176\) 0 0
\(177\) 3.09085e37 2.50333
\(178\) 0 0
\(179\) 3.07461e35 0.0206879 0.0103439 0.999947i \(-0.496707\pi\)
0.0103439 + 0.999947i \(0.496707\pi\)
\(180\) 0 0
\(181\) −3.42619e37 −1.91918 −0.959589 0.281407i \(-0.909199\pi\)
−0.959589 + 0.281407i \(0.909199\pi\)
\(182\) 0 0
\(183\) −1.19336e37 −0.557607
\(184\) 0 0
\(185\) 3.63059e37 1.41789
\(186\) 0 0
\(187\) 4.54587e37 1.48672
\(188\) 0 0
\(189\) −5.53767e37 −1.51952
\(190\) 0 0
\(191\) 7.22927e36 0.166741 0.0833707 0.996519i \(-0.473431\pi\)
0.0833707 + 0.996519i \(0.473431\pi\)
\(192\) 0 0
\(193\) −1.78372e37 −0.346442 −0.173221 0.984883i \(-0.555417\pi\)
−0.173221 + 0.984883i \(0.555417\pi\)
\(194\) 0 0
\(195\) −1.01759e38 −1.66725
\(196\) 0 0
\(197\) −7.82760e37 −1.08377 −0.541884 0.840454i \(-0.682289\pi\)
−0.541884 + 0.840454i \(0.682289\pi\)
\(198\) 0 0
\(199\) 1.29050e38 1.51245 0.756227 0.654309i \(-0.227040\pi\)
0.756227 + 0.654309i \(0.227040\pi\)
\(200\) 0 0
\(201\) −1.48162e38 −1.47232
\(202\) 0 0
\(203\) 7.63403e37 0.644272
\(204\) 0 0
\(205\) 1.79961e38 1.29193
\(206\) 0 0
\(207\) 8.76572e37 0.536138
\(208\) 0 0
\(209\) 3.79206e38 1.97907
\(210\) 0 0
\(211\) −2.33739e38 −1.04249 −0.521243 0.853408i \(-0.674531\pi\)
−0.521243 + 0.853408i \(0.674531\pi\)
\(212\) 0 0
\(213\) 1.67877e38 0.640809
\(214\) 0 0
\(215\) −2.32694e38 −0.761292
\(216\) 0 0
\(217\) −3.10513e38 −0.871956
\(218\) 0 0
\(219\) −3.47650e38 −0.839101
\(220\) 0 0
\(221\) 3.27010e38 0.679343
\(222\) 0 0
\(223\) −3.29898e38 −0.590677 −0.295338 0.955393i \(-0.595432\pi\)
−0.295338 + 0.955393i \(0.595432\pi\)
\(224\) 0 0
\(225\) 9.70759e38 1.50003
\(226\) 0 0
\(227\) −2.00649e38 −0.267924 −0.133962 0.990986i \(-0.542770\pi\)
−0.133962 + 0.990986i \(0.542770\pi\)
\(228\) 0 0
\(229\) −7.20397e38 −0.832313 −0.416156 0.909293i \(-0.636623\pi\)
−0.416156 + 0.909293i \(0.636623\pi\)
\(230\) 0 0
\(231\) −1.87924e39 −1.88098
\(232\) 0 0
\(233\) −1.44309e39 −1.25291 −0.626455 0.779458i \(-0.715495\pi\)
−0.626455 + 0.779458i \(0.715495\pi\)
\(234\) 0 0
\(235\) 3.74392e38 0.282298
\(236\) 0 0
\(237\) 4.78823e39 3.13926
\(238\) 0 0
\(239\) 3.12394e39 1.78294 0.891472 0.453076i \(-0.149673\pi\)
0.891472 + 0.453076i \(0.149673\pi\)
\(240\) 0 0
\(241\) −3.54631e39 −1.76399 −0.881993 0.471263i \(-0.843798\pi\)
−0.881993 + 0.471263i \(0.843798\pi\)
\(242\) 0 0
\(243\) 2.50140e39 1.08562
\(244\) 0 0
\(245\) 1.89853e39 0.719744
\(246\) 0 0
\(247\) 2.72784e39 0.904315
\(248\) 0 0
\(249\) −6.93391e39 −2.01228
\(250\) 0 0
\(251\) −3.92288e39 −0.997674 −0.498837 0.866696i \(-0.666239\pi\)
−0.498837 + 0.866696i \(0.666239\pi\)
\(252\) 0 0
\(253\) 1.66149e39 0.370689
\(254\) 0 0
\(255\) −1.12875e40 −2.21148
\(256\) 0 0
\(257\) 1.13099e40 1.94789 0.973944 0.226789i \(-0.0728227\pi\)
0.973944 + 0.226789i \(0.0728227\pi\)
\(258\) 0 0
\(259\) 4.82316e39 0.730946
\(260\) 0 0
\(261\) −1.64500e40 −2.19582
\(262\) 0 0
\(263\) 8.46897e39 0.996691 0.498346 0.866979i \(-0.333941\pi\)
0.498346 + 0.866979i \(0.333941\pi\)
\(264\) 0 0
\(265\) −1.13925e40 −1.18321
\(266\) 0 0
\(267\) 6.95878e39 0.638407
\(268\) 0 0
\(269\) 2.01049e40 1.63076 0.815379 0.578927i \(-0.196528\pi\)
0.815379 + 0.578927i \(0.196528\pi\)
\(270\) 0 0
\(271\) −1.70128e40 −1.22119 −0.610593 0.791945i \(-0.709069\pi\)
−0.610593 + 0.791945i \(0.709069\pi\)
\(272\) 0 0
\(273\) −1.35184e40 −0.859494
\(274\) 0 0
\(275\) 1.84002e40 1.03713
\(276\) 0 0
\(277\) −2.24387e39 −0.112223 −0.0561116 0.998425i \(-0.517870\pi\)
−0.0561116 + 0.998425i \(0.517870\pi\)
\(278\) 0 0
\(279\) 6.69101e40 2.97182
\(280\) 0 0
\(281\) 1.68371e40 0.664679 0.332339 0.943160i \(-0.392162\pi\)
0.332339 + 0.943160i \(0.392162\pi\)
\(282\) 0 0
\(283\) 8.97402e39 0.315145 0.157573 0.987507i \(-0.449633\pi\)
0.157573 + 0.987507i \(0.449633\pi\)
\(284\) 0 0
\(285\) −9.41575e40 −2.94384
\(286\) 0 0
\(287\) 2.39074e40 0.666012
\(288\) 0 0
\(289\) −3.98130e39 −0.0989032
\(290\) 0 0
\(291\) −1.48837e41 −3.29973
\(292\) 0 0
\(293\) 2.29366e40 0.454164 0.227082 0.973876i \(-0.427081\pi\)
0.227082 + 0.973876i \(0.427081\pi\)
\(294\) 0 0
\(295\) −1.00918e41 −1.78609
\(296\) 0 0
\(297\) 2.26178e41 3.58070
\(298\) 0 0
\(299\) 1.19521e40 0.169383
\(300\) 0 0
\(301\) −3.09129e40 −0.392458
\(302\) 0 0
\(303\) −5.78070e39 −0.0657927
\(304\) 0 0
\(305\) 3.89637e40 0.397844
\(306\) 0 0
\(307\) −4.83147e40 −0.442890 −0.221445 0.975173i \(-0.571077\pi\)
−0.221445 + 0.975173i \(0.571077\pi\)
\(308\) 0 0
\(309\) −6.87352e40 −0.566062
\(310\) 0 0
\(311\) 3.69262e40 0.273394 0.136697 0.990613i \(-0.456351\pi\)
0.136697 + 0.990613i \(0.456351\pi\)
\(312\) 0 0
\(313\) 5.08926e40 0.338979 0.169490 0.985532i \(-0.445788\pi\)
0.169490 + 0.985532i \(0.445788\pi\)
\(314\) 0 0
\(315\) 3.23713e41 1.94105
\(316\) 0 0
\(317\) 2.50845e41 1.35496 0.677481 0.735540i \(-0.263072\pi\)
0.677481 + 0.735540i \(0.263072\pi\)
\(318\) 0 0
\(319\) −3.11800e41 −1.51820
\(320\) 0 0
\(321\) −2.76565e41 −1.21469
\(322\) 0 0
\(323\) 3.02583e41 1.19951
\(324\) 0 0
\(325\) 1.32363e41 0.473906
\(326\) 0 0
\(327\) −5.46535e41 −1.76840
\(328\) 0 0
\(329\) 4.97372e40 0.145529
\(330\) 0 0
\(331\) −4.23320e41 −1.12075 −0.560373 0.828240i \(-0.689342\pi\)
−0.560373 + 0.828240i \(0.689342\pi\)
\(332\) 0 0
\(333\) −1.03930e42 −2.49122
\(334\) 0 0
\(335\) 4.83756e41 1.05048
\(336\) 0 0
\(337\) 2.87768e40 0.0566434 0.0283217 0.999599i \(-0.490984\pi\)
0.0283217 + 0.999599i \(0.490984\pi\)
\(338\) 0 0
\(339\) 1.08060e41 0.192916
\(340\) 0 0
\(341\) 1.26825e42 2.05474
\(342\) 0 0
\(343\) 7.04006e41 1.03567
\(344\) 0 0
\(345\) −4.12552e41 −0.551396
\(346\) 0 0
\(347\) −7.94303e41 −0.965051 −0.482525 0.875882i \(-0.660280\pi\)
−0.482525 + 0.875882i \(0.660280\pi\)
\(348\) 0 0
\(349\) 1.38764e42 1.53341 0.766705 0.641999i \(-0.221895\pi\)
0.766705 + 0.641999i \(0.221895\pi\)
\(350\) 0 0
\(351\) 1.62703e42 1.63616
\(352\) 0 0
\(353\) −1.94987e42 −1.78534 −0.892672 0.450707i \(-0.851172\pi\)
−0.892672 + 0.450707i \(0.851172\pi\)
\(354\) 0 0
\(355\) −5.48127e41 −0.457208
\(356\) 0 0
\(357\) −1.49952e42 −1.14005
\(358\) 0 0
\(359\) −6.33120e40 −0.0438963 −0.0219481 0.999759i \(-0.506987\pi\)
−0.0219481 + 0.999759i \(0.506987\pi\)
\(360\) 0 0
\(361\) 9.43304e41 0.596737
\(362\) 0 0
\(363\) 4.54639e42 2.62547
\(364\) 0 0
\(365\) 1.13509e42 0.598686
\(366\) 0 0
\(367\) −3.11073e42 −1.49924 −0.749621 0.661867i \(-0.769764\pi\)
−0.749621 + 0.661867i \(0.769764\pi\)
\(368\) 0 0
\(369\) −5.15163e42 −2.26991
\(370\) 0 0
\(371\) −1.51346e42 −0.609961
\(372\) 0 0
\(373\) −3.97777e42 −1.46705 −0.733524 0.679663i \(-0.762126\pi\)
−0.733524 + 0.679663i \(0.762126\pi\)
\(374\) 0 0
\(375\) 2.33063e42 0.786970
\(376\) 0 0
\(377\) −2.24296e42 −0.693728
\(378\) 0 0
\(379\) −4.00371e41 −0.113479 −0.0567395 0.998389i \(-0.518070\pi\)
−0.0567395 + 0.998389i \(0.518070\pi\)
\(380\) 0 0
\(381\) 9.96779e42 2.59022
\(382\) 0 0
\(383\) −4.12729e42 −0.983751 −0.491875 0.870666i \(-0.663688\pi\)
−0.491875 + 0.870666i \(0.663688\pi\)
\(384\) 0 0
\(385\) 6.13580e42 1.34205
\(386\) 0 0
\(387\) 6.66119e42 1.33758
\(388\) 0 0
\(389\) −5.15355e41 −0.0950470 −0.0475235 0.998870i \(-0.515133\pi\)
−0.0475235 + 0.998870i \(0.515133\pi\)
\(390\) 0 0
\(391\) 1.32577e42 0.224673
\(392\) 0 0
\(393\) 6.56890e42 1.02333
\(394\) 0 0
\(395\) −1.56338e43 −2.23982
\(396\) 0 0
\(397\) 3.89760e42 0.513752 0.256876 0.966444i \(-0.417307\pi\)
0.256876 + 0.966444i \(0.417307\pi\)
\(398\) 0 0
\(399\) −1.25086e43 −1.51760
\(400\) 0 0
\(401\) 1.58967e43 1.77593 0.887965 0.459911i \(-0.152119\pi\)
0.887965 + 0.459911i \(0.152119\pi\)
\(402\) 0 0
\(403\) 9.12322e42 0.938890
\(404\) 0 0
\(405\) −2.53667e43 −2.40578
\(406\) 0 0
\(407\) −1.96995e43 −1.72245
\(408\) 0 0
\(409\) 1.48346e43 1.19630 0.598150 0.801384i \(-0.295903\pi\)
0.598150 + 0.801384i \(0.295903\pi\)
\(410\) 0 0
\(411\) 1.23770e43 0.920924
\(412\) 0 0
\(413\) −1.34067e43 −0.920758
\(414\) 0 0
\(415\) 2.26396e43 1.43573
\(416\) 0 0
\(417\) −2.10541e43 −1.23337
\(418\) 0 0
\(419\) −6.90718e41 −0.0373912 −0.0186956 0.999825i \(-0.505951\pi\)
−0.0186956 + 0.999825i \(0.505951\pi\)
\(420\) 0 0
\(421\) −2.87526e43 −1.43887 −0.719437 0.694558i \(-0.755600\pi\)
−0.719437 + 0.694558i \(0.755600\pi\)
\(422\) 0 0
\(423\) −1.07175e43 −0.495994
\(424\) 0 0
\(425\) 1.46822e43 0.628600
\(426\) 0 0
\(427\) 5.17624e42 0.205095
\(428\) 0 0
\(429\) 5.52140e43 2.02537
\(430\) 0 0
\(431\) −1.01416e43 −0.344535 −0.172267 0.985050i \(-0.555109\pi\)
−0.172267 + 0.985050i \(0.555109\pi\)
\(432\) 0 0
\(433\) 3.96098e43 1.24667 0.623336 0.781954i \(-0.285777\pi\)
0.623336 + 0.781954i \(0.285777\pi\)
\(434\) 0 0
\(435\) 7.74206e43 2.25831
\(436\) 0 0
\(437\) 1.10593e43 0.299077
\(438\) 0 0
\(439\) −4.74746e43 −1.19068 −0.595342 0.803472i \(-0.702984\pi\)
−0.595342 + 0.803472i \(0.702984\pi\)
\(440\) 0 0
\(441\) −5.43481e43 −1.26458
\(442\) 0 0
\(443\) −3.78302e43 −0.816915 −0.408458 0.912777i \(-0.633933\pi\)
−0.408458 + 0.912777i \(0.633933\pi\)
\(444\) 0 0
\(445\) −2.27208e43 −0.455494
\(446\) 0 0
\(447\) 4.87486e43 0.907587
\(448\) 0 0
\(449\) −9.87539e43 −1.70801 −0.854004 0.520267i \(-0.825832\pi\)
−0.854004 + 0.520267i \(0.825832\pi\)
\(450\) 0 0
\(451\) −9.76463e43 −1.56943
\(452\) 0 0
\(453\) −1.80027e43 −0.268978
\(454\) 0 0
\(455\) 4.41384e43 0.613237
\(456\) 0 0
\(457\) 1.20565e44 1.55814 0.779069 0.626939i \(-0.215692\pi\)
0.779069 + 0.626939i \(0.215692\pi\)
\(458\) 0 0
\(459\) 1.80476e44 2.17025
\(460\) 0 0
\(461\) −9.88539e43 −1.10644 −0.553221 0.833035i \(-0.686601\pi\)
−0.553221 + 0.833035i \(0.686601\pi\)
\(462\) 0 0
\(463\) −4.50681e43 −0.469660 −0.234830 0.972036i \(-0.575453\pi\)
−0.234830 + 0.972036i \(0.575453\pi\)
\(464\) 0 0
\(465\) −3.14908e44 −3.05639
\(466\) 0 0
\(467\) 1.47361e44 1.33246 0.666228 0.745748i \(-0.267908\pi\)
0.666228 + 0.745748i \(0.267908\pi\)
\(468\) 0 0
\(469\) 6.42659e43 0.541537
\(470\) 0 0
\(471\) −5.37301e43 −0.422058
\(472\) 0 0
\(473\) 1.26259e44 0.924814
\(474\) 0 0
\(475\) 1.22476e44 0.836769
\(476\) 0 0
\(477\) 3.26124e44 2.07888
\(478\) 0 0
\(479\) −1.57341e44 −0.936067 −0.468034 0.883711i \(-0.655037\pi\)
−0.468034 + 0.883711i \(0.655037\pi\)
\(480\) 0 0
\(481\) −1.41709e44 −0.787055
\(482\) 0 0
\(483\) −5.48066e43 −0.284253
\(484\) 0 0
\(485\) 4.85961e44 2.35431
\(486\) 0 0
\(487\) −3.18979e44 −1.44389 −0.721945 0.691950i \(-0.756752\pi\)
−0.721945 + 0.691950i \(0.756752\pi\)
\(488\) 0 0
\(489\) 9.03427e43 0.382205
\(490\) 0 0
\(491\) 3.97831e44 1.57345 0.786727 0.617301i \(-0.211774\pi\)
0.786727 + 0.617301i \(0.211774\pi\)
\(492\) 0 0
\(493\) −2.48797e44 −0.920177
\(494\) 0 0
\(495\) −1.32216e45 −4.57401
\(496\) 0 0
\(497\) −7.28175e43 −0.235698
\(498\) 0 0
\(499\) 3.71007e44 1.12389 0.561945 0.827175i \(-0.310053\pi\)
0.561945 + 0.827175i \(0.310053\pi\)
\(500\) 0 0
\(501\) −4.25368e44 −1.20626
\(502\) 0 0
\(503\) 3.58796e44 0.952743 0.476371 0.879244i \(-0.341952\pi\)
0.476371 + 0.879244i \(0.341952\pi\)
\(504\) 0 0
\(505\) 1.88743e43 0.0469421
\(506\) 0 0
\(507\) −3.78324e44 −0.881522
\(508\) 0 0
\(509\) 2.88589e44 0.630139 0.315070 0.949069i \(-0.397972\pi\)
0.315070 + 0.949069i \(0.397972\pi\)
\(510\) 0 0
\(511\) 1.50795e44 0.308632
\(512\) 0 0
\(513\) 1.50549e45 2.88895
\(514\) 0 0
\(515\) 2.24424e44 0.403877
\(516\) 0 0
\(517\) −2.03144e44 −0.342934
\(518\) 0 0
\(519\) −1.64062e44 −0.259864
\(520\) 0 0
\(521\) −4.02237e44 −0.597943 −0.298971 0.954262i \(-0.596644\pi\)
−0.298971 + 0.954262i \(0.596644\pi\)
\(522\) 0 0
\(523\) −6.82047e44 −0.951780 −0.475890 0.879505i \(-0.657874\pi\)
−0.475890 + 0.879505i \(0.657874\pi\)
\(524\) 0 0
\(525\) −6.06956e44 −0.795296
\(526\) 0 0
\(527\) 1.01198e45 1.24537
\(528\) 0 0
\(529\) −8.16549e44 −0.943981
\(530\) 0 0
\(531\) 2.88890e45 3.13815
\(532\) 0 0
\(533\) −7.02426e44 −0.717136
\(534\) 0 0
\(535\) 9.02998e44 0.866662
\(536\) 0 0
\(537\) 4.14236e43 0.0373828
\(538\) 0 0
\(539\) −1.03014e45 −0.874341
\(540\) 0 0
\(541\) 2.04672e44 0.163420 0.0817098 0.996656i \(-0.473962\pi\)
0.0817098 + 0.996656i \(0.473962\pi\)
\(542\) 0 0
\(543\) −4.61603e45 −3.46794
\(544\) 0 0
\(545\) 1.78446e45 1.26173
\(546\) 0 0
\(547\) −5.26369e44 −0.350348 −0.175174 0.984538i \(-0.556049\pi\)
−0.175174 + 0.984538i \(0.556049\pi\)
\(548\) 0 0
\(549\) −1.11539e45 −0.699009
\(550\) 0 0
\(551\) −2.07541e45 −1.22491
\(552\) 0 0
\(553\) −2.07692e45 −1.15466
\(554\) 0 0
\(555\) 4.89141e45 2.56212
\(556\) 0 0
\(557\) −3.31476e45 −1.63622 −0.818109 0.575064i \(-0.804977\pi\)
−0.818109 + 0.575064i \(0.804977\pi\)
\(558\) 0 0
\(559\) 9.08255e44 0.422584
\(560\) 0 0
\(561\) 6.12455e45 2.68650
\(562\) 0 0
\(563\) 4.46448e45 1.84664 0.923321 0.384028i \(-0.125464\pi\)
0.923321 + 0.384028i \(0.125464\pi\)
\(564\) 0 0
\(565\) −3.52820e44 −0.137643
\(566\) 0 0
\(567\) −3.36991e45 −1.24022
\(568\) 0 0
\(569\) −1.55500e45 −0.539981 −0.269991 0.962863i \(-0.587021\pi\)
−0.269991 + 0.962863i \(0.587021\pi\)
\(570\) 0 0
\(571\) 4.45599e45 1.46032 0.730161 0.683275i \(-0.239445\pi\)
0.730161 + 0.683275i \(0.239445\pi\)
\(572\) 0 0
\(573\) 9.73983e44 0.301301
\(574\) 0 0
\(575\) 5.36629e44 0.156731
\(576\) 0 0
\(577\) −4.82092e45 −1.32963 −0.664813 0.747010i \(-0.731489\pi\)
−0.664813 + 0.747010i \(0.731489\pi\)
\(578\) 0 0
\(579\) −2.40316e45 −0.626017
\(580\) 0 0
\(581\) 3.00762e45 0.740144
\(582\) 0 0
\(583\) 6.18151e45 1.43735
\(584\) 0 0
\(585\) −9.51103e45 −2.09005
\(586\) 0 0
\(587\) −3.94339e45 −0.819108 −0.409554 0.912286i \(-0.634316\pi\)
−0.409554 + 0.912286i \(0.634316\pi\)
\(588\) 0 0
\(589\) 8.44171e45 1.65778
\(590\) 0 0
\(591\) −1.05460e46 −1.95836
\(592\) 0 0
\(593\) −5.04718e45 −0.886434 −0.443217 0.896414i \(-0.646163\pi\)
−0.443217 + 0.896414i \(0.646163\pi\)
\(594\) 0 0
\(595\) 4.89599e45 0.813412
\(596\) 0 0
\(597\) 1.73866e46 2.73300
\(598\) 0 0
\(599\) −7.97373e45 −1.18609 −0.593047 0.805168i \(-0.702075\pi\)
−0.593047 + 0.805168i \(0.702075\pi\)
\(600\) 0 0
\(601\) 4.22349e45 0.594624 0.297312 0.954780i \(-0.403910\pi\)
0.297312 + 0.954780i \(0.403910\pi\)
\(602\) 0 0
\(603\) −1.38482e46 −1.84568
\(604\) 0 0
\(605\) −1.48442e46 −1.87324
\(606\) 0 0
\(607\) −1.34096e46 −1.60252 −0.801258 0.598320i \(-0.795835\pi\)
−0.801258 + 0.598320i \(0.795835\pi\)
\(608\) 0 0
\(609\) 1.02852e46 1.16419
\(610\) 0 0
\(611\) −1.46133e45 −0.156700
\(612\) 0 0
\(613\) 1.30841e46 1.32937 0.664687 0.747122i \(-0.268565\pi\)
0.664687 + 0.747122i \(0.268565\pi\)
\(614\) 0 0
\(615\) 2.42458e46 2.33451
\(616\) 0 0
\(617\) 1.63656e46 1.49358 0.746788 0.665062i \(-0.231595\pi\)
0.746788 + 0.665062i \(0.231595\pi\)
\(618\) 0 0
\(619\) −2.40690e45 −0.208239 −0.104120 0.994565i \(-0.533202\pi\)
−0.104120 + 0.994565i \(0.533202\pi\)
\(620\) 0 0
\(621\) 6.59631e45 0.541115
\(622\) 0 0
\(623\) −3.01840e45 −0.234814
\(624\) 0 0
\(625\) −1.65841e46 −1.22369
\(626\) 0 0
\(627\) 5.10895e46 3.57617
\(628\) 0 0
\(629\) −1.57189e46 −1.04397
\(630\) 0 0
\(631\) −1.53069e46 −0.964722 −0.482361 0.875972i \(-0.660221\pi\)
−0.482361 + 0.875972i \(0.660221\pi\)
\(632\) 0 0
\(633\) −3.14911e46 −1.88376
\(634\) 0 0
\(635\) −3.25453e46 −1.84808
\(636\) 0 0
\(637\) −7.41038e45 −0.399521
\(638\) 0 0
\(639\) 1.56909e46 0.803310
\(640\) 0 0
\(641\) 9.03309e45 0.439217 0.219608 0.975588i \(-0.429522\pi\)
0.219608 + 0.975588i \(0.429522\pi\)
\(642\) 0 0
\(643\) −6.28642e45 −0.290351 −0.145175 0.989406i \(-0.546375\pi\)
−0.145175 + 0.989406i \(0.546375\pi\)
\(644\) 0 0
\(645\) −3.13504e46 −1.37565
\(646\) 0 0
\(647\) 4.05377e46 1.69020 0.845100 0.534608i \(-0.179541\pi\)
0.845100 + 0.534608i \(0.179541\pi\)
\(648\) 0 0
\(649\) 5.47576e46 2.16974
\(650\) 0 0
\(651\) −4.18348e46 −1.57562
\(652\) 0 0
\(653\) −1.97437e46 −0.706903 −0.353452 0.935453i \(-0.614992\pi\)
−0.353452 + 0.935453i \(0.614992\pi\)
\(654\) 0 0
\(655\) −2.14478e46 −0.730131
\(656\) 0 0
\(657\) −3.24936e46 −1.05189
\(658\) 0 0
\(659\) 1.63443e46 0.503220 0.251610 0.967829i \(-0.419040\pi\)
0.251610 + 0.967829i \(0.419040\pi\)
\(660\) 0 0
\(661\) 5.92845e45 0.173627 0.0868134 0.996225i \(-0.472332\pi\)
0.0868134 + 0.996225i \(0.472332\pi\)
\(662\) 0 0
\(663\) 4.40574e46 1.22757
\(664\) 0 0
\(665\) 4.08412e46 1.08278
\(666\) 0 0
\(667\) −9.09343e45 −0.229431
\(668\) 0 0
\(669\) −4.44465e46 −1.06735
\(670\) 0 0
\(671\) −2.11416e46 −0.483299
\(672\) 0 0
\(673\) 1.45328e46 0.316302 0.158151 0.987415i \(-0.449447\pi\)
0.158151 + 0.987415i \(0.449447\pi\)
\(674\) 0 0
\(675\) 7.30508e46 1.51395
\(676\) 0 0
\(677\) 7.75928e46 1.53147 0.765735 0.643156i \(-0.222375\pi\)
0.765735 + 0.643156i \(0.222375\pi\)
\(678\) 0 0
\(679\) 6.45589e46 1.21368
\(680\) 0 0
\(681\) −2.70331e46 −0.484136
\(682\) 0 0
\(683\) −6.20051e45 −0.105800 −0.0529000 0.998600i \(-0.516846\pi\)
−0.0529000 + 0.998600i \(0.516846\pi\)
\(684\) 0 0
\(685\) −4.04113e46 −0.657066
\(686\) 0 0
\(687\) −9.70575e46 −1.50398
\(688\) 0 0
\(689\) 4.44672e46 0.656783
\(690\) 0 0
\(691\) 1.81450e46 0.255488 0.127744 0.991807i \(-0.459226\pi\)
0.127744 + 0.991807i \(0.459226\pi\)
\(692\) 0 0
\(693\) −1.75646e47 −2.35797
\(694\) 0 0
\(695\) 6.87427e46 0.879989
\(696\) 0 0
\(697\) −7.79157e46 −0.951227
\(698\) 0 0
\(699\) −1.94424e47 −2.26400
\(700\) 0 0
\(701\) 1.34726e47 1.49660 0.748298 0.663363i \(-0.230871\pi\)
0.748298 + 0.663363i \(0.230871\pi\)
\(702\) 0 0
\(703\) −1.31124e47 −1.38969
\(704\) 0 0
\(705\) 5.04410e46 0.510110
\(706\) 0 0
\(707\) 2.50741e45 0.0241994
\(708\) 0 0
\(709\) −1.04063e47 −0.958594 −0.479297 0.877653i \(-0.659108\pi\)
−0.479297 + 0.877653i \(0.659108\pi\)
\(710\) 0 0
\(711\) 4.47539e47 3.93534
\(712\) 0 0
\(713\) 3.69875e46 0.310511
\(714\) 0 0
\(715\) −1.80276e47 −1.44507
\(716\) 0 0
\(717\) 4.20882e47 3.22177
\(718\) 0 0
\(719\) −6.64979e46 −0.486162 −0.243081 0.970006i \(-0.578158\pi\)
−0.243081 + 0.970006i \(0.578158\pi\)
\(720\) 0 0
\(721\) 2.98142e46 0.208205
\(722\) 0 0
\(723\) −4.77786e47 −3.18751
\(724\) 0 0
\(725\) −1.00705e47 −0.641911
\(726\) 0 0
\(727\) 7.19404e46 0.438183 0.219091 0.975704i \(-0.429691\pi\)
0.219091 + 0.975704i \(0.429691\pi\)
\(728\) 0 0
\(729\) 1.64405e46 0.0956999
\(730\) 0 0
\(731\) 1.00747e47 0.560526
\(732\) 0 0
\(733\) 2.85394e47 1.51785 0.758925 0.651178i \(-0.225725\pi\)
0.758925 + 0.651178i \(0.225725\pi\)
\(734\) 0 0
\(735\) 2.55785e47 1.30057
\(736\) 0 0
\(737\) −2.62484e47 −1.27611
\(738\) 0 0
\(739\) 1.92837e47 0.896512 0.448256 0.893905i \(-0.352045\pi\)
0.448256 + 0.893905i \(0.352045\pi\)
\(740\) 0 0
\(741\) 3.67516e47 1.63409
\(742\) 0 0
\(743\) −1.73889e47 −0.739532 −0.369766 0.929125i \(-0.620562\pi\)
−0.369766 + 0.929125i \(0.620562\pi\)
\(744\) 0 0
\(745\) −1.59167e47 −0.647550
\(746\) 0 0
\(747\) −6.48087e47 −2.52257
\(748\) 0 0
\(749\) 1.19961e47 0.446778
\(750\) 0 0
\(751\) −8.03243e45 −0.0286278 −0.0143139 0.999898i \(-0.504556\pi\)
−0.0143139 + 0.999898i \(0.504556\pi\)
\(752\) 0 0
\(753\) −5.28521e47 −1.80279
\(754\) 0 0
\(755\) 5.87795e46 0.191912
\(756\) 0 0
\(757\) −4.29145e47 −1.34129 −0.670643 0.741780i \(-0.733982\pi\)
−0.670643 + 0.741780i \(0.733982\pi\)
\(758\) 0 0
\(759\) 2.23849e47 0.669833
\(760\) 0 0
\(761\) 1.16851e47 0.334799 0.167400 0.985889i \(-0.446463\pi\)
0.167400 + 0.985889i \(0.446463\pi\)
\(762\) 0 0
\(763\) 2.37062e47 0.650440
\(764\) 0 0
\(765\) −1.05500e48 −2.77229
\(766\) 0 0
\(767\) 3.93903e47 0.991438
\(768\) 0 0
\(769\) 3.48880e46 0.0841185 0.0420592 0.999115i \(-0.486608\pi\)
0.0420592 + 0.999115i \(0.486608\pi\)
\(770\) 0 0
\(771\) 1.52376e48 3.51982
\(772\) 0 0
\(773\) −1.23838e47 −0.274089 −0.137044 0.990565i \(-0.543760\pi\)
−0.137044 + 0.990565i \(0.543760\pi\)
\(774\) 0 0
\(775\) 4.09617e47 0.868761
\(776\) 0 0
\(777\) 6.49813e47 1.32081
\(778\) 0 0
\(779\) −6.49955e47 −1.26624
\(780\) 0 0
\(781\) 2.97412e47 0.555414
\(782\) 0 0
\(783\) −1.23788e48 −2.21620
\(784\) 0 0
\(785\) 1.75431e47 0.301133
\(786\) 0 0
\(787\) 6.66573e47 1.09715 0.548573 0.836102i \(-0.315171\pi\)
0.548573 + 0.836102i \(0.315171\pi\)
\(788\) 0 0
\(789\) 1.14101e48 1.80101
\(790\) 0 0
\(791\) −4.68714e46 −0.0709571
\(792\) 0 0
\(793\) −1.52084e47 −0.220838
\(794\) 0 0
\(795\) −1.53488e48 −2.13805
\(796\) 0 0
\(797\) −1.00743e48 −1.34633 −0.673166 0.739492i \(-0.735066\pi\)
−0.673166 + 0.739492i \(0.735066\pi\)
\(798\) 0 0
\(799\) −1.62096e47 −0.207851
\(800\) 0 0
\(801\) 6.50412e47 0.800298
\(802\) 0 0
\(803\) −6.15898e47 −0.727281
\(804\) 0 0
\(805\) 1.78946e47 0.202811
\(806\) 0 0
\(807\) 2.70869e48 2.94677
\(808\) 0 0
\(809\) −5.01023e47 −0.523247 −0.261624 0.965170i \(-0.584258\pi\)
−0.261624 + 0.965170i \(0.584258\pi\)
\(810\) 0 0
\(811\) −1.96073e46 −0.0196596 −0.00982978 0.999952i \(-0.503129\pi\)
−0.00982978 + 0.999952i \(0.503129\pi\)
\(812\) 0 0
\(813\) −2.29209e48 −2.20667
\(814\) 0 0
\(815\) −2.94973e47 −0.272698
\(816\) 0 0
\(817\) 8.40408e47 0.746151
\(818\) 0 0
\(819\) −1.26352e48 −1.07745
\(820\) 0 0
\(821\) −2.20701e48 −1.80776 −0.903881 0.427783i \(-0.859295\pi\)
−0.903881 + 0.427783i \(0.859295\pi\)
\(822\) 0 0
\(823\) −1.22606e48 −0.964753 −0.482377 0.875964i \(-0.660226\pi\)
−0.482377 + 0.875964i \(0.660226\pi\)
\(824\) 0 0
\(825\) 2.47902e48 1.87409
\(826\) 0 0
\(827\) 5.69912e47 0.413968 0.206984 0.978344i \(-0.433635\pi\)
0.206984 + 0.978344i \(0.433635\pi\)
\(828\) 0 0
\(829\) 2.85621e47 0.199361 0.0996806 0.995019i \(-0.468218\pi\)
0.0996806 + 0.995019i \(0.468218\pi\)
\(830\) 0 0
\(831\) −3.02312e47 −0.202786
\(832\) 0 0
\(833\) −8.21987e47 −0.529934
\(834\) 0 0
\(835\) 1.38885e48 0.860652
\(836\) 0 0
\(837\) 5.03506e48 2.99940
\(838\) 0 0
\(839\) 1.07133e48 0.613553 0.306777 0.951782i \(-0.400750\pi\)
0.306777 + 0.951782i \(0.400750\pi\)
\(840\) 0 0
\(841\) −1.09579e47 −0.0603383
\(842\) 0 0
\(843\) 2.26842e48 1.20107
\(844\) 0 0
\(845\) 1.23525e48 0.628953
\(846\) 0 0
\(847\) −1.97202e48 −0.965683
\(848\) 0 0
\(849\) 1.20905e48 0.569465
\(850\) 0 0
\(851\) −5.74520e47 −0.260296
\(852\) 0 0
\(853\) −3.08488e48 −1.34456 −0.672278 0.740299i \(-0.734684\pi\)
−0.672278 + 0.740299i \(0.734684\pi\)
\(854\) 0 0
\(855\) −8.80056e48 −3.69036
\(856\) 0 0
\(857\) −1.63429e48 −0.659394 −0.329697 0.944087i \(-0.606947\pi\)
−0.329697 + 0.944087i \(0.606947\pi\)
\(858\) 0 0
\(859\) 9.43457e47 0.366298 0.183149 0.983085i \(-0.441371\pi\)
0.183149 + 0.983085i \(0.441371\pi\)
\(860\) 0 0
\(861\) 3.22100e48 1.20348
\(862\) 0 0
\(863\) 9.06251e47 0.325889 0.162945 0.986635i \(-0.447901\pi\)
0.162945 + 0.986635i \(0.447901\pi\)
\(864\) 0 0
\(865\) 5.35670e47 0.185409
\(866\) 0 0
\(867\) −5.36392e47 −0.178717
\(868\) 0 0
\(869\) 8.48285e48 2.72092
\(870\) 0 0
\(871\) −1.88820e48 −0.583107
\(872\) 0 0
\(873\) −1.39113e49 −4.13649
\(874\) 0 0
\(875\) −1.01092e48 −0.289458
\(876\) 0 0
\(877\) 1.93817e48 0.534441 0.267220 0.963635i \(-0.413895\pi\)
0.267220 + 0.963635i \(0.413895\pi\)
\(878\) 0 0
\(879\) 3.09019e48 0.820672
\(880\) 0 0
\(881\) 4.83488e48 1.23675 0.618377 0.785881i \(-0.287790\pi\)
0.618377 + 0.785881i \(0.287790\pi\)
\(882\) 0 0
\(883\) 5.63759e48 1.38913 0.694563 0.719432i \(-0.255598\pi\)
0.694563 + 0.719432i \(0.255598\pi\)
\(884\) 0 0
\(885\) −1.35964e49 −3.22746
\(886\) 0 0
\(887\) 1.24504e48 0.284737 0.142369 0.989814i \(-0.454528\pi\)
0.142369 + 0.989814i \(0.454528\pi\)
\(888\) 0 0
\(889\) −4.32357e48 −0.952717
\(890\) 0 0
\(891\) 1.37639e49 2.92253
\(892\) 0 0
\(893\) −1.35217e48 −0.276683
\(894\) 0 0
\(895\) −1.35250e47 −0.0266721
\(896\) 0 0
\(897\) 1.61028e48 0.306073
\(898\) 0 0
\(899\) −6.94116e48 −1.27174
\(900\) 0 0
\(901\) 4.93246e48 0.871173
\(902\) 0 0
\(903\) −4.16483e48 −0.709169
\(904\) 0 0
\(905\) 1.50716e49 2.47432
\(906\) 0 0
\(907\) 7.99759e48 1.26601 0.633007 0.774146i \(-0.281820\pi\)
0.633007 + 0.774146i \(0.281820\pi\)
\(908\) 0 0
\(909\) −5.40301e47 −0.0824768
\(910\) 0 0
\(911\) 8.86733e48 1.30539 0.652694 0.757621i \(-0.273639\pi\)
0.652694 + 0.757621i \(0.273639\pi\)
\(912\) 0 0
\(913\) −1.22841e49 −1.74412
\(914\) 0 0
\(915\) 5.24950e48 0.718902
\(916\) 0 0
\(917\) −2.84929e48 −0.376394
\(918\) 0 0
\(919\) −1.32606e49 −1.68989 −0.844943 0.534856i \(-0.820366\pi\)
−0.844943 + 0.534856i \(0.820366\pi\)
\(920\) 0 0
\(921\) −6.50933e48 −0.800299
\(922\) 0 0
\(923\) 2.13946e48 0.253791
\(924\) 0 0
\(925\) −6.36252e48 −0.728267
\(926\) 0 0
\(927\) −6.42443e48 −0.709608
\(928\) 0 0
\(929\) −7.45085e48 −0.794231 −0.397115 0.917769i \(-0.629989\pi\)
−0.397115 + 0.917769i \(0.629989\pi\)
\(930\) 0 0
\(931\) −6.85682e48 −0.705429
\(932\) 0 0
\(933\) 4.97499e48 0.494021
\(934\) 0 0
\(935\) −1.99969e49 −1.91678
\(936\) 0 0
\(937\) 4.58685e48 0.424435 0.212218 0.977222i \(-0.431931\pi\)
0.212218 + 0.977222i \(0.431931\pi\)
\(938\) 0 0
\(939\) 6.85665e48 0.612533
\(940\) 0 0
\(941\) 2.06166e49 1.77823 0.889117 0.457681i \(-0.151320\pi\)
0.889117 + 0.457681i \(0.151320\pi\)
\(942\) 0 0
\(943\) −2.84778e48 −0.237172
\(944\) 0 0
\(945\) 2.43598e49 1.95906
\(946\) 0 0
\(947\) −9.92376e48 −0.770729 −0.385365 0.922764i \(-0.625924\pi\)
−0.385365 + 0.922764i \(0.625924\pi\)
\(948\) 0 0
\(949\) −4.43051e48 −0.332323
\(950\) 0 0
\(951\) 3.37957e49 2.44841
\(952\) 0 0
\(953\) 5.61698e48 0.393070 0.196535 0.980497i \(-0.437031\pi\)
0.196535 + 0.980497i \(0.437031\pi\)
\(954\) 0 0
\(955\) −3.18010e48 −0.214974
\(956\) 0 0
\(957\) −4.20082e49 −2.74338
\(958\) 0 0
\(959\) −5.36856e48 −0.338728
\(960\) 0 0
\(961\) 1.18296e49 0.721166
\(962\) 0 0
\(963\) −2.58495e49 −1.52272
\(964\) 0 0
\(965\) 7.84643e48 0.446654
\(966\) 0 0
\(967\) 4.81816e48 0.265060 0.132530 0.991179i \(-0.457690\pi\)
0.132530 + 0.991179i \(0.457690\pi\)
\(968\) 0 0
\(969\) 4.07663e49 2.16750
\(970\) 0 0
\(971\) 1.74311e49 0.895793 0.447897 0.894085i \(-0.352173\pi\)
0.447897 + 0.894085i \(0.352173\pi\)
\(972\) 0 0
\(973\) 9.13231e48 0.453648
\(974\) 0 0
\(975\) 1.78330e49 0.856345
\(976\) 0 0
\(977\) −2.09476e49 −0.972464 −0.486232 0.873830i \(-0.661629\pi\)
−0.486232 + 0.873830i \(0.661629\pi\)
\(978\) 0 0
\(979\) 1.23282e49 0.553332
\(980\) 0 0
\(981\) −5.10827e49 −2.21684
\(982\) 0 0
\(983\) −1.66660e49 −0.699356 −0.349678 0.936870i \(-0.613709\pi\)
−0.349678 + 0.936870i \(0.613709\pi\)
\(984\) 0 0
\(985\) 3.44330e49 1.39726
\(986\) 0 0
\(987\) 6.70098e48 0.262969
\(988\) 0 0
\(989\) 3.68226e48 0.139758
\(990\) 0 0
\(991\) −1.93978e49 −0.712092 −0.356046 0.934468i \(-0.615875\pi\)
−0.356046 + 0.934468i \(0.615875\pi\)
\(992\) 0 0
\(993\) −5.70330e49 −2.02518
\(994\) 0 0
\(995\) −5.67682e49 −1.94995
\(996\) 0 0
\(997\) 3.04287e49 1.01114 0.505572 0.862784i \(-0.331282\pi\)
0.505572 + 0.862784i \(0.331282\pi\)
\(998\) 0 0
\(999\) −7.82089e49 −2.51435
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 4.34.a.a.1.3 3
4.3 odd 2 16.34.a.d.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.34.a.a.1.3 3 1.1 even 1 trivial
16.34.a.d.1.1 3 4.3 odd 2