Properties

Label 3.22.a.a.1.1
Level $3$
Weight $22$
Character 3.1
Self dual yes
Analytic conductor $8.384$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.38432032861\)
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 \(\chi\) \(=\) 3.1

$q$-expansion

\(f(q)\) \(=\) \(q-2844.00 q^{2} -59049.0 q^{3} +5.99118e6 q^{4} +3.10995e6 q^{5} +1.67935e8 q^{6} +3.63304e8 q^{7} -1.10746e10 q^{8} +3.48678e9 q^{9} +O(q^{10})\) \(q-2844.00 q^{2} -59049.0 q^{3} +5.99118e6 q^{4} +3.10995e6 q^{5} +1.67935e8 q^{6} +3.63304e8 q^{7} -1.10746e10 q^{8} +3.48678e9 q^{9} -8.84470e9 q^{10} +1.45818e10 q^{11} -3.53773e11 q^{12} +1.13351e11 q^{13} -1.03324e12 q^{14} -1.83639e11 q^{15} +1.89318e13 q^{16} -8.58939e12 q^{17} -9.91641e12 q^{18} -2.92029e13 q^{19} +1.86323e13 q^{20} -2.14527e13 q^{21} -4.14707e13 q^{22} -1.55899e14 q^{23} +6.53946e14 q^{24} -4.67165e14 q^{25} -3.22370e14 q^{26} -2.05891e14 q^{27} +2.17662e15 q^{28} +2.40079e15 q^{29} +5.22271e14 q^{30} +2.23982e15 q^{31} -3.06169e16 q^{32} -8.61043e14 q^{33} +2.44282e16 q^{34} +1.12986e15 q^{35} +2.08900e16 q^{36} -3.07851e16 q^{37} +8.30532e16 q^{38} -6.69325e15 q^{39} -3.44415e16 q^{40} -1.03208e17 q^{41} +6.10116e16 q^{42} -1.65557e17 q^{43} +8.73624e16 q^{44} +1.08437e16 q^{45} +4.43377e17 q^{46} -6.65872e16 q^{47} -1.11790e18 q^{48} -4.26556e17 q^{49} +1.32862e18 q^{50} +5.07195e17 q^{51} +6.79105e17 q^{52} +4.35423e17 q^{53} +5.85554e17 q^{54} +4.53488e16 q^{55} -4.02346e18 q^{56} +1.72440e18 q^{57} -6.82784e18 q^{58} +5.53437e18 q^{59} -1.10022e18 q^{60} -7.17621e18 q^{61} -6.37005e18 q^{62} +1.26676e18 q^{63} +4.73716e19 q^{64} +3.52515e17 q^{65} +2.44881e18 q^{66} -1.57554e19 q^{67} -5.14606e19 q^{68} +9.20569e18 q^{69} -3.21331e18 q^{70} +2.64579e19 q^{71} -3.86148e19 q^{72} +1.34712e19 q^{73} +8.75527e19 q^{74} +2.75856e19 q^{75} -1.74960e20 q^{76} +5.29764e18 q^{77} +1.90356e19 q^{78} -1.68861e19 q^{79} +5.88770e19 q^{80} +1.21577e19 q^{81} +2.93522e20 q^{82} -1.70688e20 q^{83} -1.28527e20 q^{84} -2.67126e19 q^{85} +4.70845e20 q^{86} -1.41764e20 q^{87} -1.61488e20 q^{88} -3.12592e20 q^{89} -3.08396e19 q^{90} +4.11808e19 q^{91} -9.34021e20 q^{92} -1.32259e20 q^{93} +1.89374e20 q^{94} -9.08197e19 q^{95} +1.80790e21 q^{96} +9.49015e20 q^{97} +1.21313e21 q^{98} +5.08437e19 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2844.00 −1.96388 −0.981939 0.189196i \(-0.939412\pi\)
−0.981939 + 0.189196i \(0.939412\pi\)
\(3\) −59049.0 −0.577350
\(4\) 5.99118e6 2.85682
\(5\) 3.10995e6 0.142419 0.0712096 0.997461i \(-0.477314\pi\)
0.0712096 + 0.997461i \(0.477314\pi\)
\(6\) 1.67935e8 1.13385
\(7\) 3.63304e8 0.486117 0.243058 0.970012i \(-0.421849\pi\)
0.243058 + 0.970012i \(0.421849\pi\)
\(8\) −1.10746e10 −3.64657
\(9\) 3.48678e9 0.333333
\(10\) −8.84470e9 −0.279694
\(11\) 1.45818e10 0.169508 0.0847538 0.996402i \(-0.472990\pi\)
0.0847538 + 0.996402i \(0.472990\pi\)
\(12\) −3.53773e11 −1.64939
\(13\) 1.13351e11 0.228044 0.114022 0.993478i \(-0.463627\pi\)
0.114022 + 0.993478i \(0.463627\pi\)
\(14\) −1.03324e12 −0.954674
\(15\) −1.83639e11 −0.0822257
\(16\) 1.89318e13 4.30460
\(17\) −8.58939e12 −1.03335 −0.516676 0.856181i \(-0.672831\pi\)
−0.516676 + 0.856181i \(0.672831\pi\)
\(18\) −9.91641e12 −0.654626
\(19\) −2.92029e13 −1.09273 −0.546366 0.837546i \(-0.683989\pi\)
−0.546366 + 0.837546i \(0.683989\pi\)
\(20\) 1.86323e13 0.406866
\(21\) −2.14527e13 −0.280660
\(22\) −4.14707e13 −0.332892
\(23\) −1.55899e14 −0.784696 −0.392348 0.919817i \(-0.628337\pi\)
−0.392348 + 0.919817i \(0.628337\pi\)
\(24\) 6.53946e14 2.10535
\(25\) −4.67165e14 −0.979717
\(26\) −3.22370e14 −0.447851
\(27\) −2.05891e14 −0.192450
\(28\) 2.17662e15 1.38875
\(29\) 2.40079e15 1.05968 0.529840 0.848097i \(-0.322252\pi\)
0.529840 + 0.848097i \(0.322252\pi\)
\(30\) 5.22271e14 0.161481
\(31\) 2.23982e15 0.490812 0.245406 0.969420i \(-0.421079\pi\)
0.245406 + 0.969420i \(0.421079\pi\)
\(32\) −3.06169e16 −4.80714
\(33\) −8.61043e14 −0.0978652
\(34\) 2.44282e16 2.02938
\(35\) 1.12986e15 0.0692323
\(36\) 2.08900e16 0.952273
\(37\) −3.07851e16 −1.05250 −0.526250 0.850330i \(-0.676402\pi\)
−0.526250 + 0.850330i \(0.676402\pi\)
\(38\) 8.30532e16 2.14599
\(39\) −6.69325e15 −0.131661
\(40\) −3.44415e16 −0.519341
\(41\) −1.03208e17 −1.20083 −0.600414 0.799689i \(-0.704998\pi\)
−0.600414 + 0.799689i \(0.704998\pi\)
\(42\) 6.10116e16 0.551182
\(43\) −1.65557e17 −1.16823 −0.584117 0.811670i \(-0.698559\pi\)
−0.584117 + 0.811670i \(0.698559\pi\)
\(44\) 8.73624e16 0.484252
\(45\) 1.08437e16 0.0474730
\(46\) 4.43377e17 1.54105
\(47\) −6.65872e16 −0.184656 −0.0923280 0.995729i \(-0.529431\pi\)
−0.0923280 + 0.995729i \(0.529431\pi\)
\(48\) −1.11790e18 −2.48526
\(49\) −4.26556e17 −0.763690
\(50\) 1.32862e18 1.92404
\(51\) 5.07195e17 0.596607
\(52\) 6.79105e17 0.651482
\(53\) 4.35423e17 0.341991 0.170995 0.985272i \(-0.445302\pi\)
0.170995 + 0.985272i \(0.445302\pi\)
\(54\) 5.85554e17 0.377949
\(55\) 4.53488e16 0.0241411
\(56\) −4.02346e18 −1.77266
\(57\) 1.72440e18 0.630889
\(58\) −6.82784e18 −2.08108
\(59\) 5.53437e18 1.40968 0.704842 0.709364i \(-0.251018\pi\)
0.704842 + 0.709364i \(0.251018\pi\)
\(60\) −1.10022e18 −0.234904
\(61\) −7.17621e18 −1.28805 −0.644023 0.765006i \(-0.722736\pi\)
−0.644023 + 0.765006i \(0.722736\pi\)
\(62\) −6.37005e18 −0.963896
\(63\) 1.26676e18 0.162039
\(64\) 4.73716e19 5.13604
\(65\) 3.52515e17 0.0324779
\(66\) 2.44881e18 0.192195
\(67\) −1.57554e19 −1.05595 −0.527977 0.849258i \(-0.677049\pi\)
−0.527977 + 0.849258i \(0.677049\pi\)
\(68\) −5.14606e19 −2.95210
\(69\) 9.20569e18 0.453045
\(70\) −3.21331e18 −0.135964
\(71\) 2.64579e19 0.964588 0.482294 0.876009i \(-0.339804\pi\)
0.482294 + 0.876009i \(0.339804\pi\)
\(72\) −3.86148e19 −1.21552
\(73\) 1.34712e19 0.366875 0.183437 0.983031i \(-0.441278\pi\)
0.183437 + 0.983031i \(0.441278\pi\)
\(74\) 8.75527e19 2.06698
\(75\) 2.75856e19 0.565640
\(76\) −1.74960e20 −3.12174
\(77\) 5.29764e18 0.0824005
\(78\) 1.90356e19 0.258567
\(79\) −1.68861e19 −0.200653 −0.100326 0.994955i \(-0.531989\pi\)
−0.100326 + 0.994955i \(0.531989\pi\)
\(80\) 5.88770e19 0.613057
\(81\) 1.21577e19 0.111111
\(82\) 2.93522e20 2.35828
\(83\) −1.70688e20 −1.20749 −0.603744 0.797178i \(-0.706325\pi\)
−0.603744 + 0.797178i \(0.706325\pi\)
\(84\) −1.28527e20 −0.801794
\(85\) −2.67126e19 −0.147169
\(86\) 4.70845e20 2.29427
\(87\) −1.41764e20 −0.611807
\(88\) −1.61488e20 −0.618121
\(89\) −3.12592e20 −1.06263 −0.531317 0.847173i \(-0.678303\pi\)
−0.531317 + 0.847173i \(0.678303\pi\)
\(90\) −3.08396e19 −0.0932313
\(91\) 4.11808e19 0.110856
\(92\) −9.34021e20 −2.24174
\(93\) −1.32259e20 −0.283371
\(94\) 1.89374e20 0.362642
\(95\) −9.08197e19 −0.155626
\(96\) 1.80790e21 2.77540
\(97\) 9.49015e20 1.30668 0.653341 0.757064i \(-0.273367\pi\)
0.653341 + 0.757064i \(0.273367\pi\)
\(98\) 1.21313e21 1.49980
\(99\) 5.08437e19 0.0565025
\(100\) −2.79887e21 −2.79887
\(101\) 1.44798e20 0.130433 0.0652166 0.997871i \(-0.479226\pi\)
0.0652166 + 0.997871i \(0.479226\pi\)
\(102\) −1.44246e21 −1.17166
\(103\) 2.19627e21 1.61025 0.805127 0.593102i \(-0.202097\pi\)
0.805127 + 0.593102i \(0.202097\pi\)
\(104\) −1.25532e21 −0.831579
\(105\) −6.67169e19 −0.0399713
\(106\) −1.23834e21 −0.671629
\(107\) −1.63087e20 −0.0801473 −0.0400737 0.999197i \(-0.512759\pi\)
−0.0400737 + 0.999197i \(0.512759\pi\)
\(108\) −1.23353e21 −0.549795
\(109\) 2.24852e20 0.0909743 0.0454871 0.998965i \(-0.485516\pi\)
0.0454871 + 0.998965i \(0.485516\pi\)
\(110\) −1.28972e20 −0.0474102
\(111\) 1.81783e21 0.607661
\(112\) 6.87800e21 2.09254
\(113\) −4.24118e21 −1.17534 −0.587670 0.809101i \(-0.699955\pi\)
−0.587670 + 0.809101i \(0.699955\pi\)
\(114\) −4.90421e21 −1.23899
\(115\) −4.84839e20 −0.111756
\(116\) 1.43836e22 3.02732
\(117\) 3.95230e20 0.0760148
\(118\) −1.57397e22 −2.76845
\(119\) −3.12056e21 −0.502330
\(120\) 2.03374e21 0.299842
\(121\) −7.18762e21 −0.971267
\(122\) 2.04091e22 2.52957
\(123\) 6.09430e21 0.693299
\(124\) 1.34192e22 1.40216
\(125\) −2.93580e21 −0.281950
\(126\) −3.60267e21 −0.318225
\(127\) 1.66312e21 0.135202 0.0676012 0.997712i \(-0.478465\pi\)
0.0676012 + 0.997712i \(0.478465\pi\)
\(128\) −7.05165e22 −5.27942
\(129\) 9.77599e21 0.674480
\(130\) −1.00255e21 −0.0637826
\(131\) 6.40663e21 0.376081 0.188040 0.982161i \(-0.439786\pi\)
0.188040 + 0.982161i \(0.439786\pi\)
\(132\) −5.15867e21 −0.279583
\(133\) −1.06095e22 −0.531196
\(134\) 4.48085e22 2.07377
\(135\) −6.40311e20 −0.0274086
\(136\) 9.51243e22 3.76819
\(137\) −1.98314e22 −0.727423 −0.363711 0.931512i \(-0.618491\pi\)
−0.363711 + 0.931512i \(0.618491\pi\)
\(138\) −2.61810e22 −0.889725
\(139\) −5.20143e21 −0.163858 −0.0819290 0.996638i \(-0.526108\pi\)
−0.0819290 + 0.996638i \(0.526108\pi\)
\(140\) 6.76918e21 0.197784
\(141\) 3.93191e21 0.106611
\(142\) −7.52461e22 −1.89433
\(143\) 1.65286e21 0.0386552
\(144\) 6.60112e22 1.43487
\(145\) 7.46633e21 0.150919
\(146\) −3.83122e22 −0.720498
\(147\) 2.51877e22 0.440917
\(148\) −1.84439e23 −3.00680
\(149\) −7.47631e22 −1.13562 −0.567808 0.823161i \(-0.692208\pi\)
−0.567808 + 0.823161i \(0.692208\pi\)
\(150\) −7.84536e22 −1.11085
\(151\) 1.11044e23 1.46635 0.733174 0.680042i \(-0.238038\pi\)
0.733174 + 0.680042i \(0.238038\pi\)
\(152\) 3.23412e23 3.98472
\(153\) −2.99493e22 −0.344451
\(154\) −1.50665e22 −0.161824
\(155\) 6.96573e21 0.0699011
\(156\) −4.01005e22 −0.376133
\(157\) 4.36563e22 0.382913 0.191457 0.981501i \(-0.438679\pi\)
0.191457 + 0.981501i \(0.438679\pi\)
\(158\) 4.80241e22 0.394058
\(159\) −2.57113e22 −0.197449
\(160\) −9.52171e22 −0.684628
\(161\) −5.66388e22 −0.381454
\(162\) −3.45764e22 −0.218209
\(163\) 2.85661e23 1.68998 0.844989 0.534783i \(-0.179607\pi\)
0.844989 + 0.534783i \(0.179607\pi\)
\(164\) −6.18336e23 −3.43055
\(165\) −2.67780e21 −0.0139379
\(166\) 4.85437e23 2.37136
\(167\) 2.66950e23 1.22435 0.612177 0.790720i \(-0.290294\pi\)
0.612177 + 0.790720i \(0.290294\pi\)
\(168\) 2.37581e23 1.02344
\(169\) −2.34216e23 −0.947996
\(170\) 7.59706e22 0.289022
\(171\) −1.01824e23 −0.364244
\(172\) −9.91884e23 −3.33743
\(173\) −2.28496e23 −0.723425 −0.361713 0.932290i \(-0.617808\pi\)
−0.361713 + 0.932290i \(0.617808\pi\)
\(174\) 4.03177e23 1.20151
\(175\) −1.69723e23 −0.476257
\(176\) 2.76061e23 0.729661
\(177\) −3.26799e23 −0.813881
\(178\) 8.89013e23 2.08688
\(179\) −1.29151e22 −0.0285852 −0.0142926 0.999898i \(-0.504550\pi\)
−0.0142926 + 0.999898i \(0.504550\pi\)
\(180\) 6.49668e22 0.135622
\(181\) −8.75338e23 −1.72405 −0.862026 0.506863i \(-0.830805\pi\)
−0.862026 + 0.506863i \(0.830805\pi\)
\(182\) −1.17118e23 −0.217708
\(183\) 4.23748e23 0.743654
\(184\) 1.72653e24 2.86145
\(185\) −9.57400e22 −0.149896
\(186\) 3.76145e23 0.556506
\(187\) −1.25249e23 −0.175161
\(188\) −3.98936e23 −0.527529
\(189\) −7.48011e22 −0.0935532
\(190\) 2.58291e23 0.305631
\(191\) 1.33961e24 1.50013 0.750066 0.661363i \(-0.230022\pi\)
0.750066 + 0.661363i \(0.230022\pi\)
\(192\) −2.79725e24 −2.96529
\(193\) −2.13970e23 −0.214783 −0.107392 0.994217i \(-0.534250\pi\)
−0.107392 + 0.994217i \(0.534250\pi\)
\(194\) −2.69900e24 −2.56616
\(195\) −2.08157e22 −0.0187511
\(196\) −2.55558e24 −2.18173
\(197\) 1.42895e24 1.15644 0.578220 0.815881i \(-0.303748\pi\)
0.578220 + 0.815881i \(0.303748\pi\)
\(198\) −1.44600e23 −0.110964
\(199\) 7.45856e23 0.542872 0.271436 0.962456i \(-0.412501\pi\)
0.271436 + 0.962456i \(0.412501\pi\)
\(200\) 5.17368e24 3.57260
\(201\) 9.30344e23 0.609656
\(202\) −4.11805e23 −0.256155
\(203\) 8.72216e23 0.515129
\(204\) 3.03870e24 1.70440
\(205\) −3.20970e23 −0.171021
\(206\) −6.24619e24 −3.16234
\(207\) −5.43587e23 −0.261565
\(208\) 2.14594e24 0.981639
\(209\) −4.25832e23 −0.185226
\(210\) 1.89743e23 0.0784988
\(211\) −2.65090e24 −1.04334 −0.521671 0.853147i \(-0.674691\pi\)
−0.521671 + 0.853147i \(0.674691\pi\)
\(212\) 2.60870e24 0.977006
\(213\) −1.56231e24 −0.556905
\(214\) 4.63819e23 0.157400
\(215\) −5.14875e23 −0.166379
\(216\) 2.28017e24 0.701782
\(217\) 8.13736e23 0.238592
\(218\) −6.39479e23 −0.178662
\(219\) −7.95464e23 −0.211815
\(220\) 2.71693e23 0.0689668
\(221\) −9.73614e23 −0.235650
\(222\) −5.16990e24 −1.19337
\(223\) 3.97174e24 0.874539 0.437270 0.899330i \(-0.355946\pi\)
0.437270 + 0.899330i \(0.355946\pi\)
\(224\) −1.11232e25 −2.33683
\(225\) −1.62890e24 −0.326572
\(226\) 1.20619e25 2.30823
\(227\) −2.14690e24 −0.392229 −0.196114 0.980581i \(-0.562832\pi\)
−0.196114 + 0.980581i \(0.562832\pi\)
\(228\) 1.03312e25 1.80234
\(229\) 3.66024e24 0.609869 0.304934 0.952373i \(-0.401365\pi\)
0.304934 + 0.952373i \(0.401365\pi\)
\(230\) 1.37888e24 0.219475
\(231\) −3.12820e23 −0.0475739
\(232\) −2.65878e25 −3.86420
\(233\) −2.90921e24 −0.404145 −0.202073 0.979371i \(-0.564768\pi\)
−0.202073 + 0.979371i \(0.564768\pi\)
\(234\) −1.12403e24 −0.149284
\(235\) −2.07083e23 −0.0262985
\(236\) 3.31574e25 4.02721
\(237\) 9.97109e23 0.115847
\(238\) 8.87487e24 0.986516
\(239\) −4.80224e24 −0.510818 −0.255409 0.966833i \(-0.582210\pi\)
−0.255409 + 0.966833i \(0.582210\pi\)
\(240\) −3.47663e24 −0.353949
\(241\) 7.86263e24 0.766282 0.383141 0.923690i \(-0.374842\pi\)
0.383141 + 0.923690i \(0.374842\pi\)
\(242\) 2.04416e25 1.90745
\(243\) −7.17898e23 −0.0641500
\(244\) −4.29940e25 −3.67972
\(245\) −1.32657e24 −0.108764
\(246\) −1.73322e25 −1.36155
\(247\) −3.31018e24 −0.249191
\(248\) −2.48052e25 −1.78978
\(249\) 1.00790e25 0.697144
\(250\) 8.34942e24 0.553715
\(251\) −1.99525e25 −1.26889 −0.634443 0.772969i \(-0.718771\pi\)
−0.634443 + 0.772969i \(0.718771\pi\)
\(252\) 7.58941e24 0.462916
\(253\) −2.27330e24 −0.133012
\(254\) −4.72991e24 −0.265521
\(255\) 1.57735e24 0.0849682
\(256\) 1.01203e26 5.23210
\(257\) 1.33580e25 0.662894 0.331447 0.943474i \(-0.392463\pi\)
0.331447 + 0.943474i \(0.392463\pi\)
\(258\) −2.78029e25 −1.32460
\(259\) −1.11843e25 −0.511638
\(260\) 2.11198e24 0.0927834
\(261\) 8.37103e24 0.353227
\(262\) −1.82204e25 −0.738576
\(263\) −5.83637e24 −0.227304 −0.113652 0.993521i \(-0.536255\pi\)
−0.113652 + 0.993521i \(0.536255\pi\)
\(264\) 9.53573e24 0.356872
\(265\) 1.35414e24 0.0487060
\(266\) 3.01735e25 1.04320
\(267\) 1.84583e25 0.613512
\(268\) −9.43938e25 −3.01667
\(269\) 5.35297e25 1.64511 0.822557 0.568683i \(-0.192547\pi\)
0.822557 + 0.568683i \(0.192547\pi\)
\(270\) 1.82104e24 0.0538271
\(271\) −1.04403e25 −0.296849 −0.148425 0.988924i \(-0.547420\pi\)
−0.148425 + 0.988924i \(0.547420\pi\)
\(272\) −1.62613e26 −4.44817
\(273\) −2.43168e24 −0.0640029
\(274\) 5.64004e25 1.42857
\(275\) −6.81213e24 −0.166069
\(276\) 5.51530e25 1.29427
\(277\) 3.14884e25 0.711399 0.355699 0.934600i \(-0.384243\pi\)
0.355699 + 0.934600i \(0.384243\pi\)
\(278\) 1.47929e25 0.321797
\(279\) 7.80977e24 0.163604
\(280\) −1.25127e25 −0.252460
\(281\) −1.15887e25 −0.225225 −0.112613 0.993639i \(-0.535922\pi\)
−0.112613 + 0.993639i \(0.535922\pi\)
\(282\) −1.11823e25 −0.209371
\(283\) −4.80399e25 −0.866652 −0.433326 0.901237i \(-0.642660\pi\)
−0.433326 + 0.901237i \(0.642660\pi\)
\(284\) 1.58514e26 2.75565
\(285\) 5.36281e24 0.0898507
\(286\) −4.70074e24 −0.0759142
\(287\) −3.74957e25 −0.583743
\(288\) −1.06755e26 −1.60238
\(289\) 4.68568e24 0.0678180
\(290\) −2.12343e25 −0.296386
\(291\) −5.60384e25 −0.754413
\(292\) 8.07087e25 1.04809
\(293\) −7.96714e25 −0.998142 −0.499071 0.866561i \(-0.666325\pi\)
−0.499071 + 0.866561i \(0.666325\pi\)
\(294\) −7.16339e25 −0.865907
\(295\) 1.72116e25 0.200766
\(296\) 3.40933e26 3.83801
\(297\) −3.00227e24 −0.0326217
\(298\) 2.12626e26 2.23021
\(299\) −1.76713e25 −0.178946
\(300\) 1.65271e26 1.61593
\(301\) −6.01476e25 −0.567898
\(302\) −3.15809e26 −2.87973
\(303\) −8.55018e24 −0.0753056
\(304\) −5.52865e26 −4.70377
\(305\) −2.23176e25 −0.183442
\(306\) 8.51760e25 0.676460
\(307\) 1.51498e26 1.16266 0.581332 0.813666i \(-0.302532\pi\)
0.581332 + 0.813666i \(0.302532\pi\)
\(308\) 3.17391e25 0.235403
\(309\) −1.29687e26 −0.929681
\(310\) −1.98105e25 −0.137277
\(311\) 1.41406e26 0.947292 0.473646 0.880715i \(-0.342938\pi\)
0.473646 + 0.880715i \(0.342938\pi\)
\(312\) 7.41253e25 0.480112
\(313\) 4.99598e25 0.312899 0.156450 0.987686i \(-0.449995\pi\)
0.156450 + 0.987686i \(0.449995\pi\)
\(314\) −1.24158e26 −0.751995
\(315\) 3.93957e24 0.0230774
\(316\) −1.01168e26 −0.573229
\(317\) 1.81535e26 0.995033 0.497517 0.867454i \(-0.334245\pi\)
0.497517 + 0.867454i \(0.334245\pi\)
\(318\) 7.31229e25 0.387765
\(319\) 3.50079e25 0.179624
\(320\) 1.47323e26 0.731470
\(321\) 9.63011e24 0.0462731
\(322\) 1.61081e26 0.749130
\(323\) 2.50835e26 1.12918
\(324\) 7.28388e25 0.317424
\(325\) −5.29536e25 −0.223419
\(326\) −8.12420e26 −3.31891
\(327\) −1.32773e25 −0.0525240
\(328\) 1.14299e27 4.37890
\(329\) −2.41914e25 −0.0897644
\(330\) 7.61566e24 0.0273723
\(331\) 1.44090e26 0.501695 0.250848 0.968027i \(-0.419291\pi\)
0.250848 + 0.968027i \(0.419291\pi\)
\(332\) −1.02262e27 −3.44958
\(333\) −1.07341e26 −0.350833
\(334\) −7.59206e26 −2.40448
\(335\) −4.89987e25 −0.150388
\(336\) −4.06139e26 −1.20813
\(337\) −3.63051e26 −1.04678 −0.523388 0.852095i \(-0.675332\pi\)
−0.523388 + 0.852095i \(0.675332\pi\)
\(338\) 6.66111e26 1.86175
\(339\) 2.50438e26 0.678583
\(340\) −1.60040e26 −0.420436
\(341\) 3.26607e25 0.0831964
\(342\) 2.89588e26 0.715331
\(343\) −3.57891e26 −0.857360
\(344\) 1.83349e27 4.26004
\(345\) 2.86292e25 0.0645222
\(346\) 6.49841e26 1.42072
\(347\) −7.09622e25 −0.150511 −0.0752554 0.997164i \(-0.523977\pi\)
−0.0752554 + 0.997164i \(0.523977\pi\)
\(348\) −8.49335e26 −1.74782
\(349\) −7.03939e26 −1.40562 −0.702810 0.711378i \(-0.748072\pi\)
−0.702810 + 0.711378i \(0.748072\pi\)
\(350\) 4.82692e26 0.935311
\(351\) −2.33379e25 −0.0438872
\(352\) −4.46451e26 −0.814846
\(353\) −1.08085e26 −0.191483 −0.0957414 0.995406i \(-0.530522\pi\)
−0.0957414 + 0.995406i \(0.530522\pi\)
\(354\) 9.29416e26 1.59836
\(355\) 8.22826e25 0.137376
\(356\) −1.87280e27 −3.03575
\(357\) 1.84266e26 0.290020
\(358\) 3.67305e25 0.0561378
\(359\) −1.67492e26 −0.248601 −0.124301 0.992245i \(-0.539669\pi\)
−0.124301 + 0.992245i \(0.539669\pi\)
\(360\) −1.20090e26 −0.173114
\(361\) 1.38602e26 0.194064
\(362\) 2.48946e27 3.38583
\(363\) 4.24422e26 0.560761
\(364\) 2.46722e26 0.316696
\(365\) 4.18949e25 0.0522500
\(366\) −1.20514e27 −1.46045
\(367\) 9.83667e26 1.15839 0.579194 0.815190i \(-0.303367\pi\)
0.579194 + 0.815190i \(0.303367\pi\)
\(368\) −2.95146e27 −3.37780
\(369\) −3.59863e26 −0.400276
\(370\) 2.72285e26 0.294378
\(371\) 1.58191e26 0.166248
\(372\) −7.92389e26 −0.809539
\(373\) 1.00058e26 0.0993824 0.0496912 0.998765i \(-0.484176\pi\)
0.0496912 + 0.998765i \(0.484176\pi\)
\(374\) 3.56208e26 0.343995
\(375\) 1.73356e26 0.162784
\(376\) 7.37429e26 0.673360
\(377\) 2.72131e26 0.241654
\(378\) 2.12734e26 0.183727
\(379\) 9.23905e25 0.0776096 0.0388048 0.999247i \(-0.487645\pi\)
0.0388048 + 0.999247i \(0.487645\pi\)
\(380\) −5.44117e26 −0.444595
\(381\) −9.82055e25 −0.0780591
\(382\) −3.80986e27 −2.94608
\(383\) −2.13677e27 −1.60757 −0.803786 0.594919i \(-0.797184\pi\)
−0.803786 + 0.594919i \(0.797184\pi\)
\(384\) 4.16393e27 3.04807
\(385\) 1.64754e25 0.0117354
\(386\) 6.08530e26 0.421809
\(387\) −5.77263e26 −0.389411
\(388\) 5.68572e27 3.73295
\(389\) −1.25581e27 −0.802516 −0.401258 0.915965i \(-0.631427\pi\)
−0.401258 + 0.915965i \(0.631427\pi\)
\(390\) 5.91998e25 0.0368249
\(391\) 1.33908e27 0.810868
\(392\) 4.72395e27 2.78485
\(393\) −3.78305e26 −0.217130
\(394\) −4.06394e27 −2.27111
\(395\) −5.25150e25 −0.0285768
\(396\) 3.04614e26 0.161417
\(397\) 1.78165e27 0.919439 0.459719 0.888064i \(-0.347950\pi\)
0.459719 + 0.888064i \(0.347950\pi\)
\(398\) −2.12121e27 −1.06614
\(399\) 6.26483e26 0.306686
\(400\) −8.84429e27 −4.21729
\(401\) 1.40181e27 0.651141 0.325570 0.945518i \(-0.394444\pi\)
0.325570 + 0.945518i \(0.394444\pi\)
\(402\) −2.64590e27 −1.19729
\(403\) 2.53885e26 0.111927
\(404\) 8.67511e26 0.372624
\(405\) 3.78097e25 0.0158243
\(406\) −2.48058e27 −1.01165
\(407\) −4.48903e26 −0.178407
\(408\) −5.61699e27 −2.17557
\(409\) 2.17493e27 0.821015 0.410507 0.911857i \(-0.365352\pi\)
0.410507 + 0.911857i \(0.365352\pi\)
\(410\) 9.12840e26 0.335864
\(411\) 1.17102e27 0.419978
\(412\) 1.31582e28 4.60021
\(413\) 2.01066e27 0.685271
\(414\) 1.54596e27 0.513683
\(415\) −5.30831e26 −0.171969
\(416\) −3.47045e27 −1.09624
\(417\) 3.07139e26 0.0946034
\(418\) 1.21107e27 0.363762
\(419\) −6.07636e27 −1.77990 −0.889952 0.456055i \(-0.849262\pi\)
−0.889952 + 0.456055i \(0.849262\pi\)
\(420\) −3.99713e26 −0.114191
\(421\) −1.89993e27 −0.529389 −0.264695 0.964332i \(-0.585271\pi\)
−0.264695 + 0.964332i \(0.585271\pi\)
\(422\) 7.53915e27 2.04900
\(423\) −2.32175e26 −0.0615520
\(424\) −4.82214e27 −1.24709
\(425\) 4.01267e27 1.01239
\(426\) 4.44321e27 1.09369
\(427\) −2.60714e27 −0.626141
\(428\) −9.77083e26 −0.228966
\(429\) −9.75999e25 −0.0223176
\(430\) 1.46430e27 0.326748
\(431\) −8.08572e24 −0.00176079 −0.000880395 1.00000i \(-0.500280\pi\)
−0.000880395 1.00000i \(0.500280\pi\)
\(432\) −3.89789e27 −0.828420
\(433\) −5.60439e27 −1.16253 −0.581267 0.813713i \(-0.697443\pi\)
−0.581267 + 0.813713i \(0.697443\pi\)
\(434\) −2.31426e27 −0.468566
\(435\) −4.40879e26 −0.0871330
\(436\) 1.34713e27 0.259897
\(437\) 4.55272e27 0.857463
\(438\) 2.26230e27 0.415979
\(439\) −8.51110e27 −1.52795 −0.763973 0.645248i \(-0.776754\pi\)
−0.763973 + 0.645248i \(0.776754\pi\)
\(440\) −5.02221e26 −0.0880322
\(441\) −1.48731e27 −0.254563
\(442\) 2.76896e27 0.462789
\(443\) −6.63134e27 −1.08234 −0.541168 0.840915i \(-0.682018\pi\)
−0.541168 + 0.840915i \(0.682018\pi\)
\(444\) 1.08909e28 1.73598
\(445\) −9.72147e26 −0.151339
\(446\) −1.12956e28 −1.71749
\(447\) 4.41468e27 0.655648
\(448\) 1.72103e28 2.49671
\(449\) 1.30394e28 1.84787 0.923933 0.382555i \(-0.124956\pi\)
0.923933 + 0.382555i \(0.124956\pi\)
\(450\) 4.63261e27 0.641348
\(451\) −1.50496e27 −0.203549
\(452\) −2.54097e28 −3.35773
\(453\) −6.55703e27 −0.846596
\(454\) 6.10577e27 0.770289
\(455\) 1.28070e26 0.0157880
\(456\) −1.90971e28 −2.30058
\(457\) 4.72949e27 0.556793 0.278397 0.960466i \(-0.410197\pi\)
0.278397 + 0.960466i \(0.410197\pi\)
\(458\) −1.04097e28 −1.19771
\(459\) 1.76848e27 0.198869
\(460\) −2.90476e27 −0.319266
\(461\) 4.92722e27 0.529349 0.264675 0.964338i \(-0.414735\pi\)
0.264675 + 0.964338i \(0.414735\pi\)
\(462\) 8.89661e26 0.0934294
\(463\) 1.20207e28 1.23404 0.617021 0.786947i \(-0.288339\pi\)
0.617021 + 0.786947i \(0.288339\pi\)
\(464\) 4.54513e28 4.56150
\(465\) −4.11319e26 −0.0403574
\(466\) 8.27379e27 0.793693
\(467\) 1.09969e28 1.03144 0.515719 0.856758i \(-0.327525\pi\)
0.515719 + 0.856758i \(0.327525\pi\)
\(468\) 2.36789e27 0.217161
\(469\) −5.72402e27 −0.513317
\(470\) 5.88944e26 0.0516471
\(471\) −2.57786e27 −0.221075
\(472\) −6.12910e28 −5.14051
\(473\) −2.41413e27 −0.198024
\(474\) −2.83578e27 −0.227510
\(475\) 1.36426e28 1.07057
\(476\) −1.86958e28 −1.43507
\(477\) 1.51823e27 0.113997
\(478\) 1.36576e28 1.00318
\(479\) −1.32717e28 −0.953684 −0.476842 0.878989i \(-0.658219\pi\)
−0.476842 + 0.878989i \(0.658219\pi\)
\(480\) 5.62247e27 0.395270
\(481\) −3.48951e27 −0.240017
\(482\) −2.23613e28 −1.50489
\(483\) 3.34446e27 0.220233
\(484\) −4.30624e28 −2.77473
\(485\) 2.95139e27 0.186096
\(486\) 2.04170e27 0.125983
\(487\) 2.62576e28 1.58563 0.792813 0.609466i \(-0.208616\pi\)
0.792813 + 0.609466i \(0.208616\pi\)
\(488\) 7.94738e28 4.69695
\(489\) −1.68680e28 −0.975710
\(490\) 3.77276e27 0.213600
\(491\) −2.54066e28 −1.40796 −0.703982 0.710218i \(-0.748596\pi\)
−0.703982 + 0.710218i \(0.748596\pi\)
\(492\) 3.65121e28 1.98063
\(493\) −2.06213e28 −1.09502
\(494\) 9.41414e27 0.489382
\(495\) 1.58121e26 0.00804704
\(496\) 4.24039e28 2.11275
\(497\) 9.61224e27 0.468903
\(498\) −2.86645e28 −1.36911
\(499\) −1.30048e28 −0.608204 −0.304102 0.952640i \(-0.598356\pi\)
−0.304102 + 0.952640i \(0.598356\pi\)
\(500\) −1.75889e28 −0.805479
\(501\) −1.57631e28 −0.706882
\(502\) 5.67449e28 2.49194
\(503\) 1.34993e27 0.0580559 0.0290280 0.999579i \(-0.490759\pi\)
0.0290280 + 0.999579i \(0.490759\pi\)
\(504\) −1.40289e28 −0.590886
\(505\) 4.50314e26 0.0185762
\(506\) 6.46525e27 0.261219
\(507\) 1.38302e28 0.547326
\(508\) 9.96406e27 0.386249
\(509\) −4.04902e28 −1.53749 −0.768746 0.639554i \(-0.779119\pi\)
−0.768746 + 0.639554i \(0.779119\pi\)
\(510\) −4.48599e27 −0.166867
\(511\) 4.89416e27 0.178344
\(512\) −1.39939e29 −4.99579
\(513\) 6.01263e27 0.210296
\(514\) −3.79902e28 −1.30184
\(515\) 6.83028e27 0.229331
\(516\) 5.85698e28 1.92687
\(517\) −9.70964e26 −0.0313006
\(518\) 3.18083e28 1.00479
\(519\) 1.34924e28 0.417670
\(520\) −3.90398e27 −0.118433
\(521\) 5.40378e28 1.60658 0.803289 0.595590i \(-0.203082\pi\)
0.803289 + 0.595590i \(0.203082\pi\)
\(522\) −2.38072e28 −0.693695
\(523\) 1.54066e28 0.439988 0.219994 0.975501i \(-0.429396\pi\)
0.219994 + 0.975501i \(0.429396\pi\)
\(524\) 3.83833e28 1.07439
\(525\) 1.00220e28 0.274967
\(526\) 1.65986e28 0.446398
\(527\) −1.92387e28 −0.507182
\(528\) −1.63011e28 −0.421270
\(529\) −1.51670e28 −0.384252
\(530\) −3.85118e27 −0.0956528
\(531\) 1.92971e28 0.469895
\(532\) −6.35637e28 −1.51753
\(533\) −1.16987e28 −0.273842
\(534\) −5.24953e28 −1.20486
\(535\) −5.07192e26 −0.0114145
\(536\) 1.74486e29 3.85061
\(537\) 7.62623e26 0.0165037
\(538\) −1.52238e29 −3.23080
\(539\) −6.21997e27 −0.129451
\(540\) −3.83622e27 −0.0783013
\(541\) −7.54478e28 −1.51034 −0.755171 0.655528i \(-0.772446\pi\)
−0.755171 + 0.655528i \(0.772446\pi\)
\(542\) 2.96923e28 0.582976
\(543\) 5.16878e28 0.995382
\(544\) 2.62981e29 4.96747
\(545\) 6.99279e26 0.0129565
\(546\) 6.91571e27 0.125694
\(547\) 7.90524e28 1.40944 0.704722 0.709483i \(-0.251072\pi\)
0.704722 + 0.709483i \(0.251072\pi\)
\(548\) −1.18813e29 −2.07812
\(549\) −2.50219e28 −0.429349
\(550\) 1.93737e28 0.326140
\(551\) −7.01101e28 −1.15795
\(552\) −1.01950e29 −1.65206
\(553\) −6.13480e27 −0.0975408
\(554\) −8.95531e28 −1.39710
\(555\) 5.65335e27 0.0865426
\(556\) −3.11627e28 −0.468113
\(557\) 1.17729e29 1.73541 0.867707 0.497075i \(-0.165593\pi\)
0.867707 + 0.497075i \(0.165593\pi\)
\(558\) −2.22110e28 −0.321299
\(559\) −1.87660e28 −0.266409
\(560\) 2.13902e28 0.298017
\(561\) 7.39583e27 0.101129
\(562\) 3.29582e28 0.442315
\(563\) −9.20807e28 −1.21291 −0.606457 0.795116i \(-0.707410\pi\)
−0.606457 + 0.795116i \(0.707410\pi\)
\(564\) 2.35568e28 0.304569
\(565\) −1.31899e28 −0.167391
\(566\) 1.36626e29 1.70200
\(567\) 4.41693e27 0.0540130
\(568\) −2.93011e29 −3.51744
\(569\) −2.71795e27 −0.0320304 −0.0160152 0.999872i \(-0.505098\pi\)
−0.0160152 + 0.999872i \(0.505098\pi\)
\(570\) −1.52518e28 −0.176456
\(571\) 1.28086e28 0.145487 0.0727434 0.997351i \(-0.476825\pi\)
0.0727434 + 0.997351i \(0.476825\pi\)
\(572\) 9.90260e27 0.110431
\(573\) −7.91029e28 −0.866102
\(574\) 1.06638e29 1.14640
\(575\) 7.28307e28 0.768780
\(576\) 1.65175e29 1.71201
\(577\) 1.49329e29 1.51984 0.759922 0.650015i \(-0.225237\pi\)
0.759922 + 0.650015i \(0.225237\pi\)
\(578\) −1.33261e28 −0.133186
\(579\) 1.26347e28 0.124005
\(580\) 4.47322e28 0.431148
\(581\) −6.20116e28 −0.586980
\(582\) 1.59373e29 1.48158
\(583\) 6.34926e27 0.0579700
\(584\) −1.49189e29 −1.33783
\(585\) 1.22914e27 0.0108260
\(586\) 2.26586e29 1.96023
\(587\) 2.62975e28 0.223468 0.111734 0.993738i \(-0.464360\pi\)
0.111734 + 0.993738i \(0.464360\pi\)
\(588\) 1.50904e29 1.25962
\(589\) −6.54093e28 −0.536326
\(590\) −4.89498e28 −0.394280
\(591\) −8.43783e28 −0.667671
\(592\) −5.82817e29 −4.53059
\(593\) 1.92294e29 1.46856 0.734278 0.678849i \(-0.237521\pi\)
0.734278 + 0.678849i \(0.237521\pi\)
\(594\) 8.53846e27 0.0640651
\(595\) −9.70478e27 −0.0715414
\(596\) −4.47919e29 −3.24425
\(597\) −4.40421e28 −0.313428
\(598\) 5.02572e28 0.351427
\(599\) −2.45874e29 −1.68940 −0.844699 0.535242i \(-0.820220\pi\)
−0.844699 + 0.535242i \(0.820220\pi\)
\(600\) −3.05501e29 −2.06264
\(601\) 7.47252e28 0.495776 0.247888 0.968789i \(-0.420264\pi\)
0.247888 + 0.968789i \(0.420264\pi\)
\(602\) 1.71060e29 1.11528
\(603\) −5.49359e28 −0.351985
\(604\) 6.65285e29 4.18909
\(605\) −2.23531e28 −0.138327
\(606\) 2.43167e28 0.147891
\(607\) −1.23466e29 −0.738016 −0.369008 0.929426i \(-0.620302\pi\)
−0.369008 + 0.929426i \(0.620302\pi\)
\(608\) 8.94104e29 5.25291
\(609\) −5.15035e28 −0.297410
\(610\) 6.34714e28 0.360259
\(611\) −7.54771e27 −0.0421098
\(612\) −1.79432e29 −0.984034
\(613\) −1.59244e29 −0.858476 −0.429238 0.903191i \(-0.641218\pi\)
−0.429238 + 0.903191i \(0.641218\pi\)
\(614\) −4.30861e29 −2.28333
\(615\) 1.89530e28 0.0987390
\(616\) −5.86694e28 −0.300479
\(617\) 1.23256e29 0.620602 0.310301 0.950638i \(-0.399570\pi\)
0.310301 + 0.950638i \(0.399570\pi\)
\(618\) 3.68831e29 1.82578
\(619\) −5.18990e28 −0.252585 −0.126292 0.991993i \(-0.540308\pi\)
−0.126292 + 0.991993i \(0.540308\pi\)
\(620\) 4.17330e28 0.199695
\(621\) 3.20983e28 0.151015
\(622\) −4.02159e29 −1.86037
\(623\) −1.13566e29 −0.516564
\(624\) −1.26715e29 −0.566750
\(625\) 2.13632e29 0.939562
\(626\) −1.42086e29 −0.614497
\(627\) 2.51450e28 0.106940
\(628\) 2.61553e29 1.09391
\(629\) 2.64425e29 1.08760
\(630\) −1.12041e28 −0.0453213
\(631\) −2.05208e28 −0.0816366 −0.0408183 0.999167i \(-0.512996\pi\)
−0.0408183 + 0.999167i \(0.512996\pi\)
\(632\) 1.87008e29 0.731694
\(633\) 1.56533e29 0.602374
\(634\) −5.16285e29 −1.95412
\(635\) 5.17222e27 0.0192554
\(636\) −1.54041e29 −0.564075
\(637\) −4.83505e28 −0.174155
\(638\) −9.95625e28 −0.352759
\(639\) 9.22528e28 0.321529
\(640\) −2.19303e29 −0.751890
\(641\) 5.14342e29 1.73477 0.867386 0.497636i \(-0.165798\pi\)
0.867386 + 0.497636i \(0.165798\pi\)
\(642\) −2.73880e28 −0.0908747
\(643\) −8.85766e28 −0.289137 −0.144569 0.989495i \(-0.546179\pi\)
−0.144569 + 0.989495i \(0.546179\pi\)
\(644\) −3.39333e29 −1.08975
\(645\) 3.04028e28 0.0960588
\(646\) −7.13376e29 −2.21757
\(647\) 5.46916e29 1.67273 0.836364 0.548174i \(-0.184677\pi\)
0.836364 + 0.548174i \(0.184677\pi\)
\(648\) −1.34642e29 −0.405174
\(649\) 8.07012e28 0.238952
\(650\) 1.50600e29 0.438768
\(651\) −4.80503e28 −0.137751
\(652\) 1.71145e30 4.82796
\(653\) −5.66153e29 −1.57161 −0.785806 0.618473i \(-0.787752\pi\)
−0.785806 + 0.618473i \(0.787752\pi\)
\(654\) 3.77606e28 0.103151
\(655\) 1.99243e28 0.0535611
\(656\) −1.95391e30 −5.16908
\(657\) 4.69713e28 0.122292
\(658\) 6.88003e28 0.176286
\(659\) −1.48653e29 −0.374867 −0.187434 0.982277i \(-0.560017\pi\)
−0.187434 + 0.982277i \(0.560017\pi\)
\(660\) −1.60432e28 −0.0398180
\(661\) −4.04669e29 −0.988517 −0.494259 0.869315i \(-0.664560\pi\)
−0.494259 + 0.869315i \(0.664560\pi\)
\(662\) −4.09792e29 −0.985268
\(663\) 5.74909e28 0.136053
\(664\) 1.89031e30 4.40319
\(665\) −3.29951e28 −0.0756524
\(666\) 3.05278e29 0.688994
\(667\) −3.74281e29 −0.831527
\(668\) 1.59935e30 3.49776
\(669\) −2.34527e29 −0.504916
\(670\) 1.39352e29 0.295344
\(671\) −1.04642e29 −0.218334
\(672\) 6.56816e29 1.34917
\(673\) −1.54590e29 −0.312625 −0.156313 0.987708i \(-0.549961\pi\)
−0.156313 + 0.987708i \(0.549961\pi\)
\(674\) 1.03252e30 2.05574
\(675\) 9.61852e28 0.188547
\(676\) −1.40323e30 −2.70825
\(677\) −9.88421e29 −1.87828 −0.939142 0.343530i \(-0.888377\pi\)
−0.939142 + 0.343530i \(0.888377\pi\)
\(678\) −7.12245e29 −1.33265
\(679\) 3.44781e29 0.635200
\(680\) 2.95832e29 0.536662
\(681\) 1.26772e29 0.226453
\(682\) −9.28870e28 −0.163388
\(683\) −1.41169e29 −0.244524 −0.122262 0.992498i \(-0.539015\pi\)
−0.122262 + 0.992498i \(0.539015\pi\)
\(684\) −6.10048e29 −1.04058
\(685\) −6.16746e28 −0.103599
\(686\) 1.01784e30 1.68375
\(687\) −2.16133e29 −0.352108
\(688\) −3.13430e30 −5.02877
\(689\) 4.93555e28 0.0779891
\(690\) −8.14216e28 −0.126714
\(691\) 7.11585e29 1.09070 0.545352 0.838207i \(-0.316396\pi\)
0.545352 + 0.838207i \(0.316396\pi\)
\(692\) −1.36896e30 −2.06669
\(693\) 1.84717e28 0.0274668
\(694\) 2.01817e29 0.295585
\(695\) −1.61762e28 −0.0233365
\(696\) 1.56999e30 2.23099
\(697\) 8.86490e29 1.24088
\(698\) 2.00200e30 2.76047
\(699\) 1.71786e29 0.233334
\(700\) −1.01684e30 −1.36058
\(701\) −8.80754e29 −1.16096 −0.580478 0.814276i \(-0.697134\pi\)
−0.580478 + 0.814276i \(0.697134\pi\)
\(702\) 6.63731e28 0.0861891
\(703\) 8.99015e29 1.15010
\(704\) 6.90765e29 0.870597
\(705\) 1.22280e28 0.0151835
\(706\) 3.07393e29 0.376049
\(707\) 5.26057e28 0.0634058
\(708\) −1.95791e30 −2.32511
\(709\) −2.79000e29 −0.326452 −0.163226 0.986589i \(-0.552190\pi\)
−0.163226 + 0.986589i \(0.552190\pi\)
\(710\) −2.34012e29 −0.269789
\(711\) −5.88783e28 −0.0668843
\(712\) 3.46185e30 3.87496
\(713\) −3.49186e29 −0.385139
\(714\) −5.24052e29 −0.569565
\(715\) 5.14032e27 0.00550524
\(716\) −7.73767e28 −0.0816626
\(717\) 2.83568e29 0.294921
\(718\) 4.76349e29 0.488223
\(719\) 1.21404e30 1.22625 0.613124 0.789987i \(-0.289913\pi\)
0.613124 + 0.789987i \(0.289913\pi\)
\(720\) 2.05291e29 0.204352
\(721\) 7.97913e29 0.782772
\(722\) −3.94185e29 −0.381118
\(723\) −4.64280e29 −0.442413
\(724\) −5.24431e30 −4.92531
\(725\) −1.12157e30 −1.03819
\(726\) −1.20706e30 −1.10127
\(727\) −6.54831e29 −0.588868 −0.294434 0.955672i \(-0.595131\pi\)
−0.294434 + 0.955672i \(0.595131\pi\)
\(728\) −4.56062e29 −0.404245
\(729\) 4.23912e28 0.0370370
\(730\) −1.19149e29 −0.102613
\(731\) 1.42204e30 1.20720
\(732\) 2.53875e30 2.12449
\(733\) 2.20665e29 0.182029 0.0910147 0.995850i \(-0.470989\pi\)
0.0910147 + 0.995850i \(0.470989\pi\)
\(734\) −2.79755e30 −2.27493
\(735\) 7.83325e28 0.0627950
\(736\) 4.77315e30 3.77214
\(737\) −2.29743e29 −0.178992
\(738\) 1.02345e30 0.786094
\(739\) −4.07297e29 −0.308422 −0.154211 0.988038i \(-0.549283\pi\)
−0.154211 + 0.988038i \(0.549283\pi\)
\(740\) −5.73596e29 −0.428226
\(741\) 1.95463e29 0.143871
\(742\) −4.49895e29 −0.326490
\(743\) 3.97218e29 0.284214 0.142107 0.989851i \(-0.454612\pi\)
0.142107 + 0.989851i \(0.454612\pi\)
\(744\) 1.46472e30 1.03333
\(745\) −2.32509e29 −0.161733
\(746\) −2.84565e29 −0.195175
\(747\) −5.95152e29 −0.402496
\(748\) −7.50390e29 −0.500403
\(749\) −5.92500e28 −0.0389610
\(750\) −4.93025e29 −0.319687
\(751\) −1.86890e30 −1.19500 −0.597498 0.801871i \(-0.703838\pi\)
−0.597498 + 0.801871i \(0.703838\pi\)
\(752\) −1.26062e30 −0.794869
\(753\) 1.17817e30 0.732592
\(754\) −7.73941e29 −0.474579
\(755\) 3.45341e29 0.208836
\(756\) −4.48147e29 −0.267265
\(757\) −2.99103e30 −1.75919 −0.879597 0.475720i \(-0.842188\pi\)
−0.879597 + 0.475720i \(0.842188\pi\)
\(758\) −2.62759e29 −0.152416
\(759\) 1.34236e29 0.0767945
\(760\) 1.00579e30 0.567501
\(761\) 1.51341e30 0.842206 0.421103 0.907013i \(-0.361643\pi\)
0.421103 + 0.907013i \(0.361643\pi\)
\(762\) 2.79297e29 0.153299
\(763\) 8.16896e28 0.0442241
\(764\) 8.02588e30 4.28561
\(765\) −9.31410e28 −0.0490564
\(766\) 6.07697e30 3.15707
\(767\) 6.27325e29 0.321470
\(768\) −5.97596e30 −3.02075
\(769\) 2.53401e30 1.26352 0.631759 0.775165i \(-0.282333\pi\)
0.631759 + 0.775165i \(0.282333\pi\)
\(770\) −4.68560e28 −0.0230469
\(771\) −7.88777e29 −0.382722
\(772\) −1.28193e30 −0.613598
\(773\) −1.69545e30 −0.800571 −0.400285 0.916390i \(-0.631089\pi\)
−0.400285 + 0.916390i \(0.631089\pi\)
\(774\) 1.64173e30 0.764756
\(775\) −1.04637e30 −0.480857
\(776\) −1.05100e31 −4.76490
\(777\) 6.60424e29 0.295394
\(778\) 3.57153e30 1.57604
\(779\) 3.01396e30 1.31218
\(780\) −1.24711e29 −0.0535685
\(781\) 3.85804e29 0.163505
\(782\) −3.80834e30 −1.59245
\(783\) −4.94301e29 −0.203936
\(784\) −8.07548e30 −3.28738
\(785\) 1.35769e29 0.0545342
\(786\) 1.07590e30 0.426417
\(787\) −1.87332e30 −0.732616 −0.366308 0.930494i \(-0.619378\pi\)
−0.366308 + 0.930494i \(0.619378\pi\)
\(788\) 8.56113e30 3.30374
\(789\) 3.44632e29 0.131234
\(790\) 1.49353e29 0.0561214
\(791\) −1.54084e30 −0.571353
\(792\) −5.63075e29 −0.206040
\(793\) −8.13429e29 −0.293732
\(794\) −5.06702e30 −1.80567
\(795\) −7.99608e28 −0.0281204
\(796\) 4.46856e30 1.55089
\(797\) −7.79664e29 −0.267051 −0.133526 0.991045i \(-0.542630\pi\)
−0.133526 + 0.991045i \(0.542630\pi\)
\(798\) −1.78172e30 −0.602294
\(799\) 5.71944e29 0.190815
\(800\) 1.43032e31 4.70963
\(801\) −1.08994e30 −0.354211
\(802\) −3.98676e30 −1.27876
\(803\) 1.96436e29 0.0621880
\(804\) 5.57386e30 1.74168
\(805\) −1.76144e29 −0.0543264
\(806\) −7.22050e29 −0.219811
\(807\) −3.16088e30 −0.949807
\(808\) −1.60358e30 −0.475633
\(809\) 8.82262e29 0.258308 0.129154 0.991625i \(-0.458774\pi\)
0.129154 + 0.991625i \(0.458774\pi\)
\(810\) −1.07531e29 −0.0310771
\(811\) −2.06044e30 −0.587815 −0.293907 0.955834i \(-0.594956\pi\)
−0.293907 + 0.955834i \(0.594956\pi\)
\(812\) 5.22561e30 1.47163
\(813\) 6.16490e29 0.171386
\(814\) 1.27668e30 0.350369
\(815\) 8.88392e29 0.240685
\(816\) 9.60212e30 2.56815
\(817\) 4.83476e30 1.27657
\(818\) −6.18551e30 −1.61237
\(819\) 1.43589e29 0.0369521
\(820\) −1.92299e30 −0.488576
\(821\) −1.83846e30 −0.461160 −0.230580 0.973053i \(-0.574062\pi\)
−0.230580 + 0.973053i \(0.574062\pi\)
\(822\) −3.33039e30 −0.824785
\(823\) 7.73766e29 0.189196 0.0945979 0.995516i \(-0.469843\pi\)
0.0945979 + 0.995516i \(0.469843\pi\)
\(824\) −2.43229e31 −5.87190
\(825\) 4.02249e29 0.0958802
\(826\) −5.71831e30 −1.34579
\(827\) −4.59989e30 −1.06891 −0.534453 0.845198i \(-0.679482\pi\)
−0.534453 + 0.845198i \(0.679482\pi\)
\(828\) −3.25673e30 −0.747245
\(829\) 7.93000e30 1.79660 0.898298 0.439386i \(-0.144804\pi\)
0.898298 + 0.439386i \(0.144804\pi\)
\(830\) 1.50968e30 0.337727
\(831\) −1.85936e30 −0.410726
\(832\) 5.36961e30 1.17124
\(833\) 3.66386e30 0.789162
\(834\) −8.73504e29 −0.185790
\(835\) 8.30202e29 0.174372
\(836\) −2.55124e30 −0.529158
\(837\) −4.61159e29 −0.0944569
\(838\) 1.72812e31 3.49551
\(839\) 4.84033e30 0.966884 0.483442 0.875376i \(-0.339386\pi\)
0.483442 + 0.875376i \(0.339386\pi\)
\(840\) 7.38865e29 0.145758
\(841\) 6.30944e29 0.122923
\(842\) 5.40340e30 1.03966
\(843\) 6.84300e29 0.130034
\(844\) −1.58820e31 −2.98064
\(845\) −7.28400e29 −0.135013
\(846\) 6.60306e29 0.120881
\(847\) −2.61129e30 −0.472149
\(848\) 8.24334e30 1.47213
\(849\) 2.83671e30 0.500362
\(850\) −1.14120e31 −1.98822
\(851\) 4.79937e30 0.825893
\(852\) −9.36009e30 −1.59098
\(853\) 2.96903e30 0.498483 0.249242 0.968441i \(-0.419819\pi\)
0.249242 + 0.968441i \(0.419819\pi\)
\(854\) 7.41472e30 1.22967
\(855\) −3.16669e29 −0.0518753
\(856\) 1.80612e30 0.292263
\(857\) −3.70009e30 −0.591444 −0.295722 0.955274i \(-0.595560\pi\)
−0.295722 + 0.955274i \(0.595560\pi\)
\(858\) 2.77574e29 0.0438291
\(859\) 7.75385e29 0.120945 0.0604726 0.998170i \(-0.480739\pi\)
0.0604726 + 0.998170i \(0.480739\pi\)
\(860\) −3.08471e30 −0.475314
\(861\) 2.21408e30 0.337024
\(862\) 2.29958e28 0.00345798
\(863\) −1.29544e31 −1.92443 −0.962217 0.272283i \(-0.912221\pi\)
−0.962217 + 0.272283i \(0.912221\pi\)
\(864\) 6.30375e30 0.925134
\(865\) −7.10610e29 −0.103030
\(866\) 1.59389e31 2.28307
\(867\) −2.76685e29 −0.0391548
\(868\) 4.87524e30 0.681615
\(869\) −2.46231e29 −0.0340122
\(870\) 1.25386e30 0.171119
\(871\) −1.78589e30 −0.240805
\(872\) −2.49015e30 −0.331744
\(873\) 3.30901e30 0.435560
\(874\) −1.29479e31 −1.68395
\(875\) −1.06659e30 −0.137060
\(876\) −4.76577e30 −0.605118
\(877\) −1.57355e31 −1.97417 −0.987083 0.160207i \(-0.948784\pi\)
−0.987083 + 0.160207i \(0.948784\pi\)
\(878\) 2.42056e31 3.00070
\(879\) 4.70452e30 0.576278
\(880\) 8.58535e29 0.103918
\(881\) −1.47526e31 −1.76450 −0.882252 0.470778i \(-0.843973\pi\)
−0.882252 + 0.470778i \(0.843973\pi\)
\(882\) 4.22991e30 0.499932
\(883\) −5.64453e30 −0.659235 −0.329617 0.944115i \(-0.606920\pi\)
−0.329617 + 0.944115i \(0.606920\pi\)
\(884\) −5.83310e30 −0.673210
\(885\) −1.01633e30 −0.115912
\(886\) 1.88595e31 2.12558
\(887\) 5.89300e30 0.656354 0.328177 0.944616i \(-0.393566\pi\)
0.328177 + 0.944616i \(0.393566\pi\)
\(888\) −2.01318e31 −2.21588
\(889\) 6.04218e29 0.0657242
\(890\) 2.76479e30 0.297212
\(891\) 1.77281e29 0.0188342
\(892\) 2.37954e31 2.49840
\(893\) 1.94454e30 0.201780
\(894\) −1.25554e31 −1.28761
\(895\) −4.01653e28 −0.00407107
\(896\) −2.56189e31 −2.56641
\(897\) 1.04347e30 0.103314
\(898\) −3.70840e31 −3.62898
\(899\) 5.37734e30 0.520104
\(900\) −9.75907e30 −0.932958
\(901\) −3.74002e30 −0.353397
\(902\) 4.28009e30 0.399746
\(903\) 3.55166e30 0.327876
\(904\) 4.69695e31 4.28596
\(905\) −2.72226e30 −0.245538
\(906\) 1.86482e31 1.66261
\(907\) 8.32782e30 0.733931 0.366965 0.930235i \(-0.380397\pi\)
0.366965 + 0.930235i \(0.380397\pi\)
\(908\) −1.28624e31 −1.12053
\(909\) 5.04879e29 0.0434777
\(910\) −3.64232e29 −0.0310058
\(911\) 1.08086e31 0.909548 0.454774 0.890607i \(-0.349720\pi\)
0.454774 + 0.890607i \(0.349720\pi\)
\(912\) 3.26461e31 2.71572
\(913\) −2.48894e30 −0.204678
\(914\) −1.34507e31 −1.09347
\(915\) 1.31783e30 0.105911
\(916\) 2.19291e31 1.74229
\(917\) 2.32755e30 0.182819
\(918\) −5.02955e30 −0.390554
\(919\) 6.55205e30 0.502996 0.251498 0.967858i \(-0.419077\pi\)
0.251498 + 0.967858i \(0.419077\pi\)
\(920\) 5.36941e30 0.407525
\(921\) −8.94582e30 −0.671265
\(922\) −1.40130e31 −1.03958
\(923\) 2.99902e30 0.219969
\(924\) −1.87416e30 −0.135910
\(925\) 1.43817e31 1.03115
\(926\) −3.41869e31 −2.42351
\(927\) 7.65791e30 0.536752
\(928\) −7.35047e31 −5.09403
\(929\) 1.04036e31 0.712883 0.356442 0.934318i \(-0.383990\pi\)
0.356442 + 0.934318i \(0.383990\pi\)
\(930\) 1.16979e30 0.0792570
\(931\) 1.24567e31 0.834509
\(932\) −1.74296e31 −1.15457
\(933\) −8.34989e30 −0.546919
\(934\) −3.12752e31 −2.02562
\(935\) −3.89518e29 −0.0249463
\(936\) −4.37702e30 −0.277193
\(937\) −2.09831e31 −1.31402 −0.657012 0.753880i \(-0.728180\pi\)
−0.657012 + 0.753880i \(0.728180\pi\)
\(938\) 1.62791e31 1.00809
\(939\) −2.95007e30 −0.180653
\(940\) −1.24067e30 −0.0751302
\(941\) 2.20967e31 1.32323 0.661617 0.749842i \(-0.269870\pi\)
0.661617 + 0.749842i \(0.269870\pi\)
\(942\) 7.33143e30 0.434165
\(943\) 1.60900e31 0.942286
\(944\) 1.04776e32 6.06812
\(945\) −2.32628e29 −0.0133238
\(946\) 6.86578e30 0.388896
\(947\) −2.59604e31 −1.45424 −0.727121 0.686509i \(-0.759142\pi\)
−0.727121 + 0.686509i \(0.759142\pi\)
\(948\) 5.97386e30 0.330954
\(949\) 1.52698e30 0.0836637
\(950\) −3.87996e31 −2.10247
\(951\) −1.07194e31 −0.574483
\(952\) 3.45590e31 1.83178
\(953\) −2.94729e31 −1.54507 −0.772535 0.634973i \(-0.781011\pi\)
−0.772535 + 0.634973i \(0.781011\pi\)
\(954\) −4.31783e30 −0.223876
\(955\) 4.16613e30 0.213648
\(956\) −2.87711e31 −1.45931
\(957\) −2.06718e30 −0.103706
\(958\) 3.77448e31 1.87292
\(959\) −7.20482e30 −0.353612
\(960\) −8.69929e30 −0.422314
\(961\) −1.58087e31 −0.759103
\(962\) 9.92417e30 0.471364
\(963\) −5.68648e29 −0.0267158
\(964\) 4.71064e31 2.18913
\(965\) −6.65436e29 −0.0305893
\(966\) −9.51166e30 −0.432510
\(967\) 6.98077e30 0.313997 0.156998 0.987599i \(-0.449818\pi\)
0.156998 + 0.987599i \(0.449818\pi\)
\(968\) 7.96002e31 3.54179
\(969\) −1.48116e31 −0.651931
\(970\) −8.39375e30 −0.365471
\(971\) −1.30234e30 −0.0560946 −0.0280473 0.999607i \(-0.508929\pi\)
−0.0280473 + 0.999607i \(0.508929\pi\)
\(972\) −4.30106e30 −0.183265
\(973\) −1.88970e30 −0.0796541
\(974\) −7.46765e31 −3.11398
\(975\) 3.12685e30 0.128991
\(976\) −1.35859e32 −5.54452
\(977\) 3.06599e31 1.23788 0.618939 0.785439i \(-0.287563\pi\)
0.618939 + 0.785439i \(0.287563\pi\)
\(978\) 4.79726e31 1.91618
\(979\) −4.55817e30 −0.180124
\(980\) −7.94771e30 −0.310719
\(981\) 7.84011e29 0.0303248
\(982\) 7.22564e31 2.76507
\(983\) 8.31325e30 0.314745 0.157373 0.987539i \(-0.449698\pi\)
0.157373 + 0.987539i \(0.449698\pi\)
\(984\) −6.74921e31 −2.52816
\(985\) 4.44398e30 0.164699
\(986\) 5.86470e31 2.15049
\(987\) 1.42848e30 0.0518255
\(988\) −1.98319e31 −0.711895
\(989\) 2.58102e31 0.916708
\(990\) −4.49697e29 −0.0158034
\(991\) −1.32568e31 −0.460962 −0.230481 0.973077i \(-0.574030\pi\)
−0.230481 + 0.973077i \(0.574030\pi\)
\(992\) −6.85764e31 −2.35940
\(993\) −8.50837e30 −0.289654
\(994\) −2.73372e31 −0.920868
\(995\) 2.31958e30 0.0773154
\(996\) 6.03849e31 1.99161
\(997\) 3.42048e31 1.11632 0.558159 0.829734i \(-0.311508\pi\)
0.558159 + 0.829734i \(0.311508\pi\)
\(998\) 3.69857e31 1.19444
\(999\) 6.33837e30 0.202554
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 3.22.a.a.1.1 1
3.2 odd 2 9.22.a.d.1.1 1
4.3 odd 2 48.22.a.e.1.1 1
5.2 odd 4 75.22.b.a.49.1 2
5.3 odd 4 75.22.b.a.49.2 2
5.4 even 2 75.22.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.a.1.1 1 1.1 even 1 trivial
9.22.a.d.1.1 1 3.2 odd 2
48.22.a.e.1.1 1 4.3 odd 2
75.22.a.c.1.1 1 5.4 even 2
75.22.b.a.49.1 2 5.2 odd 4
75.22.b.a.49.2 2 5.3 odd 4