Properties

Label 256.8.a.r.1.6
Level $256$
Weight $8$
Character 256.1
Self dual yes
Analytic conductor $79.971$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 256.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.9705665239\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 163x^{4} + 4820x^{2} - 15296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{27} \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.89807\) of defining polynomial
Character \(\chi\) \(=\) 256.1

$q$-expansion

\(f(q)\) \(=\) \(q+76.9497 q^{3} +338.443 q^{5} +438.996 q^{7} +3734.25 q^{9} +O(q^{10})\) \(q+76.9497 q^{3} +338.443 q^{5} +438.996 q^{7} +3734.25 q^{9} +1966.58 q^{11} +2210.98 q^{13} +26043.0 q^{15} -12114.9 q^{17} +32872.2 q^{19} +33780.6 q^{21} -19605.1 q^{23} +36418.4 q^{25} +119060. q^{27} +160689. q^{29} -229270. q^{31} +151328. q^{33} +148575. q^{35} +496284. q^{37} +170134. q^{39} -599971. q^{41} -88346.0 q^{43} +1.26383e6 q^{45} +820344. q^{47} -630825. q^{49} -932236. q^{51} +1.53717e6 q^{53} +665574. q^{55} +2.52950e6 q^{57} -1.82480e6 q^{59} -484582. q^{61} +1.63932e6 q^{63} +748290. q^{65} -79878.2 q^{67} -1.50860e6 q^{69} -1.27078e6 q^{71} -3.70820e6 q^{73} +2.80238e6 q^{75} +863321. q^{77} -2.55846e6 q^{79} +994857. q^{81} +1.53414e6 q^{83} -4.10019e6 q^{85} +1.23650e7 q^{87} -1.99492e6 q^{89} +970612. q^{91} -1.76423e7 q^{93} +1.11253e7 q^{95} -28917.7 q^{97} +7.34370e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 688 q^{7} + 2918 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 688 q^{7} + 2918 q^{9} + 17872 q^{15} + 1452 q^{17} + 1296 q^{23} + 39314 q^{25} - 89280 q^{31} + 53880 q^{33} + 328208 q^{39} - 521244 q^{41} + 1566432 q^{47} - 511050 q^{49} + 3270256 q^{55} + 1889896 q^{57} + 5776816 q^{63} + 1416480 q^{65} + 7597104 q^{71} - 2089564 q^{73} + 16015904 q^{79} - 723058 q^{81} + 37453776 q^{87} - 2169084 q^{89} + 48537936 q^{95} - 1088308 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 76.9497 1.64544 0.822721 0.568446i \(-0.192455\pi\)
0.822721 + 0.568446i \(0.192455\pi\)
\(4\) 0 0
\(5\) 338.443 1.21085 0.605425 0.795903i \(-0.293003\pi\)
0.605425 + 0.795903i \(0.293003\pi\)
\(6\) 0 0
\(7\) 438.996 0.483746 0.241873 0.970308i \(-0.422238\pi\)
0.241873 + 0.970308i \(0.422238\pi\)
\(8\) 0 0
\(9\) 3734.25 1.70748
\(10\) 0 0
\(11\) 1966.58 0.445489 0.222744 0.974877i \(-0.428498\pi\)
0.222744 + 0.974877i \(0.428498\pi\)
\(12\) 0 0
\(13\) 2210.98 0.279115 0.139557 0.990214i \(-0.455432\pi\)
0.139557 + 0.990214i \(0.455432\pi\)
\(14\) 0 0
\(15\) 26043.0 1.99238
\(16\) 0 0
\(17\) −12114.9 −0.598064 −0.299032 0.954243i \(-0.596664\pi\)
−0.299032 + 0.954243i \(0.596664\pi\)
\(18\) 0 0
\(19\) 32872.2 1.09949 0.549744 0.835333i \(-0.314725\pi\)
0.549744 + 0.835333i \(0.314725\pi\)
\(20\) 0 0
\(21\) 33780.6 0.795976
\(22\) 0 0
\(23\) −19605.1 −0.335986 −0.167993 0.985788i \(-0.553729\pi\)
−0.167993 + 0.985788i \(0.553729\pi\)
\(24\) 0 0
\(25\) 36418.4 0.466155
\(26\) 0 0
\(27\) 119060. 1.16411
\(28\) 0 0
\(29\) 160689. 1.22347 0.611735 0.791063i \(-0.290472\pi\)
0.611735 + 0.791063i \(0.290472\pi\)
\(30\) 0 0
\(31\) −229270. −1.38224 −0.691118 0.722742i \(-0.742881\pi\)
−0.691118 + 0.722742i \(0.742881\pi\)
\(32\) 0 0
\(33\) 151328. 0.733026
\(34\) 0 0
\(35\) 148575. 0.585744
\(36\) 0 0
\(37\) 496284. 1.61074 0.805368 0.592775i \(-0.201968\pi\)
0.805368 + 0.592775i \(0.201968\pi\)
\(38\) 0 0
\(39\) 170134. 0.459267
\(40\) 0 0
\(41\) −599971. −1.35952 −0.679762 0.733433i \(-0.737917\pi\)
−0.679762 + 0.733433i \(0.737917\pi\)
\(42\) 0 0
\(43\) −88346.0 −0.169452 −0.0847262 0.996404i \(-0.527002\pi\)
−0.0847262 + 0.996404i \(0.527002\pi\)
\(44\) 0 0
\(45\) 1.26383e6 2.06750
\(46\) 0 0
\(47\) 820344. 1.15253 0.576267 0.817262i \(-0.304509\pi\)
0.576267 + 0.817262i \(0.304509\pi\)
\(48\) 0 0
\(49\) −630825. −0.765989
\(50\) 0 0
\(51\) −932236. −0.984080
\(52\) 0 0
\(53\) 1.53717e6 1.41826 0.709131 0.705077i \(-0.249087\pi\)
0.709131 + 0.705077i \(0.249087\pi\)
\(54\) 0 0
\(55\) 665574. 0.539420
\(56\) 0 0
\(57\) 2.52950e6 1.80914
\(58\) 0 0
\(59\) −1.82480e6 −1.15673 −0.578367 0.815776i \(-0.696310\pi\)
−0.578367 + 0.815776i \(0.696310\pi\)
\(60\) 0 0
\(61\) −484582. −0.273346 −0.136673 0.990616i \(-0.543641\pi\)
−0.136673 + 0.990616i \(0.543641\pi\)
\(62\) 0 0
\(63\) 1.63932e6 0.825986
\(64\) 0 0
\(65\) 748290. 0.337966
\(66\) 0 0
\(67\) −79878.2 −0.0324464 −0.0162232 0.999868i \(-0.505164\pi\)
−0.0162232 + 0.999868i \(0.505164\pi\)
\(68\) 0 0
\(69\) −1.50860e6 −0.552845
\(70\) 0 0
\(71\) −1.27078e6 −0.421373 −0.210686 0.977554i \(-0.567570\pi\)
−0.210686 + 0.977554i \(0.567570\pi\)
\(72\) 0 0
\(73\) −3.70820e6 −1.11566 −0.557832 0.829954i \(-0.688367\pi\)
−0.557832 + 0.829954i \(0.688367\pi\)
\(74\) 0 0
\(75\) 2.80238e6 0.767031
\(76\) 0 0
\(77\) 863321. 0.215504
\(78\) 0 0
\(79\) −2.55846e6 −0.583827 −0.291914 0.956445i \(-0.594292\pi\)
−0.291914 + 0.956445i \(0.594292\pi\)
\(80\) 0 0
\(81\) 994857. 0.208000
\(82\) 0 0
\(83\) 1.53414e6 0.294505 0.147252 0.989099i \(-0.452957\pi\)
0.147252 + 0.989099i \(0.452957\pi\)
\(84\) 0 0
\(85\) −4.10019e6 −0.724166
\(86\) 0 0
\(87\) 1.23650e7 2.01315
\(88\) 0 0
\(89\) −1.99492e6 −0.299958 −0.149979 0.988689i \(-0.547921\pi\)
−0.149979 + 0.988689i \(0.547921\pi\)
\(90\) 0 0
\(91\) 970612. 0.135021
\(92\) 0 0
\(93\) −1.76423e7 −2.27439
\(94\) 0 0
\(95\) 1.11253e7 1.33131
\(96\) 0 0
\(97\) −28917.7 −0.00321708 −0.00160854 0.999999i \(-0.500512\pi\)
−0.00160854 + 0.999999i \(0.500512\pi\)
\(98\) 0 0
\(99\) 7.34370e6 0.760662
\(100\) 0 0
\(101\) −1.68077e7 −1.62324 −0.811622 0.584182i \(-0.801415\pi\)
−0.811622 + 0.584182i \(0.801415\pi\)
\(102\) 0 0
\(103\) 1.27746e7 1.15191 0.575953 0.817483i \(-0.304631\pi\)
0.575953 + 0.817483i \(0.304631\pi\)
\(104\) 0 0
\(105\) 1.14328e7 0.963807
\(106\) 0 0
\(107\) 1.35610e7 1.07016 0.535078 0.844803i \(-0.320282\pi\)
0.535078 + 0.844803i \(0.320282\pi\)
\(108\) 0 0
\(109\) 4.74206e6 0.350731 0.175366 0.984503i \(-0.443889\pi\)
0.175366 + 0.984503i \(0.443889\pi\)
\(110\) 0 0
\(111\) 3.81889e7 2.65037
\(112\) 0 0
\(113\) −8.06832e6 −0.526028 −0.263014 0.964792i \(-0.584717\pi\)
−0.263014 + 0.964792i \(0.584717\pi\)
\(114\) 0 0
\(115\) −6.63520e6 −0.406828
\(116\) 0 0
\(117\) 8.25635e6 0.476582
\(118\) 0 0
\(119\) −5.31839e6 −0.289311
\(120\) 0 0
\(121\) −1.56197e7 −0.801540
\(122\) 0 0
\(123\) −4.61676e7 −2.23702
\(124\) 0 0
\(125\) −1.41153e7 −0.646405
\(126\) 0 0
\(127\) −1.12410e7 −0.486960 −0.243480 0.969906i \(-0.578289\pi\)
−0.243480 + 0.969906i \(0.578289\pi\)
\(128\) 0 0
\(129\) −6.79820e6 −0.278824
\(130\) 0 0
\(131\) −8.81527e6 −0.342599 −0.171299 0.985219i \(-0.554797\pi\)
−0.171299 + 0.985219i \(0.554797\pi\)
\(132\) 0 0
\(133\) 1.44308e7 0.531874
\(134\) 0 0
\(135\) 4.02951e7 1.40956
\(136\) 0 0
\(137\) 3.33729e7 1.10885 0.554424 0.832234i \(-0.312939\pi\)
0.554424 + 0.832234i \(0.312939\pi\)
\(138\) 0 0
\(139\) 4.68161e7 1.47857 0.739287 0.673391i \(-0.235163\pi\)
0.739287 + 0.673391i \(0.235163\pi\)
\(140\) 0 0
\(141\) 6.31252e7 1.89643
\(142\) 0 0
\(143\) 4.34806e6 0.124343
\(144\) 0 0
\(145\) 5.43840e7 1.48144
\(146\) 0 0
\(147\) −4.85418e7 −1.26039
\(148\) 0 0
\(149\) 3.83709e7 0.950277 0.475139 0.879911i \(-0.342398\pi\)
0.475139 + 0.879911i \(0.342398\pi\)
\(150\) 0 0
\(151\) 7.17648e7 1.69626 0.848130 0.529788i \(-0.177729\pi\)
0.848130 + 0.529788i \(0.177729\pi\)
\(152\) 0 0
\(153\) −4.52400e7 −1.02118
\(154\) 0 0
\(155\) −7.75948e7 −1.67368
\(156\) 0 0
\(157\) 4.03778e7 0.832710 0.416355 0.909202i \(-0.363307\pi\)
0.416355 + 0.909202i \(0.363307\pi\)
\(158\) 0 0
\(159\) 1.18285e8 2.33367
\(160\) 0 0
\(161\) −8.60656e6 −0.162532
\(162\) 0 0
\(163\) 9.84512e7 1.78059 0.890296 0.455383i \(-0.150498\pi\)
0.890296 + 0.455383i \(0.150498\pi\)
\(164\) 0 0
\(165\) 5.12157e7 0.887584
\(166\) 0 0
\(167\) 4.24811e7 0.705810 0.352905 0.935659i \(-0.385194\pi\)
0.352905 + 0.935659i \(0.385194\pi\)
\(168\) 0 0
\(169\) −5.78601e7 −0.922095
\(170\) 0 0
\(171\) 1.22753e8 1.87735
\(172\) 0 0
\(173\) −2.65257e7 −0.389498 −0.194749 0.980853i \(-0.562389\pi\)
−0.194749 + 0.980853i \(0.562389\pi\)
\(174\) 0 0
\(175\) 1.59875e7 0.225501
\(176\) 0 0
\(177\) −1.40418e8 −1.90334
\(178\) 0 0
\(179\) −2.42148e7 −0.315570 −0.157785 0.987474i \(-0.550435\pi\)
−0.157785 + 0.987474i \(0.550435\pi\)
\(180\) 0 0
\(181\) −2.78961e7 −0.349679 −0.174839 0.984597i \(-0.555941\pi\)
−0.174839 + 0.984597i \(0.555941\pi\)
\(182\) 0 0
\(183\) −3.72884e7 −0.449775
\(184\) 0 0
\(185\) 1.67964e8 1.95036
\(186\) 0 0
\(187\) −2.38249e7 −0.266431
\(188\) 0 0
\(189\) 5.22671e7 0.563135
\(190\) 0 0
\(191\) 1.67596e7 0.174039 0.0870195 0.996207i \(-0.472266\pi\)
0.0870195 + 0.996207i \(0.472266\pi\)
\(192\) 0 0
\(193\) 8.75008e7 0.876116 0.438058 0.898947i \(-0.355667\pi\)
0.438058 + 0.898947i \(0.355667\pi\)
\(194\) 0 0
\(195\) 5.75806e7 0.556103
\(196\) 0 0
\(197\) −2.56239e7 −0.238789 −0.119394 0.992847i \(-0.538095\pi\)
−0.119394 + 0.992847i \(0.538095\pi\)
\(198\) 0 0
\(199\) −5.31884e7 −0.478444 −0.239222 0.970965i \(-0.576892\pi\)
−0.239222 + 0.970965i \(0.576892\pi\)
\(200\) 0 0
\(201\) −6.14660e6 −0.0533886
\(202\) 0 0
\(203\) 7.05419e7 0.591849
\(204\) 0 0
\(205\) −2.03056e8 −1.64618
\(206\) 0 0
\(207\) −7.32103e7 −0.573688
\(208\) 0 0
\(209\) 6.46457e7 0.489810
\(210\) 0 0
\(211\) −2.01165e7 −0.147423 −0.0737114 0.997280i \(-0.523484\pi\)
−0.0737114 + 0.997280i \(0.523484\pi\)
\(212\) 0 0
\(213\) −9.77862e7 −0.693344
\(214\) 0 0
\(215\) −2.99001e7 −0.205181
\(216\) 0 0
\(217\) −1.00649e8 −0.668651
\(218\) 0 0
\(219\) −2.85345e8 −1.83576
\(220\) 0 0
\(221\) −2.67858e7 −0.166929
\(222\) 0 0
\(223\) −1.67012e8 −1.00851 −0.504254 0.863555i \(-0.668232\pi\)
−0.504254 + 0.863555i \(0.668232\pi\)
\(224\) 0 0
\(225\) 1.35995e8 0.795949
\(226\) 0 0
\(227\) −1.37308e8 −0.779122 −0.389561 0.921001i \(-0.627373\pi\)
−0.389561 + 0.921001i \(0.627373\pi\)
\(228\) 0 0
\(229\) −2.67935e8 −1.47436 −0.737182 0.675694i \(-0.763844\pi\)
−0.737182 + 0.675694i \(0.763844\pi\)
\(230\) 0 0
\(231\) 6.64322e7 0.354599
\(232\) 0 0
\(233\) −1.98032e8 −1.02563 −0.512815 0.858499i \(-0.671397\pi\)
−0.512815 + 0.858499i \(0.671397\pi\)
\(234\) 0 0
\(235\) 2.77639e8 1.39554
\(236\) 0 0
\(237\) −1.96873e8 −0.960654
\(238\) 0 0
\(239\) 8.22277e7 0.389606 0.194803 0.980842i \(-0.437593\pi\)
0.194803 + 0.980842i \(0.437593\pi\)
\(240\) 0 0
\(241\) −2.70650e8 −1.24551 −0.622757 0.782415i \(-0.713988\pi\)
−0.622757 + 0.782415i \(0.713988\pi\)
\(242\) 0 0
\(243\) −1.83831e8 −0.821860
\(244\) 0 0
\(245\) −2.13498e8 −0.927498
\(246\) 0 0
\(247\) 7.26797e7 0.306884
\(248\) 0 0
\(249\) 1.18052e8 0.484591
\(250\) 0 0
\(251\) −2.90747e8 −1.16053 −0.580266 0.814427i \(-0.697051\pi\)
−0.580266 + 0.814427i \(0.697051\pi\)
\(252\) 0 0
\(253\) −3.85549e7 −0.149678
\(254\) 0 0
\(255\) −3.15508e8 −1.19157
\(256\) 0 0
\(257\) 4.36047e7 0.160239 0.0801193 0.996785i \(-0.474470\pi\)
0.0801193 + 0.996785i \(0.474470\pi\)
\(258\) 0 0
\(259\) 2.17867e8 0.779188
\(260\) 0 0
\(261\) 6.00053e8 2.08905
\(262\) 0 0
\(263\) −4.27678e8 −1.44968 −0.724840 0.688917i \(-0.758086\pi\)
−0.724840 + 0.688917i \(0.758086\pi\)
\(264\) 0 0
\(265\) 5.20244e8 1.71730
\(266\) 0 0
\(267\) −1.53509e8 −0.493564
\(268\) 0 0
\(269\) 8.26134e7 0.258772 0.129386 0.991594i \(-0.458699\pi\)
0.129386 + 0.991594i \(0.458699\pi\)
\(270\) 0 0
\(271\) 6.15189e7 0.187766 0.0938829 0.995583i \(-0.470072\pi\)
0.0938829 + 0.995583i \(0.470072\pi\)
\(272\) 0 0
\(273\) 7.46883e7 0.222169
\(274\) 0 0
\(275\) 7.16196e7 0.207667
\(276\) 0 0
\(277\) 4.39237e7 0.124171 0.0620854 0.998071i \(-0.480225\pi\)
0.0620854 + 0.998071i \(0.480225\pi\)
\(278\) 0 0
\(279\) −8.56153e8 −2.36013
\(280\) 0 0
\(281\) −5.80931e8 −1.56190 −0.780948 0.624596i \(-0.785264\pi\)
−0.780948 + 0.624596i \(0.785264\pi\)
\(282\) 0 0
\(283\) 6.03790e8 1.58356 0.791778 0.610810i \(-0.209156\pi\)
0.791778 + 0.610810i \(0.209156\pi\)
\(284\) 0 0
\(285\) 8.56091e8 2.19060
\(286\) 0 0
\(287\) −2.63385e8 −0.657665
\(288\) 0 0
\(289\) −2.63568e8 −0.642319
\(290\) 0 0
\(291\) −2.22520e6 −0.00529352
\(292\) 0 0
\(293\) −1.10504e8 −0.256649 −0.128325 0.991732i \(-0.540960\pi\)
−0.128325 + 0.991732i \(0.540960\pi\)
\(294\) 0 0
\(295\) −6.17591e8 −1.40063
\(296\) 0 0
\(297\) 2.34142e8 0.518599
\(298\) 0 0
\(299\) −4.33464e7 −0.0937787
\(300\) 0 0
\(301\) −3.87836e7 −0.0819719
\(302\) 0 0
\(303\) −1.29335e9 −2.67095
\(304\) 0 0
\(305\) −1.64003e8 −0.330981
\(306\) 0 0
\(307\) 7.03386e8 1.38742 0.693712 0.720252i \(-0.255974\pi\)
0.693712 + 0.720252i \(0.255974\pi\)
\(308\) 0 0
\(309\) 9.83002e8 1.89539
\(310\) 0 0
\(311\) 8.61240e8 1.62354 0.811769 0.583978i \(-0.198505\pi\)
0.811769 + 0.583978i \(0.198505\pi\)
\(312\) 0 0
\(313\) −2.42056e8 −0.446181 −0.223090 0.974798i \(-0.571615\pi\)
−0.223090 + 0.974798i \(0.571615\pi\)
\(314\) 0 0
\(315\) 5.54817e8 1.00014
\(316\) 0 0
\(317\) −5.38362e8 −0.949221 −0.474610 0.880196i \(-0.657411\pi\)
−0.474610 + 0.880196i \(0.657411\pi\)
\(318\) 0 0
\(319\) 3.16007e8 0.545042
\(320\) 0 0
\(321\) 1.04351e9 1.76088
\(322\) 0 0
\(323\) −3.98242e8 −0.657565
\(324\) 0 0
\(325\) 8.05203e7 0.130111
\(326\) 0 0
\(327\) 3.64900e8 0.577108
\(328\) 0 0
\(329\) 3.60128e8 0.557534
\(330\) 0 0
\(331\) −1.05054e9 −1.59226 −0.796131 0.605124i \(-0.793123\pi\)
−0.796131 + 0.605124i \(0.793123\pi\)
\(332\) 0 0
\(333\) 1.85325e9 2.75029
\(334\) 0 0
\(335\) −2.70342e7 −0.0392877
\(336\) 0 0
\(337\) −2.04579e8 −0.291177 −0.145589 0.989345i \(-0.546508\pi\)
−0.145589 + 0.989345i \(0.546508\pi\)
\(338\) 0 0
\(339\) −6.20855e8 −0.865548
\(340\) 0 0
\(341\) −4.50878e8 −0.615770
\(342\) 0 0
\(343\) −6.38462e8 −0.854291
\(344\) 0 0
\(345\) −5.10576e8 −0.669412
\(346\) 0 0
\(347\) 6.28852e8 0.807970 0.403985 0.914766i \(-0.367625\pi\)
0.403985 + 0.914766i \(0.367625\pi\)
\(348\) 0 0
\(349\) −9.86717e8 −1.24252 −0.621260 0.783604i \(-0.713379\pi\)
−0.621260 + 0.783604i \(0.713379\pi\)
\(350\) 0 0
\(351\) 2.63240e8 0.324921
\(352\) 0 0
\(353\) 5.59732e8 0.677281 0.338641 0.940916i \(-0.390033\pi\)
0.338641 + 0.940916i \(0.390033\pi\)
\(354\) 0 0
\(355\) −4.30087e8 −0.510219
\(356\) 0 0
\(357\) −4.09248e8 −0.476045
\(358\) 0 0
\(359\) −1.42390e8 −0.162424 −0.0812119 0.996697i \(-0.525879\pi\)
−0.0812119 + 0.996697i \(0.525879\pi\)
\(360\) 0 0
\(361\) 1.86707e8 0.208875
\(362\) 0 0
\(363\) −1.20193e9 −1.31889
\(364\) 0 0
\(365\) −1.25501e9 −1.35090
\(366\) 0 0
\(367\) 7.13452e8 0.753414 0.376707 0.926333i \(-0.377056\pi\)
0.376707 + 0.926333i \(0.377056\pi\)
\(368\) 0 0
\(369\) −2.24044e9 −2.32136
\(370\) 0 0
\(371\) 6.74812e8 0.686079
\(372\) 0 0
\(373\) −4.14729e8 −0.413794 −0.206897 0.978363i \(-0.566336\pi\)
−0.206897 + 0.978363i \(0.566336\pi\)
\(374\) 0 0
\(375\) −1.08617e9 −1.06362
\(376\) 0 0
\(377\) 3.55280e8 0.341489
\(378\) 0 0
\(379\) 1.23625e8 0.116646 0.0583229 0.998298i \(-0.481425\pi\)
0.0583229 + 0.998298i \(0.481425\pi\)
\(380\) 0 0
\(381\) −8.64994e8 −0.801264
\(382\) 0 0
\(383\) −1.35784e9 −1.23496 −0.617480 0.786586i \(-0.711846\pi\)
−0.617480 + 0.786586i \(0.711846\pi\)
\(384\) 0 0
\(385\) 2.92184e8 0.260942
\(386\) 0 0
\(387\) −3.29906e8 −0.289336
\(388\) 0 0
\(389\) −1.00573e9 −0.866281 −0.433141 0.901326i \(-0.642595\pi\)
−0.433141 + 0.901326i \(0.642595\pi\)
\(390\) 0 0
\(391\) 2.37513e8 0.200941
\(392\) 0 0
\(393\) −6.78332e8 −0.563726
\(394\) 0 0
\(395\) −8.65893e8 −0.706927
\(396\) 0 0
\(397\) −2.30080e8 −0.184549 −0.0922747 0.995734i \(-0.529414\pi\)
−0.0922747 + 0.995734i \(0.529414\pi\)
\(398\) 0 0
\(399\) 1.11044e9 0.875167
\(400\) 0 0
\(401\) −1.24791e9 −0.966446 −0.483223 0.875497i \(-0.660534\pi\)
−0.483223 + 0.875497i \(0.660534\pi\)
\(402\) 0 0
\(403\) −5.06912e8 −0.385802
\(404\) 0 0
\(405\) 3.36702e8 0.251857
\(406\) 0 0
\(407\) 9.75982e8 0.717565
\(408\) 0 0
\(409\) 1.79923e9 1.30033 0.650166 0.759792i \(-0.274699\pi\)
0.650166 + 0.759792i \(0.274699\pi\)
\(410\) 0 0
\(411\) 2.56803e9 1.82454
\(412\) 0 0
\(413\) −8.01081e8 −0.559566
\(414\) 0 0
\(415\) 5.19219e8 0.356601
\(416\) 0 0
\(417\) 3.60248e9 2.43291
\(418\) 0 0
\(419\) −2.66870e8 −0.177235 −0.0886177 0.996066i \(-0.528245\pi\)
−0.0886177 + 0.996066i \(0.528245\pi\)
\(420\) 0 0
\(421\) −2.70575e9 −1.76726 −0.883630 0.468185i \(-0.844908\pi\)
−0.883630 + 0.468185i \(0.844908\pi\)
\(422\) 0 0
\(423\) 3.06337e9 1.96792
\(424\) 0 0
\(425\) −4.41204e8 −0.278791
\(426\) 0 0
\(427\) −2.12730e8 −0.132230
\(428\) 0 0
\(429\) 3.34582e8 0.204598
\(430\) 0 0
\(431\) 1.78455e9 1.07364 0.536820 0.843697i \(-0.319626\pi\)
0.536820 + 0.843697i \(0.319626\pi\)
\(432\) 0 0
\(433\) 1.21276e9 0.717905 0.358953 0.933356i \(-0.383134\pi\)
0.358953 + 0.933356i \(0.383134\pi\)
\(434\) 0 0
\(435\) 4.18483e9 2.43762
\(436\) 0 0
\(437\) −6.44461e8 −0.369413
\(438\) 0 0
\(439\) 1.76141e9 0.993654 0.496827 0.867850i \(-0.334498\pi\)
0.496827 + 0.867850i \(0.334498\pi\)
\(440\) 0 0
\(441\) −2.35566e9 −1.30791
\(442\) 0 0
\(443\) 8.06208e8 0.440590 0.220295 0.975433i \(-0.429298\pi\)
0.220295 + 0.975433i \(0.429298\pi\)
\(444\) 0 0
\(445\) −6.75166e8 −0.363204
\(446\) 0 0
\(447\) 2.95263e9 1.56363
\(448\) 0 0
\(449\) 5.85913e8 0.305472 0.152736 0.988267i \(-0.451192\pi\)
0.152736 + 0.988267i \(0.451192\pi\)
\(450\) 0 0
\(451\) −1.17989e9 −0.605653
\(452\) 0 0
\(453\) 5.52228e9 2.79110
\(454\) 0 0
\(455\) 3.28496e8 0.163490
\(456\) 0 0
\(457\) −5.52640e8 −0.270854 −0.135427 0.990787i \(-0.543241\pi\)
−0.135427 + 0.990787i \(0.543241\pi\)
\(458\) 0 0
\(459\) −1.44240e9 −0.696213
\(460\) 0 0
\(461\) 1.74101e9 0.827652 0.413826 0.910356i \(-0.364192\pi\)
0.413826 + 0.910356i \(0.364192\pi\)
\(462\) 0 0
\(463\) −2.84431e9 −1.33181 −0.665906 0.746035i \(-0.731955\pi\)
−0.665906 + 0.746035i \(0.731955\pi\)
\(464\) 0 0
\(465\) −5.97090e9 −2.75394
\(466\) 0 0
\(467\) −1.63130e9 −0.741183 −0.370592 0.928796i \(-0.620845\pi\)
−0.370592 + 0.928796i \(0.620845\pi\)
\(468\) 0 0
\(469\) −3.50662e7 −0.0156958
\(470\) 0 0
\(471\) 3.10706e9 1.37018
\(472\) 0 0
\(473\) −1.73739e8 −0.0754891
\(474\) 0 0
\(475\) 1.19715e9 0.512532
\(476\) 0 0
\(477\) 5.74018e9 2.42165
\(478\) 0 0
\(479\) 3.69345e9 1.53553 0.767765 0.640732i \(-0.221369\pi\)
0.767765 + 0.640732i \(0.221369\pi\)
\(480\) 0 0
\(481\) 1.09727e9 0.449580
\(482\) 0 0
\(483\) −6.62272e8 −0.267437
\(484\) 0 0
\(485\) −9.78697e6 −0.00389540
\(486\) 0 0
\(487\) −1.57153e9 −0.616554 −0.308277 0.951297i \(-0.599752\pi\)
−0.308277 + 0.951297i \(0.599752\pi\)
\(488\) 0 0
\(489\) 7.57578e9 2.92986
\(490\) 0 0
\(491\) 1.63493e9 0.623323 0.311662 0.950193i \(-0.399114\pi\)
0.311662 + 0.950193i \(0.399114\pi\)
\(492\) 0 0
\(493\) −1.94673e9 −0.731713
\(494\) 0 0
\(495\) 2.48542e9 0.921047
\(496\) 0 0
\(497\) −5.57868e8 −0.203838
\(498\) 0 0
\(499\) −6.86870e8 −0.247470 −0.123735 0.992315i \(-0.539487\pi\)
−0.123735 + 0.992315i \(0.539487\pi\)
\(500\) 0 0
\(501\) 3.26890e9 1.16137
\(502\) 0 0
\(503\) −2.33472e9 −0.817990 −0.408995 0.912537i \(-0.634121\pi\)
−0.408995 + 0.912537i \(0.634121\pi\)
\(504\) 0 0
\(505\) −5.68845e9 −1.96550
\(506\) 0 0
\(507\) −4.45231e9 −1.51725
\(508\) 0 0
\(509\) −6.21342e8 −0.208842 −0.104421 0.994533i \(-0.533299\pi\)
−0.104421 + 0.994533i \(0.533299\pi\)
\(510\) 0 0
\(511\) −1.62789e9 −0.539699
\(512\) 0 0
\(513\) 3.91378e9 1.27993
\(514\) 0 0
\(515\) 4.32347e9 1.39478
\(516\) 0 0
\(517\) 1.61327e9 0.513441
\(518\) 0 0
\(519\) −2.04114e9 −0.640896
\(520\) 0 0
\(521\) 1.65562e9 0.512897 0.256448 0.966558i \(-0.417448\pi\)
0.256448 + 0.966558i \(0.417448\pi\)
\(522\) 0 0
\(523\) −4.42671e9 −1.35309 −0.676543 0.736403i \(-0.736523\pi\)
−0.676543 + 0.736403i \(0.736523\pi\)
\(524\) 0 0
\(525\) 1.23024e9 0.371049
\(526\) 0 0
\(527\) 2.77758e9 0.826665
\(528\) 0 0
\(529\) −3.02047e9 −0.887113
\(530\) 0 0
\(531\) −6.81427e9 −1.97510
\(532\) 0 0
\(533\) −1.32652e9 −0.379463
\(534\) 0 0
\(535\) 4.58961e9 1.29580
\(536\) 0 0
\(537\) −1.86332e9 −0.519252
\(538\) 0 0
\(539\) −1.24057e9 −0.341240
\(540\) 0 0
\(541\) 3.71878e9 1.00974 0.504871 0.863195i \(-0.331540\pi\)
0.504871 + 0.863195i \(0.331540\pi\)
\(542\) 0 0
\(543\) −2.14660e9 −0.575376
\(544\) 0 0
\(545\) 1.60492e9 0.424683
\(546\) 0 0
\(547\) −3.20749e9 −0.837934 −0.418967 0.908002i \(-0.637608\pi\)
−0.418967 + 0.908002i \(0.637608\pi\)
\(548\) 0 0
\(549\) −1.80955e9 −0.466732
\(550\) 0 0
\(551\) 5.28219e9 1.34519
\(552\) 0 0
\(553\) −1.12316e9 −0.282424
\(554\) 0 0
\(555\) 1.29247e10 3.20920
\(556\) 0 0
\(557\) −4.80739e9 −1.17873 −0.589367 0.807865i \(-0.700623\pi\)
−0.589367 + 0.807865i \(0.700623\pi\)
\(558\) 0 0
\(559\) −1.95331e8 −0.0472967
\(560\) 0 0
\(561\) −1.83332e9 −0.438397
\(562\) 0 0
\(563\) 3.77127e9 0.890653 0.445326 0.895368i \(-0.353088\pi\)
0.445326 + 0.895368i \(0.353088\pi\)
\(564\) 0 0
\(565\) −2.73066e9 −0.636940
\(566\) 0 0
\(567\) 4.36739e8 0.100619
\(568\) 0 0
\(569\) −2.09341e9 −0.476388 −0.238194 0.971218i \(-0.576555\pi\)
−0.238194 + 0.971218i \(0.576555\pi\)
\(570\) 0 0
\(571\) −6.86085e9 −1.54224 −0.771119 0.636691i \(-0.780303\pi\)
−0.771119 + 0.636691i \(0.780303\pi\)
\(572\) 0 0
\(573\) 1.28965e9 0.286371
\(574\) 0 0
\(575\) −7.13986e8 −0.156622
\(576\) 0 0
\(577\) 5.70742e9 1.23687 0.618435 0.785836i \(-0.287767\pi\)
0.618435 + 0.785836i \(0.287767\pi\)
\(578\) 0 0
\(579\) 6.73316e9 1.44160
\(580\) 0 0
\(581\) 6.73483e8 0.142466
\(582\) 0 0
\(583\) 3.02297e9 0.631820
\(584\) 0 0
\(585\) 2.79430e9 0.577069
\(586\) 0 0
\(587\) 1.56510e9 0.319380 0.159690 0.987167i \(-0.448951\pi\)
0.159690 + 0.987167i \(0.448951\pi\)
\(588\) 0 0
\(589\) −7.53661e9 −1.51975
\(590\) 0 0
\(591\) −1.97175e9 −0.392913
\(592\) 0 0
\(593\) −2.78410e9 −0.548268 −0.274134 0.961692i \(-0.588391\pi\)
−0.274134 + 0.961692i \(0.588391\pi\)
\(594\) 0 0
\(595\) −1.79997e9 −0.350312
\(596\) 0 0
\(597\) −4.09283e9 −0.787251
\(598\) 0 0
\(599\) −6.18885e9 −1.17657 −0.588283 0.808655i \(-0.700196\pi\)
−0.588283 + 0.808655i \(0.700196\pi\)
\(600\) 0 0
\(601\) 2.42206e9 0.455118 0.227559 0.973764i \(-0.426926\pi\)
0.227559 + 0.973764i \(0.426926\pi\)
\(602\) 0 0
\(603\) −2.98285e8 −0.0554014
\(604\) 0 0
\(605\) −5.28639e9 −0.970544
\(606\) 0 0
\(607\) −4.62465e9 −0.839302 −0.419651 0.907686i \(-0.637848\pi\)
−0.419651 + 0.907686i \(0.637848\pi\)
\(608\) 0 0
\(609\) 5.42817e9 0.973852
\(610\) 0 0
\(611\) 1.81376e9 0.321689
\(612\) 0 0
\(613\) −5.45433e9 −0.956378 −0.478189 0.878257i \(-0.658707\pi\)
−0.478189 + 0.878257i \(0.658707\pi\)
\(614\) 0 0
\(615\) −1.56251e10 −2.70869
\(616\) 0 0
\(617\) 2.32837e9 0.399074 0.199537 0.979890i \(-0.436056\pi\)
0.199537 + 0.979890i \(0.436056\pi\)
\(618\) 0 0
\(619\) 9.58626e9 1.62455 0.812273 0.583278i \(-0.198230\pi\)
0.812273 + 0.583278i \(0.198230\pi\)
\(620\) 0 0
\(621\) −2.33419e9 −0.391125
\(622\) 0 0
\(623\) −8.75763e8 −0.145104
\(624\) 0 0
\(625\) −7.62240e9 −1.24885
\(626\) 0 0
\(627\) 4.97446e9 0.805953
\(628\) 0 0
\(629\) −6.01242e9 −0.963324
\(630\) 0 0
\(631\) −1.18616e10 −1.87949 −0.939747 0.341870i \(-0.888940\pi\)
−0.939747 + 0.341870i \(0.888940\pi\)
\(632\) 0 0
\(633\) −1.54796e9 −0.242576
\(634\) 0 0
\(635\) −3.80445e9 −0.589635
\(636\) 0 0
\(637\) −1.39474e9 −0.213799
\(638\) 0 0
\(639\) −4.74542e9 −0.719484
\(640\) 0 0
\(641\) −1.06130e8 −0.0159161 −0.00795805 0.999968i \(-0.502533\pi\)
−0.00795805 + 0.999968i \(0.502533\pi\)
\(642\) 0 0
\(643\) −2.19289e9 −0.325296 −0.162648 0.986684i \(-0.552004\pi\)
−0.162648 + 0.986684i \(0.552004\pi\)
\(644\) 0 0
\(645\) −2.30080e9 −0.337614
\(646\) 0 0
\(647\) 4.23914e9 0.615337 0.307668 0.951494i \(-0.400451\pi\)
0.307668 + 0.951494i \(0.400451\pi\)
\(648\) 0 0
\(649\) −3.58862e9 −0.515312
\(650\) 0 0
\(651\) −7.74489e9 −1.10023
\(652\) 0 0
\(653\) 9.93257e9 1.39594 0.697968 0.716129i \(-0.254088\pi\)
0.697968 + 0.716129i \(0.254088\pi\)
\(654\) 0 0
\(655\) −2.98346e9 −0.414836
\(656\) 0 0
\(657\) −1.38474e10 −1.90497
\(658\) 0 0
\(659\) −1.36634e10 −1.85977 −0.929884 0.367852i \(-0.880093\pi\)
−0.929884 + 0.367852i \(0.880093\pi\)
\(660\) 0 0
\(661\) 1.03765e10 1.39747 0.698737 0.715378i \(-0.253746\pi\)
0.698737 + 0.715378i \(0.253746\pi\)
\(662\) 0 0
\(663\) −2.06115e9 −0.274671
\(664\) 0 0
\(665\) 4.88398e9 0.644019
\(666\) 0 0
\(667\) −3.15032e9 −0.411069
\(668\) 0 0
\(669\) −1.28515e10 −1.65944
\(670\) 0 0
\(671\) −9.52968e8 −0.121773
\(672\) 0 0
\(673\) 4.70776e9 0.595336 0.297668 0.954669i \(-0.403791\pi\)
0.297668 + 0.954669i \(0.403791\pi\)
\(674\) 0 0
\(675\) 4.33599e9 0.542657
\(676\) 0 0
\(677\) 9.55050e9 1.18295 0.591474 0.806324i \(-0.298546\pi\)
0.591474 + 0.806324i \(0.298546\pi\)
\(678\) 0 0
\(679\) −1.26947e7 −0.00155625
\(680\) 0 0
\(681\) −1.05658e10 −1.28200
\(682\) 0 0
\(683\) 1.06442e10 1.27832 0.639161 0.769073i \(-0.279282\pi\)
0.639161 + 0.769073i \(0.279282\pi\)
\(684\) 0 0
\(685\) 1.12948e10 1.34265
\(686\) 0 0
\(687\) −2.06175e10 −2.42598
\(688\) 0 0
\(689\) 3.39865e9 0.395858
\(690\) 0 0
\(691\) 8.41537e9 0.970287 0.485143 0.874435i \(-0.338767\pi\)
0.485143 + 0.874435i \(0.338767\pi\)
\(692\) 0 0
\(693\) 3.22386e9 0.367967
\(694\) 0 0
\(695\) 1.58445e10 1.79033
\(696\) 0 0
\(697\) 7.26858e9 0.813083
\(698\) 0 0
\(699\) −1.52385e10 −1.68761
\(700\) 0 0
\(701\) 8.95355e9 0.981707 0.490854 0.871242i \(-0.336685\pi\)
0.490854 + 0.871242i \(0.336685\pi\)
\(702\) 0 0
\(703\) 1.63139e10 1.77099
\(704\) 0 0
\(705\) 2.13643e10 2.29629
\(706\) 0 0
\(707\) −7.37853e9 −0.785239
\(708\) 0 0
\(709\) −8.11796e9 −0.855431 −0.427716 0.903913i \(-0.640682\pi\)
−0.427716 + 0.903913i \(0.640682\pi\)
\(710\) 0 0
\(711\) −9.55395e9 −0.996872
\(712\) 0 0
\(713\) 4.49486e9 0.464412
\(714\) 0 0
\(715\) 1.47157e9 0.150560
\(716\) 0 0
\(717\) 6.32739e9 0.641073
\(718\) 0 0
\(719\) 9.54339e8 0.0957528 0.0478764 0.998853i \(-0.484755\pi\)
0.0478764 + 0.998853i \(0.484755\pi\)
\(720\) 0 0
\(721\) 5.60800e9 0.557231
\(722\) 0 0
\(723\) −2.08264e10 −2.04942
\(724\) 0 0
\(725\) 5.85203e9 0.570327
\(726\) 0 0
\(727\) −1.75084e10 −1.68996 −0.844979 0.534799i \(-0.820387\pi\)
−0.844979 + 0.534799i \(0.820387\pi\)
\(728\) 0 0
\(729\) −1.63215e10 −1.56032
\(730\) 0 0
\(731\) 1.07030e9 0.101343
\(732\) 0 0
\(733\) 1.16062e10 1.08849 0.544247 0.838925i \(-0.316815\pi\)
0.544247 + 0.838925i \(0.316815\pi\)
\(734\) 0 0
\(735\) −1.64286e10 −1.52614
\(736\) 0 0
\(737\) −1.57087e8 −0.0144545
\(738\) 0 0
\(739\) 4.74800e9 0.432768 0.216384 0.976308i \(-0.430574\pi\)
0.216384 + 0.976308i \(0.430574\pi\)
\(740\) 0 0
\(741\) 5.59268e9 0.504959
\(742\) 0 0
\(743\) 4.35857e9 0.389837 0.194918 0.980819i \(-0.437556\pi\)
0.194918 + 0.980819i \(0.437556\pi\)
\(744\) 0 0
\(745\) 1.29864e10 1.15064
\(746\) 0 0
\(747\) 5.72888e9 0.502860
\(748\) 0 0
\(749\) 5.95321e9 0.517684
\(750\) 0 0
\(751\) −6.10987e9 −0.526371 −0.263186 0.964745i \(-0.584773\pi\)
−0.263186 + 0.964745i \(0.584773\pi\)
\(752\) 0 0
\(753\) −2.23729e10 −1.90959
\(754\) 0 0
\(755\) 2.42883e10 2.05391
\(756\) 0 0
\(757\) 2.29472e10 1.92262 0.961312 0.275460i \(-0.0888303\pi\)
0.961312 + 0.275460i \(0.0888303\pi\)
\(758\) 0 0
\(759\) −2.96679e9 −0.246286
\(760\) 0 0
\(761\) −7.15151e9 −0.588236 −0.294118 0.955769i \(-0.595026\pi\)
−0.294118 + 0.955769i \(0.595026\pi\)
\(762\) 0 0
\(763\) 2.08175e9 0.169665
\(764\) 0 0
\(765\) −1.53111e10 −1.23650
\(766\) 0 0
\(767\) −4.03460e9 −0.322862
\(768\) 0 0
\(769\) −1.85381e10 −1.47002 −0.735011 0.678055i \(-0.762823\pi\)
−0.735011 + 0.678055i \(0.762823\pi\)
\(770\) 0 0
\(771\) 3.35537e9 0.263663
\(772\) 0 0
\(773\) 8.25535e9 0.642846 0.321423 0.946936i \(-0.395839\pi\)
0.321423 + 0.946936i \(0.395839\pi\)
\(774\) 0 0
\(775\) −8.34966e9 −0.644336
\(776\) 0 0
\(777\) 1.67648e10 1.28211
\(778\) 0 0
\(779\) −1.97223e10 −1.49478
\(780\) 0 0
\(781\) −2.49909e9 −0.187717
\(782\) 0 0
\(783\) 1.91317e10 1.42425
\(784\) 0 0
\(785\) 1.36656e10 1.00829
\(786\) 0 0
\(787\) 1.53185e9 0.112023 0.0560113 0.998430i \(-0.482162\pi\)
0.0560113 + 0.998430i \(0.482162\pi\)
\(788\) 0 0
\(789\) −3.29097e10 −2.38536
\(790\) 0 0
\(791\) −3.54196e9 −0.254464
\(792\) 0 0
\(793\) −1.07140e9 −0.0762950
\(794\) 0 0
\(795\) 4.00326e10 2.82572
\(796\) 0 0
\(797\) −2.78553e9 −0.194896 −0.0974480 0.995241i \(-0.531068\pi\)
−0.0974480 + 0.995241i \(0.531068\pi\)
\(798\) 0 0
\(799\) −9.93837e9 −0.689289
\(800\) 0 0
\(801\) −7.44954e9 −0.512172
\(802\) 0 0
\(803\) −7.29247e9 −0.497016
\(804\) 0 0
\(805\) −2.91283e9 −0.196802
\(806\) 0 0
\(807\) 6.35708e9 0.425794
\(808\) 0 0
\(809\) −1.17657e10 −0.781267 −0.390634 0.920546i \(-0.627744\pi\)
−0.390634 + 0.920546i \(0.627744\pi\)
\(810\) 0 0
\(811\) 6.29491e9 0.414397 0.207198 0.978299i \(-0.433565\pi\)
0.207198 + 0.978299i \(0.433565\pi\)
\(812\) 0 0
\(813\) 4.73386e9 0.308957
\(814\) 0 0
\(815\) 3.33201e10 2.15603
\(816\) 0 0
\(817\) −2.90413e9 −0.186311
\(818\) 0 0
\(819\) 3.62451e9 0.230545
\(820\) 0 0
\(821\) −4.27400e9 −0.269546 −0.134773 0.990876i \(-0.543031\pi\)
−0.134773 + 0.990876i \(0.543031\pi\)
\(822\) 0 0
\(823\) 3.16411e9 0.197858 0.0989288 0.995095i \(-0.468458\pi\)
0.0989288 + 0.995095i \(0.468458\pi\)
\(824\) 0 0
\(825\) 5.51111e9 0.341704
\(826\) 0 0
\(827\) −3.47251e9 −0.213488 −0.106744 0.994287i \(-0.534043\pi\)
−0.106744 + 0.994287i \(0.534043\pi\)
\(828\) 0 0
\(829\) −5.21388e9 −0.317848 −0.158924 0.987291i \(-0.550803\pi\)
−0.158924 + 0.987291i \(0.550803\pi\)
\(830\) 0 0
\(831\) 3.37991e9 0.204316
\(832\) 0 0
\(833\) 7.64237e9 0.458111
\(834\) 0 0
\(835\) 1.43774e10 0.854629
\(836\) 0 0
\(837\) −2.72970e10 −1.60908
\(838\) 0 0
\(839\) 2.48652e10 1.45353 0.726766 0.686885i \(-0.241023\pi\)
0.726766 + 0.686885i \(0.241023\pi\)
\(840\) 0 0
\(841\) 8.57107e9 0.496877
\(842\) 0 0
\(843\) −4.47024e10 −2.57001
\(844\) 0 0
\(845\) −1.95823e10 −1.11652
\(846\) 0 0
\(847\) −6.85701e9 −0.387742
\(848\) 0 0
\(849\) 4.64614e10 2.60565
\(850\) 0 0
\(851\) −9.72969e9 −0.541185
\(852\) 0 0
\(853\) 3.01930e10 1.66565 0.832826 0.553536i \(-0.186722\pi\)
0.832826 + 0.553536i \(0.186722\pi\)
\(854\) 0 0
\(855\) 4.15448e10 2.27319
\(856\) 0 0
\(857\) −2.57761e10 −1.39889 −0.699446 0.714685i \(-0.746570\pi\)
−0.699446 + 0.714685i \(0.746570\pi\)
\(858\) 0 0
\(859\) 2.77848e10 1.49565 0.747827 0.663894i \(-0.231097\pi\)
0.747827 + 0.663894i \(0.231097\pi\)
\(860\) 0 0
\(861\) −2.02674e10 −1.08215
\(862\) 0 0
\(863\) 3.02990e10 1.60469 0.802344 0.596862i \(-0.203586\pi\)
0.802344 + 0.596862i \(0.203586\pi\)
\(864\) 0 0
\(865\) −8.97743e9 −0.471624
\(866\) 0 0
\(867\) −2.02815e10 −1.05690
\(868\) 0 0
\(869\) −5.03142e9 −0.260089
\(870\) 0 0
\(871\) −1.76609e8 −0.00905627
\(872\) 0 0
\(873\) −1.07986e8 −0.00549309
\(874\) 0 0
\(875\) −6.19656e9 −0.312696
\(876\) 0 0
\(877\) −2.37410e9 −0.118850 −0.0594252 0.998233i \(-0.518927\pi\)
−0.0594252 + 0.998233i \(0.518927\pi\)
\(878\) 0 0
\(879\) −8.50323e9 −0.422302
\(880\) 0 0
\(881\) −3.28058e10 −1.61635 −0.808175 0.588942i \(-0.799544\pi\)
−0.808175 + 0.588942i \(0.799544\pi\)
\(882\) 0 0
\(883\) 2.14967e10 1.05078 0.525388 0.850863i \(-0.323920\pi\)
0.525388 + 0.850863i \(0.323920\pi\)
\(884\) 0 0
\(885\) −4.75234e10 −2.30466
\(886\) 0 0
\(887\) 1.50080e10 0.722089 0.361045 0.932549i \(-0.382420\pi\)
0.361045 + 0.932549i \(0.382420\pi\)
\(888\) 0 0
\(889\) −4.93477e9 −0.235565
\(890\) 0 0
\(891\) 1.95647e9 0.0926617
\(892\) 0 0
\(893\) 2.69665e10 1.26720
\(894\) 0 0
\(895\) −8.19532e9 −0.382107
\(896\) 0 0
\(897\) −3.33549e9 −0.154307
\(898\) 0 0
\(899\) −3.68412e10 −1.69112
\(900\) 0 0
\(901\) −1.86226e10 −0.848212
\(902\) 0 0
\(903\) −2.98438e9 −0.134880
\(904\) 0 0
\(905\) −9.44124e9 −0.423408
\(906\) 0 0
\(907\) −1.57906e10 −0.702704 −0.351352 0.936243i \(-0.614278\pi\)
−0.351352 + 0.936243i \(0.614278\pi\)
\(908\) 0 0
\(909\) −6.27643e10 −2.77165
\(910\) 0 0
\(911\) −6.82128e9 −0.298917 −0.149459 0.988768i \(-0.547753\pi\)
−0.149459 + 0.988768i \(0.547753\pi\)
\(912\) 0 0
\(913\) 3.01701e9 0.131199
\(914\) 0 0
\(915\) −1.26200e10 −0.544609
\(916\) 0 0
\(917\) −3.86987e9 −0.165731
\(918\) 0 0
\(919\) 1.08588e10 0.461506 0.230753 0.973012i \(-0.425881\pi\)
0.230753 + 0.973012i \(0.425881\pi\)
\(920\) 0 0
\(921\) 5.41253e10 2.28293
\(922\) 0 0
\(923\) −2.80967e9 −0.117611
\(924\) 0 0
\(925\) 1.80739e10 0.750853
\(926\) 0 0
\(927\) 4.77036e10 1.96685
\(928\) 0 0
\(929\) −2.99248e9 −0.122455 −0.0612275 0.998124i \(-0.519502\pi\)
−0.0612275 + 0.998124i \(0.519502\pi\)
\(930\) 0 0
\(931\) −2.07366e10 −0.842197
\(932\) 0 0
\(933\) 6.62721e10 2.67144
\(934\) 0 0
\(935\) −8.06335e9 −0.322608
\(936\) 0 0
\(937\) −2.39333e10 −0.950417 −0.475208 0.879873i \(-0.657627\pi\)
−0.475208 + 0.879873i \(0.657627\pi\)
\(938\) 0 0
\(939\) −1.86261e10 −0.734164
\(940\) 0 0
\(941\) 3.17291e10 1.24135 0.620675 0.784068i \(-0.286859\pi\)
0.620675 + 0.784068i \(0.286859\pi\)
\(942\) 0 0
\(943\) 1.17625e10 0.456781
\(944\) 0 0
\(945\) 1.76894e10 0.681871
\(946\) 0 0
\(947\) 4.13161e10 1.58086 0.790431 0.612551i \(-0.209857\pi\)
0.790431 + 0.612551i \(0.209857\pi\)
\(948\) 0 0
\(949\) −8.19876e9 −0.311399
\(950\) 0 0
\(951\) −4.14268e10 −1.56189
\(952\) 0 0
\(953\) 4.31993e10 1.61678 0.808390 0.588647i \(-0.200339\pi\)
0.808390 + 0.588647i \(0.200339\pi\)
\(954\) 0 0
\(955\) 5.67216e9 0.210735
\(956\) 0 0
\(957\) 2.43167e10 0.896835
\(958\) 0 0
\(959\) 1.46506e10 0.536401
\(960\) 0 0
\(961\) 2.50523e10 0.910574
\(962\) 0 0
\(963\) 5.06400e10 1.82727
\(964\) 0 0
\(965\) 2.96140e10 1.06084
\(966\) 0 0
\(967\) −6.99318e9 −0.248703 −0.124352 0.992238i \(-0.539685\pi\)
−0.124352 + 0.992238i \(0.539685\pi\)
\(968\) 0 0
\(969\) −3.06446e10 −1.08198
\(970\) 0 0
\(971\) 8.48357e9 0.297380 0.148690 0.988884i \(-0.452494\pi\)
0.148690 + 0.988884i \(0.452494\pi\)
\(972\) 0 0
\(973\) 2.05521e10 0.715255
\(974\) 0 0
\(975\) 6.19601e9 0.214090
\(976\) 0 0
\(977\) −1.72197e10 −0.590739 −0.295370 0.955383i \(-0.595443\pi\)
−0.295370 + 0.955383i \(0.595443\pi\)
\(978\) 0 0
\(979\) −3.92317e9 −0.133628
\(980\) 0 0
\(981\) 1.77081e10 0.598866
\(982\) 0 0
\(983\) 3.76267e10 1.26345 0.631725 0.775192i \(-0.282347\pi\)
0.631725 + 0.775192i \(0.282347\pi\)
\(984\) 0 0
\(985\) −8.67222e9 −0.289137
\(986\) 0 0
\(987\) 2.77117e10 0.917389
\(988\) 0 0
\(989\) 1.73203e9 0.0569336
\(990\) 0 0
\(991\) −3.13036e9 −0.102173 −0.0510866 0.998694i \(-0.516268\pi\)
−0.0510866 + 0.998694i \(0.516268\pi\)
\(992\) 0 0
\(993\) −8.08387e10 −2.61997
\(994\) 0 0
\(995\) −1.80012e10 −0.579323
\(996\) 0 0
\(997\) −1.67605e9 −0.0535617 −0.0267808 0.999641i \(-0.508526\pi\)
−0.0267808 + 0.999641i \(0.508526\pi\)
\(998\) 0 0
\(999\) 5.90878e10 1.87508
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 256.8.a.r.1.6 6
4.3 odd 2 256.8.a.q.1.1 6
8.3 odd 2 256.8.a.q.1.6 6
8.5 even 2 inner 256.8.a.r.1.1 6
16.3 odd 4 32.8.b.a.17.1 6
16.5 even 4 8.8.b.a.5.6 yes 6
16.11 odd 4 32.8.b.a.17.6 6
16.13 even 4 8.8.b.a.5.5 6
48.5 odd 4 72.8.d.b.37.1 6
48.11 even 4 288.8.d.b.145.1 6
48.29 odd 4 72.8.d.b.37.2 6
48.35 even 4 288.8.d.b.145.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.b.a.5.5 6 16.13 even 4
8.8.b.a.5.6 yes 6 16.5 even 4
32.8.b.a.17.1 6 16.3 odd 4
32.8.b.a.17.6 6 16.11 odd 4
72.8.d.b.37.1 6 48.5 odd 4
72.8.d.b.37.2 6 48.29 odd 4
256.8.a.q.1.1 6 4.3 odd 2
256.8.a.q.1.6 6 8.3 odd 2
256.8.a.r.1.1 6 8.5 even 2 inner
256.8.a.r.1.6 6 1.1 even 1 trivial
288.8.d.b.145.1 6 48.11 even 4
288.8.d.b.145.6 6 48.35 even 4