Properties

Label 9.22.a.d.1.1
Level $9$
Weight $22$
Character 9.1
Self dual yes
Analytic conductor $25.153$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.1529609858\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9.1

$q$-expansion

\(f(q)\) \(=\) \(q+2844.00 q^{2} +5.99118e6 q^{4} -3.10995e6 q^{5} +3.63304e8 q^{7} +1.10746e10 q^{8} +O(q^{10})\) \(q+2844.00 q^{2} +5.99118e6 q^{4} -3.10995e6 q^{5} +3.63304e8 q^{7} +1.10746e10 q^{8} -8.84470e9 q^{10} -1.45818e10 q^{11} +1.13351e11 q^{13} +1.03324e12 q^{14} +1.89318e13 q^{16} +8.58939e12 q^{17} -2.92029e13 q^{19} -1.86323e13 q^{20} -4.14707e13 q^{22} +1.55899e14 q^{23} -4.67165e14 q^{25} +3.22370e14 q^{26} +2.17662e15 q^{28} -2.40079e15 q^{29} +2.23982e15 q^{31} +3.06169e16 q^{32} +2.44282e16 q^{34} -1.12986e15 q^{35} -3.07851e16 q^{37} -8.30532e16 q^{38} -3.44415e16 q^{40} +1.03208e17 q^{41} -1.65557e17 q^{43} -8.73624e16 q^{44} +4.43377e17 q^{46} +6.65872e16 q^{47} -4.26556e17 q^{49} -1.32862e18 q^{50} +6.79105e17 q^{52} -4.35423e17 q^{53} +4.53488e16 q^{55} +4.02346e18 q^{56} -6.82784e18 q^{58} -5.53437e18 q^{59} -7.17621e18 q^{61} +6.37005e18 q^{62} +4.73716e19 q^{64} -3.52515e17 q^{65} -1.57554e19 q^{67} +5.14606e19 q^{68} -3.21331e18 q^{70} -2.64579e19 q^{71} +1.34712e19 q^{73} -8.75527e19 q^{74} -1.74960e20 q^{76} -5.29764e18 q^{77} -1.68861e19 q^{79} -5.88770e19 q^{80} +2.93522e20 q^{82} +1.70688e20 q^{83} -2.67126e19 q^{85} -4.70845e20 q^{86} -1.61488e20 q^{88} +3.12592e20 q^{89} +4.11808e19 q^{91} +9.34021e20 q^{92} +1.89374e20 q^{94} +9.08197e19 q^{95} +9.49015e20 q^{97} -1.21313e21 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2844.00 1.96388 0.981939 0.189196i \(-0.0605882\pi\)
0.981939 + 0.189196i \(0.0605882\pi\)
\(3\) 0 0
\(4\) 5.99118e6 2.85682
\(5\) −3.10995e6 −0.142419 −0.0712096 0.997461i \(-0.522686\pi\)
−0.0712096 + 0.997461i \(0.522686\pi\)
\(6\) 0 0
\(7\) 3.63304e8 0.486117 0.243058 0.970012i \(-0.421849\pi\)
0.243058 + 0.970012i \(0.421849\pi\)
\(8\) 1.10746e10 3.64657
\(9\) 0 0
\(10\) −8.84470e9 −0.279694
\(11\) −1.45818e10 −0.169508 −0.0847538 0.996402i \(-0.527010\pi\)
−0.0847538 + 0.996402i \(0.527010\pi\)
\(12\) 0 0
\(13\) 1.13351e11 0.228044 0.114022 0.993478i \(-0.463627\pi\)
0.114022 + 0.993478i \(0.463627\pi\)
\(14\) 1.03324e12 0.954674
\(15\) 0 0
\(16\) 1.89318e13 4.30460
\(17\) 8.58939e12 1.03335 0.516676 0.856181i \(-0.327169\pi\)
0.516676 + 0.856181i \(0.327169\pi\)
\(18\) 0 0
\(19\) −2.92029e13 −1.09273 −0.546366 0.837546i \(-0.683989\pi\)
−0.546366 + 0.837546i \(0.683989\pi\)
\(20\) −1.86323e13 −0.406866
\(21\) 0 0
\(22\) −4.14707e13 −0.332892
\(23\) 1.55899e14 0.784696 0.392348 0.919817i \(-0.371663\pi\)
0.392348 + 0.919817i \(0.371663\pi\)
\(24\) 0 0
\(25\) −4.67165e14 −0.979717
\(26\) 3.22370e14 0.447851
\(27\) 0 0
\(28\) 2.17662e15 1.38875
\(29\) −2.40079e15 −1.05968 −0.529840 0.848097i \(-0.677748\pi\)
−0.529840 + 0.848097i \(0.677748\pi\)
\(30\) 0 0
\(31\) 2.23982e15 0.490812 0.245406 0.969420i \(-0.421079\pi\)
0.245406 + 0.969420i \(0.421079\pi\)
\(32\) 3.06169e16 4.80714
\(33\) 0 0
\(34\) 2.44282e16 2.02938
\(35\) −1.12986e15 −0.0692323
\(36\) 0 0
\(37\) −3.07851e16 −1.05250 −0.526250 0.850330i \(-0.676402\pi\)
−0.526250 + 0.850330i \(0.676402\pi\)
\(38\) −8.30532e16 −2.14599
\(39\) 0 0
\(40\) −3.44415e16 −0.519341
\(41\) 1.03208e17 1.20083 0.600414 0.799689i \(-0.295002\pi\)
0.600414 + 0.799689i \(0.295002\pi\)
\(42\) 0 0
\(43\) −1.65557e17 −1.16823 −0.584117 0.811670i \(-0.698559\pi\)
−0.584117 + 0.811670i \(0.698559\pi\)
\(44\) −8.73624e16 −0.484252
\(45\) 0 0
\(46\) 4.43377e17 1.54105
\(47\) 6.65872e16 0.184656 0.0923280 0.995729i \(-0.470569\pi\)
0.0923280 + 0.995729i \(0.470569\pi\)
\(48\) 0 0
\(49\) −4.26556e17 −0.763690
\(50\) −1.32862e18 −1.92404
\(51\) 0 0
\(52\) 6.79105e17 0.651482
\(53\) −4.35423e17 −0.341991 −0.170995 0.985272i \(-0.554698\pi\)
−0.170995 + 0.985272i \(0.554698\pi\)
\(54\) 0 0
\(55\) 4.53488e16 0.0241411
\(56\) 4.02346e18 1.77266
\(57\) 0 0
\(58\) −6.82784e18 −2.08108
\(59\) −5.53437e18 −1.40968 −0.704842 0.709364i \(-0.748982\pi\)
−0.704842 + 0.709364i \(0.748982\pi\)
\(60\) 0 0
\(61\) −7.17621e18 −1.28805 −0.644023 0.765006i \(-0.722736\pi\)
−0.644023 + 0.765006i \(0.722736\pi\)
\(62\) 6.37005e18 0.963896
\(63\) 0 0
\(64\) 4.73716e19 5.13604
\(65\) −3.52515e17 −0.0324779
\(66\) 0 0
\(67\) −1.57554e19 −1.05595 −0.527977 0.849258i \(-0.677049\pi\)
−0.527977 + 0.849258i \(0.677049\pi\)
\(68\) 5.14606e19 2.95210
\(69\) 0 0
\(70\) −3.21331e18 −0.135964
\(71\) −2.64579e19 −0.964588 −0.482294 0.876009i \(-0.660196\pi\)
−0.482294 + 0.876009i \(0.660196\pi\)
\(72\) 0 0
\(73\) 1.34712e19 0.366875 0.183437 0.983031i \(-0.441278\pi\)
0.183437 + 0.983031i \(0.441278\pi\)
\(74\) −8.75527e19 −2.06698
\(75\) 0 0
\(76\) −1.74960e20 −3.12174
\(77\) −5.29764e18 −0.0824005
\(78\) 0 0
\(79\) −1.68861e19 −0.200653 −0.100326 0.994955i \(-0.531989\pi\)
−0.100326 + 0.994955i \(0.531989\pi\)
\(80\) −5.88770e19 −0.613057
\(81\) 0 0
\(82\) 2.93522e20 2.35828
\(83\) 1.70688e20 1.20749 0.603744 0.797178i \(-0.293675\pi\)
0.603744 + 0.797178i \(0.293675\pi\)
\(84\) 0 0
\(85\) −2.67126e19 −0.147169
\(86\) −4.70845e20 −2.29427
\(87\) 0 0
\(88\) −1.61488e20 −0.618121
\(89\) 3.12592e20 1.06263 0.531317 0.847173i \(-0.321697\pi\)
0.531317 + 0.847173i \(0.321697\pi\)
\(90\) 0 0
\(91\) 4.11808e19 0.110856
\(92\) 9.34021e20 2.24174
\(93\) 0 0
\(94\) 1.89374e20 0.362642
\(95\) 9.08197e19 0.155626
\(96\) 0 0
\(97\) 9.49015e20 1.30668 0.653341 0.757064i \(-0.273367\pi\)
0.653341 + 0.757064i \(0.273367\pi\)
\(98\) −1.21313e21 −1.49980
\(99\) 0 0
\(100\) −2.79887e21 −2.79887
\(101\) −1.44798e20 −0.130433 −0.0652166 0.997871i \(-0.520774\pi\)
−0.0652166 + 0.997871i \(0.520774\pi\)
\(102\) 0 0
\(103\) 2.19627e21 1.61025 0.805127 0.593102i \(-0.202097\pi\)
0.805127 + 0.593102i \(0.202097\pi\)
\(104\) 1.25532e21 0.831579
\(105\) 0 0
\(106\) −1.23834e21 −0.671629
\(107\) 1.63087e20 0.0801473 0.0400737 0.999197i \(-0.487241\pi\)
0.0400737 + 0.999197i \(0.487241\pi\)
\(108\) 0 0
\(109\) 2.24852e20 0.0909743 0.0454871 0.998965i \(-0.485516\pi\)
0.0454871 + 0.998965i \(0.485516\pi\)
\(110\) 1.28972e20 0.0474102
\(111\) 0 0
\(112\) 6.87800e21 2.09254
\(113\) 4.24118e21 1.17534 0.587670 0.809101i \(-0.300045\pi\)
0.587670 + 0.809101i \(0.300045\pi\)
\(114\) 0 0
\(115\) −4.84839e20 −0.111756
\(116\) −1.43836e22 −3.02732
\(117\) 0 0
\(118\) −1.57397e22 −2.76845
\(119\) 3.12056e21 0.502330
\(120\) 0 0
\(121\) −7.18762e21 −0.971267
\(122\) −2.04091e22 −2.52957
\(123\) 0 0
\(124\) 1.34192e22 1.40216
\(125\) 2.93580e21 0.281950
\(126\) 0 0
\(127\) 1.66312e21 0.135202 0.0676012 0.997712i \(-0.478465\pi\)
0.0676012 + 0.997712i \(0.478465\pi\)
\(128\) 7.05165e22 5.27942
\(129\) 0 0
\(130\) −1.00255e21 −0.0637826
\(131\) −6.40663e21 −0.376081 −0.188040 0.982161i \(-0.560214\pi\)
−0.188040 + 0.982161i \(0.560214\pi\)
\(132\) 0 0
\(133\) −1.06095e22 −0.531196
\(134\) −4.48085e22 −2.07377
\(135\) 0 0
\(136\) 9.51243e22 3.76819
\(137\) 1.98314e22 0.727423 0.363711 0.931512i \(-0.381509\pi\)
0.363711 + 0.931512i \(0.381509\pi\)
\(138\) 0 0
\(139\) −5.20143e21 −0.163858 −0.0819290 0.996638i \(-0.526108\pi\)
−0.0819290 + 0.996638i \(0.526108\pi\)
\(140\) −6.76918e21 −0.197784
\(141\) 0 0
\(142\) −7.52461e22 −1.89433
\(143\) −1.65286e21 −0.0386552
\(144\) 0 0
\(145\) 7.46633e21 0.150919
\(146\) 3.83122e22 0.720498
\(147\) 0 0
\(148\) −1.84439e23 −3.00680
\(149\) 7.47631e22 1.13562 0.567808 0.823161i \(-0.307792\pi\)
0.567808 + 0.823161i \(0.307792\pi\)
\(150\) 0 0
\(151\) 1.11044e23 1.46635 0.733174 0.680042i \(-0.238038\pi\)
0.733174 + 0.680042i \(0.238038\pi\)
\(152\) −3.23412e23 −3.98472
\(153\) 0 0
\(154\) −1.50665e22 −0.161824
\(155\) −6.96573e21 −0.0699011
\(156\) 0 0
\(157\) 4.36563e22 0.382913 0.191457 0.981501i \(-0.438679\pi\)
0.191457 + 0.981501i \(0.438679\pi\)
\(158\) −4.80241e22 −0.394058
\(159\) 0 0
\(160\) −9.52171e22 −0.684628
\(161\) 5.66388e22 0.381454
\(162\) 0 0
\(163\) 2.85661e23 1.68998 0.844989 0.534783i \(-0.179607\pi\)
0.844989 + 0.534783i \(0.179607\pi\)
\(164\) 6.18336e23 3.43055
\(165\) 0 0
\(166\) 4.85437e23 2.37136
\(167\) −2.66950e23 −1.22435 −0.612177 0.790720i \(-0.709706\pi\)
−0.612177 + 0.790720i \(0.709706\pi\)
\(168\) 0 0
\(169\) −2.34216e23 −0.947996
\(170\) −7.59706e22 −0.289022
\(171\) 0 0
\(172\) −9.91884e23 −3.33743
\(173\) 2.28496e23 0.723425 0.361713 0.932290i \(-0.382192\pi\)
0.361713 + 0.932290i \(0.382192\pi\)
\(174\) 0 0
\(175\) −1.69723e23 −0.476257
\(176\) −2.76061e23 −0.729661
\(177\) 0 0
\(178\) 8.89013e23 2.08688
\(179\) 1.29151e22 0.0285852 0.0142926 0.999898i \(-0.495450\pi\)
0.0142926 + 0.999898i \(0.495450\pi\)
\(180\) 0 0
\(181\) −8.75338e23 −1.72405 −0.862026 0.506863i \(-0.830805\pi\)
−0.862026 + 0.506863i \(0.830805\pi\)
\(182\) 1.17118e23 0.217708
\(183\) 0 0
\(184\) 1.72653e24 2.86145
\(185\) 9.57400e22 0.149896
\(186\) 0 0
\(187\) −1.25249e23 −0.175161
\(188\) 3.98936e23 0.527529
\(189\) 0 0
\(190\) 2.58291e23 0.305631
\(191\) −1.33961e24 −1.50013 −0.750066 0.661363i \(-0.769978\pi\)
−0.750066 + 0.661363i \(0.769978\pi\)
\(192\) 0 0
\(193\) −2.13970e23 −0.214783 −0.107392 0.994217i \(-0.534250\pi\)
−0.107392 + 0.994217i \(0.534250\pi\)
\(194\) 2.69900e24 2.56616
\(195\) 0 0
\(196\) −2.55558e24 −2.18173
\(197\) −1.42895e24 −1.15644 −0.578220 0.815881i \(-0.696252\pi\)
−0.578220 + 0.815881i \(0.696252\pi\)
\(198\) 0 0
\(199\) 7.45856e23 0.542872 0.271436 0.962456i \(-0.412501\pi\)
0.271436 + 0.962456i \(0.412501\pi\)
\(200\) −5.17368e24 −3.57260
\(201\) 0 0
\(202\) −4.11805e23 −0.256155
\(203\) −8.72216e23 −0.515129
\(204\) 0 0
\(205\) −3.20970e23 −0.171021
\(206\) 6.24619e24 3.16234
\(207\) 0 0
\(208\) 2.14594e24 0.981639
\(209\) 4.25832e23 0.185226
\(210\) 0 0
\(211\) −2.65090e24 −1.04334 −0.521671 0.853147i \(-0.674691\pi\)
−0.521671 + 0.853147i \(0.674691\pi\)
\(212\) −2.60870e24 −0.977006
\(213\) 0 0
\(214\) 4.63819e23 0.157400
\(215\) 5.14875e23 0.166379
\(216\) 0 0
\(217\) 8.13736e23 0.238592
\(218\) 6.39479e23 0.178662
\(219\) 0 0
\(220\) 2.71693e23 0.0689668
\(221\) 9.73614e23 0.235650
\(222\) 0 0
\(223\) 3.97174e24 0.874539 0.437270 0.899330i \(-0.355946\pi\)
0.437270 + 0.899330i \(0.355946\pi\)
\(224\) 1.11232e25 2.33683
\(225\) 0 0
\(226\) 1.20619e25 2.30823
\(227\) 2.14690e24 0.392229 0.196114 0.980581i \(-0.437168\pi\)
0.196114 + 0.980581i \(0.437168\pi\)
\(228\) 0 0
\(229\) 3.66024e24 0.609869 0.304934 0.952373i \(-0.401365\pi\)
0.304934 + 0.952373i \(0.401365\pi\)
\(230\) −1.37888e24 −0.219475
\(231\) 0 0
\(232\) −2.65878e25 −3.86420
\(233\) 2.90921e24 0.404145 0.202073 0.979371i \(-0.435232\pi\)
0.202073 + 0.979371i \(0.435232\pi\)
\(234\) 0 0
\(235\) −2.07083e23 −0.0262985
\(236\) −3.31574e25 −4.02721
\(237\) 0 0
\(238\) 8.87487e24 0.986516
\(239\) 4.80224e24 0.510818 0.255409 0.966833i \(-0.417790\pi\)
0.255409 + 0.966833i \(0.417790\pi\)
\(240\) 0 0
\(241\) 7.86263e24 0.766282 0.383141 0.923690i \(-0.374842\pi\)
0.383141 + 0.923690i \(0.374842\pi\)
\(242\) −2.04416e25 −1.90745
\(243\) 0 0
\(244\) −4.29940e25 −3.67972
\(245\) 1.32657e24 0.108764
\(246\) 0 0
\(247\) −3.31018e24 −0.249191
\(248\) 2.48052e25 1.78978
\(249\) 0 0
\(250\) 8.34942e24 0.553715
\(251\) 1.99525e25 1.26889 0.634443 0.772969i \(-0.281229\pi\)
0.634443 + 0.772969i \(0.281229\pi\)
\(252\) 0 0
\(253\) −2.27330e24 −0.133012
\(254\) 4.72991e24 0.265521
\(255\) 0 0
\(256\) 1.01203e26 5.23210
\(257\) −1.33580e25 −0.662894 −0.331447 0.943474i \(-0.607537\pi\)
−0.331447 + 0.943474i \(0.607537\pi\)
\(258\) 0 0
\(259\) −1.11843e25 −0.511638
\(260\) −2.11198e24 −0.0927834
\(261\) 0 0
\(262\) −1.82204e25 −0.738576
\(263\) 5.83637e24 0.227304 0.113652 0.993521i \(-0.463745\pi\)
0.113652 + 0.993521i \(0.463745\pi\)
\(264\) 0 0
\(265\) 1.35414e24 0.0487060
\(266\) −3.01735e25 −1.04320
\(267\) 0 0
\(268\) −9.43938e25 −3.01667
\(269\) −5.35297e25 −1.64511 −0.822557 0.568683i \(-0.807453\pi\)
−0.822557 + 0.568683i \(0.807453\pi\)
\(270\) 0 0
\(271\) −1.04403e25 −0.296849 −0.148425 0.988924i \(-0.547420\pi\)
−0.148425 + 0.988924i \(0.547420\pi\)
\(272\) 1.62613e26 4.44817
\(273\) 0 0
\(274\) 5.64004e25 1.42857
\(275\) 6.81213e24 0.166069
\(276\) 0 0
\(277\) 3.14884e25 0.711399 0.355699 0.934600i \(-0.384243\pi\)
0.355699 + 0.934600i \(0.384243\pi\)
\(278\) −1.47929e25 −0.321797
\(279\) 0 0
\(280\) −1.25127e25 −0.252460
\(281\) 1.15887e25 0.225225 0.112613 0.993639i \(-0.464078\pi\)
0.112613 + 0.993639i \(0.464078\pi\)
\(282\) 0 0
\(283\) −4.80399e25 −0.866652 −0.433326 0.901237i \(-0.642660\pi\)
−0.433326 + 0.901237i \(0.642660\pi\)
\(284\) −1.58514e26 −2.75565
\(285\) 0 0
\(286\) −4.70074e24 −0.0759142
\(287\) 3.74957e25 0.583743
\(288\) 0 0
\(289\) 4.68568e24 0.0678180
\(290\) 2.12343e25 0.296386
\(291\) 0 0
\(292\) 8.07087e25 1.04809
\(293\) 7.96714e25 0.998142 0.499071 0.866561i \(-0.333675\pi\)
0.499071 + 0.866561i \(0.333675\pi\)
\(294\) 0 0
\(295\) 1.72116e25 0.200766
\(296\) −3.40933e26 −3.83801
\(297\) 0 0
\(298\) 2.12626e26 2.23021
\(299\) 1.76713e25 0.178946
\(300\) 0 0
\(301\) −6.01476e25 −0.567898
\(302\) 3.15809e26 2.87973
\(303\) 0 0
\(304\) −5.52865e26 −4.70377
\(305\) 2.23176e25 0.183442
\(306\) 0 0
\(307\) 1.51498e26 1.16266 0.581332 0.813666i \(-0.302532\pi\)
0.581332 + 0.813666i \(0.302532\pi\)
\(308\) −3.17391e25 −0.235403
\(309\) 0 0
\(310\) −1.98105e25 −0.137277
\(311\) −1.41406e26 −0.947292 −0.473646 0.880715i \(-0.657062\pi\)
−0.473646 + 0.880715i \(0.657062\pi\)
\(312\) 0 0
\(313\) 4.99598e25 0.312899 0.156450 0.987686i \(-0.449995\pi\)
0.156450 + 0.987686i \(0.449995\pi\)
\(314\) 1.24158e26 0.751995
\(315\) 0 0
\(316\) −1.01168e26 −0.573229
\(317\) −1.81535e26 −0.995033 −0.497517 0.867454i \(-0.665755\pi\)
−0.497517 + 0.867454i \(0.665755\pi\)
\(318\) 0 0
\(319\) 3.50079e25 0.179624
\(320\) −1.47323e26 −0.731470
\(321\) 0 0
\(322\) 1.61081e26 0.749130
\(323\) −2.50835e26 −1.12918
\(324\) 0 0
\(325\) −5.29536e25 −0.223419
\(326\) 8.12420e26 3.31891
\(327\) 0 0
\(328\) 1.14299e27 4.37890
\(329\) 2.41914e25 0.0897644
\(330\) 0 0
\(331\) 1.44090e26 0.501695 0.250848 0.968027i \(-0.419291\pi\)
0.250848 + 0.968027i \(0.419291\pi\)
\(332\) 1.02262e27 3.44958
\(333\) 0 0
\(334\) −7.59206e26 −2.40448
\(335\) 4.89987e25 0.150388
\(336\) 0 0
\(337\) −3.63051e26 −1.04678 −0.523388 0.852095i \(-0.675332\pi\)
−0.523388 + 0.852095i \(0.675332\pi\)
\(338\) −6.66111e26 −1.86175
\(339\) 0 0
\(340\) −1.60040e26 −0.420436
\(341\) −3.26607e25 −0.0831964
\(342\) 0 0
\(343\) −3.57891e26 −0.857360
\(344\) −1.83349e27 −4.26004
\(345\) 0 0
\(346\) 6.49841e26 1.42072
\(347\) 7.09622e25 0.150511 0.0752554 0.997164i \(-0.476023\pi\)
0.0752554 + 0.997164i \(0.476023\pi\)
\(348\) 0 0
\(349\) −7.03939e26 −1.40562 −0.702810 0.711378i \(-0.748072\pi\)
−0.702810 + 0.711378i \(0.748072\pi\)
\(350\) −4.82692e26 −0.935311
\(351\) 0 0
\(352\) −4.46451e26 −0.814846
\(353\) 1.08085e26 0.191483 0.0957414 0.995406i \(-0.469478\pi\)
0.0957414 + 0.995406i \(0.469478\pi\)
\(354\) 0 0
\(355\) 8.22826e25 0.137376
\(356\) 1.87280e27 3.03575
\(357\) 0 0
\(358\) 3.67305e25 0.0561378
\(359\) 1.67492e26 0.248601 0.124301 0.992245i \(-0.460331\pi\)
0.124301 + 0.992245i \(0.460331\pi\)
\(360\) 0 0
\(361\) 1.38602e26 0.194064
\(362\) −2.48946e27 −3.38583
\(363\) 0 0
\(364\) 2.46722e26 0.316696
\(365\) −4.18949e25 −0.0522500
\(366\) 0 0
\(367\) 9.83667e26 1.15839 0.579194 0.815190i \(-0.303367\pi\)
0.579194 + 0.815190i \(0.303367\pi\)
\(368\) 2.95146e27 3.37780
\(369\) 0 0
\(370\) 2.72285e26 0.294378
\(371\) −1.58191e26 −0.166248
\(372\) 0 0
\(373\) 1.00058e26 0.0993824 0.0496912 0.998765i \(-0.484176\pi\)
0.0496912 + 0.998765i \(0.484176\pi\)
\(374\) −3.56208e26 −0.343995
\(375\) 0 0
\(376\) 7.37429e26 0.673360
\(377\) −2.72131e26 −0.241654
\(378\) 0 0
\(379\) 9.23905e25 0.0776096 0.0388048 0.999247i \(-0.487645\pi\)
0.0388048 + 0.999247i \(0.487645\pi\)
\(380\) 5.44117e26 0.444595
\(381\) 0 0
\(382\) −3.80986e27 −2.94608
\(383\) 2.13677e27 1.60757 0.803786 0.594919i \(-0.202816\pi\)
0.803786 + 0.594919i \(0.202816\pi\)
\(384\) 0 0
\(385\) 1.64754e25 0.0117354
\(386\) −6.08530e26 −0.421809
\(387\) 0 0
\(388\) 5.68572e27 3.73295
\(389\) 1.25581e27 0.802516 0.401258 0.915965i \(-0.368573\pi\)
0.401258 + 0.915965i \(0.368573\pi\)
\(390\) 0 0
\(391\) 1.33908e27 0.810868
\(392\) −4.72395e27 −2.78485
\(393\) 0 0
\(394\) −4.06394e27 −2.27111
\(395\) 5.25150e25 0.0285768
\(396\) 0 0
\(397\) 1.78165e27 0.919439 0.459719 0.888064i \(-0.347950\pi\)
0.459719 + 0.888064i \(0.347950\pi\)
\(398\) 2.12121e27 1.06614
\(399\) 0 0
\(400\) −8.84429e27 −4.21729
\(401\) −1.40181e27 −0.651141 −0.325570 0.945518i \(-0.605556\pi\)
−0.325570 + 0.945518i \(0.605556\pi\)
\(402\) 0 0
\(403\) 2.53885e26 0.111927
\(404\) −8.67511e26 −0.372624
\(405\) 0 0
\(406\) −2.48058e27 −1.01165
\(407\) 4.48903e26 0.178407
\(408\) 0 0
\(409\) 2.17493e27 0.821015 0.410507 0.911857i \(-0.365352\pi\)
0.410507 + 0.911857i \(0.365352\pi\)
\(410\) −9.12840e26 −0.335864
\(411\) 0 0
\(412\) 1.31582e28 4.60021
\(413\) −2.01066e27 −0.685271
\(414\) 0 0
\(415\) −5.30831e26 −0.171969
\(416\) 3.47045e27 1.09624
\(417\) 0 0
\(418\) 1.21107e27 0.363762
\(419\) 6.07636e27 1.77990 0.889952 0.456055i \(-0.150738\pi\)
0.889952 + 0.456055i \(0.150738\pi\)
\(420\) 0 0
\(421\) −1.89993e27 −0.529389 −0.264695 0.964332i \(-0.585271\pi\)
−0.264695 + 0.964332i \(0.585271\pi\)
\(422\) −7.53915e27 −2.04900
\(423\) 0 0
\(424\) −4.82214e27 −1.24709
\(425\) −4.01267e27 −1.01239
\(426\) 0 0
\(427\) −2.60714e27 −0.626141
\(428\) 9.77083e26 0.228966
\(429\) 0 0
\(430\) 1.46430e27 0.326748
\(431\) 8.08572e24 0.00176079 0.000880395 1.00000i \(-0.499720\pi\)
0.000880395 1.00000i \(0.499720\pi\)
\(432\) 0 0
\(433\) −5.60439e27 −1.16253 −0.581267 0.813713i \(-0.697443\pi\)
−0.581267 + 0.813713i \(0.697443\pi\)
\(434\) 2.31426e27 0.468566
\(435\) 0 0
\(436\) 1.34713e27 0.259897
\(437\) −4.55272e27 −0.857463
\(438\) 0 0
\(439\) −8.51110e27 −1.52795 −0.763973 0.645248i \(-0.776754\pi\)
−0.763973 + 0.645248i \(0.776754\pi\)
\(440\) 5.02221e26 0.0880322
\(441\) 0 0
\(442\) 2.76896e27 0.462789
\(443\) 6.63134e27 1.08234 0.541168 0.840915i \(-0.317982\pi\)
0.541168 + 0.840915i \(0.317982\pi\)
\(444\) 0 0
\(445\) −9.72147e26 −0.151339
\(446\) 1.12956e28 1.71749
\(447\) 0 0
\(448\) 1.72103e28 2.49671
\(449\) −1.30394e28 −1.84787 −0.923933 0.382555i \(-0.875044\pi\)
−0.923933 + 0.382555i \(0.875044\pi\)
\(450\) 0 0
\(451\) −1.50496e27 −0.203549
\(452\) 2.54097e28 3.35773
\(453\) 0 0
\(454\) 6.10577e27 0.770289
\(455\) −1.28070e26 −0.0157880
\(456\) 0 0
\(457\) 4.72949e27 0.556793 0.278397 0.960466i \(-0.410197\pi\)
0.278397 + 0.960466i \(0.410197\pi\)
\(458\) 1.04097e28 1.19771
\(459\) 0 0
\(460\) −2.90476e27 −0.319266
\(461\) −4.92722e27 −0.529349 −0.264675 0.964338i \(-0.585265\pi\)
−0.264675 + 0.964338i \(0.585265\pi\)
\(462\) 0 0
\(463\) 1.20207e28 1.23404 0.617021 0.786947i \(-0.288339\pi\)
0.617021 + 0.786947i \(0.288339\pi\)
\(464\) −4.54513e28 −4.56150
\(465\) 0 0
\(466\) 8.27379e27 0.793693
\(467\) −1.09969e28 −1.03144 −0.515719 0.856758i \(-0.672475\pi\)
−0.515719 + 0.856758i \(0.672475\pi\)
\(468\) 0 0
\(469\) −5.72402e27 −0.513317
\(470\) −5.88944e26 −0.0516471
\(471\) 0 0
\(472\) −6.12910e28 −5.14051
\(473\) 2.41413e27 0.198024
\(474\) 0 0
\(475\) 1.36426e28 1.07057
\(476\) 1.86958e28 1.43507
\(477\) 0 0
\(478\) 1.36576e28 1.00318
\(479\) 1.32717e28 0.953684 0.476842 0.878989i \(-0.341781\pi\)
0.476842 + 0.878989i \(0.341781\pi\)
\(480\) 0 0
\(481\) −3.48951e27 −0.240017
\(482\) 2.23613e28 1.50489
\(483\) 0 0
\(484\) −4.30624e28 −2.77473
\(485\) −2.95139e27 −0.186096
\(486\) 0 0
\(487\) 2.62576e28 1.58563 0.792813 0.609466i \(-0.208616\pi\)
0.792813 + 0.609466i \(0.208616\pi\)
\(488\) −7.94738e28 −4.69695
\(489\) 0 0
\(490\) 3.77276e27 0.213600
\(491\) 2.54066e28 1.40796 0.703982 0.710218i \(-0.251404\pi\)
0.703982 + 0.710218i \(0.251404\pi\)
\(492\) 0 0
\(493\) −2.06213e28 −1.09502
\(494\) −9.41414e27 −0.489382
\(495\) 0 0
\(496\) 4.24039e28 2.11275
\(497\) −9.61224e27 −0.468903
\(498\) 0 0
\(499\) −1.30048e28 −0.608204 −0.304102 0.952640i \(-0.598356\pi\)
−0.304102 + 0.952640i \(0.598356\pi\)
\(500\) 1.75889e28 0.805479
\(501\) 0 0
\(502\) 5.67449e28 2.49194
\(503\) −1.34993e27 −0.0580559 −0.0290280 0.999579i \(-0.509241\pi\)
−0.0290280 + 0.999579i \(0.509241\pi\)
\(504\) 0 0
\(505\) 4.50314e26 0.0185762
\(506\) −6.46525e27 −0.261219
\(507\) 0 0
\(508\) 9.96406e27 0.386249
\(509\) 4.04902e28 1.53749 0.768746 0.639554i \(-0.220881\pi\)
0.768746 + 0.639554i \(0.220881\pi\)
\(510\) 0 0
\(511\) 4.89416e27 0.178344
\(512\) 1.39939e29 4.99579
\(513\) 0 0
\(514\) −3.79902e28 −1.30184
\(515\) −6.83028e27 −0.229331
\(516\) 0 0
\(517\) −9.70964e26 −0.0313006
\(518\) −3.18083e28 −1.00479
\(519\) 0 0
\(520\) −3.90398e27 −0.118433
\(521\) −5.40378e28 −1.60658 −0.803289 0.595590i \(-0.796918\pi\)
−0.803289 + 0.595590i \(0.796918\pi\)
\(522\) 0 0
\(523\) 1.54066e28 0.439988 0.219994 0.975501i \(-0.429396\pi\)
0.219994 + 0.975501i \(0.429396\pi\)
\(524\) −3.83833e28 −1.07439
\(525\) 0 0
\(526\) 1.65986e28 0.446398
\(527\) 1.92387e28 0.507182
\(528\) 0 0
\(529\) −1.51670e28 −0.384252
\(530\) 3.85118e27 0.0956528
\(531\) 0 0
\(532\) −6.35637e28 −1.51753
\(533\) 1.16987e28 0.273842
\(534\) 0 0
\(535\) −5.07192e26 −0.0114145
\(536\) −1.74486e29 −3.85061
\(537\) 0 0
\(538\) −1.52238e29 −3.23080
\(539\) 6.21997e27 0.129451
\(540\) 0 0
\(541\) −7.54478e28 −1.51034 −0.755171 0.655528i \(-0.772446\pi\)
−0.755171 + 0.655528i \(0.772446\pi\)
\(542\) −2.96923e28 −0.582976
\(543\) 0 0
\(544\) 2.62981e29 4.96747
\(545\) −6.99279e26 −0.0129565
\(546\) 0 0
\(547\) 7.90524e28 1.40944 0.704722 0.709483i \(-0.251072\pi\)
0.704722 + 0.709483i \(0.251072\pi\)
\(548\) 1.18813e29 2.07812
\(549\) 0 0
\(550\) 1.93737e28 0.326140
\(551\) 7.01101e28 1.15795
\(552\) 0 0
\(553\) −6.13480e27 −0.0975408
\(554\) 8.95531e28 1.39710
\(555\) 0 0
\(556\) −3.11627e28 −0.468113
\(557\) −1.17729e29 −1.73541 −0.867707 0.497075i \(-0.834407\pi\)
−0.867707 + 0.497075i \(0.834407\pi\)
\(558\) 0 0
\(559\) −1.87660e28 −0.266409
\(560\) −2.13902e28 −0.298017
\(561\) 0 0
\(562\) 3.29582e28 0.442315
\(563\) 9.20807e28 1.21291 0.606457 0.795116i \(-0.292590\pi\)
0.606457 + 0.795116i \(0.292590\pi\)
\(564\) 0 0
\(565\) −1.31899e28 −0.167391
\(566\) −1.36626e29 −1.70200
\(567\) 0 0
\(568\) −2.93011e29 −3.51744
\(569\) 2.71795e27 0.0320304 0.0160152 0.999872i \(-0.494902\pi\)
0.0160152 + 0.999872i \(0.494902\pi\)
\(570\) 0 0
\(571\) 1.28086e28 0.145487 0.0727434 0.997351i \(-0.476825\pi\)
0.0727434 + 0.997351i \(0.476825\pi\)
\(572\) −9.90260e27 −0.110431
\(573\) 0 0
\(574\) 1.06638e29 1.14640
\(575\) −7.28307e28 −0.768780
\(576\) 0 0
\(577\) 1.49329e29 1.51984 0.759922 0.650015i \(-0.225237\pi\)
0.759922 + 0.650015i \(0.225237\pi\)
\(578\) 1.33261e28 0.133186
\(579\) 0 0
\(580\) 4.47322e28 0.431148
\(581\) 6.20116e28 0.586980
\(582\) 0 0
\(583\) 6.34926e27 0.0579700
\(584\) 1.49189e29 1.33783
\(585\) 0 0
\(586\) 2.26586e29 1.96023
\(587\) −2.62975e28 −0.223468 −0.111734 0.993738i \(-0.535640\pi\)
−0.111734 + 0.993738i \(0.535640\pi\)
\(588\) 0 0
\(589\) −6.54093e28 −0.536326
\(590\) 4.89498e28 0.394280
\(591\) 0 0
\(592\) −5.82817e29 −4.53059
\(593\) −1.92294e29 −1.46856 −0.734278 0.678849i \(-0.762479\pi\)
−0.734278 + 0.678849i \(0.762479\pi\)
\(594\) 0 0
\(595\) −9.70478e27 −0.0715414
\(596\) 4.47919e29 3.24425
\(597\) 0 0
\(598\) 5.02572e28 0.351427
\(599\) 2.45874e29 1.68940 0.844699 0.535242i \(-0.179780\pi\)
0.844699 + 0.535242i \(0.179780\pi\)
\(600\) 0 0
\(601\) 7.47252e28 0.495776 0.247888 0.968789i \(-0.420264\pi\)
0.247888 + 0.968789i \(0.420264\pi\)
\(602\) −1.71060e29 −1.11528
\(603\) 0 0
\(604\) 6.65285e29 4.18909
\(605\) 2.23531e28 0.138327
\(606\) 0 0
\(607\) −1.23466e29 −0.738016 −0.369008 0.929426i \(-0.620302\pi\)
−0.369008 + 0.929426i \(0.620302\pi\)
\(608\) −8.94104e29 −5.25291
\(609\) 0 0
\(610\) 6.34714e28 0.360259
\(611\) 7.54771e27 0.0421098
\(612\) 0 0
\(613\) −1.59244e29 −0.858476 −0.429238 0.903191i \(-0.641218\pi\)
−0.429238 + 0.903191i \(0.641218\pi\)
\(614\) 4.30861e29 2.28333
\(615\) 0 0
\(616\) −5.86694e28 −0.300479
\(617\) −1.23256e29 −0.620602 −0.310301 0.950638i \(-0.600430\pi\)
−0.310301 + 0.950638i \(0.600430\pi\)
\(618\) 0 0
\(619\) −5.18990e28 −0.252585 −0.126292 0.991993i \(-0.540308\pi\)
−0.126292 + 0.991993i \(0.540308\pi\)
\(620\) −4.17330e28 −0.199695
\(621\) 0 0
\(622\) −4.02159e29 −1.86037
\(623\) 1.13566e29 0.516564
\(624\) 0 0
\(625\) 2.13632e29 0.939562
\(626\) 1.42086e29 0.614497
\(627\) 0 0
\(628\) 2.61553e29 1.09391
\(629\) −2.64425e29 −1.08760
\(630\) 0 0
\(631\) −2.05208e28 −0.0816366 −0.0408183 0.999167i \(-0.512996\pi\)
−0.0408183 + 0.999167i \(0.512996\pi\)
\(632\) −1.87008e29 −0.731694
\(633\) 0 0
\(634\) −5.16285e29 −1.95412
\(635\) −5.17222e27 −0.0192554
\(636\) 0 0
\(637\) −4.83505e28 −0.174155
\(638\) 9.95625e28 0.352759
\(639\) 0 0
\(640\) −2.19303e29 −0.751890
\(641\) −5.14342e29 −1.73477 −0.867386 0.497636i \(-0.834202\pi\)
−0.867386 + 0.497636i \(0.834202\pi\)
\(642\) 0 0
\(643\) −8.85766e28 −0.289137 −0.144569 0.989495i \(-0.546179\pi\)
−0.144569 + 0.989495i \(0.546179\pi\)
\(644\) 3.39333e29 1.08975
\(645\) 0 0
\(646\) −7.13376e29 −2.21757
\(647\) −5.46916e29 −1.67273 −0.836364 0.548174i \(-0.815323\pi\)
−0.836364 + 0.548174i \(0.815323\pi\)
\(648\) 0 0
\(649\) 8.07012e28 0.238952
\(650\) −1.50600e29 −0.438768
\(651\) 0 0
\(652\) 1.71145e30 4.82796
\(653\) 5.66153e29 1.57161 0.785806 0.618473i \(-0.212248\pi\)
0.785806 + 0.618473i \(0.212248\pi\)
\(654\) 0 0
\(655\) 1.99243e28 0.0535611
\(656\) 1.95391e30 5.16908
\(657\) 0 0
\(658\) 6.88003e28 0.176286
\(659\) 1.48653e29 0.374867 0.187434 0.982277i \(-0.439983\pi\)
0.187434 + 0.982277i \(0.439983\pi\)
\(660\) 0 0
\(661\) −4.04669e29 −0.988517 −0.494259 0.869315i \(-0.664560\pi\)
−0.494259 + 0.869315i \(0.664560\pi\)
\(662\) 4.09792e29 0.985268
\(663\) 0 0
\(664\) 1.89031e30 4.40319
\(665\) 3.29951e28 0.0756524
\(666\) 0 0
\(667\) −3.74281e29 −0.831527
\(668\) −1.59935e30 −3.49776
\(669\) 0 0
\(670\) 1.39352e29 0.295344
\(671\) 1.04642e29 0.218334
\(672\) 0 0
\(673\) −1.54590e29 −0.312625 −0.156313 0.987708i \(-0.549961\pi\)
−0.156313 + 0.987708i \(0.549961\pi\)
\(674\) −1.03252e30 −2.05574
\(675\) 0 0
\(676\) −1.40323e30 −2.70825
\(677\) 9.88421e29 1.87828 0.939142 0.343530i \(-0.111623\pi\)
0.939142 + 0.343530i \(0.111623\pi\)
\(678\) 0 0
\(679\) 3.44781e29 0.635200
\(680\) −2.95832e29 −0.536662
\(681\) 0 0
\(682\) −9.28870e28 −0.163388
\(683\) 1.41169e29 0.244524 0.122262 0.992498i \(-0.460985\pi\)
0.122262 + 0.992498i \(0.460985\pi\)
\(684\) 0 0
\(685\) −6.16746e28 −0.103599
\(686\) −1.01784e30 −1.68375
\(687\) 0 0
\(688\) −3.13430e30 −5.02877
\(689\) −4.93555e28 −0.0779891
\(690\) 0 0
\(691\) 7.11585e29 1.09070 0.545352 0.838207i \(-0.316396\pi\)
0.545352 + 0.838207i \(0.316396\pi\)
\(692\) 1.36896e30 2.06669
\(693\) 0 0
\(694\) 2.01817e29 0.295585
\(695\) 1.61762e28 0.0233365
\(696\) 0 0
\(697\) 8.86490e29 1.24088
\(698\) −2.00200e30 −2.76047
\(699\) 0 0
\(700\) −1.01684e30 −1.36058
\(701\) 8.80754e29 1.16096 0.580478 0.814276i \(-0.302866\pi\)
0.580478 + 0.814276i \(0.302866\pi\)
\(702\) 0 0
\(703\) 8.99015e29 1.15010
\(704\) −6.90765e29 −0.870597
\(705\) 0 0
\(706\) 3.07393e29 0.376049
\(707\) −5.26057e28 −0.0634058
\(708\) 0 0
\(709\) −2.79000e29 −0.326452 −0.163226 0.986589i \(-0.552190\pi\)
−0.163226 + 0.986589i \(0.552190\pi\)
\(710\) 2.34012e29 0.269789
\(711\) 0 0
\(712\) 3.46185e30 3.87496
\(713\) 3.49186e29 0.385139
\(714\) 0 0
\(715\) 5.14032e27 0.00550524
\(716\) 7.73767e28 0.0816626
\(717\) 0 0
\(718\) 4.76349e29 0.488223
\(719\) −1.21404e30 −1.22625 −0.613124 0.789987i \(-0.710087\pi\)
−0.613124 + 0.789987i \(0.710087\pi\)
\(720\) 0 0
\(721\) 7.97913e29 0.782772
\(722\) 3.94185e29 0.381118
\(723\) 0 0
\(724\) −5.24431e30 −4.92531
\(725\) 1.12157e30 1.03819
\(726\) 0 0
\(727\) −6.54831e29 −0.588868 −0.294434 0.955672i \(-0.595131\pi\)
−0.294434 + 0.955672i \(0.595131\pi\)
\(728\) 4.56062e29 0.404245
\(729\) 0 0
\(730\) −1.19149e29 −0.102613
\(731\) −1.42204e30 −1.20720
\(732\) 0 0
\(733\) 2.20665e29 0.182029 0.0910147 0.995850i \(-0.470989\pi\)
0.0910147 + 0.995850i \(0.470989\pi\)
\(734\) 2.79755e30 2.27493
\(735\) 0 0
\(736\) 4.77315e30 3.77214
\(737\) 2.29743e29 0.178992
\(738\) 0 0
\(739\) −4.07297e29 −0.308422 −0.154211 0.988038i \(-0.549283\pi\)
−0.154211 + 0.988038i \(0.549283\pi\)
\(740\) 5.73596e29 0.428226
\(741\) 0 0
\(742\) −4.49895e29 −0.326490
\(743\) −3.97218e29 −0.284214 −0.142107 0.989851i \(-0.545388\pi\)
−0.142107 + 0.989851i \(0.545388\pi\)
\(744\) 0 0
\(745\) −2.32509e29 −0.161733
\(746\) 2.84565e29 0.195175
\(747\) 0 0
\(748\) −7.50390e29 −0.500403
\(749\) 5.92500e28 0.0389610
\(750\) 0 0
\(751\) −1.86890e30 −1.19500 −0.597498 0.801871i \(-0.703838\pi\)
−0.597498 + 0.801871i \(0.703838\pi\)
\(752\) 1.26062e30 0.794869
\(753\) 0 0
\(754\) −7.73941e29 −0.474579
\(755\) −3.45341e29 −0.208836
\(756\) 0 0
\(757\) −2.99103e30 −1.75919 −0.879597 0.475720i \(-0.842188\pi\)
−0.879597 + 0.475720i \(0.842188\pi\)
\(758\) 2.62759e29 0.152416
\(759\) 0 0
\(760\) 1.00579e30 0.567501
\(761\) −1.51341e30 −0.842206 −0.421103 0.907013i \(-0.638357\pi\)
−0.421103 + 0.907013i \(0.638357\pi\)
\(762\) 0 0
\(763\) 8.16896e28 0.0442241
\(764\) −8.02588e30 −4.28561
\(765\) 0 0
\(766\) 6.07697e30 3.15707
\(767\) −6.27325e29 −0.321470
\(768\) 0 0
\(769\) 2.53401e30 1.26352 0.631759 0.775165i \(-0.282333\pi\)
0.631759 + 0.775165i \(0.282333\pi\)
\(770\) 4.68560e28 0.0230469
\(771\) 0 0
\(772\) −1.28193e30 −0.613598
\(773\) 1.69545e30 0.800571 0.400285 0.916390i \(-0.368911\pi\)
0.400285 + 0.916390i \(0.368911\pi\)
\(774\) 0 0
\(775\) −1.04637e30 −0.480857
\(776\) 1.05100e31 4.76490
\(777\) 0 0
\(778\) 3.57153e30 1.57604
\(779\) −3.01396e30 −1.31218
\(780\) 0 0
\(781\) 3.85804e29 0.163505
\(782\) 3.80834e30 1.59245
\(783\) 0 0
\(784\) −8.07548e30 −3.28738
\(785\) −1.35769e29 −0.0545342
\(786\) 0 0
\(787\) −1.87332e30 −0.732616 −0.366308 0.930494i \(-0.619378\pi\)
−0.366308 + 0.930494i \(0.619378\pi\)
\(788\) −8.56113e30 −3.30374
\(789\) 0 0
\(790\) 1.49353e29 0.0561214
\(791\) 1.54084e30 0.571353
\(792\) 0 0
\(793\) −8.13429e29 −0.293732
\(794\) 5.06702e30 1.80567
\(795\) 0 0
\(796\) 4.46856e30 1.55089
\(797\) 7.79664e29 0.267051 0.133526 0.991045i \(-0.457370\pi\)
0.133526 + 0.991045i \(0.457370\pi\)
\(798\) 0 0
\(799\) 5.71944e29 0.190815
\(800\) −1.43032e31 −4.70963
\(801\) 0 0
\(802\) −3.98676e30 −1.27876
\(803\) −1.96436e29 −0.0621880
\(804\) 0 0
\(805\) −1.76144e29 −0.0543264
\(806\) 7.22050e29 0.219811
\(807\) 0 0
\(808\) −1.60358e30 −0.475633
\(809\) −8.82262e29 −0.258308 −0.129154 0.991625i \(-0.541226\pi\)
−0.129154 + 0.991625i \(0.541226\pi\)
\(810\) 0 0
\(811\) −2.06044e30 −0.587815 −0.293907 0.955834i \(-0.594956\pi\)
−0.293907 + 0.955834i \(0.594956\pi\)
\(812\) −5.22561e30 −1.47163
\(813\) 0 0
\(814\) 1.27668e30 0.350369
\(815\) −8.88392e29 −0.240685
\(816\) 0 0
\(817\) 4.83476e30 1.27657
\(818\) 6.18551e30 1.61237
\(819\) 0 0
\(820\) −1.92299e30 −0.488576
\(821\) 1.83846e30 0.461160 0.230580 0.973053i \(-0.425938\pi\)
0.230580 + 0.973053i \(0.425938\pi\)
\(822\) 0 0
\(823\) 7.73766e29 0.189196 0.0945979 0.995516i \(-0.469843\pi\)
0.0945979 + 0.995516i \(0.469843\pi\)
\(824\) 2.43229e31 5.87190
\(825\) 0 0
\(826\) −5.71831e30 −1.34579
\(827\) 4.59989e30 1.06891 0.534453 0.845198i \(-0.320518\pi\)
0.534453 + 0.845198i \(0.320518\pi\)
\(828\) 0 0
\(829\) 7.93000e30 1.79660 0.898298 0.439386i \(-0.144804\pi\)
0.898298 + 0.439386i \(0.144804\pi\)
\(830\) −1.50968e30 −0.337727
\(831\) 0 0
\(832\) 5.36961e30 1.17124
\(833\) −3.66386e30 −0.789162
\(834\) 0 0
\(835\) 8.30202e29 0.174372
\(836\) 2.55124e30 0.529158
\(837\) 0 0
\(838\) 1.72812e31 3.49551
\(839\) −4.84033e30 −0.966884 −0.483442 0.875376i \(-0.660614\pi\)
−0.483442 + 0.875376i \(0.660614\pi\)
\(840\) 0 0
\(841\) 6.30944e29 0.122923
\(842\) −5.40340e30 −1.03966
\(843\) 0 0
\(844\) −1.58820e31 −2.98064
\(845\) 7.28400e29 0.135013
\(846\) 0 0
\(847\) −2.61129e30 −0.472149
\(848\) −8.24334e30 −1.47213
\(849\) 0 0
\(850\) −1.14120e31 −1.98822
\(851\) −4.79937e30 −0.825893
\(852\) 0 0
\(853\) 2.96903e30 0.498483 0.249242 0.968441i \(-0.419819\pi\)
0.249242 + 0.968441i \(0.419819\pi\)
\(854\) −7.41472e30 −1.22967
\(855\) 0 0
\(856\) 1.80612e30 0.292263
\(857\) 3.70009e30 0.591444 0.295722 0.955274i \(-0.404440\pi\)
0.295722 + 0.955274i \(0.404440\pi\)
\(858\) 0 0
\(859\) 7.75385e29 0.120945 0.0604726 0.998170i \(-0.480739\pi\)
0.0604726 + 0.998170i \(0.480739\pi\)
\(860\) 3.08471e30 0.475314
\(861\) 0 0
\(862\) 2.29958e28 0.00345798
\(863\) 1.29544e31 1.92443 0.962217 0.272283i \(-0.0877786\pi\)
0.962217 + 0.272283i \(0.0877786\pi\)
\(864\) 0 0
\(865\) −7.10610e29 −0.103030
\(866\) −1.59389e31 −2.28307
\(867\) 0 0
\(868\) 4.87524e30 0.681615
\(869\) 2.46231e29 0.0340122
\(870\) 0 0
\(871\) −1.78589e30 −0.240805
\(872\) 2.49015e30 0.331744
\(873\) 0 0
\(874\) −1.29479e31 −1.68395
\(875\) 1.06659e30 0.137060
\(876\) 0 0
\(877\) −1.57355e31 −1.97417 −0.987083 0.160207i \(-0.948784\pi\)
−0.987083 + 0.160207i \(0.948784\pi\)
\(878\) −2.42056e31 −3.00070
\(879\) 0 0
\(880\) 8.58535e29 0.103918
\(881\) 1.47526e31 1.76450 0.882252 0.470778i \(-0.156027\pi\)
0.882252 + 0.470778i \(0.156027\pi\)
\(882\) 0 0
\(883\) −5.64453e30 −0.659235 −0.329617 0.944115i \(-0.606920\pi\)
−0.329617 + 0.944115i \(0.606920\pi\)
\(884\) 5.83310e30 0.673210
\(885\) 0 0
\(886\) 1.88595e31 2.12558
\(887\) −5.89300e30 −0.656354 −0.328177 0.944616i \(-0.606434\pi\)
−0.328177 + 0.944616i \(0.606434\pi\)
\(888\) 0 0
\(889\) 6.04218e29 0.0657242
\(890\) −2.76479e30 −0.297212
\(891\) 0 0
\(892\) 2.37954e31 2.49840
\(893\) −1.94454e30 −0.201780
\(894\) 0 0
\(895\) −4.01653e28 −0.00407107
\(896\) 2.56189e31 2.56641
\(897\) 0 0
\(898\) −3.70840e31 −3.62898
\(899\) −5.37734e30 −0.520104
\(900\) 0 0
\(901\) −3.74002e30 −0.353397
\(902\) −4.28009e30 −0.399746
\(903\) 0 0
\(904\) 4.69695e31 4.28596
\(905\) 2.72226e30 0.245538
\(906\) 0 0
\(907\) 8.32782e30 0.733931 0.366965 0.930235i \(-0.380397\pi\)
0.366965 + 0.930235i \(0.380397\pi\)
\(908\) 1.28624e31 1.12053
\(909\) 0 0
\(910\) −3.64232e29 −0.0310058
\(911\) −1.08086e31 −0.909548 −0.454774 0.890607i \(-0.650280\pi\)
−0.454774 + 0.890607i \(0.650280\pi\)
\(912\) 0 0
\(913\) −2.48894e30 −0.204678
\(914\) 1.34507e31 1.09347
\(915\) 0 0
\(916\) 2.19291e31 1.74229
\(917\) −2.32755e30 −0.182819
\(918\) 0 0
\(919\) 6.55205e30 0.502996 0.251498 0.967858i \(-0.419077\pi\)
0.251498 + 0.967858i \(0.419077\pi\)
\(920\) −5.36941e30 −0.407525
\(921\) 0 0
\(922\) −1.40130e31 −1.03958
\(923\) −2.99902e30 −0.219969
\(924\) 0 0
\(925\) 1.43817e31 1.03115
\(926\) 3.41869e31 2.42351
\(927\) 0 0
\(928\) −7.35047e31 −5.09403
\(929\) −1.04036e31 −0.712883 −0.356442 0.934318i \(-0.616010\pi\)
−0.356442 + 0.934318i \(0.616010\pi\)
\(930\) 0 0
\(931\) 1.24567e31 0.834509
\(932\) 1.74296e31 1.15457
\(933\) 0 0
\(934\) −3.12752e31 −2.02562
\(935\) 3.89518e29 0.0249463
\(936\) 0 0
\(937\) −2.09831e31 −1.31402 −0.657012 0.753880i \(-0.728180\pi\)
−0.657012 + 0.753880i \(0.728180\pi\)
\(938\) −1.62791e31 −1.00809
\(939\) 0 0
\(940\) −1.24067e30 −0.0751302
\(941\) −2.20967e31 −1.32323 −0.661617 0.749842i \(-0.730130\pi\)
−0.661617 + 0.749842i \(0.730130\pi\)
\(942\) 0 0
\(943\) 1.60900e31 0.942286
\(944\) −1.04776e32 −6.06812
\(945\) 0 0
\(946\) 6.86578e30 0.388896
\(947\) 2.59604e31 1.45424 0.727121 0.686509i \(-0.240858\pi\)
0.727121 + 0.686509i \(0.240858\pi\)
\(948\) 0 0
\(949\) 1.52698e30 0.0836637
\(950\) 3.87996e31 2.10247
\(951\) 0 0
\(952\) 3.45590e31 1.83178
\(953\) 2.94729e31 1.54507 0.772535 0.634973i \(-0.218989\pi\)
0.772535 + 0.634973i \(0.218989\pi\)
\(954\) 0 0
\(955\) 4.16613e30 0.213648
\(956\) 2.87711e31 1.45931
\(957\) 0 0
\(958\) 3.77448e31 1.87292
\(959\) 7.20482e30 0.353612
\(960\) 0 0
\(961\) −1.58087e31 −0.759103
\(962\) −9.92417e30 −0.471364
\(963\) 0 0
\(964\) 4.71064e31 2.18913
\(965\) 6.65436e29 0.0305893
\(966\) 0 0
\(967\) 6.98077e30 0.313997 0.156998 0.987599i \(-0.449818\pi\)
0.156998 + 0.987599i \(0.449818\pi\)
\(968\) −7.96002e31 −3.54179
\(969\) 0 0
\(970\) −8.39375e30 −0.365471
\(971\) 1.30234e30 0.0560946 0.0280473 0.999607i \(-0.491071\pi\)
0.0280473 + 0.999607i \(0.491071\pi\)
\(972\) 0 0
\(973\) −1.88970e30 −0.0796541
\(974\) 7.46765e31 3.11398
\(975\) 0 0
\(976\) −1.35859e32 −5.54452
\(977\) −3.06599e31 −1.23788 −0.618939 0.785439i \(-0.712437\pi\)
−0.618939 + 0.785439i \(0.712437\pi\)
\(978\) 0 0
\(979\) −4.55817e30 −0.180124
\(980\) 7.94771e30 0.310719
\(981\) 0 0
\(982\) 7.22564e31 2.76507
\(983\) −8.31325e30 −0.314745 −0.157373 0.987539i \(-0.550302\pi\)
−0.157373 + 0.987539i \(0.550302\pi\)
\(984\) 0 0
\(985\) 4.44398e30 0.164699
\(986\) −5.86470e31 −2.15049
\(987\) 0 0
\(988\) −1.98319e31 −0.711895
\(989\) −2.58102e31 −0.916708
\(990\) 0 0
\(991\) −1.32568e31 −0.460962 −0.230481 0.973077i \(-0.574030\pi\)
−0.230481 + 0.973077i \(0.574030\pi\)
\(992\) 6.85764e31 2.35940
\(993\) 0 0
\(994\) −2.73372e31 −0.920868
\(995\) −2.31958e30 −0.0773154
\(996\) 0 0
\(997\) 3.42048e31 1.11632 0.558159 0.829734i \(-0.311508\pi\)
0.558159 + 0.829734i \(0.311508\pi\)
\(998\) −3.69857e31 −1.19444
\(999\) 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 9.22.a.d.1.1 1
3.2 odd 2 3.22.a.a.1.1 1
12.11 even 2 48.22.a.e.1.1 1
15.2 even 4 75.22.b.a.49.1 2
15.8 even 4 75.22.b.a.49.2 2
15.14 odd 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 3.2 odd 2
9.22.a.d.1.1 1 1.1 even 1 trivial
48.22.a.e.1.1 1 12.11 even 2
75.22.a.c.1.1 1 15.14 odd 2
75.22.b.a.49.1 2 15.2 even 4
75.22.b.a.49.2 2 15.8 even 4