Properties

Label 256.8.a.r.1.1
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 1.1
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 \(-0.807545\pi\)
−0.822721 + 0.568446i \(0.807545\pi\)
\(4\) 0 0
\(5\) −338.443 −1.21085 −0.605425 0.795903i \(-0.706997\pi\)
−0.605425 + 0.795903i \(0.706997\pi\)
\(6\) 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 \(-0.571502\pi\)
−0.222744 + 0.974877i \(0.571502\pi\)
\(12\) 0 0
\(13\) −2210.98 −0.279115 −0.139557 0.990214i \(-0.544568\pi\)
−0.139557 + 0.990214i \(0.544568\pi\)
\(14\) 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 \(-0.685275\pi\)
−0.549744 + 0.835333i \(0.685275\pi\)
\(20\) 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 \(-0.709528\pi\)
−0.611735 + 0.791063i \(0.709528\pi\)
\(30\) 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 \(-0.798032\pi\)
−0.805368 + 0.592775i \(0.798032\pi\)
\(38\) 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 \(-0.472998\pi\)
0.0847262 + 0.996404i \(0.472998\pi\)
\(44\) 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 \(-0.750913\pi\)
−0.709131 + 0.705077i \(0.750913\pi\)
\(54\) 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 \(-0.303690\pi\)
0.578367 + 0.815776i \(0.303690\pi\)
\(60\) 0 0
\(61\) 484582. 0.273346 0.136673 0.990616i \(-0.456359\pi\)
0.136673 + 0.990616i \(0.456359\pi\)
\(62\) 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 \(-0.494836\pi\)
0.0162232 + 0.999868i \(0.494836\pi\)
\(68\) 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 \(-0.547043\pi\)
−0.147252 + 0.989099i \(0.547043\pi\)
\(84\) 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 \(-0.198585\pi\)
0.811622 + 0.584182i \(0.198585\pi\)
\(102\) 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 \(-0.679718\pi\)
−0.535078 + 0.844803i \(0.679718\pi\)
\(108\) 0 0
\(109\) −4.74206e6 −0.350731 −0.175366 0.984503i \(-0.556111\pi\)
−0.175366 + 0.984503i \(0.556111\pi\)
\(110\) 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 \(-0.445203\pi\)
0.171299 + 0.985219i \(0.445203\pi\)
\(132\) 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 \(-0.764837\pi\)
−0.739287 + 0.673391i \(0.764837\pi\)
\(140\) 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 \(-0.657602\pi\)
−0.475139 + 0.879911i \(0.657602\pi\)
\(150\) 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 \(-0.636693\pi\)
−0.416355 + 0.909202i \(0.636693\pi\)
\(158\) 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 \(-0.849502\pi\)
−0.890296 + 0.455383i \(0.849502\pi\)
\(164\) 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 \(-0.437611\pi\)
0.194749 + 0.980853i \(0.437611\pi\)
\(174\) 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 \(-0.449565\pi\)
0.157785 + 0.987474i \(0.449565\pi\)
\(180\) 0 0
\(181\) 2.78961e7 0.349679 0.174839 0.984597i \(-0.444059\pi\)
0.174839 + 0.984597i \(0.444059\pi\)
\(182\) 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 \(-0.461905\pi\)
0.119394 + 0.992847i \(0.461905\pi\)
\(198\) 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 \(-0.476516\pi\)
0.0737114 + 0.997280i \(0.476516\pi\)
\(212\) 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 \(-0.372627\pi\)
0.389561 + 0.921001i \(0.372627\pi\)
\(228\) 0 0
\(229\) 2.67935e8 1.47436 0.737182 0.675694i \(-0.236156\pi\)
0.737182 + 0.675694i \(0.236156\pi\)
\(230\) 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 \(-0.302949\pi\)
0.580266 + 0.814427i \(0.302949\pi\)
\(252\) 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 \(-0.541301\pi\)
−0.129386 + 0.991594i \(0.541301\pi\)
\(270\) 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 \(-0.519775\pi\)
−0.0620854 + 0.998071i \(0.519775\pi\)
\(278\) 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 \(-0.790844\pi\)
−0.791778 + 0.610810i \(0.790844\pi\)
\(284\) 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 \(-0.459040\pi\)
0.128325 + 0.991732i \(0.459040\pi\)
\(294\) 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 \(-0.744026\pi\)
−0.693712 + 0.720252i \(0.744026\pi\)
\(308\) 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 \(-0.342589\pi\)
0.474610 + 0.880196i \(0.342589\pi\)
\(318\) 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 \(-0.206877\pi\)
0.796131 + 0.605124i \(0.206877\pi\)
\(332\) 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 \(-0.632375\pi\)
−0.403985 + 0.914766i \(0.632375\pi\)
\(348\) 0 0
\(349\) 9.86717e8 1.24252 0.621260 0.783604i \(-0.286621\pi\)
0.621260 + 0.783604i \(0.286621\pi\)
\(350\) 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 \(-0.433664\pi\)
0.206897 + 0.978363i \(0.433664\pi\)
\(374\) 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 \(-0.518575\pi\)
−0.0583229 + 0.998298i \(0.518575\pi\)
\(380\) 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 \(-0.357405\pi\)
0.433141 + 0.901326i \(0.357405\pi\)
\(390\) 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 \(-0.470586\pi\)
0.0922747 + 0.995734i \(0.470586\pi\)
\(398\) 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 \(-0.471755\pi\)
0.0886177 + 0.996066i \(0.471755\pi\)
\(420\) 0 0
\(421\) 2.70575e9 1.76726 0.883630 0.468185i \(-0.155092\pi\)
0.883630 + 0.468185i \(0.155092\pi\)
\(422\) 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 \(-0.570702\pi\)
−0.220295 + 0.975433i \(0.570702\pi\)
\(444\) 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 \(-0.635808\pi\)
−0.413826 + 0.910356i \(0.635808\pi\)
\(462\) 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 \(-0.379155\pi\)
0.370592 + 0.928796i \(0.379155\pi\)
\(468\) 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 \(-0.600886\pi\)
−0.311662 + 0.950193i \(0.600886\pi\)
\(492\) 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 \(-0.460513\pi\)
0.123735 + 0.992315i \(0.460513\pi\)
\(500\) 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 \(-0.466701\pi\)
0.104421 + 0.994533i \(0.466701\pi\)
\(510\) 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 \(-0.263477\pi\)
0.676543 + 0.736403i \(0.263477\pi\)
\(524\) 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 \(-0.668460\pi\)
−0.504871 + 0.863195i \(0.668460\pi\)
\(542\) 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 \(-0.362392\pi\)
0.418967 + 0.908002i \(0.362392\pi\)
\(548\) 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 \(-0.299377\pi\)
0.589367 + 0.807865i \(0.299377\pi\)
\(558\) 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 \(-0.646912\pi\)
−0.445326 + 0.895368i \(0.646912\pi\)
\(564\) 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 \(-0.219697\pi\)
0.771119 + 0.636691i \(0.219697\pi\)
\(572\) 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 \(-0.551049\pi\)
−0.159690 + 0.987167i \(0.551049\pi\)
\(588\) 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 \(-0.341293\pi\)
0.478189 + 0.878257i \(0.341293\pi\)
\(614\) 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 \(-0.801770\pi\)
−0.812273 + 0.583278i \(0.801770\pi\)
\(620\) 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 \(-0.447996\pi\)
0.162648 + 0.986684i \(0.447996\pi\)
\(644\) 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 \(-0.745912\pi\)
−0.697968 + 0.716129i \(0.745912\pi\)
\(654\) 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 \(-0.119907\pi\)
0.929884 + 0.367852i \(0.119907\pi\)
\(660\) 0 0
\(661\) −1.03765e10 −1.39747 −0.698737 0.715378i \(-0.746254\pi\)
−0.698737 + 0.715378i \(0.746254\pi\)
\(662\) 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 \(-0.701454\pi\)
−0.591474 + 0.806324i \(0.701454\pi\)
\(678\) 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 \(-0.720718\pi\)
−0.639161 + 0.769073i \(0.720718\pi\)
\(684\) 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 \(-0.661233\pi\)
−0.485143 + 0.874435i \(0.661233\pi\)
\(692\) 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 \(-0.663315\pi\)
−0.490854 + 0.871242i \(0.663315\pi\)
\(702\) 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 \(-0.359318\pi\)
0.427716 + 0.903913i \(0.359318\pi\)
\(710\) 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 \(-0.683185\pi\)
−0.544247 + 0.838925i \(0.683185\pi\)
\(734\) 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 \(-0.569426\pi\)
−0.216384 + 0.976308i \(0.569426\pi\)
\(740\) 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 \(-0.911170\pi\)
−0.961312 + 0.275460i \(0.911170\pi\)
\(758\) 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 \(-0.604161\pi\)
−0.321423 + 0.946936i \(0.604161\pi\)
\(774\) 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 \(-0.517838\pi\)
−0.0560113 + 0.998430i \(0.517838\pi\)
\(788\) 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 \(-0.468932\pi\)
0.0974480 + 0.995241i \(0.468932\pi\)
\(798\) 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 \(-0.566435\pi\)
−0.207198 + 0.978299i \(0.566435\pi\)
\(812\) 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 \(-0.456969\pi\)
0.134773 + 0.990876i \(0.456969\pi\)
\(822\) 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 \(-0.465957\pi\)
0.106744 + 0.994287i \(0.465957\pi\)
\(828\) 0 0
\(829\) 5.21388e9 0.317848 0.158924 0.987291i \(-0.449197\pi\)
0.158924 + 0.987291i \(0.449197\pi\)
\(830\) 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 \(-0.813278\pi\)
−0.832826 + 0.553536i \(0.813278\pi\)
\(854\) 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 \(-0.768903\pi\)
−0.747827 + 0.663894i \(0.768903\pi\)
\(860\) 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 \(-0.481073\pi\)
0.0594252 + 0.998233i \(0.481073\pi\)
\(878\) 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 \(-0.676080\pi\)
−0.525388 + 0.850863i \(0.676080\pi\)
\(884\) 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 \(-0.385722\pi\)
0.351352 + 0.936243i \(0.385722\pi\)
\(908\) 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 \(-0.713141\pi\)
−0.620675 + 0.784068i \(0.713141\pi\)
\(942\) 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 \(-0.790143\pi\)
−0.790431 + 0.612551i \(0.790143\pi\)
\(948\) 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 \(-0.547506\pi\)
−0.148690 + 0.988884i \(0.547506\pi\)
\(972\) 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 \(-0.491474\pi\)
0.0267808 + 0.999641i \(0.491474\pi\)
\(998\) 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.1.1 6
4.3 odd 2 256.8.a.q.1.6 6
8.3 odd 2 256.8.a.q.1.1 6
8.5 even 2 inner 256.8.a.r.1.6 6
16.3 odd 4 32.8.b.a.17.6 6
16.5 even 4 8.8.b.a.5.5 6
16.11 odd 4 32.8.b.a.17.1 6
16.13 even 4 8.8.b.a.5.6 yes 6
48.5 odd 4 72.8.d.b.37.2 6
48.11 even 4 288.8.d.b.145.6 6
48.29 odd 4 72.8.d.b.37.1 6
48.35 even 4 288.8.d.b.145.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.b.a.5.5 6 16.5 even 4
8.8.b.a.5.6 yes 6 16.13 even 4
32.8.b.a.17.1 6 16.11 odd 4
32.8.b.a.17.6 6 16.3 odd 4
72.8.d.b.37.1 6 48.29 odd 4
72.8.d.b.37.2 6 48.5 odd 4
256.8.a.q.1.1 6 8.3 odd 2
256.8.a.q.1.6 6 4.3 odd 2
256.8.a.r.1.1 6 1.1 even 1 trivial
256.8.a.r.1.6 6 8.5 even 2 inner
288.8.d.b.145.1 6 48.35 even 4
288.8.d.b.145.6 6 48.11 even 4