Properties

Label 9.38.a.c.1.2
Level 9
Weight 38
Character 9.1
Self dual yes
Analytic conductor 78.043
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 38 \)
Character orbit: \([\chi]\) = 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.0426343121\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{10}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-35434.6\)
Character \(\chi\) = 9.1

$q$-expansion

\(f(q)\) \(=\) \(q-322000. q^{2} -3.37552e10 q^{4} -1.53382e13 q^{5} +2.48687e15 q^{7} +5.51245e16 q^{8} +O(q^{10})\) \(q-322000. q^{2} -3.37552e10 q^{4} -1.53382e13 q^{5} +2.48687e15 q^{7} +5.51245e16 q^{8} +4.93889e18 q^{10} +1.32093e18 q^{11} +1.53342e20 q^{13} -8.00770e20 q^{14} -1.31108e22 q^{16} -1.04822e23 q^{17} -6.96548e23 q^{19} +5.17744e23 q^{20} -4.25339e23 q^{22} -1.11946e25 q^{23} +1.62501e26 q^{25} -4.93762e25 q^{26} -8.39448e25 q^{28} -1.43220e27 q^{29} +3.90372e26 q^{31} -3.35459e27 q^{32} +3.37525e28 q^{34} -3.81440e28 q^{35} -8.53226e28 q^{37} +2.24288e29 q^{38} -8.45509e29 q^{40} -4.51987e29 q^{41} -2.04528e30 q^{43} -4.45883e28 q^{44} +3.60466e30 q^{46} +3.80345e30 q^{47} -1.23776e31 q^{49} -5.23251e31 q^{50} -5.17611e30 q^{52} -2.08682e31 q^{53} -2.02607e31 q^{55} +1.37087e32 q^{56} +4.61167e32 q^{58} +1.01104e33 q^{59} -1.04937e33 q^{61} -1.25699e32 q^{62} +2.88211e33 q^{64} -2.35199e33 q^{65} +2.33599e33 q^{67} +3.53828e33 q^{68} +1.22824e34 q^{70} -1.08272e33 q^{71} -3.23026e34 q^{73} +2.74738e34 q^{74} +2.35121e34 q^{76} +3.28498e33 q^{77} -2.35299e35 q^{79} +2.01095e35 q^{80} +1.45540e35 q^{82} +1.58763e35 q^{83} +1.60777e36 q^{85} +6.58578e35 q^{86} +7.28155e34 q^{88} +3.16700e35 q^{89} +3.81342e35 q^{91} +3.77876e35 q^{92} -1.22471e36 q^{94} +1.06838e37 q^{95} +6.89438e36 q^{97} +3.98559e36 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 437562q^{2} + 346098955492q^{4} + 4099829756904q^{5} + 6605809948153184q^{7} - 140484113342159976q^{8} + O(q^{10}) \) \( 4q - 437562q^{2} + 346098955492q^{4} + 4099829756904q^{5} + 6605809948153184q^{7} - 140484113342159976q^{8} - 21511023001649316q^{10} - 20953708852195292976q^{11} + 51830892788989874168q^{13} + \)\(18\!\cdots\!12\)\(q^{14} + \)\(55\!\cdots\!40\)\(q^{16} - \)\(81\!\cdots\!28\)\(q^{17} - \)\(54\!\cdots\!32\)\(q^{19} + \)\(35\!\cdots\!76\)\(q^{20} + \)\(23\!\cdots\!76\)\(q^{22} + \)\(61\!\cdots\!88\)\(q^{23} + \)\(95\!\cdots\!16\)\(q^{25} - \)\(42\!\cdots\!88\)\(q^{26} + \)\(24\!\cdots\!48\)\(q^{28} - \)\(41\!\cdots\!36\)\(q^{29} + \)\(89\!\cdots\!64\)\(q^{31} - \)\(36\!\cdots\!84\)\(q^{32} + \)\(31\!\cdots\!24\)\(q^{34} - \)\(42\!\cdots\!40\)\(q^{35} + \)\(55\!\cdots\!24\)\(q^{37} + \)\(73\!\cdots\!68\)\(q^{38} - \)\(26\!\cdots\!52\)\(q^{40} + \)\(86\!\cdots\!76\)\(q^{41} - \)\(50\!\cdots\!80\)\(q^{43} - \)\(28\!\cdots\!36\)\(q^{44} - \)\(14\!\cdots\!96\)\(q^{46} - \)\(42\!\cdots\!20\)\(q^{47} + \)\(40\!\cdots\!20\)\(q^{49} - \)\(10\!\cdots\!14\)\(q^{50} + \)\(18\!\cdots\!68\)\(q^{52} + \)\(12\!\cdots\!88\)\(q^{53} - \)\(32\!\cdots\!24\)\(q^{55} - \)\(49\!\cdots\!80\)\(q^{56} - \)\(75\!\cdots\!56\)\(q^{58} + \)\(13\!\cdots\!88\)\(q^{59} - \)\(12\!\cdots\!60\)\(q^{61} + \)\(37\!\cdots\!92\)\(q^{62} + \)\(85\!\cdots\!28\)\(q^{64} - \)\(15\!\cdots\!68\)\(q^{65} + \)\(16\!\cdots\!48\)\(q^{67} - \)\(63\!\cdots\!84\)\(q^{68} + \)\(82\!\cdots\!60\)\(q^{70} - \)\(10\!\cdots\!88\)\(q^{71} - \)\(19\!\cdots\!48\)\(q^{73} + \)\(89\!\cdots\!12\)\(q^{74} - \)\(95\!\cdots\!68\)\(q^{76} + \)\(25\!\cdots\!92\)\(q^{77} + \)\(42\!\cdots\!20\)\(q^{79} + \)\(95\!\cdots\!36\)\(q^{80} + \)\(33\!\cdots\!48\)\(q^{82} + \)\(46\!\cdots\!24\)\(q^{83} + \)\(18\!\cdots\!12\)\(q^{85} + \)\(36\!\cdots\!04\)\(q^{86} + \)\(42\!\cdots\!56\)\(q^{88} + \)\(31\!\cdots\!52\)\(q^{89} + \)\(26\!\cdots\!24\)\(q^{91} + \)\(16\!\cdots\!16\)\(q^{92} - \)\(57\!\cdots\!48\)\(q^{94} + \)\(89\!\cdots\!56\)\(q^{95} + \)\(44\!\cdots\!48\)\(q^{97} + \)\(43\!\cdots\!78\)\(q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −322000. −0.868561 −0.434281 0.900778i \(-0.642997\pi\)
−0.434281 + 0.900778i \(0.642997\pi\)
\(3\) 0 0
\(4\) −3.37552e10 −0.245602
\(5\) −1.53382e13 −1.79816 −0.899081 0.437781i \(-0.855764\pi\)
−0.899081 + 0.437781i \(0.855764\pi\)
\(6\) 0 0
\(7\) 2.48687e15 0.577216 0.288608 0.957447i \(-0.406807\pi\)
0.288608 + 0.957447i \(0.406807\pi\)
\(8\) 5.51245e16 1.08188
\(9\) 0 0
\(10\) 4.93889e18 1.56181
\(11\) 1.32093e18 0.0716333 0.0358167 0.999358i \(-0.488597\pi\)
0.0358167 + 0.999358i \(0.488597\pi\)
\(12\) 0 0
\(13\) 1.53342e20 0.378190 0.189095 0.981959i \(-0.439445\pi\)
0.189095 + 0.981959i \(0.439445\pi\)
\(14\) −8.00770e20 −0.501348
\(15\) 0 0
\(16\) −1.31108e22 −0.694078
\(17\) −1.04822e23 −1.80778 −0.903890 0.427765i \(-0.859301\pi\)
−0.903890 + 0.427765i \(0.859301\pi\)
\(18\) 0 0
\(19\) −6.96548e23 −1.53465 −0.767325 0.641258i \(-0.778413\pi\)
−0.767325 + 0.641258i \(0.778413\pi\)
\(20\) 5.17744e23 0.441632
\(21\) 0 0
\(22\) −4.25339e23 −0.0622179
\(23\) −1.11946e25 −0.719521 −0.359761 0.933045i \(-0.617142\pi\)
−0.359761 + 0.933045i \(0.617142\pi\)
\(24\) 0 0
\(25\) 1.62501e26 2.23339
\(26\) −4.93762e25 −0.328481
\(27\) 0 0
\(28\) −8.39448e25 −0.141765
\(29\) −1.43220e27 −1.26369 −0.631845 0.775095i \(-0.717702\pi\)
−0.631845 + 0.775095i \(0.717702\pi\)
\(30\) 0 0
\(31\) 3.90372e26 0.100297 0.0501484 0.998742i \(-0.484031\pi\)
0.0501484 + 0.998742i \(0.484031\pi\)
\(32\) −3.35459e27 −0.479032
\(33\) 0 0
\(34\) 3.37525e28 1.57017
\(35\) −3.81440e28 −1.03793
\(36\) 0 0
\(37\) −8.53226e28 −0.830486 −0.415243 0.909711i \(-0.636303\pi\)
−0.415243 + 0.909711i \(0.636303\pi\)
\(38\) 2.24288e29 1.33294
\(39\) 0 0
\(40\) −8.45509e29 −1.94540
\(41\) −4.51987e29 −0.658605 −0.329302 0.944225i \(-0.606814\pi\)
−0.329302 + 0.944225i \(0.606814\pi\)
\(42\) 0 0
\(43\) −2.04528e30 −1.23477 −0.617384 0.786662i \(-0.711808\pi\)
−0.617384 + 0.786662i \(0.711808\pi\)
\(44\) −4.45883e28 −0.0175933
\(45\) 0 0
\(46\) 3.60466e30 0.624948
\(47\) 3.80345e30 0.442963 0.221481 0.975165i \(-0.428911\pi\)
0.221481 + 0.975165i \(0.428911\pi\)
\(48\) 0 0
\(49\) −1.23776e31 −0.666821
\(50\) −5.23251e31 −1.93984
\(51\) 0 0
\(52\) −5.17611e30 −0.0928841
\(53\) −2.08682e31 −0.263257 −0.131629 0.991299i \(-0.542021\pi\)
−0.131629 + 0.991299i \(0.542021\pi\)
\(54\) 0 0
\(55\) −2.02607e31 −0.128808
\(56\) 1.37087e32 0.624480
\(57\) 0 0
\(58\) 4.61167e32 1.09759
\(59\) 1.01104e33 1.75390 0.876952 0.480578i \(-0.159573\pi\)
0.876952 + 0.480578i \(0.159573\pi\)
\(60\) 0 0
\(61\) −1.04937e33 −0.982484 −0.491242 0.871023i \(-0.663457\pi\)
−0.491242 + 0.871023i \(0.663457\pi\)
\(62\) −1.25699e32 −0.0871138
\(63\) 0 0
\(64\) 2.88211e33 1.11015
\(65\) −2.35199e33 −0.680047
\(66\) 0 0
\(67\) 2.33599e33 0.385557 0.192778 0.981242i \(-0.438250\pi\)
0.192778 + 0.981242i \(0.438250\pi\)
\(68\) 3.53828e33 0.443994
\(69\) 0 0
\(70\) 1.22824e34 0.901505
\(71\) −1.08272e33 −0.0611275 −0.0305637 0.999533i \(-0.509730\pi\)
−0.0305637 + 0.999533i \(0.509730\pi\)
\(72\) 0 0
\(73\) −3.23026e34 −1.09085 −0.545424 0.838160i \(-0.683631\pi\)
−0.545424 + 0.838160i \(0.683631\pi\)
\(74\) 2.74738e34 0.721328
\(75\) 0 0
\(76\) 2.35121e34 0.376913
\(77\) 3.28498e33 0.0413479
\(78\) 0 0
\(79\) −2.35299e35 −1.84297 −0.921486 0.388411i \(-0.873024\pi\)
−0.921486 + 0.388411i \(0.873024\pi\)
\(80\) 2.01095e35 1.24807
\(81\) 0 0
\(82\) 1.45540e35 0.572038
\(83\) 1.58763e35 0.498660 0.249330 0.968419i \(-0.419790\pi\)
0.249330 + 0.968419i \(0.419790\pi\)
\(84\) 0 0
\(85\) 1.60777e36 3.25068
\(86\) 6.58578e35 1.07247
\(87\) 0 0
\(88\) 7.28155e34 0.0774987
\(89\) 3.16700e35 0.273484 0.136742 0.990607i \(-0.456337\pi\)
0.136742 + 0.990607i \(0.456337\pi\)
\(90\) 0 0
\(91\) 3.81342e35 0.218298
\(92\) 3.77876e35 0.176716
\(93\) 0 0
\(94\) −1.22471e36 −0.384740
\(95\) 1.06838e37 2.75955
\(96\) 0 0
\(97\) 6.89438e36 1.21121 0.605603 0.795767i \(-0.292932\pi\)
0.605603 + 0.795767i \(0.292932\pi\)
\(98\) 3.98559e36 0.579175
\(99\) 0 0
\(100\) −5.48524e36 −0.548524
\(101\) −8.14698e36 −0.677721 −0.338861 0.940837i \(-0.610042\pi\)
−0.338861 + 0.940837i \(0.610042\pi\)
\(102\) 0 0
\(103\) −1.09944e37 −0.636329 −0.318165 0.948036i \(-0.603066\pi\)
−0.318165 + 0.948036i \(0.603066\pi\)
\(104\) 8.45291e36 0.409157
\(105\) 0 0
\(106\) 6.71954e36 0.228655
\(107\) −2.26830e37 −0.648783 −0.324392 0.945923i \(-0.605160\pi\)
−0.324392 + 0.945923i \(0.605160\pi\)
\(108\) 0 0
\(109\) 3.74392e37 0.760216 0.380108 0.924942i \(-0.375887\pi\)
0.380108 + 0.924942i \(0.375887\pi\)
\(110\) 6.52393e36 0.111878
\(111\) 0 0
\(112\) −3.26047e37 −0.400633
\(113\) −1.81279e37 −0.188972 −0.0944858 0.995526i \(-0.530121\pi\)
−0.0944858 + 0.995526i \(0.530121\pi\)
\(114\) 0 0
\(115\) 1.71705e38 1.29382
\(116\) 4.83442e37 0.310364
\(117\) 0 0
\(118\) −3.25554e38 −1.52337
\(119\) −2.60677e38 −1.04348
\(120\) 0 0
\(121\) −3.38295e38 −0.994869
\(122\) 3.37896e38 0.853348
\(123\) 0 0
\(124\) −1.31771e37 −0.0246331
\(125\) −1.37646e39 −2.21784
\(126\) 0 0
\(127\) 3.75363e38 0.450903 0.225451 0.974254i \(-0.427614\pi\)
0.225451 + 0.974254i \(0.427614\pi\)
\(128\) −4.66986e38 −0.485198
\(129\) 0 0
\(130\) 7.57341e38 0.590663
\(131\) −1.53812e39 −1.04105 −0.520525 0.853846i \(-0.674264\pi\)
−0.520525 + 0.853846i \(0.674264\pi\)
\(132\) 0 0
\(133\) −1.73222e39 −0.885825
\(134\) −7.52189e38 −0.334879
\(135\) 0 0
\(136\) −5.77823e39 −1.95580
\(137\) 5.26351e39 1.55577 0.777885 0.628407i \(-0.216293\pi\)
0.777885 + 0.628407i \(0.216293\pi\)
\(138\) 0 0
\(139\) 3.79319e39 0.857494 0.428747 0.903425i \(-0.358955\pi\)
0.428747 + 0.903425i \(0.358955\pi\)
\(140\) 1.28756e39 0.254917
\(141\) 0 0
\(142\) 3.48635e38 0.0530930
\(143\) 2.02555e38 0.0270910
\(144\) 0 0
\(145\) 2.19673e40 2.27232
\(146\) 1.04014e40 0.947468
\(147\) 0 0
\(148\) 2.88009e39 0.203969
\(149\) 2.87234e40 1.79593 0.897967 0.440063i \(-0.145044\pi\)
0.897967 + 0.440063i \(0.145044\pi\)
\(150\) 0 0
\(151\) 1.01312e39 0.0494982 0.0247491 0.999694i \(-0.492121\pi\)
0.0247491 + 0.999694i \(0.492121\pi\)
\(152\) −3.83968e40 −1.66031
\(153\) 0 0
\(154\) −1.05776e39 −0.0359132
\(155\) −5.98759e39 −0.180350
\(156\) 0 0
\(157\) −3.12964e40 −0.743620 −0.371810 0.928309i \(-0.621263\pi\)
−0.371810 + 0.928309i \(0.621263\pi\)
\(158\) 7.57662e40 1.60073
\(159\) 0 0
\(160\) 5.14533e40 0.861378
\(161\) −2.78395e40 −0.415319
\(162\) 0 0
\(163\) −6.14970e40 −0.730102 −0.365051 0.930988i \(-0.618948\pi\)
−0.365051 + 0.930988i \(0.618948\pi\)
\(164\) 1.52569e40 0.161754
\(165\) 0 0
\(166\) −5.11216e40 −0.433117
\(167\) 1.95759e41 1.48411 0.742055 0.670339i \(-0.233851\pi\)
0.742055 + 0.670339i \(0.233851\pi\)
\(168\) 0 0
\(169\) −1.40887e41 −0.856972
\(170\) −5.17702e41 −2.82342
\(171\) 0 0
\(172\) 6.90388e40 0.303261
\(173\) −8.49345e40 −0.335144 −0.167572 0.985860i \(-0.553593\pi\)
−0.167572 + 0.985860i \(0.553593\pi\)
\(174\) 0 0
\(175\) 4.04117e41 1.28915
\(176\) −1.73184e40 −0.0497191
\(177\) 0 0
\(178\) −1.01977e41 −0.237538
\(179\) −5.07222e41 −1.06517 −0.532583 0.846378i \(-0.678778\pi\)
−0.532583 + 0.846378i \(0.678778\pi\)
\(180\) 0 0
\(181\) 2.84986e41 0.487270 0.243635 0.969867i \(-0.421660\pi\)
0.243635 + 0.969867i \(0.421660\pi\)
\(182\) −1.22792e41 −0.189605
\(183\) 0 0
\(184\) −6.17096e41 −0.778436
\(185\) 1.30869e42 1.49335
\(186\) 0 0
\(187\) −1.38462e41 −0.129497
\(188\) −1.28386e41 −0.108792
\(189\) 0 0
\(190\) −3.44017e42 −2.39684
\(191\) −1.34993e42 −0.853483 −0.426741 0.904374i \(-0.640339\pi\)
−0.426741 + 0.904374i \(0.640339\pi\)
\(192\) 0 0
\(193\) 1.53006e42 0.797806 0.398903 0.916993i \(-0.369391\pi\)
0.398903 + 0.916993i \(0.369391\pi\)
\(194\) −2.21999e42 −1.05201
\(195\) 0 0
\(196\) 4.17809e41 0.163772
\(197\) 1.58341e42 0.564896 0.282448 0.959283i \(-0.408854\pi\)
0.282448 + 0.959283i \(0.408854\pi\)
\(198\) 0 0
\(199\) 3.81225e42 1.12823 0.564117 0.825695i \(-0.309217\pi\)
0.564117 + 0.825695i \(0.309217\pi\)
\(200\) 8.95775e42 2.41626
\(201\) 0 0
\(202\) 2.62332e42 0.588642
\(203\) −3.56169e42 −0.729422
\(204\) 0 0
\(205\) 6.93266e42 1.18428
\(206\) 3.54018e42 0.552691
\(207\) 0 0
\(208\) −2.01044e42 −0.262493
\(209\) −9.20091e41 −0.109932
\(210\) 0 0
\(211\) −1.03480e43 −1.03665 −0.518325 0.855184i \(-0.673444\pi\)
−0.518325 + 0.855184i \(0.673444\pi\)
\(212\) 7.04410e41 0.0646565
\(213\) 0 0
\(214\) 7.30390e42 0.563508
\(215\) 3.13708e43 2.22031
\(216\) 0 0
\(217\) 9.70802e41 0.0578929
\(218\) −1.20554e43 −0.660294
\(219\) 0 0
\(220\) 6.83904e41 0.0316356
\(221\) −1.60736e43 −0.683684
\(222\) 0 0
\(223\) −2.53188e43 −0.911598 −0.455799 0.890083i \(-0.650646\pi\)
−0.455799 + 0.890083i \(0.650646\pi\)
\(224\) −8.34241e42 −0.276505
\(225\) 0 0
\(226\) 5.83718e42 0.164133
\(227\) −5.88632e43 −1.52534 −0.762668 0.646790i \(-0.776111\pi\)
−0.762668 + 0.646790i \(0.776111\pi\)
\(228\) 0 0
\(229\) 2.71698e43 0.598590 0.299295 0.954161i \(-0.403249\pi\)
0.299295 + 0.954161i \(0.403249\pi\)
\(230\) −5.52889e43 −1.12376
\(231\) 0 0
\(232\) −7.89492e43 −1.36716
\(233\) 1.14243e43 0.182703 0.0913513 0.995819i \(-0.470881\pi\)
0.0913513 + 0.995819i \(0.470881\pi\)
\(234\) 0 0
\(235\) −5.83380e43 −0.796520
\(236\) −3.41279e43 −0.430762
\(237\) 0 0
\(238\) 8.39379e43 0.906326
\(239\) −1.69778e43 −0.169637 −0.0848185 0.996396i \(-0.527031\pi\)
−0.0848185 + 0.996396i \(0.527031\pi\)
\(240\) 0 0
\(241\) −3.09126e43 −0.264740 −0.132370 0.991200i \(-0.542259\pi\)
−0.132370 + 0.991200i \(0.542259\pi\)
\(242\) 1.08931e44 0.864104
\(243\) 0 0
\(244\) 3.54217e43 0.241300
\(245\) 1.89850e44 1.19905
\(246\) 0 0
\(247\) −1.06810e44 −0.580390
\(248\) 2.15190e43 0.108509
\(249\) 0 0
\(250\) 4.43221e44 1.92633
\(251\) −2.54005e44 −1.02537 −0.512683 0.858578i \(-0.671348\pi\)
−0.512683 + 0.858578i \(0.671348\pi\)
\(252\) 0 0
\(253\) −1.47873e43 −0.0515417
\(254\) −1.20867e44 −0.391637
\(255\) 0 0
\(256\) −2.45744e44 −0.688722
\(257\) 3.66815e44 0.956499 0.478249 0.878224i \(-0.341271\pi\)
0.478249 + 0.878224i \(0.341271\pi\)
\(258\) 0 0
\(259\) −2.12186e44 −0.479370
\(260\) 7.93922e43 0.167021
\(261\) 0 0
\(262\) 4.95275e44 0.904216
\(263\) 2.74111e44 0.466385 0.233192 0.972431i \(-0.425083\pi\)
0.233192 + 0.972431i \(0.425083\pi\)
\(264\) 0 0
\(265\) 3.20080e44 0.473380
\(266\) 5.57775e44 0.769393
\(267\) 0 0
\(268\) −7.88521e43 −0.0946933
\(269\) −7.40366e44 −0.829906 −0.414953 0.909843i \(-0.636202\pi\)
−0.414953 + 0.909843i \(0.636202\pi\)
\(270\) 0 0
\(271\) −3.47586e44 −0.339727 −0.169863 0.985468i \(-0.554333\pi\)
−0.169863 + 0.985468i \(0.554333\pi\)
\(272\) 1.37429e45 1.25474
\(273\) 0 0
\(274\) −1.69485e45 −1.35128
\(275\) 2.14652e44 0.159985
\(276\) 0 0
\(277\) 1.55336e45 1.01250 0.506252 0.862385i \(-0.331031\pi\)
0.506252 + 0.862385i \(0.331031\pi\)
\(278\) −1.22141e45 −0.744786
\(279\) 0 0
\(280\) −2.10267e45 −1.12292
\(281\) −5.16475e44 −0.258216 −0.129108 0.991631i \(-0.541211\pi\)
−0.129108 + 0.991631i \(0.541211\pi\)
\(282\) 0 0
\(283\) 2.81373e45 1.23377 0.616884 0.787054i \(-0.288395\pi\)
0.616884 + 0.787054i \(0.288395\pi\)
\(284\) 3.65475e43 0.0150130
\(285\) 0 0
\(286\) −6.52225e43 −0.0235302
\(287\) −1.12403e45 −0.380157
\(288\) 0 0
\(289\) 7.62546e45 2.26807
\(290\) −7.07347e45 −1.97365
\(291\) 0 0
\(292\) 1.09038e45 0.267914
\(293\) 3.48209e45 0.803135 0.401568 0.915829i \(-0.368466\pi\)
0.401568 + 0.915829i \(0.368466\pi\)
\(294\) 0 0
\(295\) −1.55075e46 −3.15380
\(296\) −4.70336e45 −0.898487
\(297\) 0 0
\(298\) −9.24893e45 −1.55988
\(299\) −1.71661e45 −0.272116
\(300\) 0 0
\(301\) −5.08633e45 −0.712729
\(302\) −3.26226e44 −0.0429922
\(303\) 0 0
\(304\) 9.13228e45 1.06517
\(305\) 1.60954e46 1.76667
\(306\) 0 0
\(307\) 2.84597e45 0.276803 0.138401 0.990376i \(-0.455804\pi\)
0.138401 + 0.990376i \(0.455804\pi\)
\(308\) −1.10885e44 −0.0101551
\(309\) 0 0
\(310\) 1.92800e45 0.156645
\(311\) −8.10350e45 −0.620306 −0.310153 0.950687i \(-0.600380\pi\)
−0.310153 + 0.950687i \(0.600380\pi\)
\(312\) 0 0
\(313\) 1.80984e46 1.23046 0.615232 0.788346i \(-0.289062\pi\)
0.615232 + 0.788346i \(0.289062\pi\)
\(314\) 1.00774e46 0.645879
\(315\) 0 0
\(316\) 7.94257e45 0.452637
\(317\) 6.00023e45 0.322531 0.161266 0.986911i \(-0.448442\pi\)
0.161266 + 0.986911i \(0.448442\pi\)
\(318\) 0 0
\(319\) −1.89183e45 −0.0905223
\(320\) −4.42063e46 −1.99622
\(321\) 0 0
\(322\) 8.96430e45 0.360730
\(323\) 7.30132e46 2.77431
\(324\) 0 0
\(325\) 2.49182e46 0.844646
\(326\) 1.98020e46 0.634138
\(327\) 0 0
\(328\) −2.49155e46 −0.712532
\(329\) 9.45867e45 0.255686
\(330\) 0 0
\(331\) 1.64524e46 0.397567 0.198784 0.980043i \(-0.436301\pi\)
0.198784 + 0.980043i \(0.436301\pi\)
\(332\) −5.35908e45 −0.122472
\(333\) 0 0
\(334\) −6.30341e46 −1.28904
\(335\) −3.58299e46 −0.693293
\(336\) 0 0
\(337\) 6.90318e46 1.19645 0.598227 0.801327i \(-0.295872\pi\)
0.598227 + 0.801327i \(0.295872\pi\)
\(338\) 4.53655e46 0.744333
\(339\) 0 0
\(340\) −5.42708e46 −0.798373
\(341\) 5.15654e44 0.00718459
\(342\) 0 0
\(343\) −7.69430e46 −0.962117
\(344\) −1.12745e47 −1.33587
\(345\) 0 0
\(346\) 2.73489e46 0.291093
\(347\) −3.75519e46 −0.378911 −0.189455 0.981889i \(-0.560672\pi\)
−0.189455 + 0.981889i \(0.560672\pi\)
\(348\) 0 0
\(349\) −1.14417e47 −1.03805 −0.519026 0.854758i \(-0.673705\pi\)
−0.519026 + 0.854758i \(0.673705\pi\)
\(350\) −1.30126e47 −1.11971
\(351\) 0 0
\(352\) −4.43118e45 −0.0343147
\(353\) −9.61675e46 −0.706637 −0.353318 0.935503i \(-0.614947\pi\)
−0.353318 + 0.935503i \(0.614947\pi\)
\(354\) 0 0
\(355\) 1.66070e46 0.109917
\(356\) −1.06903e46 −0.0671682
\(357\) 0 0
\(358\) 1.63325e47 0.925161
\(359\) 3.14753e45 0.0169326 0.00846629 0.999964i \(-0.497305\pi\)
0.00846629 + 0.999964i \(0.497305\pi\)
\(360\) 0 0
\(361\) 2.79171e47 1.35515
\(362\) −9.17654e46 −0.423224
\(363\) 0 0
\(364\) −1.28723e46 −0.0536143
\(365\) 4.95463e47 1.96152
\(366\) 0 0
\(367\) −6.18183e46 −0.221205 −0.110603 0.993865i \(-0.535278\pi\)
−0.110603 + 0.993865i \(0.535278\pi\)
\(368\) 1.46770e47 0.499404
\(369\) 0 0
\(370\) −4.21399e47 −1.29706
\(371\) −5.18963e46 −0.151957
\(372\) 0 0
\(373\) −6.43130e47 −1.70485 −0.852424 0.522852i \(-0.824868\pi\)
−0.852424 + 0.522852i \(0.824868\pi\)
\(374\) 4.45847e46 0.112476
\(375\) 0 0
\(376\) 2.09663e47 0.479233
\(377\) −2.19617e47 −0.477915
\(378\) 0 0
\(379\) 8.24615e47 1.62714 0.813572 0.581464i \(-0.197520\pi\)
0.813572 + 0.581464i \(0.197520\pi\)
\(380\) −3.60634e47 −0.677751
\(381\) 0 0
\(382\) 4.34678e47 0.741302
\(383\) 1.05734e48 1.71806 0.859032 0.511922i \(-0.171066\pi\)
0.859032 + 0.511922i \(0.171066\pi\)
\(384\) 0 0
\(385\) −5.03856e46 −0.0743503
\(386\) −4.92678e47 −0.692944
\(387\) 0 0
\(388\) −2.32721e47 −0.297474
\(389\) 3.81308e46 0.0464738 0.0232369 0.999730i \(-0.492603\pi\)
0.0232369 + 0.999730i \(0.492603\pi\)
\(390\) 0 0
\(391\) 1.17343e48 1.30074
\(392\) −6.82309e47 −0.721421
\(393\) 0 0
\(394\) −5.09858e47 −0.490647
\(395\) 3.60906e48 3.31396
\(396\) 0 0
\(397\) −3.87592e47 −0.324153 −0.162077 0.986778i \(-0.551819\pi\)
−0.162077 + 0.986778i \(0.551819\pi\)
\(398\) −1.22754e48 −0.979941
\(399\) 0 0
\(400\) −2.13051e48 −1.55015
\(401\) −1.24335e48 −0.863820 −0.431910 0.901917i \(-0.642160\pi\)
−0.431910 + 0.901917i \(0.642160\pi\)
\(402\) 0 0
\(403\) 5.98605e46 0.0379312
\(404\) 2.75003e47 0.166450
\(405\) 0 0
\(406\) 1.14686e48 0.633548
\(407\) −1.12705e47 −0.0594905
\(408\) 0 0
\(409\) 1.71559e48 0.827053 0.413527 0.910492i \(-0.364297\pi\)
0.413527 + 0.910492i \(0.364297\pi\)
\(410\) −2.23231e48 −1.02862
\(411\) 0 0
\(412\) 3.71118e47 0.156284
\(413\) 2.51432e48 1.01238
\(414\) 0 0
\(415\) −2.43514e48 −0.896672
\(416\) −5.14400e47 −0.181165
\(417\) 0 0
\(418\) 2.96269e47 0.0954827
\(419\) −2.35400e48 −0.725851 −0.362925 0.931818i \(-0.618222\pi\)
−0.362925 + 0.931818i \(0.618222\pi\)
\(420\) 0 0
\(421\) 3.28974e48 0.928844 0.464422 0.885614i \(-0.346262\pi\)
0.464422 + 0.885614i \(0.346262\pi\)
\(422\) 3.33206e48 0.900393
\(423\) 0 0
\(424\) −1.15035e48 −0.284813
\(425\) −1.70336e49 −4.03748
\(426\) 0 0
\(427\) −2.60964e48 −0.567106
\(428\) 7.65669e47 0.159342
\(429\) 0 0
\(430\) −1.01014e49 −1.92848
\(431\) 4.39374e48 0.803534 0.401767 0.915742i \(-0.368396\pi\)
0.401767 + 0.915742i \(0.368396\pi\)
\(432\) 0 0
\(433\) 1.65722e48 0.278198 0.139099 0.990279i \(-0.455579\pi\)
0.139099 + 0.990279i \(0.455579\pi\)
\(434\) −3.12598e47 −0.0502835
\(435\) 0 0
\(436\) −1.26377e48 −0.186710
\(437\) 7.79757e48 1.10421
\(438\) 0 0
\(439\) 4.30529e48 0.560285 0.280142 0.959958i \(-0.409618\pi\)
0.280142 + 0.959958i \(0.409618\pi\)
\(440\) −1.11686e48 −0.139355
\(441\) 0 0
\(442\) 5.17569e48 0.593822
\(443\) 7.44863e48 0.819611 0.409805 0.912173i \(-0.365597\pi\)
0.409805 + 0.912173i \(0.365597\pi\)
\(444\) 0 0
\(445\) −4.85760e48 −0.491769
\(446\) 8.15264e48 0.791778
\(447\) 0 0
\(448\) 7.16741e48 0.640795
\(449\) 1.42651e49 1.22382 0.611909 0.790928i \(-0.290402\pi\)
0.611909 + 0.790928i \(0.290402\pi\)
\(450\) 0 0
\(451\) −5.97043e47 −0.0471780
\(452\) 6.11912e47 0.0464118
\(453\) 0 0
\(454\) 1.89539e49 1.32485
\(455\) −5.84910e48 −0.392535
\(456\) 0 0
\(457\) 2.02914e49 1.25563 0.627817 0.778361i \(-0.283948\pi\)
0.627817 + 0.778361i \(0.283948\pi\)
\(458\) −8.74865e48 −0.519912
\(459\) 0 0
\(460\) −5.79594e48 −0.317763
\(461\) −1.29918e49 −0.684232 −0.342116 0.939658i \(-0.611144\pi\)
−0.342116 + 0.939658i \(0.611144\pi\)
\(462\) 0 0
\(463\) −1.83380e49 −0.891463 −0.445731 0.895167i \(-0.647056\pi\)
−0.445731 + 0.895167i \(0.647056\pi\)
\(464\) 1.87772e49 0.877099
\(465\) 0 0
\(466\) −3.67861e48 −0.158688
\(467\) −1.14081e49 −0.472987 −0.236494 0.971633i \(-0.575998\pi\)
−0.236494 + 0.971633i \(0.575998\pi\)
\(468\) 0 0
\(469\) 5.80931e48 0.222550
\(470\) 1.87848e49 0.691826
\(471\) 0 0
\(472\) 5.57330e49 1.89752
\(473\) −2.70167e48 −0.0884506
\(474\) 0 0
\(475\) −1.13189e50 −3.42747
\(476\) 8.79922e48 0.256281
\(477\) 0 0
\(478\) 5.46685e48 0.147340
\(479\) 2.24706e48 0.0582649 0.0291324 0.999576i \(-0.490726\pi\)
0.0291324 + 0.999576i \(0.490726\pi\)
\(480\) 0 0
\(481\) −1.30836e49 −0.314082
\(482\) 9.95383e48 0.229943
\(483\) 0 0
\(484\) 1.14192e49 0.244341
\(485\) −1.05747e50 −2.17794
\(486\) 0 0
\(487\) −6.62105e49 −1.26369 −0.631846 0.775094i \(-0.717702\pi\)
−0.631846 + 0.775094i \(0.717702\pi\)
\(488\) −5.78458e49 −1.06293
\(489\) 0 0
\(490\) −6.11317e49 −1.04145
\(491\) −1.27151e49 −0.208599 −0.104300 0.994546i \(-0.533260\pi\)
−0.104300 + 0.994546i \(0.533260\pi\)
\(492\) 0 0
\(493\) 1.50125e50 2.28447
\(494\) 3.43929e49 0.504104
\(495\) 0 0
\(496\) −5.11807e48 −0.0696138
\(497\) −2.69258e48 −0.0352838
\(498\) 0 0
\(499\) −1.07378e49 −0.130633 −0.0653166 0.997865i \(-0.520806\pi\)
−0.0653166 + 0.997865i \(0.520806\pi\)
\(500\) 4.64629e49 0.544704
\(501\) 0 0
\(502\) 8.17896e49 0.890593
\(503\) −8.93827e48 −0.0938093 −0.0469046 0.998899i \(-0.514936\pi\)
−0.0469046 + 0.998899i \(0.514936\pi\)
\(504\) 0 0
\(505\) 1.24960e50 1.21865
\(506\) 4.76150e48 0.0447671
\(507\) 0 0
\(508\) −1.26705e49 −0.110743
\(509\) −1.92539e50 −1.62270 −0.811352 0.584558i \(-0.801268\pi\)
−0.811352 + 0.584558i \(0.801268\pi\)
\(510\) 0 0
\(511\) −8.03322e49 −0.629655
\(512\) 1.43312e50 1.08340
\(513\) 0 0
\(514\) −1.18114e50 −0.830778
\(515\) 1.68634e50 1.14422
\(516\) 0 0
\(517\) 5.02409e48 0.0317309
\(518\) 6.83238e49 0.416362
\(519\) 0 0
\(520\) −1.29652e50 −0.735731
\(521\) 1.93636e50 1.06045 0.530223 0.847858i \(-0.322108\pi\)
0.530223 + 0.847858i \(0.322108\pi\)
\(522\) 0 0
\(523\) 1.09154e50 0.556875 0.278438 0.960454i \(-0.410183\pi\)
0.278438 + 0.960454i \(0.410183\pi\)
\(524\) 5.19197e49 0.255684
\(525\) 0 0
\(526\) −8.82635e49 −0.405084
\(527\) −4.09194e49 −0.181314
\(528\) 0 0
\(529\) −1.16745e50 −0.482289
\(530\) −1.03066e50 −0.411159
\(531\) 0 0
\(532\) 5.84716e49 0.217560
\(533\) −6.93088e49 −0.249078
\(534\) 0 0
\(535\) 3.47915e50 1.16662
\(536\) 1.28770e50 0.417126
\(537\) 0 0
\(538\) 2.38397e50 0.720824
\(539\) −1.63500e49 −0.0477666
\(540\) 0 0
\(541\) 2.05140e50 0.559630 0.279815 0.960054i \(-0.409727\pi\)
0.279815 + 0.960054i \(0.409727\pi\)
\(542\) 1.11922e50 0.295073
\(543\) 0 0
\(544\) 3.51633e50 0.865984
\(545\) −5.74250e50 −1.36699
\(546\) 0 0
\(547\) −5.42161e50 −1.20604 −0.603021 0.797725i \(-0.706036\pi\)
−0.603021 + 0.797725i \(0.706036\pi\)
\(548\) −1.77671e50 −0.382100
\(549\) 0 0
\(550\) −6.91178e49 −0.138957
\(551\) 9.97595e50 1.93932
\(552\) 0 0
\(553\) −5.85157e50 −1.06379
\(554\) −5.00182e50 −0.879422
\(555\) 0 0
\(556\) −1.28040e50 −0.210602
\(557\) −7.40435e50 −1.17806 −0.589028 0.808113i \(-0.700489\pi\)
−0.589028 + 0.808113i \(0.700489\pi\)
\(558\) 0 0
\(559\) −3.13628e50 −0.466977
\(560\) 5.00097e50 0.720404
\(561\) 0 0
\(562\) 1.66305e50 0.224276
\(563\) 5.85968e50 0.764660 0.382330 0.924026i \(-0.375122\pi\)
0.382330 + 0.924026i \(0.375122\pi\)
\(564\) 0 0
\(565\) 2.78049e50 0.339802
\(566\) −9.06020e50 −1.07160
\(567\) 0 0
\(568\) −5.96843e49 −0.0661327
\(569\) 6.38178e50 0.684486 0.342243 0.939611i \(-0.388813\pi\)
0.342243 + 0.939611i \(0.388813\pi\)
\(570\) 0 0
\(571\) 3.57695e50 0.359538 0.179769 0.983709i \(-0.442465\pi\)
0.179769 + 0.983709i \(0.442465\pi\)
\(572\) −6.83728e48 −0.00665360
\(573\) 0 0
\(574\) 3.61938e50 0.330190
\(575\) −1.81913e51 −1.60697
\(576\) 0 0
\(577\) −1.18260e51 −0.979680 −0.489840 0.871812i \(-0.662945\pi\)
−0.489840 + 0.871812i \(0.662945\pi\)
\(578\) −2.45540e51 −1.96996
\(579\) 0 0
\(580\) −7.41513e50 −0.558086
\(581\) 3.94822e50 0.287835
\(582\) 0 0
\(583\) −2.75654e49 −0.0188580
\(584\) −1.78066e51 −1.18017
\(585\) 0 0
\(586\) −1.12123e51 −0.697572
\(587\) 2.16256e51 1.30366 0.651828 0.758367i \(-0.274002\pi\)
0.651828 + 0.758367i \(0.274002\pi\)
\(588\) 0 0
\(589\) −2.71913e50 −0.153920
\(590\) 4.99341e51 2.73927
\(591\) 0 0
\(592\) 1.11864e51 0.576422
\(593\) 1.31685e51 0.657695 0.328847 0.944383i \(-0.393340\pi\)
0.328847 + 0.944383i \(0.393340\pi\)
\(594\) 0 0
\(595\) 3.99832e51 1.87635
\(596\) −9.69566e50 −0.441084
\(597\) 0 0
\(598\) 5.52746e50 0.236349
\(599\) −2.09368e51 −0.867990 −0.433995 0.900915i \(-0.642896\pi\)
−0.433995 + 0.900915i \(0.642896\pi\)
\(600\) 0 0
\(601\) 4.42052e50 0.172304 0.0861521 0.996282i \(-0.472543\pi\)
0.0861521 + 0.996282i \(0.472543\pi\)
\(602\) 1.63780e51 0.619048
\(603\) 0 0
\(604\) −3.41983e49 −0.0121568
\(605\) 5.18883e51 1.78894
\(606\) 0 0
\(607\) 4.41338e51 1.43146 0.715732 0.698375i \(-0.246093\pi\)
0.715732 + 0.698375i \(0.246093\pi\)
\(608\) 2.33663e51 0.735147
\(609\) 0 0
\(610\) −5.18271e51 −1.53446
\(611\) 5.83230e50 0.167524
\(612\) 0 0
\(613\) −3.52904e51 −0.954198 −0.477099 0.878850i \(-0.658312\pi\)
−0.477099 + 0.878850i \(0.658312\pi\)
\(614\) −9.16402e50 −0.240420
\(615\) 0 0
\(616\) 1.81083e50 0.0447336
\(617\) −6.46795e51 −1.55057 −0.775284 0.631613i \(-0.782393\pi\)
−0.775284 + 0.631613i \(0.782393\pi\)
\(618\) 0 0
\(619\) −1.51171e51 −0.341344 −0.170672 0.985328i \(-0.554594\pi\)
−0.170672 + 0.985328i \(0.554594\pi\)
\(620\) 2.02113e50 0.0442942
\(621\) 0 0
\(622\) 2.60932e51 0.538773
\(623\) 7.87590e50 0.157860
\(624\) 0 0
\(625\) 9.28899e51 1.75464
\(626\) −5.82766e51 −1.06873
\(627\) 0 0
\(628\) 1.05642e51 0.182634
\(629\) 8.94365e51 1.50134
\(630\) 0 0
\(631\) −7.44924e51 −1.17915 −0.589575 0.807714i \(-0.700705\pi\)
−0.589575 + 0.807714i \(0.700705\pi\)
\(632\) −1.29707e52 −1.99388
\(633\) 0 0
\(634\) −1.93207e51 −0.280138
\(635\) −5.75739e51 −0.810797
\(636\) 0 0
\(637\) −1.89801e51 −0.252185
\(638\) 6.09170e50 0.0786241
\(639\) 0 0
\(640\) 7.16271e51 0.872465
\(641\) 2.96058e51 0.350351 0.175176 0.984537i \(-0.443951\pi\)
0.175176 + 0.984537i \(0.443951\pi\)
\(642\) 0 0
\(643\) −1.66529e52 −1.86032 −0.930159 0.367157i \(-0.880331\pi\)
−0.930159 + 0.367157i \(0.880331\pi\)
\(644\) 9.39728e50 0.102003
\(645\) 0 0
\(646\) −2.35102e52 −2.40966
\(647\) 3.47499e51 0.346118 0.173059 0.984911i \(-0.444635\pi\)
0.173059 + 0.984911i \(0.444635\pi\)
\(648\) 0 0
\(649\) 1.33551e51 0.125638
\(650\) −8.02365e51 −0.733627
\(651\) 0 0
\(652\) 2.07585e51 0.179314
\(653\) −1.71299e51 −0.143834 −0.0719169 0.997411i \(-0.522912\pi\)
−0.0719169 + 0.997411i \(0.522912\pi\)
\(654\) 0 0
\(655\) 2.35920e52 1.87198
\(656\) 5.92590e51 0.457123
\(657\) 0 0
\(658\) −3.04569e51 −0.222078
\(659\) −7.36499e51 −0.522146 −0.261073 0.965319i \(-0.584076\pi\)
−0.261073 + 0.965319i \(0.584076\pi\)
\(660\) 0 0
\(661\) 1.02642e52 0.688017 0.344009 0.938967i \(-0.388215\pi\)
0.344009 + 0.938967i \(0.388215\pi\)
\(662\) −5.29765e51 −0.345311
\(663\) 0 0
\(664\) 8.75173e51 0.539491
\(665\) 2.65691e52 1.59286
\(666\) 0 0
\(667\) 1.60329e52 0.909251
\(668\) −6.60788e51 −0.364500
\(669\) 0 0
\(670\) 1.15372e52 0.602168
\(671\) −1.38614e51 −0.0703786
\(672\) 0 0
\(673\) −1.32994e52 −0.639077 −0.319539 0.947573i \(-0.603528\pi\)
−0.319539 + 0.947573i \(0.603528\pi\)
\(674\) −2.22282e52 −1.03919
\(675\) 0 0
\(676\) 4.75567e51 0.210474
\(677\) −8.79193e51 −0.378611 −0.189306 0.981918i \(-0.560624\pi\)
−0.189306 + 0.981918i \(0.560624\pi\)
\(678\) 0 0
\(679\) 1.71454e52 0.699128
\(680\) 8.86276e52 3.51685
\(681\) 0 0
\(682\) −1.66040e50 −0.00624025
\(683\) −2.88301e52 −1.05454 −0.527269 0.849698i \(-0.676784\pi\)
−0.527269 + 0.849698i \(0.676784\pi\)
\(684\) 0 0
\(685\) −8.07328e52 −2.79753
\(686\) 2.47756e52 0.835657
\(687\) 0 0
\(688\) 2.68151e52 0.857026
\(689\) −3.19997e51 −0.0995614
\(690\) 0 0
\(691\) −6.30524e52 −1.85933 −0.929667 0.368401i \(-0.879905\pi\)
−0.929667 + 0.368401i \(0.879905\pi\)
\(692\) 2.86698e51 0.0823119
\(693\) 0 0
\(694\) 1.20917e52 0.329107
\(695\) −5.81807e52 −1.54191
\(696\) 0 0
\(697\) 4.73780e52 1.19061
\(698\) 3.68422e52 0.901612
\(699\) 0 0
\(700\) −1.36411e52 −0.316617
\(701\) 4.33208e52 0.979294 0.489647 0.871921i \(-0.337126\pi\)
0.489647 + 0.871921i \(0.337126\pi\)
\(702\) 0 0
\(703\) 5.94313e52 1.27451
\(704\) 3.80706e51 0.0795235
\(705\) 0 0
\(706\) 3.09659e52 0.613757
\(707\) −2.02605e52 −0.391192
\(708\) 0 0
\(709\) −2.56002e52 −0.469124 −0.234562 0.972101i \(-0.575366\pi\)
−0.234562 + 0.972101i \(0.575366\pi\)
\(710\) −5.34743e51 −0.0954698
\(711\) 0 0
\(712\) 1.74579e52 0.295877
\(713\) −4.37005e51 −0.0721656
\(714\) 0 0
\(715\) −3.10682e51 −0.0487141
\(716\) 1.71214e52 0.261606
\(717\) 0 0
\(718\) −1.01350e51 −0.0147070
\(719\) −6.18204e52 −0.874274 −0.437137 0.899395i \(-0.644008\pi\)
−0.437137 + 0.899395i \(0.644008\pi\)
\(720\) 0 0
\(721\) −2.73415e52 −0.367300
\(722\) −8.98931e52 −1.17703
\(723\) 0 0
\(724\) −9.61978e51 −0.119674
\(725\) −2.32733e53 −2.82231
\(726\) 0 0
\(727\) −3.15811e52 −0.363949 −0.181975 0.983303i \(-0.558249\pi\)
−0.181975 + 0.983303i \(0.558249\pi\)
\(728\) 2.10213e52 0.236172
\(729\) 0 0
\(730\) −1.59539e53 −1.70370
\(731\) 2.14389e53 2.23219
\(732\) 0 0
\(733\) 1.64324e53 1.62659 0.813296 0.581850i \(-0.197671\pi\)
0.813296 + 0.581850i \(0.197671\pi\)
\(734\) 1.99055e52 0.192130
\(735\) 0 0
\(736\) 3.75533e52 0.344674
\(737\) 3.08569e51 0.0276187
\(738\) 0 0
\(739\) 9.02443e52 0.768241 0.384120 0.923283i \(-0.374505\pi\)
0.384120 + 0.923283i \(0.374505\pi\)
\(740\) −4.41753e52 −0.366769
\(741\) 0 0
\(742\) 1.67106e52 0.131984
\(743\) −5.81589e51 −0.0448047 −0.0224023 0.999749i \(-0.507131\pi\)
−0.0224023 + 0.999749i \(0.507131\pi\)
\(744\) 0 0
\(745\) −4.40565e53 −3.22938
\(746\) 2.07088e53 1.48076
\(747\) 0 0
\(748\) 4.67382e51 0.0318048
\(749\) −5.64095e52 −0.374488
\(750\) 0 0
\(751\) −2.03346e53 −1.28498 −0.642491 0.766293i \(-0.722099\pi\)
−0.642491 + 0.766293i \(0.722099\pi\)
\(752\) −4.98661e52 −0.307451
\(753\) 0 0
\(754\) 7.07165e52 0.415098
\(755\) −1.55395e52 −0.0890058
\(756\) 0 0
\(757\) 8.46759e52 0.461834 0.230917 0.972973i \(-0.425827\pi\)
0.230917 + 0.972973i \(0.425827\pi\)
\(758\) −2.65526e53 −1.41327
\(759\) 0 0
\(760\) 5.88938e53 2.98551
\(761\) 6.60711e52 0.326885 0.163443 0.986553i \(-0.447740\pi\)
0.163443 + 0.986553i \(0.447740\pi\)
\(762\) 0 0
\(763\) 9.31064e52 0.438809
\(764\) 4.55673e52 0.209617
\(765\) 0 0
\(766\) −3.40465e53 −1.49224
\(767\) 1.55035e53 0.663309
\(768\) 0 0
\(769\) 4.12388e53 1.68139 0.840696 0.541507i \(-0.182146\pi\)
0.840696 + 0.541507i \(0.182146\pi\)
\(770\) 1.62241e52 0.0645778
\(771\) 0 0
\(772\) −5.16475e52 −0.195943
\(773\) 1.80579e53 0.668879 0.334439 0.942417i \(-0.391453\pi\)
0.334439 + 0.942417i \(0.391453\pi\)
\(774\) 0 0
\(775\) 6.34356e52 0.224002
\(776\) 3.80049e53 1.31038
\(777\) 0 0
\(778\) −1.22781e52 −0.0403653
\(779\) 3.14831e53 1.01073
\(780\) 0 0
\(781\) −1.43020e51 −0.00437877
\(782\) −3.77846e53 −1.12977
\(783\) 0 0
\(784\) 1.62280e53 0.462826
\(785\) 4.80030e53 1.33715
\(786\) 0 0
\(787\) 6.04656e53 1.60685 0.803425 0.595405i \(-0.203009\pi\)
0.803425 + 0.595405i \(0.203009\pi\)
\(788\) −5.34484e52 −0.138739
\(789\) 0 0
\(790\) −1.16212e54 −2.87838
\(791\) −4.50817e52 −0.109078
\(792\) 0 0
\(793\) −1.60913e53 −0.371566
\(794\) 1.24804e53 0.281547
\(795\) 0 0
\(796\) −1.28683e53 −0.277096
\(797\) −6.79705e53 −1.43002 −0.715009 0.699116i \(-0.753577\pi\)
−0.715009 + 0.699116i \(0.753577\pi\)
\(798\) 0 0
\(799\) −3.98684e53 −0.800779
\(800\) −5.45122e53 −1.06987
\(801\) 0 0
\(802\) 4.00359e53 0.750280
\(803\) −4.26694e52 −0.0781411
\(804\) 0 0
\(805\) 4.27007e53 0.746812
\(806\) −1.92751e52 −0.0329456
\(807\) 0 0
\(808\) −4.49098e53 −0.733214
\(809\) −1.11799e54 −1.78397 −0.891987 0.452061i \(-0.850689\pi\)
−0.891987 + 0.452061i \(0.850689\pi\)
\(810\) 0 0
\(811\) −9.12670e53 −1.39132 −0.695662 0.718369i \(-0.744889\pi\)
−0.695662 + 0.718369i \(0.744889\pi\)
\(812\) 1.20226e53 0.179147
\(813\) 0 0
\(814\) 3.62910e52 0.0516711
\(815\) 9.43253e53 1.31284
\(816\) 0 0
\(817\) 1.42463e54 1.89494
\(818\) −5.52421e53 −0.718346
\(819\) 0 0
\(820\) −2.34014e53 −0.290861
\(821\) −4.11474e53 −0.500027 −0.250014 0.968242i \(-0.580435\pi\)
−0.250014 + 0.968242i \(0.580435\pi\)
\(822\) 0 0
\(823\) 2.41018e53 0.279996 0.139998 0.990152i \(-0.455290\pi\)
0.139998 + 0.990152i \(0.455290\pi\)
\(824\) −6.06059e53 −0.688432
\(825\) 0 0
\(826\) −8.09610e53 −0.879316
\(827\) −1.22285e54 −1.29873 −0.649367 0.760475i \(-0.724966\pi\)
−0.649367 + 0.760475i \(0.724966\pi\)
\(828\) 0 0
\(829\) 5.14834e53 0.522888 0.261444 0.965219i \(-0.415801\pi\)
0.261444 + 0.965219i \(0.415801\pi\)
\(830\) 7.84113e53 0.778815
\(831\) 0 0
\(832\) 4.41949e53 0.419846
\(833\) 1.29744e54 1.20547
\(834\) 0 0
\(835\) −3.00258e54 −2.66867
\(836\) 3.10579e52 0.0269995
\(837\) 0 0
\(838\) 7.57988e53 0.630446
\(839\) 1.47001e54 1.19598 0.597990 0.801504i \(-0.295966\pi\)
0.597990 + 0.801504i \(0.295966\pi\)
\(840\) 0 0
\(841\) 7.66718e53 0.596911
\(842\) −1.05929e54 −0.806757
\(843\) 0 0
\(844\) 3.49300e53 0.254603
\(845\) 2.16095e54 1.54098
\(846\) 0 0
\(847\) −8.41294e53 −0.574255
\(848\) 2.73598e53 0.182721
\(849\) 0 0
\(850\) 5.48480e54 3.50680
\(851\) 9.55152e53 0.597552
\(852\) 0 0
\(853\) −2.39398e54 −1.43405 −0.717024 0.697048i \(-0.754496\pi\)
−0.717024 + 0.697048i \(0.754496\pi\)
\(854\) 8.40302e53 0.492566
\(855\) 0 0
\(856\) −1.25039e54 −0.701907
\(857\) −1.12045e54 −0.615526 −0.307763 0.951463i \(-0.599580\pi\)
−0.307763 + 0.951463i \(0.599580\pi\)
\(858\) 0 0
\(859\) 2.95876e54 1.55681 0.778405 0.627762i \(-0.216029\pi\)
0.778405 + 0.627762i \(0.216029\pi\)
\(860\) −1.05893e54 −0.545313
\(861\) 0 0
\(862\) −1.41478e54 −0.697919
\(863\) 2.90233e52 0.0140135 0.00700675 0.999975i \(-0.497770\pi\)
0.00700675 + 0.999975i \(0.497770\pi\)
\(864\) 0 0
\(865\) 1.30274e54 0.602643
\(866\) −5.33623e53 −0.241632
\(867\) 0 0
\(868\) −3.27697e52 −0.0142186
\(869\) −3.10813e53 −0.132018
\(870\) 0 0
\(871\) 3.58207e53 0.145814
\(872\) 2.06382e54 0.822463
\(873\) 0 0
\(874\) −2.51082e54 −0.959077
\(875\) −3.42308e54 −1.28017
\(876\) 0 0
\(877\) −9.87214e53 −0.353931 −0.176966 0.984217i \(-0.556628\pi\)
−0.176966 + 0.984217i \(0.556628\pi\)
\(878\) −1.38630e54 −0.486642
\(879\) 0 0
\(880\) 2.65633e53 0.0894031
\(881\) 3.43429e54 1.13183 0.565916 0.824463i \(-0.308523\pi\)
0.565916 + 0.824463i \(0.308523\pi\)
\(882\) 0 0
\(883\) −3.21368e54 −1.01561 −0.507807 0.861471i \(-0.669544\pi\)
−0.507807 + 0.861471i \(0.669544\pi\)
\(884\) 5.42568e53 0.167914
\(885\) 0 0
\(886\) −2.39846e54 −0.711882
\(887\) 3.18523e54 0.925879 0.462940 0.886390i \(-0.346795\pi\)
0.462940 + 0.886390i \(0.346795\pi\)
\(888\) 0 0
\(889\) 9.33479e53 0.260269
\(890\) 1.56414e54 0.427132
\(891\) 0 0
\(892\) 8.54643e53 0.223890
\(893\) −2.64929e54 −0.679793
\(894\) 0 0
\(895\) 7.77987e54 1.91534
\(896\) −1.16133e54 −0.280064
\(897\) 0 0
\(898\) −4.59335e54 −1.06296
\(899\) −5.59090e53 −0.126744
\(900\) 0 0
\(901\) 2.18743e54 0.475911
\(902\) 1.92248e53 0.0409770
\(903\) 0 0
\(904\) −9.99291e53 −0.204445
\(905\) −4.37117e54 −0.876192
\(906\) 0 0
\(907\) −1.66261e54 −0.319931 −0.159966 0.987123i \(-0.551138\pi\)
−0.159966 + 0.987123i \(0.551138\pi\)
\(908\) 1.98694e54 0.374625
\(909\) 0 0
\(910\) 1.88341e54 0.340940
\(911\) −1.12509e55 −1.99570 −0.997851 0.0655227i \(-0.979129\pi\)
−0.997851 + 0.0655227i \(0.979129\pi\)
\(912\) 0 0
\(913\) 2.09715e53 0.0357207
\(914\) −6.53384e54 −1.09060
\(915\) 0 0
\(916\) −9.17122e53 −0.147015
\(917\) −3.82510e54 −0.600912
\(918\) 0 0
\(919\) −1.81936e54 −0.274525 −0.137263 0.990535i \(-0.543830\pi\)
−0.137263 + 0.990535i \(0.543830\pi\)
\(920\) 9.46514e54 1.39976
\(921\) 0 0
\(922\) 4.18336e54 0.594297
\(923\) −1.66027e53 −0.0231178
\(924\) 0 0
\(925\) −1.38650e55 −1.85480
\(926\) 5.90482e54 0.774290
\(927\) 0 0
\(928\) 4.80444e54 0.605348
\(929\) −6.12532e52 −0.00756550 −0.00378275 0.999993i \(-0.501204\pi\)
−0.00378275 + 0.999993i \(0.501204\pi\)
\(930\) 0 0
\(931\) 8.62160e54 1.02334
\(932\) −3.85630e53 −0.0448721
\(933\) 0 0
\(934\) 3.67339e54 0.410818
\(935\) 2.12376e54 0.232857
\(936\) 0 0
\(937\) −1.54519e55 −1.62855 −0.814274 0.580481i \(-0.802865\pi\)
−0.814274 + 0.580481i \(0.802865\pi\)
\(938\) −1.87059e54 −0.193298
\(939\) 0 0
\(940\) 1.96922e54 0.195627
\(941\) 1.06842e55 1.04072 0.520361 0.853946i \(-0.325797\pi\)
0.520361 + 0.853946i \(0.325797\pi\)
\(942\) 0 0
\(943\) 5.05981e54 0.473880
\(944\) −1.32555e55 −1.21735
\(945\) 0 0
\(946\) 8.69935e53 0.0768247
\(947\) −3.77859e54 −0.327232 −0.163616 0.986524i \(-0.552316\pi\)
−0.163616 + 0.986524i \(0.552316\pi\)
\(948\) 0 0
\(949\) −4.95335e54 −0.412548
\(950\) 3.64469e55 2.97697
\(951\) 0 0
\(952\) −1.43697e55 −1.12892
\(953\) −5.53742e53 −0.0426667 −0.0213333 0.999772i \(-0.506791\pi\)
−0.0213333 + 0.999772i \(0.506791\pi\)
\(954\) 0 0
\(955\) 2.07055e55 1.53470
\(956\) 5.73090e53 0.0416631
\(957\) 0 0
\(958\) −7.23551e53 −0.0506066
\(959\) 1.30897e55 0.898016
\(960\) 0 0
\(961\) −1.49966e55 −0.989941
\(962\) 4.21290e54 0.272799
\(963\) 0 0
\(964\) 1.04346e54 0.0650206
\(965\) −2.34683e55 −1.43459
\(966\) 0 0
\(967\) 1.91207e55 1.12490 0.562449 0.826832i \(-0.309859\pi\)
0.562449 + 0.826832i \(0.309859\pi\)
\(968\) −1.86483e55 −1.07633
\(969\) 0 0
\(970\) 3.40506e55 1.89168
\(971\) −8.30735e54 −0.452800 −0.226400 0.974034i \(-0.572696\pi\)
−0.226400 + 0.974034i \(0.572696\pi\)
\(972\) 0 0
\(973\) 9.43317e54 0.494960
\(974\) 2.13197e55 1.09759
\(975\) 0 0
\(976\) 1.37580e55 0.681921
\(977\) 3.32881e55 1.61897 0.809486 0.587140i \(-0.199746\pi\)
0.809486 + 0.587140i \(0.199746\pi\)
\(978\) 0 0
\(979\) 4.18338e53 0.0195906
\(980\) −6.40844e54 −0.294490
\(981\) 0 0
\(982\) 4.09426e54 0.181181
\(983\) 5.66371e54 0.245958 0.122979 0.992409i \(-0.460755\pi\)
0.122979 + 0.992409i \(0.460755\pi\)
\(984\) 0 0
\(985\) −2.42867e55 −1.01578
\(986\) −4.83403e55 −1.98420
\(987\) 0 0
\(988\) 3.60541e54 0.142545
\(989\) 2.28960e55 0.888442
\(990\) 0 0
\(991\) −3.93647e55 −1.47145 −0.735723 0.677283i \(-0.763157\pi\)
−0.735723 + 0.677283i \(0.763157\pi\)
\(992\) −1.30954e54 −0.0480453
\(993\) 0 0
\(994\) 8.67009e53 0.0306461
\(995\) −5.84730e55 −2.02875
\(996\) 0 0
\(997\) 1.67431e53 0.00559727 0.00279863 0.999996i \(-0.499109\pi\)
0.00279863 + 0.999996i \(0.499109\pi\)
\(998\) 3.45755e54 0.113463
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9.38.a.c.1.2 4
3.2 odd 2 3.38.a.b.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.38.a.b.1.3 4 3.2 odd 2
9.38.a.c.1.2 4 1.1 even 1 trivial