Properties

Label 2.42.a.a.1.1
Level 2
Weight 42
Character 2.1
Self dual yes
Analytic conductor 21.294
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 42 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.2943340913\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.04858e6 q^{2} +5.04352e9 q^{3} +1.09951e12 q^{4} -4.85042e13 q^{5} -5.28851e15 q^{6} -1.19392e17 q^{7} -1.15292e18 q^{8} -1.10359e19 q^{9} +O(q^{10})\) \(q-1.04858e6 q^{2} +5.04352e9 q^{3} +1.09951e12 q^{4} -4.85042e13 q^{5} -5.28851e15 q^{6} -1.19392e17 q^{7} -1.15292e18 q^{8} -1.10359e19 q^{9} +5.08603e19 q^{10} +3.15381e21 q^{11} +5.54541e21 q^{12} -1.14103e22 q^{13} +1.25192e23 q^{14} -2.44632e23 q^{15} +1.20893e24 q^{16} -2.67238e25 q^{17} +1.15720e25 q^{18} +6.79752e25 q^{19} -5.33309e25 q^{20} -6.02158e26 q^{21} -3.30701e27 q^{22} -1.35051e28 q^{23} -5.81478e27 q^{24} -4.31221e28 q^{25} +1.19646e28 q^{26} -2.39612e29 q^{27} -1.31273e29 q^{28} +1.36715e29 q^{29} +2.56515e29 q^{30} +3.06142e30 q^{31} -1.26765e30 q^{32} +1.59063e31 q^{33} +2.80219e31 q^{34} +5.79103e30 q^{35} -1.21341e31 q^{36} -2.21949e32 q^{37} -7.12772e31 q^{38} -5.75480e31 q^{39} +5.59215e31 q^{40} -5.01985e32 q^{41} +6.31408e32 q^{42} -3.11848e33 q^{43} +3.46765e33 q^{44} +5.35289e32 q^{45} +1.41611e34 q^{46} +1.31555e34 q^{47} +6.09724e33 q^{48} -3.03131e34 q^{49} +4.52168e34 q^{50} -1.34782e35 q^{51} -1.25458e34 q^{52} -3.23999e35 q^{53} +2.51251e35 q^{54} -1.52973e35 q^{55} +1.37650e35 q^{56} +3.42834e35 q^{57} -1.43357e35 q^{58} +3.45957e36 q^{59} -2.68975e35 q^{60} -9.78043e35 q^{61} -3.21013e36 q^{62} +1.31761e36 q^{63} +1.32923e36 q^{64} +5.53447e35 q^{65} -1.66790e37 q^{66} +1.66275e37 q^{67} -2.93831e37 q^{68} -6.81131e37 q^{69} -6.07234e36 q^{70} +1.16969e38 q^{71} +1.27236e37 q^{72} +1.90709e38 q^{73} +2.32731e38 q^{74} -2.17487e38 q^{75} +7.47395e37 q^{76} -3.76541e38 q^{77} +6.03435e37 q^{78} -5.61362e38 q^{79} -5.86380e37 q^{80} -8.05974e38 q^{81} +5.26369e38 q^{82} -6.05771e38 q^{83} -6.62079e38 q^{84} +1.29621e39 q^{85} +3.26996e39 q^{86} +6.89527e38 q^{87} -3.63609e39 q^{88} +1.19154e40 q^{89} -5.61291e38 q^{90} +1.36230e39 q^{91} -1.48490e40 q^{92} +1.54403e40 q^{93} -1.37946e40 q^{94} -3.29708e39 q^{95} -6.39342e39 q^{96} -6.35760e40 q^{97} +3.17856e40 q^{98} -3.48052e40 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.04858e6 −0.707107
\(3\) 5.04352e9 0.835118 0.417559 0.908650i \(-0.362886\pi\)
0.417559 + 0.908650i \(0.362886\pi\)
\(4\) 1.09951e12 0.500000
\(5\) −4.85042e13 −0.227454 −0.113727 0.993512i \(-0.536279\pi\)
−0.113727 + 0.993512i \(0.536279\pi\)
\(6\) −5.28851e15 −0.590517
\(7\) −1.19392e17 −0.565545 −0.282772 0.959187i \(-0.591254\pi\)
−0.282772 + 0.959187i \(0.591254\pi\)
\(8\) −1.15292e18 −0.353553
\(9\) −1.10359e19 −0.302578
\(10\) 5.08603e19 0.160835
\(11\) 3.15381e21 1.41347 0.706733 0.707480i \(-0.250168\pi\)
0.706733 + 0.707480i \(0.250168\pi\)
\(12\) 5.54541e21 0.417559
\(13\) −1.14103e22 −0.166517 −0.0832585 0.996528i \(-0.526533\pi\)
−0.0832585 + 0.996528i \(0.526533\pi\)
\(14\) 1.25192e23 0.399901
\(15\) −2.44632e23 −0.189951
\(16\) 1.20893e24 0.250000
\(17\) −2.67238e25 −1.59476 −0.797379 0.603479i \(-0.793781\pi\)
−0.797379 + 0.603479i \(0.793781\pi\)
\(18\) 1.15720e25 0.213955
\(19\) 6.79752e25 0.414860 0.207430 0.978250i \(-0.433490\pi\)
0.207430 + 0.978250i \(0.433490\pi\)
\(20\) −5.33309e25 −0.113727
\(21\) −6.02158e26 −0.472297
\(22\) −3.30701e27 −0.999471
\(23\) −1.35051e28 −1.64088 −0.820439 0.571734i \(-0.806271\pi\)
−0.820439 + 0.571734i \(0.806271\pi\)
\(24\) −5.81478e27 −0.295259
\(25\) −4.31221e28 −0.948265
\(26\) 1.19646e28 0.117745
\(27\) −2.39612e29 −1.08781
\(28\) −1.31273e29 −0.282772
\(29\) 1.36715e29 0.143436 0.0717181 0.997425i \(-0.477152\pi\)
0.0717181 + 0.997425i \(0.477152\pi\)
\(30\) 2.56515e29 0.134316
\(31\) 3.06142e30 0.818481 0.409241 0.912427i \(-0.365794\pi\)
0.409241 + 0.912427i \(0.365794\pi\)
\(32\) −1.26765e30 −0.176777
\(33\) 1.59063e31 1.18041
\(34\) 2.80219e31 1.12766
\(35\) 5.79103e30 0.128636
\(36\) −1.21341e31 −0.151289
\(37\) −2.21949e32 −1.57804 −0.789021 0.614366i \(-0.789412\pi\)
−0.789021 + 0.614366i \(0.789412\pi\)
\(38\) −7.12772e31 −0.293350
\(39\) −5.75480e31 −0.139061
\(40\) 5.59215e31 0.0804173
\(41\) −5.01985e32 −0.435133 −0.217566 0.976046i \(-0.569812\pi\)
−0.217566 + 0.976046i \(0.569812\pi\)
\(42\) 6.31408e32 0.333964
\(43\) −3.11848e33 −1.01821 −0.509107 0.860703i \(-0.670024\pi\)
−0.509107 + 0.860703i \(0.670024\pi\)
\(44\) 3.46765e33 0.706733
\(45\) 5.35289e32 0.0688227
\(46\) 1.41611e34 1.16028
\(47\) 1.31555e34 0.693588 0.346794 0.937941i \(-0.387270\pi\)
0.346794 + 0.937941i \(0.387270\pi\)
\(48\) 6.09724e33 0.208779
\(49\) −3.03131e34 −0.680159
\(50\) 4.52168e34 0.670524
\(51\) −1.34782e35 −1.33181
\(52\) −1.25458e34 −0.0832585
\(53\) −3.23999e35 −1.45508 −0.727542 0.686064i \(-0.759337\pi\)
−0.727542 + 0.686064i \(0.759337\pi\)
\(54\) 2.51251e35 0.769195
\(55\) −1.52973e35 −0.321499
\(56\) 1.37650e35 0.199950
\(57\) 3.42834e35 0.346457
\(58\) −1.43357e35 −0.101425
\(59\) 3.45957e36 1.72408 0.862038 0.506844i \(-0.169188\pi\)
0.862038 + 0.506844i \(0.169188\pi\)
\(60\) −2.68975e35 −0.0949756
\(61\) −9.78043e35 −0.246091 −0.123046 0.992401i \(-0.539266\pi\)
−0.123046 + 0.992401i \(0.539266\pi\)
\(62\) −3.21013e36 −0.578753
\(63\) 1.31761e36 0.171122
\(64\) 1.32923e36 0.125000
\(65\) 5.53447e35 0.0378750
\(66\) −1.66790e37 −0.834676
\(67\) 1.66275e37 0.611357 0.305678 0.952135i \(-0.401117\pi\)
0.305678 + 0.952135i \(0.401117\pi\)
\(68\) −2.93831e37 −0.797379
\(69\) −6.81131e37 −1.37033
\(70\) −6.07234e36 −0.0909591
\(71\) 1.16969e38 1.31001 0.655004 0.755625i \(-0.272667\pi\)
0.655004 + 0.755625i \(0.272667\pi\)
\(72\) 1.27236e37 0.106978
\(73\) 1.90709e38 1.20851 0.604257 0.796790i \(-0.293470\pi\)
0.604257 + 0.796790i \(0.293470\pi\)
\(74\) 2.32731e38 1.11584
\(75\) −2.17487e38 −0.791913
\(76\) 7.47395e37 0.207430
\(77\) −3.76541e38 −0.799378
\(78\) 6.03435e37 0.0983311
\(79\) −5.61362e38 −0.704511 −0.352256 0.935904i \(-0.614585\pi\)
−0.352256 + 0.935904i \(0.614585\pi\)
\(80\) −5.86380e37 −0.0568636
\(81\) −8.05974e38 −0.605868
\(82\) 5.26369e38 0.307685
\(83\) −6.05771e38 −0.276190 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(84\) −6.62079e38 −0.236148
\(85\) 1.29621e39 0.362735
\(86\) 3.26996e39 0.719987
\(87\) 6.89527e38 0.119786
\(88\) −3.63609e39 −0.499736
\(89\) 1.19154e40 1.29902 0.649508 0.760355i \(-0.274975\pi\)
0.649508 + 0.760355i \(0.274975\pi\)
\(90\) −5.61291e38 −0.0486650
\(91\) 1.36230e39 0.0941728
\(92\) −1.48490e40 −0.820439
\(93\) 1.54403e40 0.683528
\(94\) −1.37946e40 −0.490441
\(95\) −3.29708e39 −0.0943618
\(96\) −6.39342e39 −0.147629
\(97\) −6.35760e40 −1.18706 −0.593530 0.804812i \(-0.702266\pi\)
−0.593530 + 0.804812i \(0.702266\pi\)
\(98\) 3.17856e40 0.480945
\(99\) −3.48052e40 −0.427684
\(100\) −4.74132e40 −0.474132
\(101\) 6.20987e39 0.0506401 0.0253200 0.999679i \(-0.491940\pi\)
0.0253200 + 0.999679i \(0.491940\pi\)
\(102\) 1.41329e41 0.941732
\(103\) 1.73412e41 0.946056 0.473028 0.881047i \(-0.343161\pi\)
0.473028 + 0.881047i \(0.343161\pi\)
\(104\) 1.31552e40 0.0588726
\(105\) 2.92072e40 0.107426
\(106\) 3.39737e41 1.02890
\(107\) 1.77847e41 0.444304 0.222152 0.975012i \(-0.428692\pi\)
0.222152 + 0.975012i \(0.428692\pi\)
\(108\) −2.63456e41 −0.543903
\(109\) −1.02737e42 −1.75583 −0.877916 0.478815i \(-0.841066\pi\)
−0.877916 + 0.478815i \(0.841066\pi\)
\(110\) 1.60404e41 0.227334
\(111\) −1.11941e42 −1.31785
\(112\) −1.44337e41 −0.141386
\(113\) 1.36076e42 1.11089 0.555447 0.831552i \(-0.312547\pi\)
0.555447 + 0.831552i \(0.312547\pi\)
\(114\) −3.59488e41 −0.244982
\(115\) 6.55053e41 0.373225
\(116\) 1.50320e41 0.0717181
\(117\) 1.25923e41 0.0503844
\(118\) −3.62763e42 −1.21911
\(119\) 3.19062e42 0.901907
\(120\) 2.82041e41 0.0671579
\(121\) 4.96799e42 0.997886
\(122\) 1.02555e42 0.174013
\(123\) −2.53177e42 −0.363387
\(124\) 3.36607e42 0.409241
\(125\) 4.29732e42 0.443141
\(126\) −1.38161e42 −0.121001
\(127\) −1.25701e43 −0.936185 −0.468092 0.883679i \(-0.655059\pi\)
−0.468092 + 0.883679i \(0.655059\pi\)
\(128\) −1.39380e42 −0.0883883
\(129\) −1.57281e43 −0.850329
\(130\) −5.80332e41 −0.0267817
\(131\) −1.14997e43 −0.453551 −0.226776 0.973947i \(-0.572818\pi\)
−0.226776 + 0.973947i \(0.572818\pi\)
\(132\) 1.74891e43 0.590205
\(133\) −8.11573e42 −0.234622
\(134\) −1.74352e43 −0.432295
\(135\) 1.16222e43 0.247426
\(136\) 3.08104e43 0.563832
\(137\) −1.99120e43 −0.313577 −0.156788 0.987632i \(-0.550114\pi\)
−0.156788 + 0.987632i \(0.550114\pi\)
\(138\) 7.14217e43 0.968967
\(139\) 8.57436e43 1.00322 0.501612 0.865093i \(-0.332740\pi\)
0.501612 + 0.865093i \(0.332740\pi\)
\(140\) 6.36731e42 0.0643178
\(141\) 6.63501e43 0.579228
\(142\) −1.22651e44 −0.926316
\(143\) −3.59859e43 −0.235366
\(144\) −1.33416e43 −0.0756446
\(145\) −6.63127e42 −0.0326252
\(146\) −1.99972e44 −0.854548
\(147\) −1.52885e44 −0.568013
\(148\) −2.44036e44 −0.789021
\(149\) 2.85509e43 0.0804083 0.0402041 0.999191i \(-0.487199\pi\)
0.0402041 + 0.999191i \(0.487199\pi\)
\(150\) 2.28052e44 0.559967
\(151\) 1.89816e44 0.406729 0.203365 0.979103i \(-0.434812\pi\)
0.203365 + 0.979103i \(0.434812\pi\)
\(152\) −7.83701e43 −0.146675
\(153\) 2.94922e44 0.482539
\(154\) 3.94832e44 0.565246
\(155\) −1.48492e44 −0.186167
\(156\) −6.32747e43 −0.0695306
\(157\) 3.84758e44 0.370890 0.185445 0.982655i \(-0.440627\pi\)
0.185445 + 0.982655i \(0.440627\pi\)
\(158\) 5.88631e44 0.498165
\(159\) −1.63409e45 −1.21517
\(160\) 6.14864e43 0.0402086
\(161\) 1.61240e45 0.927990
\(162\) 8.45125e44 0.428413
\(163\) −4.75006e44 −0.212253 −0.106126 0.994353i \(-0.533845\pi\)
−0.106126 + 0.994353i \(0.533845\pi\)
\(164\) −5.51938e44 −0.217566
\(165\) −7.71522e44 −0.268489
\(166\) 6.35197e44 0.195296
\(167\) −4.32345e45 −1.17528 −0.587642 0.809121i \(-0.699944\pi\)
−0.587642 + 0.809121i \(0.699944\pi\)
\(168\) 6.94241e44 0.166982
\(169\) −4.56526e45 −0.972272
\(170\) −1.35918e45 −0.256492
\(171\) −7.50170e44 −0.125528
\(172\) −3.42880e45 −0.509107
\(173\) 1.44924e46 1.91072 0.955358 0.295452i \(-0.0954703\pi\)
0.955358 + 0.295452i \(0.0954703\pi\)
\(174\) −7.23021e44 −0.0847016
\(175\) 5.14845e45 0.536286
\(176\) 3.81272e45 0.353366
\(177\) 1.74484e46 1.43981
\(178\) −1.24942e46 −0.918543
\(179\) 5.67669e45 0.372056 0.186028 0.982544i \(-0.440439\pi\)
0.186028 + 0.982544i \(0.440439\pi\)
\(180\) 5.88557e44 0.0344114
\(181\) −2.20895e46 −1.15286 −0.576429 0.817147i \(-0.695554\pi\)
−0.576429 + 0.817147i \(0.695554\pi\)
\(182\) −1.42848e45 −0.0665902
\(183\) −4.93278e45 −0.205515
\(184\) 1.55703e46 0.580138
\(185\) 1.07655e46 0.358932
\(186\) −1.61904e46 −0.483327
\(187\) −8.42816e46 −2.25414
\(188\) 1.44646e46 0.346794
\(189\) 2.86079e46 0.615203
\(190\) 3.45724e45 0.0667238
\(191\) −6.23395e46 −1.08039 −0.540194 0.841541i \(-0.681649\pi\)
−0.540194 + 0.841541i \(0.681649\pi\)
\(192\) 6.70398e45 0.104390
\(193\) 1.06448e47 1.49009 0.745043 0.667017i \(-0.232429\pi\)
0.745043 + 0.667017i \(0.232429\pi\)
\(194\) 6.66643e46 0.839378
\(195\) 2.79132e45 0.0316301
\(196\) −3.33296e46 −0.340080
\(197\) 8.49296e46 0.780731 0.390365 0.920660i \(-0.372349\pi\)
0.390365 + 0.920660i \(0.372349\pi\)
\(198\) 3.64959e46 0.302418
\(199\) −1.82716e47 −1.36549 −0.682746 0.730656i \(-0.739214\pi\)
−0.682746 + 0.730656i \(0.739214\pi\)
\(200\) 4.97164e46 0.335262
\(201\) 8.38613e46 0.510555
\(202\) −6.51152e45 −0.0358079
\(203\) −1.63228e46 −0.0811196
\(204\) −1.48194e47 −0.665905
\(205\) 2.43484e46 0.0989728
\(206\) −1.81836e47 −0.668963
\(207\) 1.49041e47 0.496494
\(208\) −1.37942e46 −0.0416292
\(209\) 2.14381e47 0.586391
\(210\) −3.06259e46 −0.0759616
\(211\) −7.87226e47 −1.77137 −0.885684 0.464288i \(-0.846310\pi\)
−0.885684 + 0.464288i \(0.846310\pi\)
\(212\) −3.56240e47 −0.727542
\(213\) 5.89934e47 1.09401
\(214\) −1.86487e47 −0.314170
\(215\) 1.51259e47 0.231597
\(216\) 2.76254e47 0.384598
\(217\) −3.65511e47 −0.462888
\(218\) 1.07727e48 1.24156
\(219\) 9.61842e47 1.00925
\(220\) −1.68196e47 −0.160749
\(221\) 3.04926e47 0.265554
\(222\) 1.17378e48 0.931861
\(223\) −1.32160e48 −0.956862 −0.478431 0.878125i \(-0.658794\pi\)
−0.478431 + 0.878125i \(0.658794\pi\)
\(224\) 1.51348e47 0.0999751
\(225\) 4.75893e47 0.286924
\(226\) −1.42686e48 −0.785521
\(227\) −1.43456e48 −0.721419 −0.360710 0.932678i \(-0.617465\pi\)
−0.360710 + 0.932678i \(0.617465\pi\)
\(228\) 3.76950e47 0.173229
\(229\) −3.68923e48 −1.54991 −0.774957 0.632014i \(-0.782229\pi\)
−0.774957 + 0.632014i \(0.782229\pi\)
\(230\) −6.86873e47 −0.263910
\(231\) −1.89909e48 −0.667575
\(232\) −1.57622e47 −0.0507123
\(233\) 6.32561e48 1.86340 0.931700 0.363228i \(-0.118325\pi\)
0.931700 + 0.363228i \(0.118325\pi\)
\(234\) −1.32040e47 −0.0356272
\(235\) −6.38098e47 −0.157760
\(236\) 3.80384e48 0.862038
\(237\) −2.83124e48 −0.588350
\(238\) −3.34560e48 −0.637744
\(239\) 8.39385e48 1.46826 0.734132 0.679006i \(-0.237589\pi\)
0.734132 + 0.679006i \(0.237589\pi\)
\(240\) −2.95742e47 −0.0474878
\(241\) 2.76168e48 0.407215 0.203608 0.979053i \(-0.434733\pi\)
0.203608 + 0.979053i \(0.434733\pi\)
\(242\) −5.20932e48 −0.705612
\(243\) 4.67443e48 0.581835
\(244\) −1.07537e48 −0.123046
\(245\) 1.47031e48 0.154705
\(246\) 2.65475e48 0.256953
\(247\) −7.75618e47 −0.0690812
\(248\) −3.52958e48 −0.289377
\(249\) −3.05522e48 −0.230651
\(250\) −4.50606e48 −0.313348
\(251\) 5.47506e48 0.350814 0.175407 0.984496i \(-0.443876\pi\)
0.175407 + 0.984496i \(0.443876\pi\)
\(252\) 1.44872e48 0.0855608
\(253\) −4.25924e49 −2.31933
\(254\) 1.31807e49 0.661983
\(255\) 6.53748e48 0.302926
\(256\) 1.46150e48 0.0625000
\(257\) −2.41403e49 −0.953045 −0.476522 0.879162i \(-0.658103\pi\)
−0.476522 + 0.879162i \(0.658103\pi\)
\(258\) 1.64921e49 0.601274
\(259\) 2.64991e49 0.892453
\(260\) 6.08522e47 0.0189375
\(261\) −1.50878e48 −0.0434007
\(262\) 1.20583e49 0.320709
\(263\) −1.00767e49 −0.247871 −0.123936 0.992290i \(-0.539552\pi\)
−0.123936 + 0.992290i \(0.539552\pi\)
\(264\) −1.83387e49 −0.417338
\(265\) 1.57153e49 0.330965
\(266\) 8.50996e48 0.165903
\(267\) 6.00957e49 1.08483
\(268\) 1.82822e49 0.305678
\(269\) −3.47580e49 −0.538435 −0.269217 0.963079i \(-0.586765\pi\)
−0.269217 + 0.963079i \(0.586765\pi\)
\(270\) −1.21868e49 −0.174957
\(271\) 9.20219e49 1.22468 0.612338 0.790596i \(-0.290229\pi\)
0.612338 + 0.790596i \(0.290229\pi\)
\(272\) −3.23070e49 −0.398689
\(273\) 6.87080e48 0.0786454
\(274\) 2.08793e49 0.221732
\(275\) −1.35999e50 −1.34034
\(276\) −7.48911e49 −0.685163
\(277\) −1.12019e50 −0.951605 −0.475803 0.879552i \(-0.657842\pi\)
−0.475803 + 0.879552i \(0.657842\pi\)
\(278\) −8.99087e49 −0.709387
\(279\) −3.37857e49 −0.247655
\(280\) −6.67661e48 −0.0454796
\(281\) 1.13941e50 0.721441 0.360720 0.932674i \(-0.382531\pi\)
0.360720 + 0.932674i \(0.382531\pi\)
\(282\) −6.95731e49 −0.409576
\(283\) 2.06091e50 1.12833 0.564167 0.825661i \(-0.309197\pi\)
0.564167 + 0.825661i \(0.309197\pi\)
\(284\) 1.28609e50 0.655004
\(285\) −1.66289e49 −0.0788032
\(286\) 3.77340e49 0.166429
\(287\) 5.99332e49 0.246087
\(288\) 1.39897e49 0.0534888
\(289\) 4.33354e50 1.54325
\(290\) 6.95339e48 0.0230695
\(291\) −3.20647e50 −0.991335
\(292\) 2.09686e50 0.604257
\(293\) −3.36281e50 −0.903475 −0.451738 0.892151i \(-0.649196\pi\)
−0.451738 + 0.892151i \(0.649196\pi\)
\(294\) 1.60311e50 0.401646
\(295\) −1.67804e50 −0.392149
\(296\) 2.55890e50 0.557922
\(297\) −7.55691e50 −1.53758
\(298\) −2.99378e49 −0.0568573
\(299\) 1.54097e50 0.273234
\(300\) −2.39129e50 −0.395956
\(301\) 3.72323e50 0.575846
\(302\) −1.99037e50 −0.287601
\(303\) 3.13196e49 0.0422904
\(304\) 8.21770e49 0.103715
\(305\) 4.74392e49 0.0559746
\(306\) −3.09248e50 −0.341207
\(307\) −6.52252e50 −0.673099 −0.336550 0.941666i \(-0.609260\pi\)
−0.336550 + 0.941666i \(0.609260\pi\)
\(308\) −4.14011e50 −0.399689
\(309\) 8.74608e50 0.790068
\(310\) 1.55705e50 0.131640
\(311\) −7.73007e50 −0.611781 −0.305890 0.952067i \(-0.598954\pi\)
−0.305890 + 0.952067i \(0.598954\pi\)
\(312\) 6.63484e49 0.0491656
\(313\) 2.76877e51 1.92144 0.960722 0.277511i \(-0.0895095\pi\)
0.960722 + 0.277511i \(0.0895095\pi\)
\(314\) −4.03448e50 −0.262259
\(315\) −6.39095e49 −0.0389223
\(316\) −6.17225e50 −0.352256
\(317\) 7.05058e50 0.377146 0.188573 0.982059i \(-0.439614\pi\)
0.188573 + 0.982059i \(0.439614\pi\)
\(318\) 1.71347e51 0.859252
\(319\) 4.31174e50 0.202742
\(320\) −6.44731e49 −0.0284318
\(321\) 8.96976e50 0.371046
\(322\) −1.69073e51 −0.656188
\(323\) −1.81655e51 −0.661601
\(324\) −8.86178e50 −0.302934
\(325\) 4.92036e50 0.157902
\(326\) 4.98079e50 0.150085
\(327\) −5.18155e51 −1.46633
\(328\) 5.78749e50 0.153843
\(329\) −1.57067e51 −0.392255
\(330\) 8.08999e50 0.189851
\(331\) −2.05137e51 −0.452449 −0.226224 0.974075i \(-0.572638\pi\)
−0.226224 + 0.974075i \(0.572638\pi\)
\(332\) −6.66052e50 −0.138095
\(333\) 2.44942e51 0.477481
\(334\) 4.53347e51 0.831052
\(335\) −8.06506e50 −0.139056
\(336\) −7.27964e50 −0.118074
\(337\) −1.03725e52 −1.58296 −0.791479 0.611196i \(-0.790689\pi\)
−0.791479 + 0.611196i \(0.790689\pi\)
\(338\) 4.78702e51 0.687500
\(339\) 6.86301e51 0.927728
\(340\) 1.42520e51 0.181367
\(341\) 9.65514e51 1.15689
\(342\) 7.86611e50 0.0887615
\(343\) 8.94019e51 0.950205
\(344\) 3.59536e51 0.359993
\(345\) 3.30377e51 0.311687
\(346\) −1.51964e52 −1.35108
\(347\) 2.85107e50 0.0238921 0.0119461 0.999929i \(-0.496197\pi\)
0.0119461 + 0.999929i \(0.496197\pi\)
\(348\) 7.58143e50 0.0598931
\(349\) −9.31673e51 −0.693973 −0.346986 0.937870i \(-0.612795\pi\)
−0.346986 + 0.937870i \(0.612795\pi\)
\(350\) −5.39854e51 −0.379212
\(351\) 2.73405e51 0.181138
\(352\) −3.99793e51 −0.249868
\(353\) −1.81721e52 −1.07158 −0.535788 0.844353i \(-0.679985\pi\)
−0.535788 + 0.844353i \(0.679985\pi\)
\(354\) −1.82960e52 −1.01810
\(355\) −5.67348e51 −0.297967
\(356\) 1.31012e52 0.649508
\(357\) 1.60919e52 0.753198
\(358\) −5.95244e51 −0.263083
\(359\) 1.90791e52 0.796384 0.398192 0.917302i \(-0.369638\pi\)
0.398192 + 0.917302i \(0.369638\pi\)
\(360\) −6.17147e50 −0.0243325
\(361\) −2.22265e52 −0.827891
\(362\) 2.31626e52 0.815193
\(363\) 2.50562e52 0.833352
\(364\) 1.49787e51 0.0470864
\(365\) −9.25016e51 −0.274882
\(366\) 5.17239e51 0.145321
\(367\) −2.11760e52 −0.562586 −0.281293 0.959622i \(-0.590763\pi\)
−0.281293 + 0.959622i \(0.590763\pi\)
\(368\) −1.63266e52 −0.410220
\(369\) 5.53988e51 0.131662
\(370\) −1.12884e52 −0.253804
\(371\) 3.86830e52 0.822915
\(372\) 1.69768e52 0.341764
\(373\) −2.64566e52 −0.504083 −0.252042 0.967716i \(-0.581102\pi\)
−0.252042 + 0.967716i \(0.581102\pi\)
\(374\) 8.83757e52 1.59391
\(375\) 2.16736e52 0.370075
\(376\) −1.51673e52 −0.245221
\(377\) −1.55996e51 −0.0238845
\(378\) −2.99975e52 −0.435014
\(379\) −6.28388e52 −0.863225 −0.431613 0.902059i \(-0.642055\pi\)
−0.431613 + 0.902059i \(0.642055\pi\)
\(380\) −3.62518e51 −0.0471809
\(381\) −6.33974e52 −0.781825
\(382\) 6.53677e52 0.763949
\(383\) −7.96221e52 −0.881980 −0.440990 0.897512i \(-0.645373\pi\)
−0.440990 + 0.897512i \(0.645373\pi\)
\(384\) −7.02964e51 −0.0738147
\(385\) 1.82638e52 0.181822
\(386\) −1.11618e53 −1.05365
\(387\) 3.44153e52 0.308090
\(388\) −6.99026e52 −0.593530
\(389\) −1.09705e53 −0.883607 −0.441803 0.897112i \(-0.645661\pi\)
−0.441803 + 0.897112i \(0.645661\pi\)
\(390\) −2.92691e51 −0.0223658
\(391\) 3.60906e53 2.61680
\(392\) 3.49486e52 0.240473
\(393\) −5.79991e52 −0.378769
\(394\) −8.90551e52 −0.552060
\(395\) 2.72284e52 0.160244
\(396\) −3.82688e52 −0.213842
\(397\) 3.48161e53 1.84746 0.923729 0.383046i \(-0.125125\pi\)
0.923729 + 0.383046i \(0.125125\pi\)
\(398\) 1.91592e53 0.965548
\(399\) −4.09318e52 −0.195937
\(400\) −5.21314e52 −0.237066
\(401\) 2.78180e52 0.120190 0.0600948 0.998193i \(-0.480860\pi\)
0.0600948 + 0.998193i \(0.480860\pi\)
\(402\) −8.79350e52 −0.361017
\(403\) −3.49318e52 −0.136291
\(404\) 6.82782e51 0.0253200
\(405\) 3.90931e52 0.137807
\(406\) 1.71157e52 0.0573602
\(407\) −6.99986e53 −2.23051
\(408\) 1.55393e53 0.470866
\(409\) 4.34541e52 0.125228 0.0626141 0.998038i \(-0.480056\pi\)
0.0626141 + 0.998038i \(0.480056\pi\)
\(410\) −2.55311e52 −0.0699843
\(411\) −1.00427e53 −0.261873
\(412\) 1.90669e53 0.473028
\(413\) −4.13047e53 −0.975042
\(414\) −1.56281e53 −0.351074
\(415\) 2.93824e52 0.0628206
\(416\) 1.44643e52 0.0294363
\(417\) 4.32449e53 0.837811
\(418\) −2.24795e53 −0.414641
\(419\) −5.48344e53 −0.963087 −0.481544 0.876422i \(-0.659924\pi\)
−0.481544 + 0.876422i \(0.659924\pi\)
\(420\) 3.21136e52 0.0537129
\(421\) 3.35909e53 0.535105 0.267552 0.963543i \(-0.413785\pi\)
0.267552 + 0.963543i \(0.413785\pi\)
\(422\) 8.25466e53 1.25255
\(423\) −1.45183e53 −0.209865
\(424\) 3.73545e53 0.514450
\(425\) 1.15238e54 1.51225
\(426\) −6.18591e53 −0.773583
\(427\) 1.16771e53 0.139176
\(428\) 1.95545e53 0.222152
\(429\) −1.81496e53 −0.196558
\(430\) −1.58607e53 −0.163764
\(431\) −7.52113e53 −0.740456 −0.370228 0.928941i \(-0.620720\pi\)
−0.370228 + 0.928941i \(0.620720\pi\)
\(432\) −2.89673e53 −0.271952
\(433\) 1.01171e54 0.905848 0.452924 0.891549i \(-0.350381\pi\)
0.452924 + 0.891549i \(0.350381\pi\)
\(434\) 3.83266e53 0.327311
\(435\) −3.34449e52 −0.0272459
\(436\) −1.12960e54 −0.877916
\(437\) −9.18010e53 −0.680735
\(438\) −1.00856e54 −0.713648
\(439\) 1.17795e54 0.795436 0.397718 0.917508i \(-0.369802\pi\)
0.397718 + 0.917508i \(0.369802\pi\)
\(440\) 1.76366e53 0.113667
\(441\) 3.34533e53 0.205801
\(442\) −3.19738e53 −0.187775
\(443\) 5.72043e53 0.320739 0.160370 0.987057i \(-0.448731\pi\)
0.160370 + 0.987057i \(0.448731\pi\)
\(444\) −1.23080e54 −0.658925
\(445\) −5.77948e53 −0.295467
\(446\) 1.38579e54 0.676604
\(447\) 1.43997e53 0.0671504
\(448\) −1.58700e53 −0.0706931
\(449\) −2.89270e54 −1.23099 −0.615493 0.788142i \(-0.711043\pi\)
−0.615493 + 0.788142i \(0.711043\pi\)
\(450\) −4.99010e53 −0.202886
\(451\) −1.58316e54 −0.615045
\(452\) 1.49617e54 0.555447
\(453\) 9.57340e53 0.339667
\(454\) 1.50424e54 0.510121
\(455\) −6.60774e52 −0.0214200
\(456\) −3.95261e53 −0.122491
\(457\) −3.35307e53 −0.0993483 −0.0496742 0.998765i \(-0.515818\pi\)
−0.0496742 + 0.998765i \(0.515818\pi\)
\(458\) 3.86843e54 1.09596
\(459\) 6.40334e54 1.73479
\(460\) 7.20238e53 0.186612
\(461\) 6.94064e53 0.172001 0.0860004 0.996295i \(-0.472591\pi\)
0.0860004 + 0.996295i \(0.472591\pi\)
\(462\) 1.99134e54 0.472047
\(463\) −8.64109e53 −0.195956 −0.0979779 0.995189i \(-0.531237\pi\)
−0.0979779 + 0.995189i \(0.531237\pi\)
\(464\) 1.65279e53 0.0358590
\(465\) −7.48921e53 −0.155471
\(466\) −6.63289e54 −1.31762
\(467\) −5.75866e54 −1.09478 −0.547388 0.836879i \(-0.684378\pi\)
−0.547388 + 0.836879i \(0.684378\pi\)
\(468\) 1.38454e53 0.0251922
\(469\) −1.98520e54 −0.345750
\(470\) 6.69094e53 0.111553
\(471\) 1.94053e54 0.309737
\(472\) −3.98862e54 −0.609553
\(473\) −9.83508e54 −1.43921
\(474\) 2.96877e54 0.416026
\(475\) −2.93123e54 −0.393397
\(476\) 3.50812e54 0.450953
\(477\) 3.57563e54 0.440277
\(478\) −8.80159e54 −1.03822
\(479\) 1.06241e55 1.20065 0.600323 0.799757i \(-0.295039\pi\)
0.600323 + 0.799757i \(0.295039\pi\)
\(480\) 3.10108e53 0.0335789
\(481\) 2.53251e54 0.262771
\(482\) −2.89583e54 −0.287945
\(483\) 8.13219e54 0.774981
\(484\) 5.46237e54 0.498943
\(485\) 3.08370e54 0.270002
\(486\) −4.90149e54 −0.411420
\(487\) 2.53731e54 0.204188 0.102094 0.994775i \(-0.467446\pi\)
0.102094 + 0.994775i \(0.467446\pi\)
\(488\) 1.12761e54 0.0870065
\(489\) −2.39570e54 −0.177256
\(490\) −1.54173e54 −0.109393
\(491\) 1.74013e55 1.18416 0.592080 0.805880i \(-0.298307\pi\)
0.592080 + 0.805880i \(0.298307\pi\)
\(492\) −2.78371e54 −0.181694
\(493\) −3.65355e54 −0.228746
\(494\) 8.13294e53 0.0488478
\(495\) 1.68820e54 0.0972786
\(496\) 3.70103e54 0.204620
\(497\) −1.39652e55 −0.740868
\(498\) 3.20363e54 0.163095
\(499\) 1.71532e54 0.0838079 0.0419039 0.999122i \(-0.486658\pi\)
0.0419039 + 0.999122i \(0.486658\pi\)
\(500\) 4.72495e54 0.221571
\(501\) −2.18054e55 −0.981501
\(502\) −5.74101e54 −0.248063
\(503\) −3.16291e55 −1.31203 −0.656014 0.754749i \(-0.727759\pi\)
−0.656014 + 0.754749i \(0.727759\pi\)
\(504\) −1.51910e54 −0.0605006
\(505\) −3.01205e53 −0.0115183
\(506\) 4.46614e55 1.64001
\(507\) −2.30250e55 −0.811962
\(508\) −1.38209e55 −0.468092
\(509\) 3.59116e55 1.16821 0.584104 0.811679i \(-0.301446\pi\)
0.584104 + 0.811679i \(0.301446\pi\)
\(510\) −6.85504e54 −0.214201
\(511\) −2.27692e55 −0.683469
\(512\) −1.53250e54 −0.0441942
\(513\) −1.62877e55 −0.451288
\(514\) 2.53129e55 0.673904
\(515\) −8.41122e54 −0.215185
\(516\) −1.72932e55 −0.425165
\(517\) 4.14900e55 0.980364
\(518\) −2.77863e55 −0.631060
\(519\) 7.30928e55 1.59567
\(520\) −6.38081e53 −0.0133908
\(521\) 1.03534e55 0.208885 0.104443 0.994531i \(-0.466694\pi\)
0.104443 + 0.994531i \(0.466694\pi\)
\(522\) 1.58207e54 0.0306889
\(523\) 1.16796e54 0.0217843 0.0108922 0.999941i \(-0.496533\pi\)
0.0108922 + 0.999941i \(0.496533\pi\)
\(524\) −1.26441e55 −0.226776
\(525\) 2.59663e55 0.447862
\(526\) 1.05662e55 0.175272
\(527\) −8.18127e55 −1.30528
\(528\) 1.92295e55 0.295103
\(529\) 1.14648e56 1.69248
\(530\) −1.64787e55 −0.234028
\(531\) −3.81796e55 −0.521668
\(532\) −8.92334e54 −0.117311
\(533\) 5.72780e54 0.0724569
\(534\) −6.30149e55 −0.767092
\(535\) −8.62635e54 −0.101059
\(536\) −1.91703e55 −0.216147
\(537\) 2.86305e55 0.310710
\(538\) 3.64464e55 0.380731
\(539\) −9.56017e55 −0.961382
\(540\) 1.27787e55 0.123713
\(541\) −1.27601e56 −1.18935 −0.594677 0.803964i \(-0.702720\pi\)
−0.594677 + 0.803964i \(0.702720\pi\)
\(542\) −9.64920e55 −0.865976
\(543\) −1.11409e56 −0.962772
\(544\) 3.38764e55 0.281916
\(545\) 4.98317e55 0.399372
\(546\) −7.20456e54 −0.0556107
\(547\) 1.90287e56 1.41472 0.707359 0.706854i \(-0.249886\pi\)
0.707359 + 0.706854i \(0.249886\pi\)
\(548\) −2.18935e55 −0.156788
\(549\) 1.07936e55 0.0744619
\(550\) 1.42605e56 0.947763
\(551\) 9.29326e54 0.0595060
\(552\) 7.85290e55 0.484484
\(553\) 6.70224e55 0.398433
\(554\) 1.17461e56 0.672887
\(555\) 5.42958e55 0.299751
\(556\) 9.42761e55 0.501612
\(557\) 1.52473e55 0.0781921 0.0390960 0.999235i \(-0.487552\pi\)
0.0390960 + 0.999235i \(0.487552\pi\)
\(558\) 3.54268e55 0.175118
\(559\) 3.55828e55 0.169550
\(560\) 7.00093e54 0.0321589
\(561\) −4.25076e56 −1.88247
\(562\) −1.19476e56 −0.510136
\(563\) −8.99832e55 −0.370458 −0.185229 0.982695i \(-0.559303\pi\)
−0.185229 + 0.982695i \(0.559303\pi\)
\(564\) 7.29527e55 0.289614
\(565\) −6.60025e55 −0.252678
\(566\) −2.16102e56 −0.797853
\(567\) 9.62272e55 0.342646
\(568\) −1.34856e56 −0.463158
\(569\) 3.81740e56 1.26464 0.632319 0.774708i \(-0.282103\pi\)
0.632319 + 0.774708i \(0.282103\pi\)
\(570\) 1.74367e55 0.0557223
\(571\) −2.86684e56 −0.883819 −0.441910 0.897060i \(-0.645699\pi\)
−0.441910 + 0.897060i \(0.645699\pi\)
\(572\) −3.95669e55 −0.117683
\(573\) −3.14410e56 −0.902251
\(574\) −6.28445e55 −0.174010
\(575\) 5.82367e56 1.55599
\(576\) −1.46693e55 −0.0378223
\(577\) −2.66238e55 −0.0662469 −0.0331235 0.999451i \(-0.510545\pi\)
−0.0331235 + 0.999451i \(0.510545\pi\)
\(578\) −4.54404e56 −1.09124
\(579\) 5.36870e56 1.24440
\(580\) −7.29116e54 −0.0163126
\(581\) 7.23245e55 0.156198
\(582\) 3.36222e56 0.700980
\(583\) −1.02183e57 −2.05671
\(584\) −2.19872e56 −0.427274
\(585\) −6.10781e54 −0.0114602
\(586\) 3.52616e56 0.638853
\(587\) 7.89073e56 1.38050 0.690250 0.723571i \(-0.257501\pi\)
0.690250 + 0.723571i \(0.257501\pi\)
\(588\) −1.68098e56 −0.284006
\(589\) 2.08101e56 0.339555
\(590\) 1.75955e56 0.277291
\(591\) 4.28344e56 0.652002
\(592\) −2.68320e56 −0.394510
\(593\) 2.06394e56 0.293141 0.146570 0.989200i \(-0.453177\pi\)
0.146570 + 0.989200i \(0.453177\pi\)
\(594\) 7.92399e56 1.08723
\(595\) −1.54758e56 −0.205143
\(596\) 3.13920e55 0.0402041
\(597\) −9.21532e56 −1.14035
\(598\) −1.61582e56 −0.193206
\(599\) 2.76254e56 0.319196 0.159598 0.987182i \(-0.448980\pi\)
0.159598 + 0.987182i \(0.448980\pi\)
\(600\) 2.50745e56 0.279983
\(601\) −8.31600e56 −0.897404 −0.448702 0.893681i \(-0.648113\pi\)
−0.448702 + 0.893681i \(0.648113\pi\)
\(602\) −3.90409e56 −0.407185
\(603\) −1.83501e56 −0.184983
\(604\) 2.08705e56 0.203365
\(605\) −2.40968e56 −0.226973
\(606\) −3.28410e55 −0.0299039
\(607\) −2.29755e56 −0.202254 −0.101127 0.994874i \(-0.532245\pi\)
−0.101127 + 0.994874i \(0.532245\pi\)
\(608\) −8.61688e55 −0.0733376
\(609\) −8.23243e55 −0.0677444
\(610\) −4.97436e55 −0.0395800
\(611\) −1.50108e56 −0.115494
\(612\) 3.24270e56 0.241270
\(613\) −2.25026e57 −1.61917 −0.809585 0.587002i \(-0.800308\pi\)
−0.809585 + 0.587002i \(0.800308\pi\)
\(614\) 6.83936e56 0.475953
\(615\) 1.22801e56 0.0826540
\(616\) 4.34122e56 0.282623
\(617\) −3.86090e56 −0.243132 −0.121566 0.992583i \(-0.538792\pi\)
−0.121566 + 0.992583i \(0.538792\pi\)
\(618\) −9.17093e56 −0.558663
\(619\) 1.66300e57 0.980017 0.490008 0.871718i \(-0.336994\pi\)
0.490008 + 0.871718i \(0.336994\pi\)
\(620\) −1.63269e56 −0.0930835
\(621\) 3.23598e57 1.78496
\(622\) 8.10557e56 0.432594
\(623\) −1.42261e57 −0.734652
\(624\) −6.95713e55 −0.0347653
\(625\) 1.75253e57 0.847470
\(626\) −2.90327e57 −1.35867
\(627\) 1.08123e57 0.489705
\(628\) 4.23046e56 0.185445
\(629\) 5.93132e57 2.51659
\(630\) 6.70140e55 0.0275223
\(631\) −1.11322e57 −0.442569 −0.221285 0.975209i \(-0.571025\pi\)
−0.221285 + 0.975209i \(0.571025\pi\)
\(632\) 6.47207e56 0.249082
\(633\) −3.97039e57 −1.47930
\(634\) −7.39307e56 −0.266683
\(635\) 6.09701e56 0.212939
\(636\) −1.79670e57 −0.607583
\(637\) 3.45881e56 0.113258
\(638\) −4.52119e56 −0.143360
\(639\) −1.29086e57 −0.396380
\(640\) 6.76050e55 0.0201043
\(641\) −4.23006e57 −1.21831 −0.609153 0.793053i \(-0.708491\pi\)
−0.609153 + 0.793053i \(0.708491\pi\)
\(642\) −9.40548e56 −0.262369
\(643\) −3.65416e57 −0.987330 −0.493665 0.869652i \(-0.664343\pi\)
−0.493665 + 0.869652i \(0.664343\pi\)
\(644\) 1.77286e57 0.463995
\(645\) 7.62879e56 0.193411
\(646\) 1.90479e57 0.467823
\(647\) −8.11040e57 −1.92976 −0.964882 0.262682i \(-0.915393\pi\)
−0.964882 + 0.262682i \(0.915393\pi\)
\(648\) 9.29225e56 0.214207
\(649\) 1.09108e58 2.43692
\(650\) −5.15937e56 −0.111654
\(651\) −1.84346e57 −0.386566
\(652\) −5.22274e56 −0.106126
\(653\) 1.52356e57 0.300013 0.150006 0.988685i \(-0.452071\pi\)
0.150006 + 0.988685i \(0.452071\pi\)
\(654\) 5.43325e57 1.03685
\(655\) 5.57786e56 0.103162
\(656\) −6.06863e56 −0.108783
\(657\) −2.10465e57 −0.365670
\(658\) 1.64697e57 0.277366
\(659\) 4.79803e57 0.783271 0.391635 0.920120i \(-0.371909\pi\)
0.391635 + 0.920120i \(0.371909\pi\)
\(660\) −8.48297e56 −0.134245
\(661\) −7.23138e56 −0.110941 −0.0554704 0.998460i \(-0.517666\pi\)
−0.0554704 + 0.998460i \(0.517666\pi\)
\(662\) 2.15101e57 0.319930
\(663\) 1.53790e57 0.221769
\(664\) 6.98406e56 0.0976478
\(665\) 3.93647e56 0.0533658
\(666\) −2.56840e57 −0.337630
\(667\) −1.84635e57 −0.235361
\(668\) −4.75369e57 −0.587642
\(669\) −6.66549e57 −0.799092
\(670\) 8.45683e56 0.0983273
\(671\) −3.08456e57 −0.347842
\(672\) 7.63326e56 0.0834910
\(673\) 4.08444e57 0.433334 0.216667 0.976246i \(-0.430481\pi\)
0.216667 + 0.976246i \(0.430481\pi\)
\(674\) 1.08763e58 1.11932
\(675\) 1.03326e58 1.03153
\(676\) −5.01955e57 −0.486136
\(677\) −5.75968e57 −0.541166 −0.270583 0.962697i \(-0.587217\pi\)
−0.270583 + 0.962697i \(0.587217\pi\)
\(678\) −7.19638e57 −0.656003
\(679\) 7.59050e57 0.671336
\(680\) −1.49443e57 −0.128246
\(681\) −7.23521e57 −0.602470
\(682\) −1.01242e58 −0.818048
\(683\) −4.27246e56 −0.0335007 −0.0167503 0.999860i \(-0.505332\pi\)
−0.0167503 + 0.999860i \(0.505332\pi\)
\(684\) −8.24821e56 −0.0627639
\(685\) 9.65817e56 0.0713244
\(686\) −9.37447e57 −0.671897
\(687\) −1.86067e58 −1.29436
\(688\) −3.77001e57 −0.254554
\(689\) 3.69692e57 0.242296
\(690\) −3.46425e57 −0.220396
\(691\) 2.15615e58 1.33161 0.665807 0.746124i \(-0.268088\pi\)
0.665807 + 0.746124i \(0.268088\pi\)
\(692\) 1.59346e58 0.955358
\(693\) 4.15548e57 0.241875
\(694\) −2.98957e56 −0.0168943
\(695\) −4.15892e57 −0.228188
\(696\) −7.94970e56 −0.0423508
\(697\) 1.34149e58 0.693931
\(698\) 9.76930e57 0.490713
\(699\) 3.19033e58 1.55616
\(700\) 5.66078e57 0.268143
\(701\) −1.34583e58 −0.619113 −0.309557 0.950881i \(-0.600181\pi\)
−0.309557 + 0.950881i \(0.600181\pi\)
\(702\) −2.86685e57 −0.128084
\(703\) −1.50871e58 −0.654667
\(704\) 4.19213e57 0.176683
\(705\) −3.21826e57 −0.131748
\(706\) 1.90548e58 0.757719
\(707\) −7.41412e56 −0.0286392
\(708\) 1.91847e58 0.719903
\(709\) −4.49893e58 −1.64006 −0.820032 0.572317i \(-0.806045\pi\)
−0.820032 + 0.572317i \(0.806045\pi\)
\(710\) 5.94908e57 0.210695
\(711\) 6.19516e57 0.213170
\(712\) −1.37376e58 −0.459271
\(713\) −4.13447e58 −1.34303
\(714\) −1.68736e58 −0.532592
\(715\) 1.74547e57 0.0535350
\(716\) 6.24158e57 0.186028
\(717\) 4.23345e58 1.22617
\(718\) −2.00059e58 −0.563128
\(719\) −3.60264e58 −0.985550 −0.492775 0.870157i \(-0.664017\pi\)
−0.492775 + 0.870157i \(0.664017\pi\)
\(720\) 6.47125e56 0.0172057
\(721\) −2.07041e58 −0.535037
\(722\) 2.33062e58 0.585407
\(723\) 1.39286e58 0.340073
\(724\) −2.42877e58 −0.576429
\(725\) −5.89546e57 −0.136015
\(726\) −2.62733e58 −0.589269
\(727\) −1.21072e58 −0.263991 −0.131995 0.991250i \(-0.542138\pi\)
−0.131995 + 0.991250i \(0.542138\pi\)
\(728\) −1.57063e57 −0.0332951
\(729\) 5.29718e58 1.09177
\(730\) 9.69950e57 0.194371
\(731\) 8.33375e58 1.62381
\(732\) −5.42365e57 −0.102758
\(733\) −4.56071e58 −0.840234 −0.420117 0.907470i \(-0.638011\pi\)
−0.420117 + 0.907470i \(0.638011\pi\)
\(734\) 2.22046e58 0.397808
\(735\) 7.41554e57 0.129197
\(736\) 1.71197e58 0.290069
\(737\) 5.24401e58 0.864132
\(738\) −5.80898e57 −0.0930989
\(739\) 5.98997e58 0.933713 0.466857 0.884333i \(-0.345386\pi\)
0.466857 + 0.884333i \(0.345386\pi\)
\(740\) 1.18368e58 0.179466
\(741\) −3.91184e57 −0.0576910
\(742\) −4.05620e58 −0.581889
\(743\) −1.60800e58 −0.224395 −0.112198 0.993686i \(-0.535789\pi\)
−0.112198 + 0.993686i \(0.535789\pi\)
\(744\) −1.78015e58 −0.241664
\(745\) −1.38484e57 −0.0182892
\(746\) 2.77417e58 0.356441
\(747\) 6.68525e57 0.0835690
\(748\) −9.26686e58 −1.12707
\(749\) −2.12336e58 −0.251274
\(750\) −2.27264e58 −0.261683
\(751\) −1.29410e59 −1.44994 −0.724970 0.688781i \(-0.758146\pi\)
−0.724970 + 0.688781i \(0.758146\pi\)
\(752\) 1.59040e58 0.173397
\(753\) 2.76135e58 0.292971
\(754\) 1.63574e57 0.0168889
\(755\) −9.20687e57 −0.0925123
\(756\) 3.14547e58 0.307602
\(757\) 1.04679e59 0.996312 0.498156 0.867088i \(-0.334011\pi\)
0.498156 + 0.867088i \(0.334011\pi\)
\(758\) 6.58913e58 0.610392
\(759\) −2.14816e59 −1.93691
\(760\) 3.80128e57 0.0333619
\(761\) −5.22023e58 −0.445969 −0.222984 0.974822i \(-0.571580\pi\)
−0.222984 + 0.974822i \(0.571580\pi\)
\(762\) 6.64770e58 0.552834
\(763\) 1.22660e59 0.993002
\(764\) −6.85430e58 −0.540194
\(765\) −1.43049e58 −0.109756
\(766\) 8.34899e58 0.623654
\(767\) −3.94748e58 −0.287088
\(768\) 7.37111e57 0.0521949
\(769\) −1.14958e59 −0.792592 −0.396296 0.918123i \(-0.629705\pi\)
−0.396296 + 0.918123i \(0.629705\pi\)
\(770\) −1.91510e58 −0.128568
\(771\) −1.21752e59 −0.795904
\(772\) 1.17040e59 0.745043
\(773\) 2.16390e59 1.34140 0.670699 0.741729i \(-0.265994\pi\)
0.670699 + 0.741729i \(0.265994\pi\)
\(774\) −3.60871e58 −0.217852
\(775\) −1.32015e59 −0.776137
\(776\) 7.32981e58 0.419689
\(777\) 1.33649e59 0.745304
\(778\) 1.15034e59 0.624804
\(779\) −3.41225e58 −0.180519
\(780\) 3.06909e57 0.0158150
\(781\) 3.68897e59 1.85165
\(782\) −3.78438e59 −1.85036
\(783\) −3.27587e58 −0.156031
\(784\) −3.66463e58 −0.170040
\(785\) −1.86624e58 −0.0843605
\(786\) 6.08165e58 0.267830
\(787\) 3.26265e59 1.39987 0.699937 0.714204i \(-0.253211\pi\)
0.699937 + 0.714204i \(0.253211\pi\)
\(788\) 9.33811e58 0.390365
\(789\) −5.08221e58 −0.207002
\(790\) −2.85511e58 −0.113310
\(791\) −1.62464e59 −0.628261
\(792\) 4.01277e58 0.151209
\(793\) 1.11598e58 0.0409784
\(794\) −3.65073e59 −1.30635
\(795\) 7.92603e58 0.276395
\(796\) −2.00899e59 −0.682746
\(797\) −5.37864e59 −1.78146 −0.890731 0.454530i \(-0.849807\pi\)
−0.890731 + 0.454530i \(0.849807\pi\)
\(798\) 4.29201e58 0.138548
\(799\) −3.51565e59 −1.10611
\(800\) 5.46637e58 0.167631
\(801\) −1.31498e59 −0.393054
\(802\) −2.91693e58 −0.0849868
\(803\) 6.01458e59 1.70819
\(804\) 9.22065e58 0.255277
\(805\) −7.82083e58 −0.211075
\(806\) 3.66286e58 0.0963722
\(807\) −1.75303e59 −0.449657
\(808\) −7.15949e57 −0.0179040
\(809\) −7.04425e58 −0.171747 −0.0858737 0.996306i \(-0.527368\pi\)
−0.0858737 + 0.996306i \(0.527368\pi\)
\(810\) −4.09921e58 −0.0974445
\(811\) 6.75289e59 1.56517 0.782586 0.622543i \(-0.213900\pi\)
0.782586 + 0.622543i \(0.213900\pi\)
\(812\) −1.79471e58 −0.0405598
\(813\) 4.64114e59 1.02275
\(814\) 7.33988e59 1.57721
\(815\) 2.30398e58 0.0482778
\(816\) −1.62941e59 −0.332953
\(817\) −2.11979e59 −0.422417
\(818\) −4.55649e58 −0.0885498
\(819\) −1.50343e58 −0.0284946
\(820\) 2.67713e58 0.0494864
\(821\) 1.04795e59 0.188932 0.0944659 0.995528i \(-0.469886\pi\)
0.0944659 + 0.995528i \(0.469886\pi\)
\(822\) 1.05305e59 0.185172
\(823\) 2.14107e57 0.00367227 0.00183614 0.999998i \(-0.499416\pi\)
0.00183614 + 0.999998i \(0.499416\pi\)
\(824\) −1.99931e59 −0.334481
\(825\) −6.85912e59 −1.11934
\(826\) 4.33111e59 0.689459
\(827\) −3.28295e58 −0.0509802 −0.0254901 0.999675i \(-0.508115\pi\)
−0.0254901 + 0.999675i \(0.508115\pi\)
\(828\) 1.63872e59 0.248247
\(829\) −7.20495e59 −1.06479 −0.532394 0.846497i \(-0.678707\pi\)
−0.532394 + 0.846497i \(0.678707\pi\)
\(830\) −3.08097e58 −0.0444208
\(831\) −5.64970e59 −0.794703
\(832\) −1.51669e58 −0.0208146
\(833\) 8.10080e59 1.08469
\(834\) −4.53456e59 −0.592422
\(835\) 2.09706e59 0.267324
\(836\) 2.35714e59 0.293195
\(837\) −7.33554e59 −0.890349
\(838\) 5.74980e59 0.681006
\(839\) −2.37196e59 −0.274149 −0.137074 0.990561i \(-0.543770\pi\)
−0.137074 + 0.990561i \(0.543770\pi\)
\(840\) −3.36736e58 −0.0379808
\(841\) −8.89794e59 −0.979426
\(842\) −3.52226e59 −0.378376
\(843\) 5.74664e59 0.602488
\(844\) −8.65564e59 −0.885684
\(845\) 2.21434e59 0.221148
\(846\) 1.52236e59 0.148397
\(847\) −5.93141e59 −0.564349
\(848\) −3.91690e59 −0.363771
\(849\) 1.03942e60 0.942292
\(850\) −1.20836e60 −1.06932
\(851\) 2.99744e60 2.58937
\(852\) 6.48640e59 0.547006
\(853\) 1.03258e59 0.0850096 0.0425048 0.999096i \(-0.486466\pi\)
0.0425048 + 0.999096i \(0.486466\pi\)
\(854\) −1.22443e59 −0.0984121
\(855\) 3.63864e58 0.0285518
\(856\) −2.05044e59 −0.157085
\(857\) 1.28058e60 0.957852 0.478926 0.877855i \(-0.341026\pi\)
0.478926 + 0.877855i \(0.341026\pi\)
\(858\) 1.90312e59 0.138988
\(859\) −1.16067e60 −0.827657 −0.413828 0.910355i \(-0.635809\pi\)
−0.413828 + 0.910355i \(0.635809\pi\)
\(860\) 1.66311e59 0.115799
\(861\) 3.02274e59 0.205512
\(862\) 7.88648e59 0.523581
\(863\) −1.19897e60 −0.777296 −0.388648 0.921386i \(-0.627058\pi\)
−0.388648 + 0.921386i \(0.627058\pi\)
\(864\) 3.03744e59 0.192299
\(865\) −7.02943e59 −0.434600
\(866\) −1.06086e60 −0.640531
\(867\) 2.18563e60 1.28880
\(868\) −4.01883e59 −0.231444
\(869\) −1.77043e60 −0.995803
\(870\) 3.50696e58 0.0192657
\(871\) −1.89725e59 −0.101801
\(872\) 1.18447e60 0.620780
\(873\) 7.01621e59 0.359179
\(874\) 9.62604e59 0.481352
\(875\) −5.13067e59 −0.250616
\(876\) 1.05756e60 0.504626
\(877\) −1.62618e60 −0.758014 −0.379007 0.925394i \(-0.623734\pi\)
−0.379007 + 0.925394i \(0.623734\pi\)
\(878\) −1.23517e60 −0.562459
\(879\) −1.69604e60 −0.754508
\(880\) −1.84933e59 −0.0803747
\(881\) 5.20975e59 0.221213 0.110607 0.993864i \(-0.464721\pi\)
0.110607 + 0.993864i \(0.464721\pi\)
\(882\) −3.50784e59 −0.145524
\(883\) −1.27377e60 −0.516292 −0.258146 0.966106i \(-0.583112\pi\)
−0.258146 + 0.966106i \(0.583112\pi\)
\(884\) 3.35270e59 0.132777
\(885\) −8.46322e59 −0.327490
\(886\) −5.99831e59 −0.226797
\(887\) −1.40271e60 −0.518243 −0.259122 0.965845i \(-0.583433\pi\)
−0.259122 + 0.965845i \(0.583433\pi\)
\(888\) 1.29059e60 0.465931
\(889\) 1.50077e60 0.529455
\(890\) 6.06023e59 0.208927
\(891\) −2.54189e60 −0.856374
\(892\) −1.45311e60 −0.478431
\(893\) 8.94249e59 0.287742
\(894\) −1.50992e59 −0.0474825
\(895\) −2.75343e59 −0.0846256
\(896\) 1.66409e59 0.0499876
\(897\) 7.77190e59 0.228183
\(898\) 3.03321e60 0.870439
\(899\) 4.18544e59 0.117400
\(900\) 5.23249e59 0.143462
\(901\) 8.65846e60 2.32050
\(902\) 1.66007e60 0.434903
\(903\) 1.87782e60 0.480899
\(904\) −1.56885e60 −0.392761
\(905\) 1.07143e60 0.262222
\(906\) −1.00384e60 −0.240181
\(907\) 1.32324e60 0.309519 0.154760 0.987952i \(-0.450540\pi\)
0.154760 + 0.987952i \(0.450540\pi\)
\(908\) −1.57731e60 −0.360710
\(909\) −6.85317e58 −0.0153226
\(910\) 6.92872e58 0.0151462
\(911\) −4.51973e60 −0.966019 −0.483009 0.875615i \(-0.660456\pi\)
−0.483009 + 0.875615i \(0.660456\pi\)
\(912\) 4.14461e59 0.0866143
\(913\) −1.91049e60 −0.390385
\(914\) 3.51595e59 0.0702499
\(915\) 2.39260e59 0.0467454
\(916\) −4.05635e60 −0.774957
\(917\) 1.37298e60 0.256504
\(918\) −6.71438e60 −1.22668
\(919\) −3.08144e60 −0.550536 −0.275268 0.961368i \(-0.588767\pi\)
−0.275268 + 0.961368i \(0.588767\pi\)
\(920\) −7.55224e59 −0.131955
\(921\) −3.28965e60 −0.562117
\(922\) −7.27779e59 −0.121623
\(923\) −1.33465e60 −0.218139
\(924\) −2.08807e60 −0.333788
\(925\) 9.57092e60 1.49640
\(926\) 9.06084e59 0.138562
\(927\) −1.91377e60 −0.286256
\(928\) −1.73307e59 −0.0253562
\(929\) −3.19320e60 −0.456987 −0.228494 0.973545i \(-0.573380\pi\)
−0.228494 + 0.973545i \(0.573380\pi\)
\(930\) 7.85301e59 0.109935
\(931\) −2.06054e60 −0.282171
\(932\) 6.95509e60 0.931700
\(933\) −3.89868e60 −0.510909
\(934\) 6.03839e60 0.774124
\(935\) 4.08801e60 0.512713
\(936\) −1.45180e59 −0.0178136
\(937\) −1.07427e60 −0.128959 −0.0644794 0.997919i \(-0.520539\pi\)
−0.0644794 + 0.997919i \(0.520539\pi\)
\(938\) 2.08164e60 0.244482
\(939\) 1.39643e61 1.60463
\(940\) −7.01596e59 −0.0788799
\(941\) −1.47841e61 −1.62633 −0.813164 0.582035i \(-0.802257\pi\)
−0.813164 + 0.582035i \(0.802257\pi\)
\(942\) −2.03480e60 −0.219017
\(943\) 6.77934e60 0.714000
\(944\) 4.18237e60 0.431019
\(945\) −1.38760e60 −0.139931
\(946\) 1.03128e61 1.01768
\(947\) −8.72067e60 −0.842122 −0.421061 0.907032i \(-0.638342\pi\)
−0.421061 + 0.907032i \(0.638342\pi\)
\(948\) −3.11298e60 −0.294175
\(949\) −2.17604e60 −0.201238
\(950\) 3.07362e60 0.278174
\(951\) 3.55597e60 0.314962
\(952\) −3.67853e60 −0.318872
\(953\) 6.02037e59 0.0510762 0.0255381 0.999674i \(-0.491870\pi\)
0.0255381 + 0.999674i \(0.491870\pi\)
\(954\) −3.74932e60 −0.311323
\(955\) 3.02373e60 0.245739
\(956\) 9.22914e60 0.734132
\(957\) 2.17464e60 0.169314
\(958\) −1.11402e61 −0.848986
\(959\) 2.37735e60 0.177342
\(960\) −3.25171e59 −0.0237439
\(961\) −4.61807e60 −0.330089
\(962\) −2.65553e60 −0.185807
\(963\) −1.96271e60 −0.134437
\(964\) 3.03650e60 0.203608
\(965\) −5.16316e60 −0.338926
\(966\) −8.52721e60 −0.547994
\(967\) 1.85730e61 1.16853 0.584266 0.811562i \(-0.301383\pi\)
0.584266 + 0.811562i \(0.301383\pi\)
\(968\) −5.72771e60 −0.352806
\(969\) −9.16182e60 −0.552515
\(970\) −3.23350e60 −0.190920
\(971\) −2.80101e61 −1.61928 −0.809638 0.586929i \(-0.800337\pi\)
−0.809638 + 0.586929i \(0.800337\pi\)
\(972\) 5.13959e60 0.290918
\(973\) −1.02371e61 −0.567369
\(974\) −2.66057e60 −0.144383
\(975\) 2.48159e60 0.131867
\(976\) −1.18238e60 −0.0615229
\(977\) 2.94713e60 0.150162 0.0750812 0.997177i \(-0.476078\pi\)
0.0750812 + 0.997177i \(0.476078\pi\)
\(978\) 2.51207e60 0.125339
\(979\) 3.75790e61 1.83611
\(980\) 1.61662e60 0.0773526
\(981\) 1.13380e61 0.531277
\(982\) −1.82465e61 −0.837327
\(983\) 1.11718e61 0.502084 0.251042 0.967976i \(-0.419227\pi\)
0.251042 + 0.967976i \(0.419227\pi\)
\(984\) 2.91893e60 0.128477
\(985\) −4.11944e60 −0.177581
\(986\) 3.83103e60 0.161748
\(987\) −7.92170e60 −0.327579
\(988\) −8.52801e59 −0.0345406
\(989\) 4.21153e61 1.67077
\(990\) −1.77021e60 −0.0687864
\(991\) −9.78682e60 −0.372505 −0.186252 0.982502i \(-0.559634\pi\)
−0.186252 + 0.982502i \(0.559634\pi\)
\(992\) −3.88081e60 −0.144688
\(993\) −1.03461e61 −0.377848
\(994\) 1.46436e61 0.523873
\(995\) 8.86250e60 0.310587
\(996\) −3.35925e60 −0.115325
\(997\) −1.62230e61 −0.545609 −0.272805 0.962069i \(-0.587951\pi\)
−0.272805 + 0.962069i \(0.587951\pi\)
\(998\) −1.79865e60 −0.0592611
\(999\) 5.31817e61 1.71660
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 2.42.a.a.1.1 1
3.2 odd 2 18.42.a.b.1.1 1
4.3 odd 2 16.42.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.42.a.a.1.1 1 1.1 even 1 trivial
16.42.a.a.1.1 1 4.3 odd 2
18.42.a.b.1.1 1 3.2 odd 2