Properties

Label 32.8.b.a.17.6
Level $32$
Weight $8$
Character 32.17
Analytic conductor $9.996$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 32.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.99632081549\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{30} \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 17.6
Root \(5.57668 - 0.949035i\) of defining polynomial
Character \(\chi\) \(=\) 32.17
Dual form 32.8.b.a.17.1

$q$-expansion

\(f(q)\) \(=\) \(q+76.9497i q^{3} +338.443i q^{5} +438.996 q^{7} -3734.25 q^{9} +O(q^{10})\) \(q+76.9497i q^{3} +338.443i q^{5} +438.996 q^{7} -3734.25 q^{9} -1966.58i q^{11} -2210.98i q^{13} -26043.0 q^{15} -12114.9 q^{17} +32872.2i q^{19} +33780.6i q^{21} -19605.1 q^{23} -36418.4 q^{25} -119060. i q^{27} -160689. i q^{29} +229270. q^{31} +151328. q^{33} +148575. i q^{35} +496284. i q^{37} +170134. q^{39} +599971. q^{41} +88346.0i q^{43} -1.26383e6i q^{45} -820344. q^{47} -630825. q^{49} -932236. i q^{51} +1.53717e6i q^{53} +665574. q^{55} -2.52950e6 q^{57} +1.82480e6i q^{59} +484582. i q^{61} -1.63932e6 q^{63} +748290. q^{65} -79878.2i q^{67} -1.50860e6i q^{69} -1.27078e6 q^{71} +3.70820e6 q^{73} -2.80238e6i q^{75} -863321. i q^{77} +2.55846e6 q^{79} +994857. q^{81} +1.53414e6i q^{83} -4.10019e6i q^{85} +1.23650e7 q^{87} +1.99492e6 q^{89} -970612. i q^{91} +1.76423e7i q^{93} -1.11253e7 q^{95} -28917.7 q^{97} +7.34370e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 688 q^{7} - 2918 q^{9}+O(q^{10}) \) 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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) 76.9497i 1.64544i 0.568446 + 0.822721i \(0.307545\pi\)
−0.568446 + 0.822721i \(0.692455\pi\)
\(4\) 0 0
\(5\) 338.443i 1.21085i 0.795903 + 0.605425i \(0.206997\pi\)
−0.795903 + 0.605425i \(0.793003\pi\)
\(6\) 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
\(11\) − 1966.58i − 0.445489i −0.974877 0.222744i \(-0.928498\pi\)
0.974877 0.222744i \(-0.0715015\pi\)
\(12\) 0 0
\(13\) − 2210.98i − 0.279115i −0.990214 0.139557i \(-0.955432\pi\)
0.990214 0.139557i \(-0.0445680\pi\)
\(14\) 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
\(19\) 32872.2i 1.09949i 0.835333 + 0.549744i \(0.185275\pi\)
−0.835333 + 0.549744i \(0.814725\pi\)
\(20\) 0 0
\(21\) 33780.6i 0.795976i
\(22\) 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.i − 1.16411i
\(28\) 0 0
\(29\) − 160689.i − 1.22347i −0.791063 0.611735i \(-0.790472\pi\)
0.791063 0.611735i \(-0.209528\pi\)
\(30\) 0 0
\(31\) 229270. 1.38224 0.691118 0.722742i \(-0.257119\pi\)
0.691118 + 0.722742i \(0.257119\pi\)
\(32\) 0 0
\(33\) 151328. 0.733026
\(34\) 0 0
\(35\) 148575.i 0.585744i
\(36\) 0 0
\(37\) 496284.i 1.61074i 0.592775 + 0.805368i \(0.298032\pi\)
−0.592775 + 0.805368i \(0.701968\pi\)
\(38\) 0 0
\(39\) 170134. 0.459267
\(40\) 0 0
\(41\) 599971. 1.35952 0.679762 0.733433i \(-0.262083\pi\)
0.679762 + 0.733433i \(0.262083\pi\)
\(42\) 0 0
\(43\) 88346.0i 0.169452i 0.996404 + 0.0847262i \(0.0270015\pi\)
−0.996404 + 0.0847262i \(0.972998\pi\)
\(44\) 0 0
\(45\) − 1.26383e6i − 2.06750i
\(46\) 0 0
\(47\) −820344. −1.15253 −0.576267 0.817262i \(-0.695491\pi\)
−0.576267 + 0.817262i \(0.695491\pi\)
\(48\) 0 0
\(49\) −630825. −0.765989
\(50\) 0 0
\(51\) − 932236.i − 0.984080i
\(52\) 0 0
\(53\) 1.53717e6i 1.41826i 0.705077 + 0.709131i \(0.250913\pi\)
−0.705077 + 0.709131i \(0.749087\pi\)
\(54\) 0 0
\(55\) 665574. 0.539420
\(56\) 0 0
\(57\) −2.52950e6 −1.80914
\(58\) 0 0
\(59\) 1.82480e6i 1.15673i 0.815776 + 0.578367i \(0.196310\pi\)
−0.815776 + 0.578367i \(0.803690\pi\)
\(60\) 0 0
\(61\) 484582.i 0.273346i 0.990616 + 0.136673i \(0.0436410\pi\)
−0.990616 + 0.136673i \(0.956359\pi\)
\(62\) 0 0
\(63\) −1.63932e6 −0.825986
\(64\) 0 0
\(65\) 748290. 0.337966
\(66\) 0 0
\(67\) − 79878.2i − 0.0324464i −0.999868 0.0162232i \(-0.994836\pi\)
0.999868 0.0162232i \(-0.00516423\pi\)
\(68\) 0 0
\(69\) − 1.50860e6i − 0.552845i
\(70\) 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 \(-0.311633\pi\)
0.557832 + 0.829954i \(0.311633\pi\)
\(74\) 0 0
\(75\) − 2.80238e6i − 0.767031i
\(76\) 0 0
\(77\) − 863321.i − 0.215504i
\(78\) 0 0
\(79\) 2.55846e6 0.583827 0.291914 0.956445i \(-0.405708\pi\)
0.291914 + 0.956445i \(0.405708\pi\)
\(80\) 0 0
\(81\) 994857. 0.208000
\(82\) 0 0
\(83\) 1.53414e6i 0.294505i 0.989099 + 0.147252i \(0.0470430\pi\)
−0.989099 + 0.147252i \(0.952957\pi\)
\(84\) 0 0
\(85\) − 4.10019e6i − 0.724166i
\(86\) 0 0
\(87\) 1.23650e7 2.01315
\(88\) 0 0
\(89\) 1.99492e6 0.299958 0.149979 0.988689i \(-0.452079\pi\)
0.149979 + 0.988689i \(0.452079\pi\)
\(90\) 0 0
\(91\) − 970612.i − 0.135021i
\(92\) 0 0
\(93\) 1.76423e7i 2.27439i
\(94\) 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
\(99\) 7.34370e6i 0.760662i
\(100\) 0 0
\(101\) − 1.68077e7i − 1.62324i −0.584182 0.811622i \(-0.698585\pi\)
0.584182 0.811622i \(-0.301415\pi\)
\(102\) 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
\(107\) − 1.35610e7i − 1.07016i −0.844803 0.535078i \(-0.820282\pi\)
0.844803 0.535078i \(-0.179718\pi\)
\(108\) 0 0
\(109\) − 4.74206e6i − 0.350731i −0.984503 0.175366i \(-0.943889\pi\)
0.984503 0.175366i \(-0.0561108\pi\)
\(110\) 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
\(115\) − 6.63520e6i − 0.406828i
\(116\) 0 0
\(117\) 8.25635e6i 0.476582i
\(118\) 0 0
\(119\) −5.31839e6 −0.289311
\(120\) 0 0
\(121\) 1.56197e7 0.801540
\(122\) 0 0
\(123\) 4.61676e7i 2.23702i
\(124\) 0 0
\(125\) 1.41153e7i 0.646405i
\(126\) 0 0
\(127\) 1.12410e7 0.486960 0.243480 0.969906i \(-0.421711\pi\)
0.243480 + 0.969906i \(0.421711\pi\)
\(128\) 0 0
\(129\) −6.79820e6 −0.278824
\(130\) 0 0
\(131\) − 8.81527e6i − 0.342599i −0.985219 0.171299i \(-0.945203\pi\)
0.985219 0.171299i \(-0.0547966\pi\)
\(132\) 0 0
\(133\) 1.44308e7i 0.531874i
\(134\) 0 0
\(135\) 4.02951e7 1.40956
\(136\) 0 0
\(137\) −3.33729e7 −1.10885 −0.554424 0.832234i \(-0.687061\pi\)
−0.554424 + 0.832234i \(0.687061\pi\)
\(138\) 0 0
\(139\) − 4.68161e7i − 1.47857i −0.673391 0.739287i \(-0.735163\pi\)
0.673391 0.739287i \(-0.264837\pi\)
\(140\) 0 0
\(141\) − 6.31252e7i − 1.89643i
\(142\) 0 0
\(143\) −4.34806e6 −0.124343
\(144\) 0 0
\(145\) 5.43840e7 1.48144
\(146\) 0 0
\(147\) − 4.85418e7i − 1.26039i
\(148\) 0 0
\(149\) 3.83709e7i 0.950277i 0.879911 + 0.475139i \(0.157602\pi\)
−0.879911 + 0.475139i \(0.842398\pi\)
\(150\) 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
\(155\) 7.75948e7i 1.67368i
\(156\) 0 0
\(157\) − 4.03778e7i − 0.832710i −0.909202 0.416355i \(-0.863307\pi\)
0.909202 0.416355i \(-0.136693\pi\)
\(158\) 0 0
\(159\) −1.18285e8 −2.33367
\(160\) 0 0
\(161\) −8.60656e6 −0.162532
\(162\) 0 0
\(163\) 9.84512e7i 1.78059i 0.455383 + 0.890296i \(0.349502\pi\)
−0.455383 + 0.890296i \(0.650498\pi\)
\(164\) 0 0
\(165\) 5.12157e7i 0.887584i
\(166\) 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
\(171\) − 1.22753e8i − 1.87735i
\(172\) 0 0
\(173\) 2.65257e7i 0.389498i 0.980853 + 0.194749i \(0.0623893\pi\)
−0.980853 + 0.194749i \(0.937611\pi\)
\(174\) 0 0
\(175\) −1.59875e7 −0.225501
\(176\) 0 0
\(177\) −1.40418e8 −1.90334
\(178\) 0 0
\(179\) − 2.42148e7i − 0.315570i −0.987474 0.157785i \(-0.949565\pi\)
0.987474 0.157785i \(-0.0504353\pi\)
\(180\) 0 0
\(181\) − 2.78961e7i − 0.349679i −0.984597 0.174839i \(-0.944059\pi\)
0.984597 0.174839i \(-0.0559406\pi\)
\(182\) 0 0
\(183\) −3.72884e7 −0.449775
\(184\) 0 0
\(185\) −1.67964e8 −1.95036
\(186\) 0 0
\(187\) 2.38249e7i 0.266431i
\(188\) 0 0
\(189\) − 5.22671e7i − 0.563135i
\(190\) 0 0
\(191\) −1.67596e7 −0.174039 −0.0870195 0.996207i \(-0.527734\pi\)
−0.0870195 + 0.996207i \(0.527734\pi\)
\(192\) 0 0
\(193\) 8.75008e7 0.876116 0.438058 0.898947i \(-0.355667\pi\)
0.438058 + 0.898947i \(0.355667\pi\)
\(194\) 0 0
\(195\) 5.75806e7i 0.556103i
\(196\) 0 0
\(197\) − 2.56239e7i − 0.238789i −0.992847 0.119394i \(-0.961905\pi\)
0.992847 0.119394i \(-0.0380953\pi\)
\(198\) 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
\(203\) − 7.05419e7i − 0.591849i
\(204\) 0 0
\(205\) 2.03056e8i 1.64618i
\(206\) 0 0
\(207\) 7.32103e7 0.573688
\(208\) 0 0
\(209\) 6.46457e7 0.489810
\(210\) 0 0
\(211\) − 2.01165e7i − 0.147423i −0.997280 0.0737114i \(-0.976516\pi\)
0.997280 0.0737114i \(-0.0234844\pi\)
\(212\) 0 0
\(213\) − 9.77862e7i − 0.693344i
\(214\) 0 0
\(215\) −2.99001e7 −0.205181
\(216\) 0 0
\(217\) 1.00649e8 0.668651
\(218\) 0 0
\(219\) 2.85345e8i 1.83576i
\(220\) 0 0
\(221\) 2.67858e7i 0.166929i
\(222\) 0 0
\(223\) 1.67012e8 1.00851 0.504254 0.863555i \(-0.331768\pi\)
0.504254 + 0.863555i \(0.331768\pi\)
\(224\) 0 0
\(225\) 1.35995e8 0.795949
\(226\) 0 0
\(227\) − 1.37308e8i − 0.779122i −0.921001 0.389561i \(-0.872627\pi\)
0.921001 0.389561i \(-0.127373\pi\)
\(228\) 0 0
\(229\) − 2.67935e8i − 1.47436i −0.675694 0.737182i \(-0.736156\pi\)
0.675694 0.737182i \(-0.263844\pi\)
\(230\) 0 0
\(231\) 6.64322e7 0.354599
\(232\) 0 0
\(233\) 1.98032e8 1.02563 0.512815 0.858499i \(-0.328603\pi\)
0.512815 + 0.858499i \(0.328603\pi\)
\(234\) 0 0
\(235\) − 2.77639e8i − 1.39554i
\(236\) 0 0
\(237\) 1.96873e8i 0.960654i
\(238\) 0 0
\(239\) −8.22277e7 −0.389606 −0.194803 0.980842i \(-0.562407\pi\)
−0.194803 + 0.980842i \(0.562407\pi\)
\(240\) 0 0
\(241\) −2.70650e8 −1.24551 −0.622757 0.782415i \(-0.713988\pi\)
−0.622757 + 0.782415i \(0.713988\pi\)
\(242\) 0 0
\(243\) − 1.83831e8i − 0.821860i
\(244\) 0 0
\(245\) − 2.13498e8i − 0.927498i
\(246\) 0 0
\(247\) 7.26797e7 0.306884
\(248\) 0 0
\(249\) −1.18052e8 −0.484591
\(250\) 0 0
\(251\) 2.90747e8i 1.16053i 0.814427 + 0.580266i \(0.197051\pi\)
−0.814427 + 0.580266i \(0.802949\pi\)
\(252\) 0 0
\(253\) 3.85549e7i 0.149678i
\(254\) 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
\(259\) 2.17867e8i 0.779188i
\(260\) 0 0
\(261\) 6.00053e8i 2.08905i
\(262\) 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
\(267\) 1.53509e8i 0.493564i
\(268\) 0 0
\(269\) − 8.26134e7i − 0.258772i −0.991594 0.129386i \(-0.958699\pi\)
0.991594 0.129386i \(-0.0413007\pi\)
\(270\) 0 0
\(271\) −6.15189e7 −0.187766 −0.0938829 0.995583i \(-0.529928\pi\)
−0.0938829 + 0.995583i \(0.529928\pi\)
\(272\) 0 0
\(273\) 7.46883e7 0.222169
\(274\) 0 0
\(275\) 7.16196e7i 0.207667i
\(276\) 0 0
\(277\) 4.39237e7i 0.124171i 0.998071 + 0.0620854i \(0.0197751\pi\)
−0.998071 + 0.0620854i \(0.980225\pi\)
\(278\) 0 0
\(279\) −8.56153e8 −2.36013
\(280\) 0 0
\(281\) 5.80931e8 1.56190 0.780948 0.624596i \(-0.214736\pi\)
0.780948 + 0.624596i \(0.214736\pi\)
\(282\) 0 0
\(283\) − 6.03790e8i − 1.58356i −0.610810 0.791778i \(-0.709156\pi\)
0.610810 0.791778i \(-0.290844\pi\)
\(284\) 0 0
\(285\) − 8.56091e8i − 2.19060i
\(286\) 0 0
\(287\) 2.63385e8 0.657665
\(288\) 0 0
\(289\) −2.63568e8 −0.642319
\(290\) 0 0
\(291\) − 2.22520e6i − 0.00529352i
\(292\) 0 0
\(293\) − 1.10504e8i − 0.256649i −0.991732 0.128325i \(-0.959040\pi\)
0.991732 0.128325i \(-0.0409600\pi\)
\(294\) 0 0
\(295\) −6.17591e8 −1.40063
\(296\) 0 0
\(297\) −2.34142e8 −0.518599
\(298\) 0 0
\(299\) 4.33464e7i 0.0937787i
\(300\) 0 0
\(301\) 3.87836e7i 0.0819719i
\(302\) 0 0
\(303\) 1.29335e9 2.67095
\(304\) 0 0
\(305\) −1.64003e8 −0.330981
\(306\) 0 0
\(307\) 7.03386e8i 1.38742i 0.720252 + 0.693712i \(0.244026\pi\)
−0.720252 + 0.693712i \(0.755974\pi\)
\(308\) 0 0
\(309\) 9.83002e8i 1.89539i
\(310\) 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 \(-0.428385\pi\)
0.223090 + 0.974798i \(0.428385\pi\)
\(314\) 0 0
\(315\) − 5.54817e8i − 1.00014i
\(316\) 0 0
\(317\) 5.38362e8i 0.949221i 0.880196 + 0.474610i \(0.157411\pi\)
−0.880196 + 0.474610i \(0.842589\pi\)
\(318\) 0 0
\(319\) −3.16007e8 −0.545042
\(320\) 0 0
\(321\) 1.04351e9 1.76088
\(322\) 0 0
\(323\) − 3.98242e8i − 0.657565i
\(324\) 0 0
\(325\) 8.05203e7i 0.130111i
\(326\) 0 0
\(327\) 3.64900e8 0.577108
\(328\) 0 0
\(329\) −3.60128e8 −0.557534
\(330\) 0 0
\(331\) 1.05054e9i 1.59226i 0.605124 + 0.796131i \(0.293123\pi\)
−0.605124 + 0.796131i \(0.706877\pi\)
\(332\) 0 0
\(333\) − 1.85325e9i − 2.75029i
\(334\) 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
\(339\) − 6.20855e8i − 0.865548i
\(340\) 0 0
\(341\) − 4.50878e8i − 0.615770i
\(342\) 0 0
\(343\) −6.38462e8 −0.854291
\(344\) 0 0
\(345\) 5.10576e8 0.669412
\(346\) 0 0
\(347\) − 6.28852e8i − 0.807970i −0.914766 0.403985i \(-0.867625\pi\)
0.914766 0.403985i \(-0.132375\pi\)
\(348\) 0 0
\(349\) 9.86717e8i 1.24252i 0.783604 + 0.621260i \(0.213379\pi\)
−0.783604 + 0.621260i \(0.786621\pi\)
\(350\) 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
\(355\) − 4.30087e8i − 0.510219i
\(356\) 0 0
\(357\) − 4.09248e8i − 0.476045i
\(358\) 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
\(363\) 1.20193e9i 1.31889i
\(364\) 0 0
\(365\) 1.25501e9i 1.35090i
\(366\) 0 0
\(367\) −7.13452e8 −0.753414 −0.376707 0.926333i \(-0.622944\pi\)
−0.376707 + 0.926333i \(0.622944\pi\)
\(368\) 0 0
\(369\) −2.24044e9 −2.32136
\(370\) 0 0
\(371\) 6.74812e8i 0.686079i
\(372\) 0 0
\(373\) − 4.14729e8i − 0.413794i −0.978363 0.206897i \(-0.933664\pi\)
0.978363 0.206897i \(-0.0663364\pi\)
\(374\) 0 0
\(375\) −1.08617e9 −1.06362
\(376\) 0 0
\(377\) −3.55280e8 −0.341489
\(378\) 0 0
\(379\) − 1.23625e8i − 0.116646i −0.998298 0.0583229i \(-0.981425\pi\)
0.998298 0.0583229i \(-0.0185753\pi\)
\(380\) 0 0
\(381\) 8.64994e8i 0.801264i
\(382\) 0 0
\(383\) 1.35784e9 1.23496 0.617480 0.786586i \(-0.288154\pi\)
0.617480 + 0.786586i \(0.288154\pi\)
\(384\) 0 0
\(385\) 2.92184e8 0.260942
\(386\) 0 0
\(387\) − 3.29906e8i − 0.289336i
\(388\) 0 0
\(389\) − 1.00573e9i − 0.866281i −0.901326 0.433141i \(-0.857405\pi\)
0.901326 0.433141i \(-0.142595\pi\)
\(390\) 0 0
\(391\) 2.37513e8 0.200941
\(392\) 0 0
\(393\) 6.78332e8 0.563726
\(394\) 0 0
\(395\) 8.65893e8i 0.706927i
\(396\) 0 0
\(397\) 2.30080e8i 0.184549i 0.995734 + 0.0922747i \(0.0294138\pi\)
−0.995734 + 0.0922747i \(0.970586\pi\)
\(398\) 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
\(403\) − 5.06912e8i − 0.385802i
\(404\) 0 0
\(405\) 3.36702e8i 0.251857i
\(406\) 0 0
\(407\) 9.75982e8 0.717565
\(408\) 0 0
\(409\) −1.79923e9 −1.30033 −0.650166 0.759792i \(-0.725301\pi\)
−0.650166 + 0.759792i \(0.725301\pi\)
\(410\) 0 0
\(411\) − 2.56803e9i − 1.82454i
\(412\) 0 0
\(413\) 8.01081e8i 0.559566i
\(414\) 0 0
\(415\) −5.19219e8 −0.356601
\(416\) 0 0
\(417\) 3.60248e9 2.43291
\(418\) 0 0
\(419\) − 2.66870e8i − 0.177235i −0.996066 0.0886177i \(-0.971755\pi\)
0.996066 0.0886177i \(-0.0282449\pi\)
\(420\) 0 0
\(421\) − 2.70575e9i − 1.76726i −0.468185 0.883630i \(-0.655092\pi\)
0.468185 0.883630i \(-0.344908\pi\)
\(422\) 0 0
\(423\) 3.06337e9 1.96792
\(424\) 0 0
\(425\) 4.41204e8 0.278791
\(426\) 0 0
\(427\) 2.12730e8i 0.132230i
\(428\) 0 0
\(429\) − 3.34582e8i − 0.204598i
\(430\) 0 0
\(431\) −1.78455e9 −1.07364 −0.536820 0.843697i \(-0.680374\pi\)
−0.536820 + 0.843697i \(0.680374\pi\)
\(432\) 0 0
\(433\) 1.21276e9 0.717905 0.358953 0.933356i \(-0.383134\pi\)
0.358953 + 0.933356i \(0.383134\pi\)
\(434\) 0 0
\(435\) 4.18483e9i 2.43762i
\(436\) 0 0
\(437\) − 6.44461e8i − 0.369413i
\(438\) 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
\(443\) − 8.06208e8i − 0.440590i −0.975433 0.220295i \(-0.929298\pi\)
0.975433 0.220295i \(-0.0707019\pi\)
\(444\) 0 0
\(445\) 6.75166e8i 0.363204i
\(446\) 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
\(451\) − 1.17989e9i − 0.605653i
\(452\) 0 0
\(453\) 5.52228e9i 2.79110i
\(454\) 0 0
\(455\) 3.28496e8 0.163490
\(456\) 0 0
\(457\) 5.52640e8 0.270854 0.135427 0.990787i \(-0.456759\pi\)
0.135427 + 0.990787i \(0.456759\pi\)
\(458\) 0 0
\(459\) 1.44240e9i 0.696213i
\(460\) 0 0
\(461\) − 1.74101e9i − 0.827652i −0.910356 0.413826i \(-0.864192\pi\)
0.910356 0.413826i \(-0.135808\pi\)
\(462\) 0 0
\(463\) 2.84431e9 1.33181 0.665906 0.746035i \(-0.268045\pi\)
0.665906 + 0.746035i \(0.268045\pi\)
\(464\) 0 0
\(465\) −5.97090e9 −2.75394
\(466\) 0 0
\(467\) − 1.63130e9i − 0.741183i −0.928796 0.370592i \(-0.879155\pi\)
0.928796 0.370592i \(-0.120845\pi\)
\(468\) 0 0
\(469\) − 3.50662e7i − 0.0156958i
\(470\) 0 0
\(471\) 3.10706e9 1.37018
\(472\) 0 0
\(473\) 1.73739e8 0.0754891
\(474\) 0 0
\(475\) − 1.19715e9i − 0.512532i
\(476\) 0 0
\(477\) − 5.74018e9i − 2.42165i
\(478\) 0 0
\(479\) −3.69345e9 −1.53553 −0.767765 0.640732i \(-0.778631\pi\)
−0.767765 + 0.640732i \(0.778631\pi\)
\(480\) 0 0
\(481\) 1.09727e9 0.449580
\(482\) 0 0
\(483\) − 6.62272e8i − 0.267437i
\(484\) 0 0
\(485\) − 9.78697e6i − 0.00389540i
\(486\) 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
\(491\) − 1.63493e9i − 0.623323i −0.950193 0.311662i \(-0.899114\pi\)
0.950193 0.311662i \(-0.100886\pi\)
\(492\) 0 0
\(493\) 1.94673e9i 0.731713i
\(494\) 0 0
\(495\) −2.48542e9 −0.921047
\(496\) 0 0
\(497\) −5.57868e8 −0.203838
\(498\) 0 0
\(499\) − 6.86870e8i − 0.247470i −0.992315 0.123735i \(-0.960513\pi\)
0.992315 0.123735i \(-0.0394873\pi\)
\(500\) 0 0
\(501\) 3.26890e9i 1.16137i
\(502\) 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
\(507\) 4.45231e9i 1.51725i
\(508\) 0 0
\(509\) 6.21342e8i 0.208842i 0.994533 + 0.104421i \(0.0332990\pi\)
−0.994533 + 0.104421i \(0.966701\pi\)
\(510\) 0 0
\(511\) 1.62789e9 0.539699
\(512\) 0 0
\(513\) 3.91378e9 1.27993
\(514\) 0 0
\(515\) 4.32347e9i 1.39478i
\(516\) 0 0
\(517\) 1.61327e9i 0.513441i
\(518\) 0 0
\(519\) −2.04114e9 −0.640896
\(520\) 0 0
\(521\) −1.65562e9 −0.512897 −0.256448 0.966558i \(-0.582552\pi\)
−0.256448 + 0.966558i \(0.582552\pi\)
\(522\) 0 0
\(523\) 4.42671e9i 1.35309i 0.736403 + 0.676543i \(0.236523\pi\)
−0.736403 + 0.676543i \(0.763477\pi\)
\(524\) 0 0
\(525\) − 1.23024e9i − 0.371049i
\(526\) 0 0
\(527\) −2.77758e9 −0.826665
\(528\) 0 0
\(529\) −3.02047e9 −0.887113
\(530\) 0 0
\(531\) − 6.81427e9i − 1.97510i
\(532\) 0 0
\(533\) − 1.32652e9i − 0.379463i
\(534\) 0 0
\(535\) 4.58961e9 1.29580
\(536\) 0 0
\(537\) 1.86332e9 0.519252
\(538\) 0 0
\(539\) 1.24057e9i 0.341240i
\(540\) 0 0
\(541\) − 3.71878e9i − 1.00974i −0.863195 0.504871i \(-0.831540\pi\)
0.863195 0.504871i \(-0.168460\pi\)
\(542\) 0 0
\(543\) 2.14660e9 0.575376
\(544\) 0 0
\(545\) 1.60492e9 0.424683
\(546\) 0 0
\(547\) − 3.20749e9i − 0.837934i −0.908002 0.418967i \(-0.862392\pi\)
0.908002 0.418967i \(-0.137608\pi\)
\(548\) 0 0
\(549\) − 1.80955e9i − 0.466732i
\(550\) 0 0
\(551\) 5.28219e9 1.34519
\(552\) 0 0
\(553\) 1.12316e9 0.282424
\(554\) 0 0
\(555\) − 1.29247e10i − 3.20920i
\(556\) 0 0
\(557\) 4.80739e9i 1.17873i 0.807865 + 0.589367i \(0.200623\pi\)
−0.807865 + 0.589367i \(0.799377\pi\)
\(558\) 0 0
\(559\) 1.95331e8 0.0472967
\(560\) 0 0
\(561\) −1.83332e9 −0.438397
\(562\) 0 0
\(563\) 3.77127e9i 0.890653i 0.895368 + 0.445326i \(0.146912\pi\)
−0.895368 + 0.445326i \(0.853088\pi\)
\(564\) 0 0
\(565\) − 2.73066e9i − 0.636940i
\(566\) 0 0
\(567\) 4.36739e8 0.100619
\(568\) 0 0
\(569\) 2.09341e9 0.476388 0.238194 0.971218i \(-0.423445\pi\)
0.238194 + 0.971218i \(0.423445\pi\)
\(570\) 0 0
\(571\) 6.86085e9i 1.54224i 0.636691 + 0.771119i \(0.280303\pi\)
−0.636691 + 0.771119i \(0.719697\pi\)
\(572\) 0 0
\(573\) − 1.28965e9i − 0.286371i
\(574\) 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
\(579\) 6.73316e9i 1.44160i
\(580\) 0 0
\(581\) 6.73483e8i 0.142466i
\(582\) 0 0
\(583\) 3.02297e9 0.631820
\(584\) 0 0
\(585\) −2.79430e9 −0.577069
\(586\) 0 0
\(587\) − 1.56510e9i − 0.319380i −0.987167 0.159690i \(-0.948951\pi\)
0.987167 0.159690i \(-0.0510495\pi\)
\(588\) 0 0
\(589\) 7.53661e9i 1.51975i
\(590\) 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
\(595\) − 1.79997e9i − 0.350312i
\(596\) 0 0
\(597\) − 4.09283e9i − 0.787251i
\(598\) 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 \(-0.573074\pi\)
−0.227559 + 0.973764i \(0.573074\pi\)
\(602\) 0 0
\(603\) 2.98285e8i 0.0554014i
\(604\) 0 0
\(605\) 5.28639e9i 0.970544i
\(606\) 0 0
\(607\) 4.62465e9 0.839302 0.419651 0.907686i \(-0.362152\pi\)
0.419651 + 0.907686i \(0.362152\pi\)
\(608\) 0 0
\(609\) 5.42817e9 0.973852
\(610\) 0 0
\(611\) 1.81376e9i 0.321689i
\(612\) 0 0
\(613\) − 5.45433e9i − 0.956378i −0.878257 0.478189i \(-0.841293\pi\)
0.878257 0.478189i \(-0.158707\pi\)
\(614\) 0 0
\(615\) −1.56251e10 −2.70869
\(616\) 0 0
\(617\) −2.32837e9 −0.399074 −0.199537 0.979890i \(-0.563944\pi\)
−0.199537 + 0.979890i \(0.563944\pi\)
\(618\) 0 0
\(619\) − 9.58626e9i − 1.62455i −0.583278 0.812273i \(-0.698230\pi\)
0.583278 0.812273i \(-0.301770\pi\)
\(620\) 0 0
\(621\) 2.33419e9i 0.391125i
\(622\) 0 0
\(623\) 8.75763e8 0.145104
\(624\) 0 0
\(625\) −7.62240e9 −1.24885
\(626\) 0 0
\(627\) 4.97446e9i 0.805953i
\(628\) 0 0
\(629\) − 6.01242e9i − 0.963324i
\(630\) 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
\(635\) 3.80445e9i 0.589635i
\(636\) 0 0
\(637\) 1.39474e9i 0.213799i
\(638\) 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
\(643\) − 2.19289e9i − 0.325296i −0.986684 0.162648i \(-0.947996\pi\)
0.986684 0.162648i \(-0.0520035\pi\)
\(644\) 0 0
\(645\) − 2.30080e9i − 0.337614i
\(646\) 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
\(651\) 7.74489e9i 1.10023i
\(652\) 0 0
\(653\) − 9.93257e9i − 1.39594i −0.716129 0.697968i \(-0.754088\pi\)
0.716129 0.697968i \(-0.245912\pi\)
\(654\) 0 0
\(655\) 2.98346e9 0.414836
\(656\) 0 0
\(657\) −1.38474e10 −1.90497
\(658\) 0 0
\(659\) − 1.36634e10i − 1.85977i −0.367852 0.929884i \(-0.619907\pi\)
0.367852 0.929884i \(-0.380093\pi\)
\(660\) 0 0
\(661\) 1.03765e10i 1.39747i 0.715378 + 0.698737i \(0.246254\pi\)
−0.715378 + 0.698737i \(0.753746\pi\)
\(662\) 0 0
\(663\) −2.06115e9 −0.274671
\(664\) 0 0
\(665\) −4.88398e9 −0.644019
\(666\) 0 0
\(667\) 3.15032e9i 0.411069i
\(668\) 0 0
\(669\) 1.28515e10i 1.65944i
\(670\) 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
\(675\) 4.33599e9i 0.542657i
\(676\) 0 0
\(677\) 9.55050e9i 1.18295i 0.806324 + 0.591474i \(0.201454\pi\)
−0.806324 + 0.591474i \(0.798546\pi\)
\(678\) 0 0
\(679\) −1.26947e7 −0.00155625
\(680\) 0 0
\(681\) 1.05658e10 1.28200
\(682\) 0 0
\(683\) − 1.06442e10i − 1.27832i −0.769073 0.639161i \(-0.779282\pi\)
0.769073 0.639161i \(-0.220718\pi\)
\(684\) 0 0
\(685\) − 1.12948e10i − 1.34265i
\(686\) 0 0
\(687\) 2.06175e10 2.42598
\(688\) 0 0
\(689\) 3.39865e9 0.395858
\(690\) 0 0
\(691\) 8.41537e9i 0.970287i 0.874435 + 0.485143i \(0.161233\pi\)
−0.874435 + 0.485143i \(0.838767\pi\)
\(692\) 0 0
\(693\) 3.22386e9i 0.367967i
\(694\) 0 0
\(695\) 1.58445e10 1.79033
\(696\) 0 0
\(697\) −7.26858e9 −0.813083
\(698\) 0 0
\(699\) 1.52385e10i 1.68761i
\(700\) 0 0
\(701\) − 8.95355e9i − 0.981707i −0.871242 0.490854i \(-0.836685\pi\)
0.871242 0.490854i \(-0.163315\pi\)
\(702\) 0 0
\(703\) −1.63139e10 −1.77099
\(704\) 0 0
\(705\) 2.13643e10 2.29629
\(706\) 0 0
\(707\) − 7.37853e9i − 0.785239i
\(708\) 0 0
\(709\) − 8.11796e9i − 0.855431i −0.903913 0.427716i \(-0.859318\pi\)
0.903913 0.427716i \(-0.140682\pi\)
\(710\) 0 0
\(711\) −9.55395e9 −0.996872
\(712\) 0 0
\(713\) −4.49486e9 −0.464412
\(714\) 0 0
\(715\) − 1.47157e9i − 0.150560i
\(716\) 0 0
\(717\) − 6.32739e9i − 0.641073i
\(718\) 0 0
\(719\) −9.54339e8 −0.0957528 −0.0478764 0.998853i \(-0.515245\pi\)
−0.0478764 + 0.998853i \(0.515245\pi\)
\(720\) 0 0
\(721\) 5.60800e9 0.557231
\(722\) 0 0
\(723\) − 2.08264e10i − 2.04942i
\(724\) 0 0
\(725\) 5.85203e9i 0.570327i
\(726\) 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
\(731\) − 1.07030e9i − 0.101343i
\(732\) 0 0
\(733\) − 1.16062e10i − 1.08849i −0.838925 0.544247i \(-0.816815\pi\)
0.838925 0.544247i \(-0.183185\pi\)
\(734\) 0 0
\(735\) 1.64286e10 1.52614
\(736\) 0 0
\(737\) −1.57087e8 −0.0144545
\(738\) 0 0
\(739\) 4.74800e9i 0.432768i 0.976308 + 0.216384i \(0.0694263\pi\)
−0.976308 + 0.216384i \(0.930574\pi\)
\(740\) 0 0
\(741\) 5.59268e9i 0.504959i
\(742\) 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
\(747\) − 5.72888e9i − 0.502860i
\(748\) 0 0
\(749\) − 5.95321e9i − 0.517684i
\(750\) 0 0
\(751\) 6.10987e9 0.526371 0.263186 0.964745i \(-0.415227\pi\)
0.263186 + 0.964745i \(0.415227\pi\)
\(752\) 0 0
\(753\) −2.23729e10 −1.90959
\(754\) 0 0
\(755\) 2.42883e10i 2.05391i
\(756\) 0 0
\(757\) 2.29472e10i 1.92262i 0.275460 + 0.961312i \(0.411170\pi\)
−0.275460 + 0.961312i \(0.588830\pi\)
\(758\) 0 0
\(759\) −2.96679e9 −0.246286
\(760\) 0 0
\(761\) 7.15151e9 0.588236 0.294118 0.955769i \(-0.404974\pi\)
0.294118 + 0.955769i \(0.404974\pi\)
\(762\) 0 0
\(763\) − 2.08175e9i − 0.169665i
\(764\) 0 0
\(765\) 1.53111e10i 1.23650i
\(766\) 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
\(771\) 3.35537e9i 0.263663i
\(772\) 0 0
\(773\) 8.25535e9i 0.642846i 0.946936 + 0.321423i \(0.104161\pi\)
−0.946936 + 0.321423i \(0.895839\pi\)
\(774\) 0 0
\(775\) −8.34966e9 −0.644336
\(776\) 0 0
\(777\) −1.67648e10 −1.28211
\(778\) 0 0
\(779\) 1.97223e10i 1.49478i
\(780\) 0 0
\(781\) 2.49909e9i 0.187717i
\(782\) 0 0
\(783\) −1.91317e10 −1.42425
\(784\) 0 0
\(785\) 1.36656e10 1.00829
\(786\) 0 0
\(787\) 1.53185e9i 0.112023i 0.998430 + 0.0560113i \(0.0178383\pi\)
−0.998430 + 0.0560113i \(0.982162\pi\)
\(788\) 0 0
\(789\) − 3.29097e10i − 2.38536i
\(790\) 0 0
\(791\) −3.54196e9 −0.254464
\(792\) 0 0
\(793\) 1.07140e9 0.0762950
\(794\) 0 0
\(795\) − 4.00326e10i − 2.82572i
\(796\) 0 0
\(797\) 2.78553e9i 0.194896i 0.995241 + 0.0974480i \(0.0310680\pi\)
−0.995241 + 0.0974480i \(0.968932\pi\)
\(798\) 0 0
\(799\) 9.93837e9 0.689289
\(800\) 0 0
\(801\) −7.44954e9 −0.512172
\(802\) 0 0
\(803\) − 7.29247e9i − 0.497016i
\(804\) 0 0
\(805\) − 2.91283e9i − 0.196802i
\(806\) 0 0
\(807\) 6.35708e9 0.425794
\(808\) 0 0
\(809\) 1.17657e10 0.781267 0.390634 0.920546i \(-0.372256\pi\)
0.390634 + 0.920546i \(0.372256\pi\)
\(810\) 0 0
\(811\) − 6.29491e9i − 0.414397i −0.978299 0.207198i \(-0.933565\pi\)
0.978299 0.207198i \(-0.0664346\pi\)
\(812\) 0 0
\(813\) − 4.73386e9i − 0.308957i
\(814\) 0 0
\(815\) −3.33201e10 −2.15603
\(816\) 0 0
\(817\) −2.90413e9 −0.186311
\(818\) 0 0
\(819\) 3.62451e9i 0.230545i
\(820\) 0 0
\(821\) − 4.27400e9i − 0.269546i −0.990876 0.134773i \(-0.956969\pi\)
0.990876 0.134773i \(-0.0430306\pi\)
\(822\) 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
\(827\) 3.47251e9i 0.213488i 0.994287 + 0.106744i \(0.0340426\pi\)
−0.994287 + 0.106744i \(0.965957\pi\)
\(828\) 0 0
\(829\) 5.21388e9i 0.317848i 0.987291 + 0.158924i \(0.0508026\pi\)
−0.987291 + 0.158924i \(0.949197\pi\)
\(830\) 0 0
\(831\) −3.37991e9 −0.204316
\(832\) 0 0
\(833\) 7.64237e9 0.458111
\(834\) 0 0
\(835\) 1.43774e10i 0.854629i
\(836\) 0 0
\(837\) − 2.72970e10i − 1.60908i
\(838\) 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
\(843\) 4.47024e10i 2.57001i
\(844\) 0 0
\(845\) 1.95823e10i 1.11652i
\(846\) 0 0
\(847\) 6.85701e9 0.387742
\(848\) 0 0
\(849\) 4.64614e10 2.60565
\(850\) 0 0
\(851\) − 9.72969e9i − 0.541185i
\(852\) 0 0
\(853\) 3.01930e10i 1.66565i 0.553536 + 0.832826i \(0.313278\pi\)
−0.553536 + 0.832826i \(0.686722\pi\)
\(854\) 0 0
\(855\) 4.15448e10 2.27319
\(856\) 0 0
\(857\) 2.57761e10 1.39889 0.699446 0.714685i \(-0.253430\pi\)
0.699446 + 0.714685i \(0.253430\pi\)
\(858\) 0 0
\(859\) − 2.77848e10i − 1.49565i −0.663894 0.747827i \(-0.731097\pi\)
0.663894 0.747827i \(-0.268903\pi\)
\(860\) 0 0
\(861\) 2.02674e10i 1.08215i
\(862\) 0 0
\(863\) −3.02990e10 −1.60469 −0.802344 0.596862i \(-0.796414\pi\)
−0.802344 + 0.596862i \(0.796414\pi\)
\(864\) 0 0
\(865\) −8.97743e9 −0.471624
\(866\) 0 0
\(867\) − 2.02815e10i − 1.05690i
\(868\) 0 0
\(869\) − 5.03142e9i − 0.260089i
\(870\) 0 0
\(871\) −1.76609e8 −0.00905627
\(872\) 0 0
\(873\) 1.07986e8 0.00549309
\(874\) 0 0
\(875\) 6.19656e9i 0.312696i
\(876\) 0 0
\(877\) 2.37410e9i 0.118850i 0.998233 + 0.0594252i \(0.0189268\pi\)
−0.998233 + 0.0594252i \(0.981073\pi\)
\(878\) 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
\(883\) 2.14967e10i 1.05078i 0.850863 + 0.525388i \(0.176080\pi\)
−0.850863 + 0.525388i \(0.823920\pi\)
\(884\) 0 0
\(885\) − 4.75234e10i − 2.30466i
\(886\) 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
\(891\) − 1.95647e9i − 0.0926617i
\(892\) 0 0
\(893\) − 2.69665e10i − 1.26720i
\(894\) 0 0
\(895\) 8.19532e9 0.382107
\(896\) 0 0
\(897\) −3.33549e9 −0.154307
\(898\) 0 0
\(899\) − 3.68412e10i − 1.69112i
\(900\) 0 0
\(901\) − 1.86226e10i − 0.848212i
\(902\) 0 0
\(903\) −2.98438e9 −0.134880
\(904\) 0 0
\(905\) 9.44124e9 0.423408
\(906\) 0 0
\(907\) 1.57906e10i 0.702704i 0.936243 + 0.351352i \(0.114278\pi\)
−0.936243 + 0.351352i \(0.885722\pi\)
\(908\) 0 0
\(909\) 6.27643e10i 2.77165i
\(910\) 0 0
\(911\) 6.82128e9 0.298917 0.149459 0.988768i \(-0.452247\pi\)
0.149459 + 0.988768i \(0.452247\pi\)
\(912\) 0 0
\(913\) 3.01701e9 0.131199
\(914\) 0 0
\(915\) − 1.26200e10i − 0.544609i
\(916\) 0 0
\(917\) − 3.86987e9i − 0.165731i
\(918\) 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
\(923\) 2.80967e9i 0.117611i
\(924\) 0 0
\(925\) − 1.80739e10i − 0.750853i
\(926\) 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
\(931\) − 2.07366e10i − 0.842197i
\(932\) 0 0
\(933\) 6.62721e10i 2.67144i
\(934\) 0 0
\(935\) −8.06335e9 −0.322608
\(936\) 0 0
\(937\) 2.39333e10 0.950417 0.475208 0.879873i \(-0.342373\pi\)
0.475208 + 0.879873i \(0.342373\pi\)
\(938\) 0 0
\(939\) 1.86261e10i 0.734164i
\(940\) 0 0
\(941\) − 3.17291e10i − 1.24135i −0.784068 0.620675i \(-0.786859\pi\)
0.784068 0.620675i \(-0.213141\pi\)
\(942\) 0 0
\(943\) −1.17625e10 −0.456781
\(944\) 0 0
\(945\) 1.76894e10 0.681871
\(946\) 0 0
\(947\) 4.13161e10i 1.58086i 0.612551 + 0.790431i \(0.290143\pi\)
−0.612551 + 0.790431i \(0.709857\pi\)
\(948\) 0 0
\(949\) − 8.19876e9i − 0.311399i
\(950\) 0 0
\(951\) −4.14268e10 −1.56189
\(952\) 0 0
\(953\) −4.31993e10 −1.61678 −0.808390 0.588647i \(-0.799661\pi\)
−0.808390 + 0.588647i \(0.799661\pi\)
\(954\) 0 0
\(955\) − 5.67216e9i − 0.210735i
\(956\) 0 0
\(957\) − 2.43167e10i − 0.896835i
\(958\) 0 0
\(959\) −1.46506e10 −0.536401
\(960\) 0 0
\(961\) 2.50523e10 0.910574
\(962\) 0 0
\(963\) 5.06400e10i 1.82727i
\(964\) 0 0
\(965\) 2.96140e10i 1.06084i
\(966\) 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
\(971\) − 8.48357e9i − 0.297380i −0.988884 0.148690i \(-0.952494\pi\)
0.988884 0.148690i \(-0.0475056\pi\)
\(972\) 0 0
\(973\) − 2.05521e10i − 0.715255i
\(974\) 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
\(979\) − 3.92317e9i − 0.133628i
\(980\) 0 0
\(981\) 1.77081e10i 0.598866i
\(982\) 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
\(987\) − 2.77117e10i − 0.917389i
\(988\) 0 0
\(989\) − 1.73203e9i − 0.0569336i
\(990\) 0 0
\(991\) 3.13036e9 0.102173 0.0510866 0.998694i \(-0.483732\pi\)
0.0510866 + 0.998694i \(0.483732\pi\)
\(992\) 0 0
\(993\) −8.08387e10 −2.61997
\(994\) 0 0
\(995\) − 1.80012e10i − 0.579323i
\(996\) 0 0
\(997\) − 1.67605e9i − 0.0535617i −0.999641 0.0267808i \(-0.991474\pi\)
0.999641 0.0267808i \(-0.00852563\pi\)
\(998\) 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 32.8.b.a.17.6 6
3.2 odd 2 288.8.d.b.145.1 6
4.3 odd 2 8.8.b.a.5.6 yes 6
8.3 odd 2 8.8.b.a.5.5 6
8.5 even 2 inner 32.8.b.a.17.1 6
12.11 even 2 72.8.d.b.37.1 6
16.3 odd 4 256.8.a.r.1.6 6
16.5 even 4 256.8.a.q.1.6 6
16.11 odd 4 256.8.a.r.1.1 6
16.13 even 4 256.8.a.q.1.1 6
24.5 odd 2 288.8.d.b.145.6 6
24.11 even 2 72.8.d.b.37.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.b.a.5.5 6 8.3 odd 2
8.8.b.a.5.6 yes 6 4.3 odd 2
32.8.b.a.17.1 6 8.5 even 2 inner
32.8.b.a.17.6 6 1.1 even 1 trivial
72.8.d.b.37.1 6 12.11 even 2
72.8.d.b.37.2 6 24.11 even 2
256.8.a.q.1.1 6 16.13 even 4
256.8.a.q.1.6 6 16.5 even 4
256.8.a.r.1.1 6 16.11 odd 4
256.8.a.r.1.6 6 16.3 odd 4
288.8.d.b.145.1 6 3.2 odd 2
288.8.d.b.145.6 6 24.5 odd 2