Properties

Label 4.34.a.a.1.1
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 1.1
Root \(-6032.20\) of defining polynomial
Character \(\chi\) \(=\) 4.1

$q$-expansion

\(f(q)\) \(=\) \(q-6.39317e7 q^{3} -2.18954e11 q^{5} -4.92542e13 q^{7} -1.47179e15 q^{9} +O(q^{10})\) \(q-6.39317e7 q^{3} -2.18954e11 q^{5} -4.92542e13 q^{7} -1.47179e15 q^{9} -2.23089e17 q^{11} +7.57082e17 q^{13} +1.39981e19 q^{15} -6.49754e18 q^{17} -7.92387e20 q^{19} +3.14891e21 q^{21} +2.99746e22 q^{23} -6.84746e22 q^{25} +4.49495e23 q^{27} +1.97632e24 q^{29} +4.54327e24 q^{31} +1.42624e25 q^{33} +1.07844e25 q^{35} +1.28067e26 q^{37} -4.84016e25 q^{39} -3.19347e26 q^{41} -1.35211e27 q^{43} +3.22255e26 q^{45} -3.49210e27 q^{47} -5.30502e27 q^{49} +4.15399e26 q^{51} +3.47590e28 q^{53} +4.88461e28 q^{55} +5.06587e28 q^{57} +1.26922e29 q^{59} -3.88135e29 q^{61} +7.24920e28 q^{63} -1.65766e29 q^{65} -1.28458e30 q^{67} -1.91633e30 q^{69} -2.52414e30 q^{71} +8.98899e30 q^{73} +4.37770e30 q^{75} +1.09881e31 q^{77} +9.32137e30 q^{79} -2.05552e31 q^{81} +1.92342e31 q^{83} +1.42266e30 q^{85} -1.26349e32 q^{87} -6.91532e31 q^{89} -3.72895e31 q^{91} -2.90459e32 q^{93} +1.73496e32 q^{95} +8.57153e32 q^{97} +3.28341e32 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 92491788q^{3} - 53880683886q^{5} + 4541009914392q^{7} + 6032364433690023q^{9} + O(q^{10}) \) \( 3q + 92491788q^{3} - 53880683886q^{5} + 4541009914392q^{7} + 6032364433690023q^{9} + 227617657302449700q^{11} + 272970442217358762q^{13} - 32142473481241105848q^{15} + 93037188311816716854q^{17} + \)\(13\!\cdots\!28\)\(q^{19} - \)\(22\!\cdots\!52\)\(q^{21} + \)\(11\!\cdots\!96\)\(q^{23} + \)\(25\!\cdots\!21\)\(q^{25} + \)\(11\!\cdots\!24\)\(q^{27} + \)\(28\!\cdots\!78\)\(q^{29} + \)\(15\!\cdots\!28\)\(q^{31} + \)\(51\!\cdots\!80\)\(q^{33} + \)\(10\!\cdots\!48\)\(q^{35} + \)\(34\!\cdots\!42\)\(q^{37} + \)\(13\!\cdots\!68\)\(q^{39} - \)\(67\!\cdots\!62\)\(q^{41} - \)\(15\!\cdots\!20\)\(q^{43} - \)\(82\!\cdots\!86\)\(q^{45} - \)\(72\!\cdots\!76\)\(q^{47} - \)\(47\!\cdots\!09\)\(q^{49} + \)\(24\!\cdots\!36\)\(q^{51} + \)\(54\!\cdots\!34\)\(q^{53} + \)\(72\!\cdots\!80\)\(q^{55} + \)\(27\!\cdots\!48\)\(q^{57} + \)\(20\!\cdots\!76\)\(q^{59} + \)\(59\!\cdots\!66\)\(q^{61} - \)\(12\!\cdots\!68\)\(q^{63} - \)\(22\!\cdots\!12\)\(q^{65} - \)\(29\!\cdots\!28\)\(q^{67} - \)\(15\!\cdots\!16\)\(q^{69} + \)\(16\!\cdots\!44\)\(q^{71} + \)\(13\!\cdots\!02\)\(q^{73} + \)\(20\!\cdots\!28\)\(q^{75} + \)\(20\!\cdots\!20\)\(q^{77} + \)\(43\!\cdots\!04\)\(q^{79} + \)\(60\!\cdots\!27\)\(q^{81} - \)\(71\!\cdots\!64\)\(q^{83} - \)\(13\!\cdots\!84\)\(q^{85} - \)\(25\!\cdots\!92\)\(q^{87} - \)\(19\!\cdots\!98\)\(q^{89} - \)\(38\!\cdots\!08\)\(q^{91} + \)\(55\!\cdots\!88\)\(q^{93} - \)\(20\!\cdots\!76\)\(q^{95} + \)\(56\!\cdots\!22\)\(q^{97} + \)\(22\!\cdots\!00\)\(q^{99} + O(q^{100}) \)

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
\(3\) −6.39317e7 −0.857464 −0.428732 0.903432i \(-0.641040\pi\)
−0.428732 + 0.903432i \(0.641040\pi\)
\(4\) 0 0
\(5\) −2.18954e11 −0.641723 −0.320861 0.947126i \(-0.603972\pi\)
−0.320861 + 0.947126i \(0.603972\pi\)
\(6\) 0 0
\(7\) −4.92542e13 −0.560178 −0.280089 0.959974i \(-0.590364\pi\)
−0.280089 + 0.959974i \(0.590364\pi\)
\(8\) 0 0
\(9\) −1.47179e15 −0.264756
\(10\) 0 0
\(11\) −2.23089e17 −1.46386 −0.731928 0.681382i \(-0.761379\pi\)
−0.731928 + 0.681382i \(0.761379\pi\)
\(12\) 0 0
\(13\) 7.57082e17 0.315557 0.157778 0.987475i \(-0.449567\pi\)
0.157778 + 0.987475i \(0.449567\pi\)
\(14\) 0 0
\(15\) 1.39981e19 0.550254
\(16\) 0 0
\(17\) −6.49754e18 −0.0323848 −0.0161924 0.999869i \(-0.505154\pi\)
−0.0161924 + 0.999869i \(0.505154\pi\)
\(18\) 0 0
\(19\) −7.92387e20 −0.630236 −0.315118 0.949053i \(-0.602044\pi\)
−0.315118 + 0.949053i \(0.602044\pi\)
\(20\) 0 0
\(21\) 3.14891e21 0.480332
\(22\) 0 0
\(23\) 2.99746e22 1.01916 0.509581 0.860423i \(-0.329800\pi\)
0.509581 + 0.860423i \(0.329800\pi\)
\(24\) 0 0
\(25\) −6.84746e22 −0.588192
\(26\) 0 0
\(27\) 4.49495e23 1.08448
\(28\) 0 0
\(29\) 1.97632e24 1.46653 0.733263 0.679945i \(-0.237996\pi\)
0.733263 + 0.679945i \(0.237996\pi\)
\(30\) 0 0
\(31\) 4.54327e24 1.12176 0.560881 0.827897i \(-0.310463\pi\)
0.560881 + 0.827897i \(0.310463\pi\)
\(32\) 0 0
\(33\) 1.42624e25 1.25520
\(34\) 0 0
\(35\) 1.07844e25 0.359479
\(36\) 0 0
\(37\) 1.28067e26 1.70651 0.853253 0.521498i \(-0.174627\pi\)
0.853253 + 0.521498i \(0.174627\pi\)
\(38\) 0 0
\(39\) −4.84016e25 −0.270579
\(40\) 0 0
\(41\) −3.19347e26 −0.782221 −0.391111 0.920344i \(-0.627909\pi\)
−0.391111 + 0.920344i \(0.627909\pi\)
\(42\) 0 0
\(43\) −1.35211e27 −1.50932 −0.754662 0.656114i \(-0.772199\pi\)
−0.754662 + 0.656114i \(0.772199\pi\)
\(44\) 0 0
\(45\) 3.22255e26 0.169900
\(46\) 0 0
\(47\) −3.49210e27 −0.898404 −0.449202 0.893430i \(-0.648292\pi\)
−0.449202 + 0.893430i \(0.648292\pi\)
\(48\) 0 0
\(49\) −5.30502e27 −0.686201
\(50\) 0 0
\(51\) 4.15399e26 0.0277688
\(52\) 0 0
\(53\) 3.47590e28 1.23173 0.615864 0.787852i \(-0.288807\pi\)
0.615864 + 0.787852i \(0.288807\pi\)
\(54\) 0 0
\(55\) 4.88461e28 0.939389
\(56\) 0 0
\(57\) 5.06587e28 0.540404
\(58\) 0 0
\(59\) 1.26922e29 0.766443 0.383222 0.923656i \(-0.374815\pi\)
0.383222 + 0.923656i \(0.374815\pi\)
\(60\) 0 0
\(61\) −3.88135e29 −1.35220 −0.676099 0.736811i \(-0.736331\pi\)
−0.676099 + 0.736811i \(0.736331\pi\)
\(62\) 0 0
\(63\) 7.24920e28 0.148310
\(64\) 0 0
\(65\) −1.65766e29 −0.202500
\(66\) 0 0
\(67\) −1.28458e30 −0.951758 −0.475879 0.879511i \(-0.657870\pi\)
−0.475879 + 0.879511i \(0.657870\pi\)
\(68\) 0 0
\(69\) −1.91633e30 −0.873895
\(70\) 0 0
\(71\) −2.52414e30 −0.718373 −0.359187 0.933266i \(-0.616946\pi\)
−0.359187 + 0.933266i \(0.616946\pi\)
\(72\) 0 0
\(73\) 8.98899e30 1.61765 0.808823 0.588052i \(-0.200105\pi\)
0.808823 + 0.588052i \(0.200105\pi\)
\(74\) 0 0
\(75\) 4.37770e30 0.504353
\(76\) 0 0
\(77\) 1.09881e31 0.820019
\(78\) 0 0
\(79\) 9.32137e30 0.455651 0.227826 0.973702i \(-0.426838\pi\)
0.227826 + 0.973702i \(0.426838\pi\)
\(80\) 0 0
\(81\) −2.05552e31 −0.665149
\(82\) 0 0
\(83\) 1.92342e31 0.416185 0.208093 0.978109i \(-0.433274\pi\)
0.208093 + 0.978109i \(0.433274\pi\)
\(84\) 0 0
\(85\) 1.42266e30 0.0207821
\(86\) 0 0
\(87\) −1.26349e32 −1.25749
\(88\) 0 0
\(89\) −6.91532e31 −0.473018 −0.236509 0.971629i \(-0.576003\pi\)
−0.236509 + 0.971629i \(0.576003\pi\)
\(90\) 0 0
\(91\) −3.72895e31 −0.176768
\(92\) 0 0
\(93\) −2.90459e32 −0.961870
\(94\) 0 0
\(95\) 1.73496e32 0.404436
\(96\) 0 0
\(97\) 8.57153e32 1.41685 0.708426 0.705785i \(-0.249405\pi\)
0.708426 + 0.705785i \(0.249405\pi\)
\(98\) 0 0
\(99\) 3.28341e32 0.387564
\(100\) 0 0
\(101\) 1.67206e33 1.41889 0.709444 0.704761i \(-0.248946\pi\)
0.709444 + 0.704761i \(0.248946\pi\)
\(102\) 0 0
\(103\) −1.55505e33 −0.954842 −0.477421 0.878675i \(-0.658428\pi\)
−0.477421 + 0.878675i \(0.658428\pi\)
\(104\) 0 0
\(105\) −6.89465e32 −0.308240
\(106\) 0 0
\(107\) −3.58097e32 −0.117265 −0.0586325 0.998280i \(-0.518674\pi\)
−0.0586325 + 0.998280i \(0.518674\pi\)
\(108\) 0 0
\(109\) 3.52686e33 0.850847 0.425423 0.904995i \(-0.360125\pi\)
0.425423 + 0.904995i \(0.360125\pi\)
\(110\) 0 0
\(111\) −8.18752e33 −1.46327
\(112\) 0 0
\(113\) −4.01423e33 −0.534328 −0.267164 0.963651i \(-0.586087\pi\)
−0.267164 + 0.963651i \(0.586087\pi\)
\(114\) 0 0
\(115\) −6.56304e33 −0.654020
\(116\) 0 0
\(117\) −1.11427e33 −0.0835455
\(118\) 0 0
\(119\) 3.20031e32 0.0181413
\(120\) 0 0
\(121\) 2.65434e34 1.14287
\(122\) 0 0
\(123\) 2.04164e34 0.670726
\(124\) 0 0
\(125\) 4.04823e34 1.01918
\(126\) 0 0
\(127\) −3.61417e34 −0.700241 −0.350121 0.936705i \(-0.613859\pi\)
−0.350121 + 0.936705i \(0.613859\pi\)
\(128\) 0 0
\(129\) 8.64428e34 1.29419
\(130\) 0 0
\(131\) −1.50991e35 −1.75377 −0.876887 0.480696i \(-0.840384\pi\)
−0.876887 + 0.480696i \(0.840384\pi\)
\(132\) 0 0
\(133\) 3.90284e34 0.353044
\(134\) 0 0
\(135\) −9.84186e34 −0.695937
\(136\) 0 0
\(137\) 3.42195e34 0.189838 0.0949192 0.995485i \(-0.469741\pi\)
0.0949192 + 0.995485i \(0.469741\pi\)
\(138\) 0 0
\(139\) −3.08655e35 −1.34812 −0.674060 0.738677i \(-0.735451\pi\)
−0.674060 + 0.738677i \(0.735451\pi\)
\(140\) 0 0
\(141\) 2.23256e35 0.770349
\(142\) 0 0
\(143\) −1.68896e35 −0.461930
\(144\) 0 0
\(145\) −4.32722e35 −0.941103
\(146\) 0 0
\(147\) 3.39159e35 0.588393
\(148\) 0 0
\(149\) 1.12808e36 1.56591 0.782955 0.622078i \(-0.213711\pi\)
0.782955 + 0.622078i \(0.213711\pi\)
\(150\) 0 0
\(151\) 1.27117e36 1.41607 0.708033 0.706179i \(-0.249583\pi\)
0.708033 + 0.706179i \(0.249583\pi\)
\(152\) 0 0
\(153\) 9.56304e33 0.00857407
\(154\) 0 0
\(155\) −9.94766e35 −0.719860
\(156\) 0 0
\(157\) 7.68303e35 0.449975 0.224987 0.974362i \(-0.427766\pi\)
0.224987 + 0.974362i \(0.427766\pi\)
\(158\) 0 0
\(159\) −2.22220e36 −1.05616
\(160\) 0 0
\(161\) −1.47637e36 −0.570912
\(162\) 0 0
\(163\) 1.17206e36 0.369704 0.184852 0.982766i \(-0.440819\pi\)
0.184852 + 0.982766i \(0.440819\pi\)
\(164\) 0 0
\(165\) −3.12282e36 −0.805492
\(166\) 0 0
\(167\) 5.02289e35 0.106202 0.0531009 0.998589i \(-0.483089\pi\)
0.0531009 + 0.998589i \(0.483089\pi\)
\(168\) 0 0
\(169\) −5.18296e36 −0.900424
\(170\) 0 0
\(171\) 1.16623e36 0.166859
\(172\) 0 0
\(173\) 5.22082e36 0.616564 0.308282 0.951295i \(-0.400246\pi\)
0.308282 + 0.951295i \(0.400246\pi\)
\(174\) 0 0
\(175\) 3.37266e36 0.329492
\(176\) 0 0
\(177\) −8.11437e36 −0.657197
\(178\) 0 0
\(179\) 2.47756e37 1.66705 0.833527 0.552479i \(-0.186318\pi\)
0.833527 + 0.552479i \(0.186318\pi\)
\(180\) 0 0
\(181\) 8.26307e36 0.462855 0.231427 0.972852i \(-0.425660\pi\)
0.231427 + 0.972852i \(0.425660\pi\)
\(182\) 0 0
\(183\) 2.48141e37 1.15946
\(184\) 0 0
\(185\) −2.80407e37 −1.09510
\(186\) 0 0
\(187\) 1.44953e36 0.0474067
\(188\) 0 0
\(189\) −2.21395e37 −0.607503
\(190\) 0 0
\(191\) −2.54554e37 −0.587125 −0.293562 0.955940i \(-0.594841\pi\)
−0.293562 + 0.955940i \(0.594841\pi\)
\(192\) 0 0
\(193\) −7.64000e37 −1.48388 −0.741938 0.670469i \(-0.766093\pi\)
−0.741938 + 0.670469i \(0.766093\pi\)
\(194\) 0 0
\(195\) 1.05977e37 0.173636
\(196\) 0 0
\(197\) 2.00607e37 0.277749 0.138875 0.990310i \(-0.455651\pi\)
0.138875 + 0.990310i \(0.455651\pi\)
\(198\) 0 0
\(199\) 2.98496e36 0.0349835 0.0174918 0.999847i \(-0.494432\pi\)
0.0174918 + 0.999847i \(0.494432\pi\)
\(200\) 0 0
\(201\) 8.21253e37 0.816098
\(202\) 0 0
\(203\) −9.73419e37 −0.821515
\(204\) 0 0
\(205\) 6.99223e37 0.501969
\(206\) 0 0
\(207\) −4.41164e37 −0.269829
\(208\) 0 0
\(209\) 1.76773e38 0.922574
\(210\) 0 0
\(211\) 1.29116e38 0.575863 0.287931 0.957651i \(-0.407032\pi\)
0.287931 + 0.957651i \(0.407032\pi\)
\(212\) 0 0
\(213\) 1.61372e38 0.615979
\(214\) 0 0
\(215\) 2.96050e38 0.968568
\(216\) 0 0
\(217\) −2.23775e38 −0.628386
\(218\) 0 0
\(219\) −5.74681e38 −1.38707
\(220\) 0 0
\(221\) −4.91917e36 −0.0102193
\(222\) 0 0
\(223\) −5.28250e38 −0.945822 −0.472911 0.881110i \(-0.656797\pi\)
−0.472911 + 0.881110i \(0.656797\pi\)
\(224\) 0 0
\(225\) 1.00780e38 0.155727
\(226\) 0 0
\(227\) −4.73287e38 −0.631972 −0.315986 0.948764i \(-0.602335\pi\)
−0.315986 + 0.948764i \(0.602335\pi\)
\(228\) 0 0
\(229\) −4.97730e38 −0.575054 −0.287527 0.957773i \(-0.592833\pi\)
−0.287527 + 0.957773i \(0.592833\pi\)
\(230\) 0 0
\(231\) −7.02486e38 −0.703137
\(232\) 0 0
\(233\) 2.20413e39 1.91366 0.956831 0.290644i \(-0.0938696\pi\)
0.956831 + 0.290644i \(0.0938696\pi\)
\(234\) 0 0
\(235\) 7.64608e38 0.576526
\(236\) 0 0
\(237\) −5.95931e38 −0.390704
\(238\) 0 0
\(239\) 2.24449e38 0.128101 0.0640504 0.997947i \(-0.479598\pi\)
0.0640504 + 0.997947i \(0.479598\pi\)
\(240\) 0 0
\(241\) 2.40422e39 1.19590 0.597949 0.801534i \(-0.295983\pi\)
0.597949 + 0.801534i \(0.295983\pi\)
\(242\) 0 0
\(243\) −1.18464e39 −0.514142
\(244\) 0 0
\(245\) 1.16155e39 0.440351
\(246\) 0 0
\(247\) −5.99902e38 −0.198875
\(248\) 0 0
\(249\) −1.22968e39 −0.356864
\(250\) 0 0
\(251\) 7.59890e39 1.93257 0.966283 0.257481i \(-0.0828925\pi\)
0.966283 + 0.257481i \(0.0828925\pi\)
\(252\) 0 0
\(253\) −6.68698e39 −1.49191
\(254\) 0 0
\(255\) −9.09531e37 −0.0178199
\(256\) 0 0
\(257\) −7.53622e39 −1.29795 −0.648974 0.760811i \(-0.724802\pi\)
−0.648974 + 0.760811i \(0.724802\pi\)
\(258\) 0 0
\(259\) −6.30782e39 −0.955946
\(260\) 0 0
\(261\) −2.90873e39 −0.388271
\(262\) 0 0
\(263\) 4.45996e39 0.524882 0.262441 0.964948i \(-0.415473\pi\)
0.262441 + 0.964948i \(0.415473\pi\)
\(264\) 0 0
\(265\) −7.61060e39 −0.790428
\(266\) 0 0
\(267\) 4.42109e39 0.405596
\(268\) 0 0
\(269\) 1.47220e40 1.19414 0.597069 0.802190i \(-0.296332\pi\)
0.597069 + 0.802190i \(0.296332\pi\)
\(270\) 0 0
\(271\) 1.39791e40 1.00343 0.501714 0.865034i \(-0.332703\pi\)
0.501714 + 0.865034i \(0.332703\pi\)
\(272\) 0 0
\(273\) 2.38398e39 0.151572
\(274\) 0 0
\(275\) 1.52759e40 0.861028
\(276\) 0 0
\(277\) −2.35630e40 −1.17846 −0.589228 0.807967i \(-0.700568\pi\)
−0.589228 + 0.807967i \(0.700568\pi\)
\(278\) 0 0
\(279\) −6.68676e39 −0.296993
\(280\) 0 0
\(281\) 3.71685e40 1.46731 0.733654 0.679523i \(-0.237813\pi\)
0.733654 + 0.679523i \(0.237813\pi\)
\(282\) 0 0
\(283\) −9.67224e39 −0.339665 −0.169833 0.985473i \(-0.554323\pi\)
−0.169833 + 0.985473i \(0.554323\pi\)
\(284\) 0 0
\(285\) −1.10919e40 −0.346790
\(286\) 0 0
\(287\) 1.57292e40 0.438183
\(288\) 0 0
\(289\) −4.02123e40 −0.998951
\(290\) 0 0
\(291\) −5.47993e40 −1.21490
\(292\) 0 0
\(293\) 5.46285e40 1.08169 0.540845 0.841122i \(-0.318104\pi\)
0.540845 + 0.841122i \(0.318104\pi\)
\(294\) 0 0
\(295\) −2.77901e40 −0.491844
\(296\) 0 0
\(297\) −1.00277e41 −1.58753
\(298\) 0 0
\(299\) 2.26932e40 0.321604
\(300\) 0 0
\(301\) 6.65971e40 0.845490
\(302\) 0 0
\(303\) −1.06897e41 −1.21665
\(304\) 0 0
\(305\) 8.49836e40 0.867736
\(306\) 0 0
\(307\) 1.30064e40 0.119227 0.0596135 0.998222i \(-0.481013\pi\)
0.0596135 + 0.998222i \(0.481013\pi\)
\(308\) 0 0
\(309\) 9.94173e40 0.818742
\(310\) 0 0
\(311\) 2.40564e41 1.78108 0.890542 0.454901i \(-0.150325\pi\)
0.890542 + 0.454901i \(0.150325\pi\)
\(312\) 0 0
\(313\) 3.63493e40 0.242111 0.121055 0.992646i \(-0.461372\pi\)
0.121055 + 0.992646i \(0.461372\pi\)
\(314\) 0 0
\(315\) −1.58724e40 −0.0951740
\(316\) 0 0
\(317\) 1.26788e41 0.684859 0.342429 0.939544i \(-0.388750\pi\)
0.342429 + 0.939544i \(0.388750\pi\)
\(318\) 0 0
\(319\) −4.40894e41 −2.14678
\(320\) 0 0
\(321\) 2.28938e40 0.100550
\(322\) 0 0
\(323\) 5.14856e39 0.0204101
\(324\) 0 0
\(325\) −5.18409e40 −0.185608
\(326\) 0 0
\(327\) −2.25478e41 −0.729570
\(328\) 0 0
\(329\) 1.72001e41 0.503266
\(330\) 0 0
\(331\) −2.46365e41 −0.652254 −0.326127 0.945326i \(-0.605744\pi\)
−0.326127 + 0.945326i \(0.605744\pi\)
\(332\) 0 0
\(333\) −1.88488e41 −0.451807
\(334\) 0 0
\(335\) 2.81263e41 0.610764
\(336\) 0 0
\(337\) −3.62980e41 −0.714478 −0.357239 0.934013i \(-0.616282\pi\)
−0.357239 + 0.934013i \(0.616282\pi\)
\(338\) 0 0
\(339\) 2.56637e41 0.458167
\(340\) 0 0
\(341\) −1.01355e42 −1.64210
\(342\) 0 0
\(343\) 6.42078e41 0.944572
\(344\) 0 0
\(345\) 4.19587e41 0.560798
\(346\) 0 0
\(347\) 1.21611e42 1.47754 0.738768 0.673960i \(-0.235408\pi\)
0.738768 + 0.673960i \(0.235408\pi\)
\(348\) 0 0
\(349\) 1.93880e41 0.214246 0.107123 0.994246i \(-0.465836\pi\)
0.107123 + 0.994246i \(0.465836\pi\)
\(350\) 0 0
\(351\) 3.40304e41 0.342216
\(352\) 0 0
\(353\) 2.41824e41 0.221420 0.110710 0.993853i \(-0.464688\pi\)
0.110710 + 0.993853i \(0.464688\pi\)
\(354\) 0 0
\(355\) 5.52669e41 0.460996
\(356\) 0 0
\(357\) −2.04601e40 −0.0155555
\(358\) 0 0
\(359\) −1.31667e41 −0.0912888 −0.0456444 0.998958i \(-0.514534\pi\)
−0.0456444 + 0.998958i \(0.514534\pi\)
\(360\) 0 0
\(361\) −9.52894e41 −0.602803
\(362\) 0 0
\(363\) −1.69697e42 −0.979973
\(364\) 0 0
\(365\) −1.96817e42 −1.03808
\(366\) 0 0
\(367\) 7.81021e41 0.376420 0.188210 0.982129i \(-0.439731\pi\)
0.188210 + 0.982129i \(0.439731\pi\)
\(368\) 0 0
\(369\) 4.70013e41 0.207098
\(370\) 0 0
\(371\) −1.71202e42 −0.689986
\(372\) 0 0
\(373\) −9.64138e41 −0.355585 −0.177793 0.984068i \(-0.556896\pi\)
−0.177793 + 0.984068i \(0.556896\pi\)
\(374\) 0 0
\(375\) −2.58811e42 −0.873909
\(376\) 0 0
\(377\) 1.49623e42 0.462772
\(378\) 0 0
\(379\) 4.62182e42 1.30998 0.654992 0.755636i \(-0.272672\pi\)
0.654992 + 0.755636i \(0.272672\pi\)
\(380\) 0 0
\(381\) 2.31060e42 0.600432
\(382\) 0 0
\(383\) −4.15063e41 −0.0989313 −0.0494657 0.998776i \(-0.515752\pi\)
−0.0494657 + 0.998776i \(0.515752\pi\)
\(384\) 0 0
\(385\) −2.40588e42 −0.526225
\(386\) 0 0
\(387\) 1.99003e42 0.399602
\(388\) 0 0
\(389\) −9.10627e41 −0.167947 −0.0839735 0.996468i \(-0.526761\pi\)
−0.0839735 + 0.996468i \(0.526761\pi\)
\(390\) 0 0
\(391\) −1.94761e41 −0.0330054
\(392\) 0 0
\(393\) 9.65309e42 1.50380
\(394\) 0 0
\(395\) −2.04095e42 −0.292402
\(396\) 0 0
\(397\) 1.34295e43 1.77018 0.885088 0.465423i \(-0.154098\pi\)
0.885088 + 0.465423i \(0.154098\pi\)
\(398\) 0 0
\(399\) −2.49515e42 −0.302722
\(400\) 0 0
\(401\) −1.37037e43 −1.53093 −0.765465 0.643477i \(-0.777491\pi\)
−0.765465 + 0.643477i \(0.777491\pi\)
\(402\) 0 0
\(403\) 3.43963e42 0.353980
\(404\) 0 0
\(405\) 4.50064e42 0.426841
\(406\) 0 0
\(407\) −2.85702e43 −2.49808
\(408\) 0 0
\(409\) −5.72611e42 −0.461770 −0.230885 0.972981i \(-0.574162\pi\)
−0.230885 + 0.972981i \(0.574162\pi\)
\(410\) 0 0
\(411\) −2.18771e42 −0.162780
\(412\) 0 0
\(413\) −6.25146e42 −0.429344
\(414\) 0 0
\(415\) −4.21141e42 −0.267075
\(416\) 0 0
\(417\) 1.97328e43 1.15596
\(418\) 0 0
\(419\) −2.34537e42 −0.126964 −0.0634818 0.997983i \(-0.520220\pi\)
−0.0634818 + 0.997983i \(0.520220\pi\)
\(420\) 0 0
\(421\) 5.43051e42 0.271760 0.135880 0.990725i \(-0.456614\pi\)
0.135880 + 0.990725i \(0.456614\pi\)
\(422\) 0 0
\(423\) 5.13965e42 0.237858
\(424\) 0 0
\(425\) 4.44916e41 0.0190485
\(426\) 0 0
\(427\) 1.91173e43 0.757471
\(428\) 0 0
\(429\) 1.07978e43 0.396088
\(430\) 0 0
\(431\) 2.04863e43 0.695967 0.347983 0.937501i \(-0.386867\pi\)
0.347983 + 0.937501i \(0.386867\pi\)
\(432\) 0 0
\(433\) −2.78864e43 −0.877691 −0.438846 0.898562i \(-0.644613\pi\)
−0.438846 + 0.898562i \(0.644613\pi\)
\(434\) 0 0
\(435\) 2.76647e43 0.806962
\(436\) 0 0
\(437\) −2.37514e43 −0.642312
\(438\) 0 0
\(439\) 2.25782e43 0.566272 0.283136 0.959080i \(-0.408625\pi\)
0.283136 + 0.959080i \(0.408625\pi\)
\(440\) 0 0
\(441\) 7.80789e42 0.181676
\(442\) 0 0
\(443\) 1.19378e43 0.257789 0.128894 0.991658i \(-0.458857\pi\)
0.128894 + 0.991658i \(0.458857\pi\)
\(444\) 0 0
\(445\) 1.51414e43 0.303546
\(446\) 0 0
\(447\) −7.21202e43 −1.34271
\(448\) 0 0
\(449\) −6.42445e43 −1.11115 −0.555573 0.831468i \(-0.687501\pi\)
−0.555573 + 0.831468i \(0.687501\pi\)
\(450\) 0 0
\(451\) 7.12428e43 1.14506
\(452\) 0 0
\(453\) −8.12680e43 −1.21423
\(454\) 0 0
\(455\) 8.16467e42 0.113436
\(456\) 0 0
\(457\) −6.72354e43 −0.868922 −0.434461 0.900691i \(-0.643061\pi\)
−0.434461 + 0.900691i \(0.643061\pi\)
\(458\) 0 0
\(459\) −2.92061e42 −0.0351208
\(460\) 0 0
\(461\) −2.63019e43 −0.294389 −0.147195 0.989108i \(-0.547024\pi\)
−0.147195 + 0.989108i \(0.547024\pi\)
\(462\) 0 0
\(463\) −8.76936e43 −0.913864 −0.456932 0.889502i \(-0.651052\pi\)
−0.456932 + 0.889502i \(0.651052\pi\)
\(464\) 0 0
\(465\) 6.35971e43 0.617254
\(466\) 0 0
\(467\) 1.35357e44 1.22392 0.611958 0.790890i \(-0.290382\pi\)
0.611958 + 0.790890i \(0.290382\pi\)
\(468\) 0 0
\(469\) 6.32709e43 0.533153
\(470\) 0 0
\(471\) −4.91190e43 −0.385837
\(472\) 0 0
\(473\) 3.01641e44 2.20943
\(474\) 0 0
\(475\) 5.42584e43 0.370700
\(476\) 0 0
\(477\) −5.11580e43 −0.326107
\(478\) 0 0
\(479\) −7.85163e43 −0.467115 −0.233558 0.972343i \(-0.575037\pi\)
−0.233558 + 0.972343i \(0.575037\pi\)
\(480\) 0 0
\(481\) 9.69569e43 0.538499
\(482\) 0 0
\(483\) 9.43871e43 0.489536
\(484\) 0 0
\(485\) −1.87677e44 −0.909226
\(486\) 0 0
\(487\) −3.82489e44 −1.73138 −0.865688 0.500584i \(-0.833119\pi\)
−0.865688 + 0.500584i \(0.833119\pi\)
\(488\) 0 0
\(489\) −7.49317e43 −0.317007
\(490\) 0 0
\(491\) 4.02288e44 1.59108 0.795541 0.605900i \(-0.207187\pi\)
0.795541 + 0.605900i \(0.207187\pi\)
\(492\) 0 0
\(493\) −1.28412e43 −0.0474932
\(494\) 0 0
\(495\) −7.18914e43 −0.248709
\(496\) 0 0
\(497\) 1.24324e44 0.402416
\(498\) 0 0
\(499\) −4.25677e44 −1.28950 −0.644750 0.764393i \(-0.723039\pi\)
−0.644750 + 0.764393i \(0.723039\pi\)
\(500\) 0 0
\(501\) −3.21122e43 −0.0910643
\(502\) 0 0
\(503\) −5.24439e44 −1.39259 −0.696295 0.717756i \(-0.745170\pi\)
−0.696295 + 0.717756i \(0.745170\pi\)
\(504\) 0 0
\(505\) −3.66103e44 −0.910533
\(506\) 0 0
\(507\) 3.31355e44 0.772081
\(508\) 0 0
\(509\) 8.39856e44 1.83384 0.916920 0.399071i \(-0.130667\pi\)
0.916920 + 0.399071i \(0.130667\pi\)
\(510\) 0 0
\(511\) −4.42745e44 −0.906169
\(512\) 0 0
\(513\) −3.56174e44 −0.683479
\(514\) 0 0
\(515\) 3.40485e44 0.612744
\(516\) 0 0
\(517\) 7.79048e44 1.31513
\(518\) 0 0
\(519\) −3.33776e44 −0.528681
\(520\) 0 0
\(521\) 8.55665e44 1.27198 0.635991 0.771696i \(-0.280591\pi\)
0.635991 + 0.771696i \(0.280591\pi\)
\(522\) 0 0
\(523\) 8.81639e44 1.23030 0.615152 0.788408i \(-0.289094\pi\)
0.615152 + 0.788408i \(0.289094\pi\)
\(524\) 0 0
\(525\) −2.15620e44 −0.282527
\(526\) 0 0
\(527\) −2.95201e43 −0.0363281
\(528\) 0 0
\(529\) 3.34689e43 0.0386921
\(530\) 0 0
\(531\) −1.86804e44 −0.202920
\(532\) 0 0
\(533\) −2.41772e44 −0.246835
\(534\) 0 0
\(535\) 7.84067e43 0.0752516
\(536\) 0 0
\(537\) −1.58395e45 −1.42944
\(538\) 0 0
\(539\) 1.18349e45 1.00450
\(540\) 0 0
\(541\) −2.37260e45 −1.89439 −0.947197 0.320651i \(-0.896098\pi\)
−0.947197 + 0.320651i \(0.896098\pi\)
\(542\) 0 0
\(543\) −5.28272e44 −0.396881
\(544\) 0 0
\(545\) −7.72220e44 −0.546007
\(546\) 0 0
\(547\) 1.48406e45 0.987778 0.493889 0.869525i \(-0.335575\pi\)
0.493889 + 0.869525i \(0.335575\pi\)
\(548\) 0 0
\(549\) 5.71254e44 0.358002
\(550\) 0 0
\(551\) −1.56601e45 −0.924257
\(552\) 0 0
\(553\) −4.59116e44 −0.255246
\(554\) 0 0
\(555\) 1.79269e45 0.939011
\(556\) 0 0
\(557\) 3.92126e44 0.193560 0.0967799 0.995306i \(-0.469146\pi\)
0.0967799 + 0.995306i \(0.469146\pi\)
\(558\) 0 0
\(559\) −1.02366e45 −0.476278
\(560\) 0 0
\(561\) −9.26708e43 −0.0406495
\(562\) 0 0
\(563\) 1.13519e45 0.469548 0.234774 0.972050i \(-0.424565\pi\)
0.234774 + 0.972050i \(0.424565\pi\)
\(564\) 0 0
\(565\) 8.78930e44 0.342890
\(566\) 0 0
\(567\) 1.01243e45 0.372601
\(568\) 0 0
\(569\) −1.83884e45 −0.638544 −0.319272 0.947663i \(-0.603438\pi\)
−0.319272 + 0.947663i \(0.603438\pi\)
\(570\) 0 0
\(571\) 3.16473e45 1.03715 0.518574 0.855033i \(-0.326463\pi\)
0.518574 + 0.855033i \(0.326463\pi\)
\(572\) 0 0
\(573\) 1.62741e45 0.503438
\(574\) 0 0
\(575\) −2.05249e45 −0.599463
\(576\) 0 0
\(577\) 6.15172e45 1.69666 0.848332 0.529464i \(-0.177607\pi\)
0.848332 + 0.529464i \(0.177607\pi\)
\(578\) 0 0
\(579\) 4.88438e45 1.27237
\(580\) 0 0
\(581\) −9.47367e44 −0.233137
\(582\) 0 0
\(583\) −7.75433e45 −1.80307
\(584\) 0 0
\(585\) 2.43973e44 0.0536130
\(586\) 0 0
\(587\) −6.28489e45 −1.30548 −0.652738 0.757584i \(-0.726380\pi\)
−0.652738 + 0.757584i \(0.726380\pi\)
\(588\) 0 0
\(589\) −3.60003e45 −0.706974
\(590\) 0 0
\(591\) −1.28251e45 −0.238160
\(592\) 0 0
\(593\) −3.26071e45 −0.572677 −0.286338 0.958129i \(-0.592438\pi\)
−0.286338 + 0.958129i \(0.592438\pi\)
\(594\) 0 0
\(595\) −7.00720e43 −0.0116417
\(596\) 0 0
\(597\) −1.90834e44 −0.0299971
\(598\) 0 0
\(599\) −6.21811e45 −0.924944 −0.462472 0.886634i \(-0.653037\pi\)
−0.462472 + 0.886634i \(0.653037\pi\)
\(600\) 0 0
\(601\) 8.29634e45 1.16804 0.584020 0.811739i \(-0.301479\pi\)
0.584020 + 0.811739i \(0.301479\pi\)
\(602\) 0 0
\(603\) 1.89063e45 0.251983
\(604\) 0 0
\(605\) −5.81178e45 −0.733408
\(606\) 0 0
\(607\) 6.16593e45 0.736861 0.368430 0.929655i \(-0.379895\pi\)
0.368430 + 0.929655i \(0.379895\pi\)
\(608\) 0 0
\(609\) 6.22324e45 0.704419
\(610\) 0 0
\(611\) −2.64380e45 −0.283498
\(612\) 0 0
\(613\) 5.34569e45 0.543133 0.271566 0.962420i \(-0.412458\pi\)
0.271566 + 0.962420i \(0.412458\pi\)
\(614\) 0 0
\(615\) −4.47025e45 −0.430420
\(616\) 0 0
\(617\) 2.09959e45 0.191615 0.0958077 0.995400i \(-0.469457\pi\)
0.0958077 + 0.995400i \(0.469457\pi\)
\(618\) 0 0
\(619\) 6.78373e45 0.586912 0.293456 0.955973i \(-0.405195\pi\)
0.293456 + 0.955973i \(0.405195\pi\)
\(620\) 0 0
\(621\) 1.34734e46 1.10526
\(622\) 0 0
\(623\) 3.40609e45 0.264974
\(624\) 0 0
\(625\) −8.92271e44 −0.0658380
\(626\) 0 0
\(627\) −1.13014e46 −0.791074
\(628\) 0 0
\(629\) −8.32118e44 −0.0552649
\(630\) 0 0
\(631\) 6.16884e45 0.388794 0.194397 0.980923i \(-0.437725\pi\)
0.194397 + 0.980923i \(0.437725\pi\)
\(632\) 0 0
\(633\) −8.25460e45 −0.493782
\(634\) 0 0
\(635\) 7.91337e45 0.449361
\(636\) 0 0
\(637\) −4.01633e45 −0.216535
\(638\) 0 0
\(639\) 3.71501e45 0.190193
\(640\) 0 0
\(641\) 8.65757e45 0.420958 0.210479 0.977598i \(-0.432498\pi\)
0.210479 + 0.977598i \(0.432498\pi\)
\(642\) 0 0
\(643\) −3.14944e46 −1.45463 −0.727314 0.686305i \(-0.759231\pi\)
−0.727314 + 0.686305i \(0.759231\pi\)
\(644\) 0 0
\(645\) −1.89270e46 −0.830512
\(646\) 0 0
\(647\) −1.65528e46 −0.690163 −0.345081 0.938573i \(-0.612149\pi\)
−0.345081 + 0.938573i \(0.612149\pi\)
\(648\) 0 0
\(649\) −2.83149e46 −1.12196
\(650\) 0 0
\(651\) 1.43063e46 0.538818
\(652\) 0 0
\(653\) −1.91669e46 −0.686252 −0.343126 0.939289i \(-0.611486\pi\)
−0.343126 + 0.939289i \(0.611486\pi\)
\(654\) 0 0
\(655\) 3.30600e46 1.12544
\(656\) 0 0
\(657\) −1.32299e46 −0.428281
\(658\) 0 0
\(659\) 1.47214e46 0.453251 0.226626 0.973982i \(-0.427231\pi\)
0.226626 + 0.973982i \(0.427231\pi\)
\(660\) 0 0
\(661\) 2.36391e46 0.692319 0.346160 0.938176i \(-0.387486\pi\)
0.346160 + 0.938176i \(0.387486\pi\)
\(662\) 0 0
\(663\) 3.14491e44 0.00876264
\(664\) 0 0
\(665\) −8.54541e45 −0.226556
\(666\) 0 0
\(667\) 5.92392e46 1.49463
\(668\) 0 0
\(669\) 3.37719e46 0.811008
\(670\) 0 0
\(671\) 8.65885e46 1.97942
\(672\) 0 0
\(673\) −6.32698e46 −1.37704 −0.688522 0.725216i \(-0.741740\pi\)
−0.688522 + 0.725216i \(0.741740\pi\)
\(674\) 0 0
\(675\) −3.07790e46 −0.637884
\(676\) 0 0
\(677\) 7.67075e44 0.0151400 0.00756998 0.999971i \(-0.497590\pi\)
0.00756998 + 0.999971i \(0.497590\pi\)
\(678\) 0 0
\(679\) −4.22184e46 −0.793689
\(680\) 0 0
\(681\) 3.02581e46 0.541893
\(682\) 0 0
\(683\) −1.85282e46 −0.316148 −0.158074 0.987427i \(-0.550528\pi\)
−0.158074 + 0.987427i \(0.550528\pi\)
\(684\) 0 0
\(685\) −7.49248e45 −0.121824
\(686\) 0 0
\(687\) 3.18208e46 0.493088
\(688\) 0 0
\(689\) 2.63154e46 0.388680
\(690\) 0 0
\(691\) −1.06135e47 −1.49441 −0.747207 0.664591i \(-0.768606\pi\)
−0.747207 + 0.664591i \(0.768606\pi\)
\(692\) 0 0
\(693\) −1.61722e46 −0.217105
\(694\) 0 0
\(695\) 6.75811e46 0.865119
\(696\) 0 0
\(697\) 2.07497e45 0.0253321
\(698\) 0 0
\(699\) −1.40914e47 −1.64090
\(700\) 0 0
\(701\) 8.20778e46 0.911756 0.455878 0.890042i \(-0.349325\pi\)
0.455878 + 0.890042i \(0.349325\pi\)
\(702\) 0 0
\(703\) −1.01478e47 −1.07550
\(704\) 0 0
\(705\) −4.88827e46 −0.494351
\(706\) 0 0
\(707\) −8.23558e46 −0.794830
\(708\) 0 0
\(709\) 5.53905e45 0.0510238 0.0255119 0.999675i \(-0.491878\pi\)
0.0255119 + 0.999675i \(0.491878\pi\)
\(710\) 0 0
\(711\) −1.37191e46 −0.120636
\(712\) 0 0
\(713\) 1.36182e47 1.14326
\(714\) 0 0
\(715\) 3.69805e46 0.296431
\(716\) 0 0
\(717\) −1.43494e46 −0.109842
\(718\) 0 0
\(719\) 2.50176e47 1.82902 0.914512 0.404559i \(-0.132575\pi\)
0.914512 + 0.404559i \(0.132575\pi\)
\(720\) 0 0
\(721\) 7.65930e46 0.534881
\(722\) 0 0
\(723\) −1.53706e47 −1.02544
\(724\) 0 0
\(725\) −1.35327e47 −0.862599
\(726\) 0 0
\(727\) 3.23979e47 1.97333 0.986664 0.162769i \(-0.0520424\pi\)
0.986664 + 0.162769i \(0.0520424\pi\)
\(728\) 0 0
\(729\) 1.90004e47 1.10601
\(730\) 0 0
\(731\) 8.78539e45 0.0488792
\(732\) 0 0
\(733\) −1.15065e47 −0.611968 −0.305984 0.952037i \(-0.598985\pi\)
−0.305984 + 0.952037i \(0.598985\pi\)
\(734\) 0 0
\(735\) −7.42601e46 −0.377585
\(736\) 0 0
\(737\) 2.86575e47 1.39324
\(738\) 0 0
\(739\) −2.15723e47 −1.00291 −0.501456 0.865183i \(-0.667202\pi\)
−0.501456 + 0.865183i \(0.667202\pi\)
\(740\) 0 0
\(741\) 3.83528e46 0.170528
\(742\) 0 0
\(743\) −4.13812e47 −1.75990 −0.879949 0.475069i \(-0.842423\pi\)
−0.879949 + 0.475069i \(0.842423\pi\)
\(744\) 0 0
\(745\) −2.46998e47 −1.00488
\(746\) 0 0
\(747\) −2.83088e46 −0.110187
\(748\) 0 0
\(749\) 1.76378e46 0.0656892
\(750\) 0 0
\(751\) 3.22086e47 1.14792 0.573962 0.818882i \(-0.305406\pi\)
0.573962 + 0.818882i \(0.305406\pi\)
\(752\) 0 0
\(753\) −4.85811e47 −1.65711
\(754\) 0 0
\(755\) −2.78327e47 −0.908722
\(756\) 0 0
\(757\) 2.08055e47 0.650274 0.325137 0.945667i \(-0.394590\pi\)
0.325137 + 0.945667i \(0.394590\pi\)
\(758\) 0 0
\(759\) 4.27511e47 1.27926
\(760\) 0 0
\(761\) −9.02741e46 −0.258652 −0.129326 0.991602i \(-0.541281\pi\)
−0.129326 + 0.991602i \(0.541281\pi\)
\(762\) 0 0
\(763\) −1.73713e47 −0.476625
\(764\) 0 0
\(765\) −2.09386e45 −0.00550218
\(766\) 0 0
\(767\) 9.60906e46 0.241856
\(768\) 0 0
\(769\) −1.05116e47 −0.253444 −0.126722 0.991938i \(-0.540446\pi\)
−0.126722 + 0.991938i \(0.540446\pi\)
\(770\) 0 0
\(771\) 4.81804e47 1.11294
\(772\) 0 0
\(773\) −6.02385e47 −1.33325 −0.666627 0.745392i \(-0.732262\pi\)
−0.666627 + 0.745392i \(0.732262\pi\)
\(774\) 0 0
\(775\) −3.11098e47 −0.659811
\(776\) 0 0
\(777\) 4.03270e47 0.819689
\(778\) 0 0
\(779\) 2.53047e47 0.492984
\(780\) 0 0
\(781\) 5.63106e47 1.05159
\(782\) 0 0
\(783\) 8.88344e47 1.59042
\(784\) 0 0
\(785\) −1.68223e47 −0.288759
\(786\) 0 0
\(787\) 3.16244e47 0.520523 0.260261 0.965538i \(-0.416191\pi\)
0.260261 + 0.965538i \(0.416191\pi\)
\(788\) 0 0
\(789\) −2.85133e47 −0.450067
\(790\) 0 0
\(791\) 1.97718e47 0.299318
\(792\) 0 0
\(793\) −2.93850e47 −0.426695
\(794\) 0 0
\(795\) 4.86559e47 0.677763
\(796\) 0 0
\(797\) −4.41105e47 −0.589495 −0.294747 0.955575i \(-0.595235\pi\)
−0.294747 + 0.955575i \(0.595235\pi\)
\(798\) 0 0
\(799\) 2.26900e46 0.0290947
\(800\) 0 0
\(801\) 1.01779e47 0.125234
\(802\) 0 0
\(803\) −2.00534e48 −2.36800
\(804\) 0 0
\(805\) 3.23257e47 0.366367
\(806\) 0 0
\(807\) −9.41204e47 −1.02393
\(808\) 0 0
\(809\) −1.10768e48 −1.15681 −0.578406 0.815749i \(-0.696325\pi\)
−0.578406 + 0.815749i \(0.696325\pi\)
\(810\) 0 0
\(811\) 9.46114e47 0.948635 0.474318 0.880354i \(-0.342695\pi\)
0.474318 + 0.880354i \(0.342695\pi\)
\(812\) 0 0
\(813\) −8.93709e47 −0.860403
\(814\) 0 0
\(815\) −2.56627e47 −0.237247
\(816\) 0 0
\(817\) 1.07139e48 0.951230
\(818\) 0 0
\(819\) 5.48824e46 0.0468003
\(820\) 0 0
\(821\) 2.23829e48 1.83338 0.916692 0.399594i \(-0.130849\pi\)
0.916692 + 0.399594i \(0.130849\pi\)
\(822\) 0 0
\(823\) −1.31671e48 −1.03608 −0.518040 0.855356i \(-0.673338\pi\)
−0.518040 + 0.855356i \(0.673338\pi\)
\(824\) 0 0
\(825\) −9.76615e47 −0.738301
\(826\) 0 0
\(827\) −5.96543e47 −0.433312 −0.216656 0.976248i \(-0.569515\pi\)
−0.216656 + 0.976248i \(0.569515\pi\)
\(828\) 0 0
\(829\) 1.00487e48 0.701392 0.350696 0.936489i \(-0.385945\pi\)
0.350696 + 0.936489i \(0.385945\pi\)
\(830\) 0 0
\(831\) 1.50642e48 1.01048
\(832\) 0 0
\(833\) 3.44695e46 0.0222225
\(834\) 0 0
\(835\) −1.09978e47 −0.0681521
\(836\) 0 0
\(837\) 2.04218e48 1.21653
\(838\) 0 0
\(839\) −4.96184e47 −0.284165 −0.142083 0.989855i \(-0.545380\pi\)
−0.142083 + 0.989855i \(0.545380\pi\)
\(840\) 0 0
\(841\) 2.08975e48 1.15070
\(842\) 0 0
\(843\) −2.37625e48 −1.25816
\(844\) 0 0
\(845\) 1.13483e48 0.577822
\(846\) 0 0
\(847\) −1.30737e48 −0.640212
\(848\) 0 0
\(849\) 6.18363e47 0.291251
\(850\) 0 0
\(851\) 3.83874e48 1.73921
\(852\) 0 0
\(853\) −3.70417e48 −1.61448 −0.807238 0.590226i \(-0.799039\pi\)
−0.807238 + 0.590226i \(0.799039\pi\)
\(854\) 0 0
\(855\) −2.55350e47 −0.107077
\(856\) 0 0
\(857\) 4.82096e47 0.194514 0.0972569 0.995259i \(-0.468993\pi\)
0.0972569 + 0.995259i \(0.468993\pi\)
\(858\) 0 0
\(859\) 2.42649e48 0.942087 0.471043 0.882110i \(-0.343878\pi\)
0.471043 + 0.882110i \(0.343878\pi\)
\(860\) 0 0
\(861\) −1.00559e48 −0.375726
\(862\) 0 0
\(863\) −4.96761e48 −1.78636 −0.893181 0.449698i \(-0.851532\pi\)
−0.893181 + 0.449698i \(0.851532\pi\)
\(864\) 0 0
\(865\) −1.14312e48 −0.395663
\(866\) 0 0
\(867\) 2.57084e48 0.856565
\(868\) 0 0
\(869\) −2.07949e48 −0.667008
\(870\) 0 0
\(871\) −9.72531e47 −0.300334
\(872\) 0 0
\(873\) −1.26155e48 −0.375120
\(874\) 0 0
\(875\) −1.99393e48 −0.570921
\(876\) 0 0
\(877\) −5.66923e48 −1.56326 −0.781629 0.623743i \(-0.785611\pi\)
−0.781629 + 0.623743i \(0.785611\pi\)
\(878\) 0 0
\(879\) −3.49249e48 −0.927511
\(880\) 0 0
\(881\) 4.15832e48 1.06369 0.531845 0.846842i \(-0.321499\pi\)
0.531845 + 0.846842i \(0.321499\pi\)
\(882\) 0 0
\(883\) 2.01243e48 0.495870 0.247935 0.968777i \(-0.420248\pi\)
0.247935 + 0.968777i \(0.420248\pi\)
\(884\) 0 0
\(885\) 1.77667e48 0.421738
\(886\) 0 0
\(887\) 7.27157e48 1.66299 0.831493 0.555535i \(-0.187487\pi\)
0.831493 + 0.555535i \(0.187487\pi\)
\(888\) 0 0
\(889\) 1.78013e48 0.392259
\(890\) 0 0
\(891\) 4.58563e48 0.973681
\(892\) 0 0
\(893\) 2.76709e48 0.566206
\(894\) 0 0
\(895\) −5.42472e48 −1.06979
\(896\) 0 0
\(897\) −1.45082e48 −0.275763
\(898\) 0 0
\(899\) 8.97894e48 1.64509
\(900\) 0 0
\(901\) −2.25848e47 −0.0398893
\(902\) 0 0
\(903\) −4.25767e48 −0.724977
\(904\) 0 0
\(905\) −1.80923e48 −0.297024
\(906\) 0 0
\(907\) 7.77548e48 1.23085 0.615427 0.788194i \(-0.288983\pi\)
0.615427 + 0.788194i \(0.288983\pi\)
\(908\) 0 0
\(909\) −2.46092e48 −0.375659
\(910\) 0 0
\(911\) −2.43513e48 −0.358484 −0.179242 0.983805i \(-0.557364\pi\)
−0.179242 + 0.983805i \(0.557364\pi\)
\(912\) 0 0
\(913\) −4.29094e48 −0.609235
\(914\) 0 0
\(915\) −5.43315e48 −0.744052
\(916\) 0 0
\(917\) 7.43692e48 0.982425
\(918\) 0 0
\(919\) 2.51449e48 0.320438 0.160219 0.987081i \(-0.448780\pi\)
0.160219 + 0.987081i \(0.448780\pi\)
\(920\) 0 0
\(921\) −8.31524e47 −0.102233
\(922\) 0 0
\(923\) −1.91098e48 −0.226687
\(924\) 0 0
\(925\) −8.76931e48 −1.00375
\(926\) 0 0
\(927\) 2.28872e48 0.252800
\(928\) 0 0
\(929\) −3.90806e48 −0.416584 −0.208292 0.978067i \(-0.566790\pi\)
−0.208292 + 0.978067i \(0.566790\pi\)
\(930\) 0 0
\(931\) 4.20363e48 0.432468
\(932\) 0 0
\(933\) −1.53797e49 −1.52721
\(934\) 0 0
\(935\) −3.17379e47 −0.0304220
\(936\) 0 0
\(937\) −6.90657e48 −0.639086 −0.319543 0.947572i \(-0.603529\pi\)
−0.319543 + 0.947572i \(0.603529\pi\)
\(938\) 0 0
\(939\) −2.32387e48 −0.207601
\(940\) 0 0
\(941\) 1.15202e49 0.993646 0.496823 0.867852i \(-0.334500\pi\)
0.496823 + 0.867852i \(0.334500\pi\)
\(942\) 0 0
\(943\) −9.57229e48 −0.797210
\(944\) 0 0
\(945\) 4.84753e48 0.389848
\(946\) 0 0
\(947\) −2.25510e48 −0.175142 −0.0875711 0.996158i \(-0.527910\pi\)
−0.0875711 + 0.996158i \(0.527910\pi\)
\(948\) 0 0
\(949\) 6.80540e48 0.510459
\(950\) 0 0
\(951\) −8.10579e48 −0.587242
\(952\) 0 0
\(953\) 1.18218e49 0.827277 0.413638 0.910441i \(-0.364258\pi\)
0.413638 + 0.910441i \(0.364258\pi\)
\(954\) 0 0
\(955\) 5.57357e48 0.376771
\(956\) 0 0
\(957\) 2.81871e49 1.84079
\(958\) 0 0
\(959\) −1.68545e48 −0.106343
\(960\) 0 0
\(961\) 4.23783e48 0.258349
\(962\) 0 0
\(963\) 5.27045e47 0.0310466
\(964\) 0 0
\(965\) 1.67281e49 0.952237
\(966\) 0 0
\(967\) 3.25640e48 0.179144 0.0895718 0.995980i \(-0.471450\pi\)
0.0895718 + 0.995980i \(0.471450\pi\)
\(968\) 0 0
\(969\) −3.29157e47 −0.0175009
\(970\) 0 0
\(971\) −2.66132e49 −1.36767 −0.683833 0.729639i \(-0.739688\pi\)
−0.683833 + 0.729639i \(0.739688\pi\)
\(972\) 0 0
\(973\) 1.52025e49 0.755187
\(974\) 0 0
\(975\) 3.31428e48 0.159152
\(976\) 0 0
\(977\) −1.77551e49 −0.824256 −0.412128 0.911126i \(-0.635214\pi\)
−0.412128 + 0.911126i \(0.635214\pi\)
\(978\) 0 0
\(979\) 1.54273e49 0.692430
\(980\) 0 0
\(981\) −5.19081e48 −0.225267
\(982\) 0 0
\(983\) −2.83065e49 −1.18782 −0.593912 0.804530i \(-0.702417\pi\)
−0.593912 + 0.804530i \(0.702417\pi\)
\(984\) 0 0
\(985\) −4.39236e48 −0.178238
\(986\) 0 0
\(987\) −1.09963e49 −0.431532
\(988\) 0 0
\(989\) −4.05289e49 −1.53825
\(990\) 0 0
\(991\) −1.61657e49 −0.593442 −0.296721 0.954964i \(-0.595893\pi\)
−0.296721 + 0.954964i \(0.595893\pi\)
\(992\) 0 0
\(993\) 1.57505e49 0.559284
\(994\) 0 0
\(995\) −6.53569e47 −0.0224497
\(996\) 0 0
\(997\) −8.74014e48 −0.290434 −0.145217 0.989400i \(-0.546388\pi\)
−0.145217 + 0.989400i \(0.546388\pi\)
\(998\) 0 0
\(999\) 5.75653e49 1.85067
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.1.1 3
4.3 odd 2 16.34.a.d.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.34.a.a.1.1 3 1.1 even 1 trivial
16.34.a.d.1.3 3 4.3 odd 2