Properties

Label 32.8.b.a.17.5
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 17.5
Root \(-4.85268 - 2.90715i\) of defining polynomial
Character \(\chi\) \(=\) 32.17
Dual form 32.8.b.a.17.2

$q$-expansion

\(f(q)\) \(=\) \(q+40.2163i q^{3} -324.492i q^{5} +956.960 q^{7} +569.651 q^{9} +O(q^{10})\) \(q+40.2163i q^{3} -324.492i q^{5} +956.960 q^{7} +569.651 q^{9} +5452.20i q^{11} +6289.38i q^{13} +13049.8 q^{15} +34587.3 q^{17} -14595.6i q^{19} +38485.4i q^{21} +24667.5 q^{23} -27169.8 q^{25} +110862. i q^{27} -171116. i q^{29} -111688. q^{31} -219267. q^{33} -310526. i q^{35} +103636. i q^{37} -252935. q^{39} +71691.3 q^{41} -328419. i q^{43} -184847. i q^{45} -119043. q^{47} +92230.3 q^{49} +1.39097e6i q^{51} +1.04011e6i q^{53} +1.76919e6 q^{55} +586982. q^{57} -225984. i q^{59} -1.55268e6i q^{61} +545133. q^{63} +2.04085e6 q^{65} +316375. i q^{67} +992033. i q^{69} -538965. q^{71} -2.68512e6 q^{73} -1.09267e6i q^{75} +5.21754e6i q^{77} -8.22632e6 q^{79} -3.21264e6 q^{81} -5.89510e6i q^{83} -1.12233e7i q^{85} +6.88164e6 q^{87} +437005. q^{89} +6.01868e6i q^{91} -4.49169e6i q^{93} -4.73616e6 q^{95} -7.84322e6 q^{97} +3.10585e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 688 q^{7} - 2918 q^{9}+O(q^{10}) \) 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
\(3\) 40.2163i 0.859959i 0.902839 + 0.429979i \(0.141479\pi\)
−0.902839 + 0.429979i \(0.858521\pi\)
\(4\) 0 0
\(5\) − 324.492i − 1.16094i −0.814283 0.580468i \(-0.802870\pi\)
0.814283 0.580468i \(-0.197130\pi\)
\(6\) 0 0
\(7\) 956.960 1.05451 0.527255 0.849707i \(-0.323221\pi\)
0.527255 + 0.849707i \(0.323221\pi\)
\(8\) 0 0
\(9\) 569.651 0.260471
\(10\) 0 0
\(11\) 5452.20i 1.23509i 0.786537 + 0.617544i \(0.211872\pi\)
−0.786537 + 0.617544i \(0.788128\pi\)
\(12\) 0 0
\(13\) 6289.38i 0.793973i 0.917824 + 0.396987i \(0.129944\pi\)
−0.917824 + 0.396987i \(0.870056\pi\)
\(14\) 0 0
\(15\) 13049.8 0.998357
\(16\) 0 0
\(17\) 34587.3 1.70744 0.853720 0.520733i \(-0.174341\pi\)
0.853720 + 0.520733i \(0.174341\pi\)
\(18\) 0 0
\(19\) − 14595.6i − 0.488186i −0.969752 0.244093i \(-0.921510\pi\)
0.969752 0.244093i \(-0.0784903\pi\)
\(20\) 0 0
\(21\) 38485.4i 0.906835i
\(22\) 0 0
\(23\) 24667.5 0.422743 0.211372 0.977406i \(-0.432207\pi\)
0.211372 + 0.977406i \(0.432207\pi\)
\(24\) 0 0
\(25\) −27169.8 −0.347774
\(26\) 0 0
\(27\) 110862.i 1.08395i
\(28\) 0 0
\(29\) − 171116.i − 1.30286i −0.758710 0.651429i \(-0.774170\pi\)
0.758710 0.651429i \(-0.225830\pi\)
\(30\) 0 0
\(31\) −111688. −0.673352 −0.336676 0.941620i \(-0.609303\pi\)
−0.336676 + 0.941620i \(0.609303\pi\)
\(32\) 0 0
\(33\) −219267. −1.06212
\(34\) 0 0
\(35\) − 310526.i − 1.22422i
\(36\) 0 0
\(37\) 103636.i 0.336360i 0.985756 + 0.168180i \(0.0537890\pi\)
−0.985756 + 0.168180i \(0.946211\pi\)
\(38\) 0 0
\(39\) −252935. −0.682784
\(40\) 0 0
\(41\) 71691.3 0.162451 0.0812256 0.996696i \(-0.474117\pi\)
0.0812256 + 0.996696i \(0.474117\pi\)
\(42\) 0 0
\(43\) − 328419.i − 0.629925i −0.949104 0.314962i \(-0.898008\pi\)
0.949104 0.314962i \(-0.101992\pi\)
\(44\) 0 0
\(45\) − 184847.i − 0.302391i
\(46\) 0 0
\(47\) −119043. −0.167248 −0.0836241 0.996497i \(-0.526650\pi\)
−0.0836241 + 0.996497i \(0.526650\pi\)
\(48\) 0 0
\(49\) 92230.3 0.111992
\(50\) 0 0
\(51\) 1.39097e6i 1.46833i
\(52\) 0 0
\(53\) 1.04011e6i 0.959648i 0.877365 + 0.479824i \(0.159300\pi\)
−0.877365 + 0.479824i \(0.840700\pi\)
\(54\) 0 0
\(55\) 1.76919e6 1.43386
\(56\) 0 0
\(57\) 586982. 0.419820
\(58\) 0 0
\(59\) − 225984.i − 0.143250i −0.997432 0.0716250i \(-0.977182\pi\)
0.997432 0.0716250i \(-0.0228185\pi\)
\(60\) 0 0
\(61\) − 1.55268e6i − 0.875843i −0.899013 0.437922i \(-0.855715\pi\)
0.899013 0.437922i \(-0.144285\pi\)
\(62\) 0 0
\(63\) 545133. 0.274670
\(64\) 0 0
\(65\) 2.04085e6 0.921753
\(66\) 0 0
\(67\) 316375.i 0.128511i 0.997933 + 0.0642555i \(0.0204673\pi\)
−0.997933 + 0.0642555i \(0.979533\pi\)
\(68\) 0 0
\(69\) 992033.i 0.363542i
\(70\) 0 0
\(71\) −538965. −0.178713 −0.0893566 0.996000i \(-0.528481\pi\)
−0.0893566 + 0.996000i \(0.528481\pi\)
\(72\) 0 0
\(73\) −2.68512e6 −0.807856 −0.403928 0.914791i \(-0.632355\pi\)
−0.403928 + 0.914791i \(0.632355\pi\)
\(74\) 0 0
\(75\) − 1.09267e6i − 0.299071i
\(76\) 0 0
\(77\) 5.21754e6i 1.30241i
\(78\) 0 0
\(79\) −8.22632e6 −1.87720 −0.938600 0.345007i \(-0.887877\pi\)
−0.938600 + 0.345007i \(0.887877\pi\)
\(80\) 0 0
\(81\) −3.21264e6 −0.671684
\(82\) 0 0
\(83\) − 5.89510e6i − 1.13167i −0.824520 0.565833i \(-0.808555\pi\)
0.824520 0.565833i \(-0.191445\pi\)
\(84\) 0 0
\(85\) − 1.12233e7i − 1.98223i
\(86\) 0 0
\(87\) 6.88164e6 1.12040
\(88\) 0 0
\(89\) 437005. 0.0657085 0.0328542 0.999460i \(-0.489540\pi\)
0.0328542 + 0.999460i \(0.489540\pi\)
\(90\) 0 0
\(91\) 6.01868e6i 0.837253i
\(92\) 0 0
\(93\) − 4.49169e6i − 0.579055i
\(94\) 0 0
\(95\) −4.73616e6 −0.566753
\(96\) 0 0
\(97\) −7.84322e6 −0.872556 −0.436278 0.899812i \(-0.643704\pi\)
−0.436278 + 0.899812i \(0.643704\pi\)
\(98\) 0 0
\(99\) 3.10585e6i 0.321705i
\(100\) 0 0
\(101\) 6.19757e6i 0.598545i 0.954168 + 0.299272i \(0.0967439\pi\)
−0.954168 + 0.299272i \(0.903256\pi\)
\(102\) 0 0
\(103\) 6.59816e6 0.594966 0.297483 0.954727i \(-0.403853\pi\)
0.297483 + 0.954727i \(0.403853\pi\)
\(104\) 0 0
\(105\) 1.24882e7 1.05278
\(106\) 0 0
\(107\) 512845.i 0.0404709i 0.999795 + 0.0202354i \(0.00644158\pi\)
−0.999795 + 0.0202354i \(0.993558\pi\)
\(108\) 0 0
\(109\) 1.95882e7i 1.44878i 0.689393 + 0.724388i \(0.257877\pi\)
−0.689393 + 0.724388i \(0.742123\pi\)
\(110\) 0 0
\(111\) −4.16785e6 −0.289256
\(112\) 0 0
\(113\) 1.88876e7 1.23141 0.615705 0.787977i \(-0.288871\pi\)
0.615705 + 0.787977i \(0.288871\pi\)
\(114\) 0 0
\(115\) − 8.00438e6i − 0.490778i
\(116\) 0 0
\(117\) 3.58275e6i 0.206807i
\(118\) 0 0
\(119\) 3.30987e7 1.80051
\(120\) 0 0
\(121\) −1.02394e7 −0.525441
\(122\) 0 0
\(123\) 2.88316e6i 0.139701i
\(124\) 0 0
\(125\) − 1.65345e7i − 0.757193i
\(126\) 0 0
\(127\) −3.96314e7 −1.71683 −0.858413 0.512959i \(-0.828549\pi\)
−0.858413 + 0.512959i \(0.828549\pi\)
\(128\) 0 0
\(129\) 1.32078e7 0.541709
\(130\) 0 0
\(131\) 3.65337e7i 1.41986i 0.704274 + 0.709928i \(0.251273\pi\)
−0.704274 + 0.709928i \(0.748727\pi\)
\(132\) 0 0
\(133\) − 1.39675e7i − 0.514797i
\(134\) 0 0
\(135\) 3.59739e7 1.25840
\(136\) 0 0
\(137\) −2.56967e7 −0.853799 −0.426899 0.904299i \(-0.640394\pi\)
−0.426899 + 0.904299i \(0.640394\pi\)
\(138\) 0 0
\(139\) − 5.23001e7i − 1.65177i −0.563836 0.825886i \(-0.690675\pi\)
0.563836 0.825886i \(-0.309325\pi\)
\(140\) 0 0
\(141\) − 4.78747e6i − 0.143827i
\(142\) 0 0
\(143\) −3.42910e7 −0.980626
\(144\) 0 0
\(145\) −5.55256e7 −1.51254
\(146\) 0 0
\(147\) 3.70916e6i 0.0963085i
\(148\) 0 0
\(149\) 1.80406e7i 0.446785i 0.974729 + 0.223392i \(0.0717131\pi\)
−0.974729 + 0.223392i \(0.928287\pi\)
\(150\) 0 0
\(151\) −3.87385e7 −0.915637 −0.457818 0.889046i \(-0.651369\pi\)
−0.457818 + 0.889046i \(0.651369\pi\)
\(152\) 0 0
\(153\) 1.97027e7 0.444739
\(154\) 0 0
\(155\) 3.62420e7i 0.781719i
\(156\) 0 0
\(157\) − 5.12341e7i − 1.05660i −0.849058 0.528300i \(-0.822830\pi\)
0.849058 0.528300i \(-0.177170\pi\)
\(158\) 0 0
\(159\) −4.18292e7 −0.825258
\(160\) 0 0
\(161\) 2.36058e7 0.445787
\(162\) 0 0
\(163\) − 8.57572e7i − 1.55101i −0.631343 0.775504i \(-0.717496\pi\)
0.631343 0.775504i \(-0.282504\pi\)
\(164\) 0 0
\(165\) 7.11504e7i 1.23306i
\(166\) 0 0
\(167\) 1.05871e8 1.75901 0.879503 0.475893i \(-0.157875\pi\)
0.879503 + 0.475893i \(0.157875\pi\)
\(168\) 0 0
\(169\) 2.31923e7 0.369606
\(170\) 0 0
\(171\) − 8.31441e6i − 0.127158i
\(172\) 0 0
\(173\) − 1.98148e7i − 0.290956i −0.989361 0.145478i \(-0.953528\pi\)
0.989361 0.145478i \(-0.0464720\pi\)
\(174\) 0 0
\(175\) −2.60005e7 −0.366731
\(176\) 0 0
\(177\) 9.08822e6 0.123189
\(178\) 0 0
\(179\) 2.97800e7i 0.388096i 0.980992 + 0.194048i \(0.0621618\pi\)
−0.980992 + 0.194048i \(0.937838\pi\)
\(180\) 0 0
\(181\) − 3.96227e6i − 0.0496671i −0.999692 0.0248335i \(-0.992094\pi\)
0.999692 0.0248335i \(-0.00790558\pi\)
\(182\) 0 0
\(183\) 6.24429e7 0.753189
\(184\) 0 0
\(185\) 3.36290e7 0.390493
\(186\) 0 0
\(187\) 1.88577e8i 2.10884i
\(188\) 0 0
\(189\) 1.06091e8i 1.14304i
\(190\) 0 0
\(191\) 4.80105e7 0.498562 0.249281 0.968431i \(-0.419806\pi\)
0.249281 + 0.968431i \(0.419806\pi\)
\(192\) 0 0
\(193\) −4.72502e6 −0.0473100 −0.0236550 0.999720i \(-0.507530\pi\)
−0.0236550 + 0.999720i \(0.507530\pi\)
\(194\) 0 0
\(195\) 8.20754e7i 0.792669i
\(196\) 0 0
\(197\) − 1.14882e8i − 1.07058i −0.844668 0.535290i \(-0.820202\pi\)
0.844668 0.535290i \(-0.179798\pi\)
\(198\) 0 0
\(199\) −1.20933e7 −0.108782 −0.0543911 0.998520i \(-0.517322\pi\)
−0.0543911 + 0.998520i \(0.517322\pi\)
\(200\) 0 0
\(201\) −1.27234e7 −0.110514
\(202\) 0 0
\(203\) − 1.63751e8i − 1.37388i
\(204\) 0 0
\(205\) − 2.32632e7i − 0.188596i
\(206\) 0 0
\(207\) 1.40518e7 0.110112
\(208\) 0 0
\(209\) 7.95784e7 0.602953
\(210\) 0 0
\(211\) − 1.95850e8i − 1.43527i −0.696418 0.717636i \(-0.745224\pi\)
0.696418 0.717636i \(-0.254776\pi\)
\(212\) 0 0
\(213\) − 2.16752e7i − 0.153686i
\(214\) 0 0
\(215\) −1.06569e8 −0.731302
\(216\) 0 0
\(217\) −1.06881e8 −0.710057
\(218\) 0 0
\(219\) − 1.07986e8i − 0.694723i
\(220\) 0 0
\(221\) 2.17532e8i 1.35566i
\(222\) 0 0
\(223\) −1.08024e8 −0.652311 −0.326156 0.945316i \(-0.605753\pi\)
−0.326156 + 0.945316i \(0.605753\pi\)
\(224\) 0 0
\(225\) −1.54773e7 −0.0905851
\(226\) 0 0
\(227\) 1.61144e8i 0.914374i 0.889371 + 0.457187i \(0.151143\pi\)
−0.889371 + 0.457187i \(0.848857\pi\)
\(228\) 0 0
\(229\) − 5.27173e7i − 0.290088i −0.989425 0.145044i \(-0.953668\pi\)
0.989425 0.145044i \(-0.0463323\pi\)
\(230\) 0 0
\(231\) −2.09830e8 −1.12002
\(232\) 0 0
\(233\) −1.79423e8 −0.929249 −0.464625 0.885508i \(-0.653811\pi\)
−0.464625 + 0.885508i \(0.653811\pi\)
\(234\) 0 0
\(235\) 3.86285e7i 0.194165i
\(236\) 0 0
\(237\) − 3.30832e8i − 1.61431i
\(238\) 0 0
\(239\) 8.42441e7 0.399160 0.199580 0.979882i \(-0.436042\pi\)
0.199580 + 0.979882i \(0.436042\pi\)
\(240\) 0 0
\(241\) −2.12302e8 −0.977000 −0.488500 0.872564i \(-0.662456\pi\)
−0.488500 + 0.872564i \(0.662456\pi\)
\(242\) 0 0
\(243\) 1.13255e8i 0.506333i
\(244\) 0 0
\(245\) − 2.99280e7i − 0.130016i
\(246\) 0 0
\(247\) 9.17975e7 0.387607
\(248\) 0 0
\(249\) 2.37079e8 0.973186
\(250\) 0 0
\(251\) − 1.18102e8i − 0.471411i −0.971825 0.235706i \(-0.924260\pi\)
0.971825 0.235706i \(-0.0757401\pi\)
\(252\) 0 0
\(253\) 1.34492e8i 0.522125i
\(254\) 0 0
\(255\) 4.51359e8 1.70463
\(256\) 0 0
\(257\) 1.27463e8 0.468402 0.234201 0.972188i \(-0.424753\pi\)
0.234201 + 0.972188i \(0.424753\pi\)
\(258\) 0 0
\(259\) 9.91755e7i 0.354695i
\(260\) 0 0
\(261\) − 9.74762e7i − 0.339357i
\(262\) 0 0
\(263\) −4.33125e8 −1.46814 −0.734071 0.679073i \(-0.762382\pi\)
−0.734071 + 0.679073i \(0.762382\pi\)
\(264\) 0 0
\(265\) 3.37506e8 1.11409
\(266\) 0 0
\(267\) 1.75747e7i 0.0565066i
\(268\) 0 0
\(269\) 3.44748e8i 1.07986i 0.841709 + 0.539931i \(0.181550\pi\)
−0.841709 + 0.539931i \(0.818450\pi\)
\(270\) 0 0
\(271\) 4.42513e8 1.35062 0.675311 0.737533i \(-0.264010\pi\)
0.675311 + 0.737533i \(0.264010\pi\)
\(272\) 0 0
\(273\) −2.42049e8 −0.720003
\(274\) 0 0
\(275\) − 1.48135e8i − 0.429531i
\(276\) 0 0
\(277\) 3.18148e8i 0.899395i 0.893181 + 0.449697i \(0.148468\pi\)
−0.893181 + 0.449697i \(0.851532\pi\)
\(278\) 0 0
\(279\) −6.36234e7 −0.175389
\(280\) 0 0
\(281\) 1.28497e8 0.345478 0.172739 0.984968i \(-0.444738\pi\)
0.172739 + 0.984968i \(0.444738\pi\)
\(282\) 0 0
\(283\) 3.98970e8i 1.04637i 0.852218 + 0.523187i \(0.175257\pi\)
−0.852218 + 0.523187i \(0.824743\pi\)
\(284\) 0 0
\(285\) − 1.90471e8i − 0.487384i
\(286\) 0 0
\(287\) 6.86057e7 0.171306
\(288\) 0 0
\(289\) 7.85942e8 1.91535
\(290\) 0 0
\(291\) − 3.15425e8i − 0.750362i
\(292\) 0 0
\(293\) 2.00958e8i 0.466732i 0.972389 + 0.233366i \(0.0749741\pi\)
−0.972389 + 0.233366i \(0.925026\pi\)
\(294\) 0 0
\(295\) −7.33298e7 −0.166304
\(296\) 0 0
\(297\) −6.04444e8 −1.33878
\(298\) 0 0
\(299\) 1.55143e8i 0.335647i
\(300\) 0 0
\(301\) − 3.14284e8i − 0.664262i
\(302\) 0 0
\(303\) −2.49243e8 −0.514724
\(304\) 0 0
\(305\) −5.03830e8 −1.01680
\(306\) 0 0
\(307\) − 1.58918e7i − 0.0313465i −0.999877 0.0156733i \(-0.995011\pi\)
0.999877 0.0156733i \(-0.00498916\pi\)
\(308\) 0 0
\(309\) 2.65353e8i 0.511646i
\(310\) 0 0
\(311\) −4.87710e8 −0.919391 −0.459695 0.888077i \(-0.652041\pi\)
−0.459695 + 0.888077i \(0.652041\pi\)
\(312\) 0 0
\(313\) −3.24731e8 −0.598576 −0.299288 0.954163i \(-0.596749\pi\)
−0.299288 + 0.954163i \(0.596749\pi\)
\(314\) 0 0
\(315\) − 1.76891e8i − 0.318874i
\(316\) 0 0
\(317\) − 1.06084e9i − 1.87043i −0.354086 0.935213i \(-0.615208\pi\)
0.354086 0.935213i \(-0.384792\pi\)
\(318\) 0 0
\(319\) 9.32958e8 1.60914
\(320\) 0 0
\(321\) −2.06247e7 −0.0348033
\(322\) 0 0
\(323\) − 5.04824e8i − 0.833548i
\(324\) 0 0
\(325\) − 1.70881e8i − 0.276123i
\(326\) 0 0
\(327\) −7.87763e8 −1.24589
\(328\) 0 0
\(329\) −1.13919e8 −0.176365
\(330\) 0 0
\(331\) 2.88487e8i 0.437249i 0.975809 + 0.218624i \(0.0701569\pi\)
−0.975809 + 0.218624i \(0.929843\pi\)
\(332\) 0 0
\(333\) 5.90363e7i 0.0876121i
\(334\) 0 0
\(335\) 1.02661e8 0.149193
\(336\) 0 0
\(337\) −1.10595e8 −0.157410 −0.0787051 0.996898i \(-0.525079\pi\)
−0.0787051 + 0.996898i \(0.525079\pi\)
\(338\) 0 0
\(339\) 7.59590e8i 1.05896i
\(340\) 0 0
\(341\) − 6.08948e8i − 0.831649i
\(342\) 0 0
\(343\) −6.99837e8 −0.936414
\(344\) 0 0
\(345\) 3.21907e8 0.422049
\(346\) 0 0
\(347\) − 1.10651e9i − 1.42168i −0.703352 0.710841i \(-0.748314\pi\)
0.703352 0.710841i \(-0.251686\pi\)
\(348\) 0 0
\(349\) 1.38337e9i 1.74201i 0.491278 + 0.871003i \(0.336530\pi\)
−0.491278 + 0.871003i \(0.663470\pi\)
\(350\) 0 0
\(351\) −6.97254e8 −0.860630
\(352\) 0 0
\(353\) −2.47617e8 −0.299618 −0.149809 0.988715i \(-0.547866\pi\)
−0.149809 + 0.988715i \(0.547866\pi\)
\(354\) 0 0
\(355\) 1.74890e8i 0.207475i
\(356\) 0 0
\(357\) 1.33111e9i 1.54837i
\(358\) 0 0
\(359\) 1.38641e9 1.58148 0.790738 0.612155i \(-0.209697\pi\)
0.790738 + 0.612155i \(0.209697\pi\)
\(360\) 0 0
\(361\) 6.80839e8 0.761674
\(362\) 0 0
\(363\) − 4.11789e8i − 0.451858i
\(364\) 0 0
\(365\) 8.71299e8i 0.937869i
\(366\) 0 0
\(367\) 7.49367e8 0.791341 0.395670 0.918393i \(-0.370512\pi\)
0.395670 + 0.918393i \(0.370512\pi\)
\(368\) 0 0
\(369\) 4.08390e7 0.0423139
\(370\) 0 0
\(371\) 9.95340e8i 1.01196i
\(372\) 0 0
\(373\) − 1.49519e9i − 1.49181i −0.666051 0.745906i \(-0.732017\pi\)
0.666051 0.745906i \(-0.267983\pi\)
\(374\) 0 0
\(375\) 6.64957e8 0.651155
\(376\) 0 0
\(377\) 1.07621e9 1.03443
\(378\) 0 0
\(379\) − 7.92096e7i − 0.0747379i −0.999302 0.0373689i \(-0.988102\pi\)
0.999302 0.0373689i \(-0.0118977\pi\)
\(380\) 0 0
\(381\) − 1.59383e9i − 1.47640i
\(382\) 0 0
\(383\) −4.80285e8 −0.436820 −0.218410 0.975857i \(-0.570087\pi\)
−0.218410 + 0.975857i \(0.570087\pi\)
\(384\) 0 0
\(385\) 1.69305e9 1.51202
\(386\) 0 0
\(387\) − 1.87084e8i − 0.164077i
\(388\) 0 0
\(389\) − 1.07150e9i − 0.922928i −0.887159 0.461464i \(-0.847324\pi\)
0.887159 0.461464i \(-0.152676\pi\)
\(390\) 0 0
\(391\) 8.53180e8 0.721809
\(392\) 0 0
\(393\) −1.46925e9 −1.22102
\(394\) 0 0
\(395\) 2.66937e9i 2.17931i
\(396\) 0 0
\(397\) 2.03185e9i 1.62976i 0.579627 + 0.814882i \(0.303198\pi\)
−0.579627 + 0.814882i \(0.696802\pi\)
\(398\) 0 0
\(399\) 5.61719e8 0.442705
\(400\) 0 0
\(401\) −2.57759e9 −1.99622 −0.998111 0.0614301i \(-0.980434\pi\)
−0.998111 + 0.0614301i \(0.980434\pi\)
\(402\) 0 0
\(403\) − 7.02451e8i − 0.534624i
\(404\) 0 0
\(405\) 1.04248e9i 0.779782i
\(406\) 0 0
\(407\) −5.65044e8 −0.415434
\(408\) 0 0
\(409\) 3.30242e8 0.238672 0.119336 0.992854i \(-0.461923\pi\)
0.119336 + 0.992854i \(0.461923\pi\)
\(410\) 0 0
\(411\) − 1.03343e9i − 0.734232i
\(412\) 0 0
\(413\) − 2.16257e8i − 0.151059i
\(414\) 0 0
\(415\) −1.91291e9 −1.31379
\(416\) 0 0
\(417\) 2.10331e9 1.42046
\(418\) 0 0
\(419\) − 5.80021e7i − 0.0385207i −0.999815 0.0192604i \(-0.993869\pi\)
0.999815 0.0192604i \(-0.00613114\pi\)
\(420\) 0 0
\(421\) 1.90609e8i 0.124496i 0.998061 + 0.0622480i \(0.0198270\pi\)
−0.998061 + 0.0622480i \(0.980173\pi\)
\(422\) 0 0
\(423\) −6.78129e7 −0.0435633
\(424\) 0 0
\(425\) −9.39731e8 −0.593803
\(426\) 0 0
\(427\) − 1.48585e9i − 0.923586i
\(428\) 0 0
\(429\) − 1.37906e9i − 0.843298i
\(430\) 0 0
\(431\) −2.42923e9 −1.46150 −0.730749 0.682646i \(-0.760829\pi\)
−0.730749 + 0.682646i \(0.760829\pi\)
\(432\) 0 0
\(433\) −2.37902e9 −1.40828 −0.704141 0.710060i \(-0.748668\pi\)
−0.704141 + 0.710060i \(0.748668\pi\)
\(434\) 0 0
\(435\) − 2.23303e9i − 1.30072i
\(436\) 0 0
\(437\) − 3.60037e8i − 0.206378i
\(438\) 0 0
\(439\) 1.33161e9 0.751194 0.375597 0.926783i \(-0.377438\pi\)
0.375597 + 0.926783i \(0.377438\pi\)
\(440\) 0 0
\(441\) 5.25390e7 0.0291707
\(442\) 0 0
\(443\) − 5.02643e8i − 0.274692i −0.990523 0.137346i \(-0.956143\pi\)
0.990523 0.137346i \(-0.0438573\pi\)
\(444\) 0 0
\(445\) − 1.41805e8i − 0.0762834i
\(446\) 0 0
\(447\) −7.25525e8 −0.384217
\(448\) 0 0
\(449\) 3.14785e9 1.64116 0.820580 0.571531i \(-0.193650\pi\)
0.820580 + 0.571531i \(0.193650\pi\)
\(450\) 0 0
\(451\) 3.90876e8i 0.200641i
\(452\) 0 0
\(453\) − 1.55792e9i − 0.787410i
\(454\) 0 0
\(455\) 1.95301e9 0.971998
\(456\) 0 0
\(457\) −2.68422e9 −1.31556 −0.657782 0.753209i \(-0.728505\pi\)
−0.657782 + 0.753209i \(0.728505\pi\)
\(458\) 0 0
\(459\) 3.83442e9i 1.85078i
\(460\) 0 0
\(461\) 1.30434e9i 0.620065i 0.950726 + 0.310033i \(0.100340\pi\)
−0.950726 + 0.310033i \(0.899660\pi\)
\(462\) 0 0
\(463\) −2.86853e9 −1.34315 −0.671577 0.740934i \(-0.734383\pi\)
−0.671577 + 0.740934i \(0.734383\pi\)
\(464\) 0 0
\(465\) −1.45752e9 −0.672246
\(466\) 0 0
\(467\) − 5.96519e8i − 0.271029i −0.990775 0.135514i \(-0.956731\pi\)
0.990775 0.135514i \(-0.0432687\pi\)
\(468\) 0 0
\(469\) 3.02758e8i 0.135516i
\(470\) 0 0
\(471\) 2.06045e9 0.908632
\(472\) 0 0
\(473\) 1.79061e9 0.778012
\(474\) 0 0
\(475\) 3.96561e8i 0.169778i
\(476\) 0 0
\(477\) 5.92497e8i 0.249961i
\(478\) 0 0
\(479\) 2.16068e9 0.898289 0.449144 0.893459i \(-0.351729\pi\)
0.449144 + 0.893459i \(0.351729\pi\)
\(480\) 0 0
\(481\) −6.51805e8 −0.267061
\(482\) 0 0
\(483\) 9.49337e8i 0.383359i
\(484\) 0 0
\(485\) 2.54506e9i 1.01298i
\(486\) 0 0
\(487\) −1.41934e8 −0.0556847 −0.0278424 0.999612i \(-0.508864\pi\)
−0.0278424 + 0.999612i \(0.508864\pi\)
\(488\) 0 0
\(489\) 3.44884e9 1.33380
\(490\) 0 0
\(491\) − 2.38677e9i − 0.909966i −0.890500 0.454983i \(-0.849645\pi\)
0.890500 0.454983i \(-0.150355\pi\)
\(492\) 0 0
\(493\) − 5.91843e9i − 2.22455i
\(494\) 0 0
\(495\) 1.00782e9 0.373479
\(496\) 0 0
\(497\) −5.15769e8 −0.188455
\(498\) 0 0
\(499\) 5.23900e9i 1.88754i 0.330601 + 0.943771i \(0.392749\pi\)
−0.330601 + 0.943771i \(0.607251\pi\)
\(500\) 0 0
\(501\) 4.25772e9i 1.51267i
\(502\) 0 0
\(503\) 3.63292e9 1.27282 0.636411 0.771350i \(-0.280418\pi\)
0.636411 + 0.771350i \(0.280418\pi\)
\(504\) 0 0
\(505\) 2.01106e9 0.694872
\(506\) 0 0
\(507\) 9.32706e8i 0.317846i
\(508\) 0 0
\(509\) 2.58693e9i 0.869505i 0.900550 + 0.434753i \(0.143164\pi\)
−0.900550 + 0.434753i \(0.856836\pi\)
\(510\) 0 0
\(511\) −2.56955e9 −0.851892
\(512\) 0 0
\(513\) 1.61811e9 0.529171
\(514\) 0 0
\(515\) − 2.14105e9i − 0.690718i
\(516\) 0 0
\(517\) − 6.49047e8i − 0.206566i
\(518\) 0 0
\(519\) 7.96876e8 0.250210
\(520\) 0 0
\(521\) 1.08542e8 0.0336253 0.0168127 0.999859i \(-0.494648\pi\)
0.0168127 + 0.999859i \(0.494648\pi\)
\(522\) 0 0
\(523\) − 6.10725e9i − 1.86676i −0.358884 0.933382i \(-0.616843\pi\)
0.358884 0.933382i \(-0.383157\pi\)
\(524\) 0 0
\(525\) − 1.04564e9i − 0.315374i
\(526\) 0 0
\(527\) −3.86300e9 −1.14971
\(528\) 0 0
\(529\) −2.79634e9 −0.821288
\(530\) 0 0
\(531\) − 1.28732e8i − 0.0373125i
\(532\) 0 0
\(533\) 4.50894e8i 0.128982i
\(534\) 0 0
\(535\) 1.66414e8 0.0469841
\(536\) 0 0
\(537\) −1.19764e9 −0.333747
\(538\) 0 0
\(539\) 5.02858e8i 0.138320i
\(540\) 0 0
\(541\) 5.39345e8i 0.146445i 0.997316 + 0.0732227i \(0.0233284\pi\)
−0.997316 + 0.0732227i \(0.976672\pi\)
\(542\) 0 0
\(543\) 1.59348e8 0.0427116
\(544\) 0 0
\(545\) 6.35620e9 1.68194
\(546\) 0 0
\(547\) − 8.82287e7i − 0.0230491i −0.999934 0.0115246i \(-0.996332\pi\)
0.999934 0.0115246i \(-0.00366846\pi\)
\(548\) 0 0
\(549\) − 8.84483e8i − 0.228132i
\(550\) 0 0
\(551\) −2.49754e9 −0.636037
\(552\) 0 0
\(553\) −7.87226e9 −1.97953
\(554\) 0 0
\(555\) 1.35243e9i 0.335807i
\(556\) 0 0
\(557\) − 5.57233e8i − 0.136629i −0.997664 0.0683147i \(-0.978238\pi\)
0.997664 0.0683147i \(-0.0217622\pi\)
\(558\) 0 0
\(559\) 2.06555e9 0.500143
\(560\) 0 0
\(561\) −7.58386e9 −1.81351
\(562\) 0 0
\(563\) 1.17012e9i 0.276344i 0.990408 + 0.138172i \(0.0441227\pi\)
−0.990408 + 0.138172i \(0.955877\pi\)
\(564\) 0 0
\(565\) − 6.12887e9i − 1.42959i
\(566\) 0 0
\(567\) −3.07437e9 −0.708297
\(568\) 0 0
\(569\) −2.39181e9 −0.544295 −0.272147 0.962256i \(-0.587734\pi\)
−0.272147 + 0.962256i \(0.587734\pi\)
\(570\) 0 0
\(571\) 3.15823e9i 0.709933i 0.934879 + 0.354966i \(0.115508\pi\)
−0.934879 + 0.354966i \(0.884492\pi\)
\(572\) 0 0
\(573\) 1.93080e9i 0.428743i
\(574\) 0 0
\(575\) −6.70211e8 −0.147019
\(576\) 0 0
\(577\) 4.03435e9 0.874296 0.437148 0.899390i \(-0.355989\pi\)
0.437148 + 0.899390i \(0.355989\pi\)
\(578\) 0 0
\(579\) − 1.90023e8i − 0.0406846i
\(580\) 0 0
\(581\) − 5.64138e9i − 1.19335i
\(582\) 0 0
\(583\) −5.67087e9 −1.18525
\(584\) 0 0
\(585\) 1.16257e9 0.240090
\(586\) 0 0
\(587\) 2.72240e9i 0.555544i 0.960647 + 0.277772i \(0.0895960\pi\)
−0.960647 + 0.277772i \(0.910404\pi\)
\(588\) 0 0
\(589\) 1.63016e9i 0.328721i
\(590\) 0 0
\(591\) 4.62012e9 0.920655
\(592\) 0 0
\(593\) −2.29251e9 −0.451460 −0.225730 0.974190i \(-0.572477\pi\)
−0.225730 + 0.974190i \(0.572477\pi\)
\(594\) 0 0
\(595\) − 1.07402e10i − 2.09028i
\(596\) 0 0
\(597\) − 4.86346e8i − 0.0935482i
\(598\) 0 0
\(599\) 3.58734e9 0.681991 0.340995 0.940065i \(-0.389236\pi\)
0.340995 + 0.940065i \(0.389236\pi\)
\(600\) 0 0
\(601\) 8.20369e9 1.54152 0.770759 0.637127i \(-0.219877\pi\)
0.770759 + 0.637127i \(0.219877\pi\)
\(602\) 0 0
\(603\) 1.80223e8i 0.0334734i
\(604\) 0 0
\(605\) 3.32259e9i 0.610004i
\(606\) 0 0
\(607\) 4.60087e9 0.834986 0.417493 0.908680i \(-0.362909\pi\)
0.417493 + 0.908680i \(0.362909\pi\)
\(608\) 0 0
\(609\) 6.58546e9 1.18148
\(610\) 0 0
\(611\) − 7.48707e8i − 0.132791i
\(612\) 0 0
\(613\) 8.55728e9i 1.50046i 0.661178 + 0.750229i \(0.270057\pi\)
−0.661178 + 0.750229i \(0.729943\pi\)
\(614\) 0 0
\(615\) 9.35561e8 0.162184
\(616\) 0 0
\(617\) −2.58089e9 −0.442355 −0.221178 0.975234i \(-0.570990\pi\)
−0.221178 + 0.975234i \(0.570990\pi\)
\(618\) 0 0
\(619\) − 5.26641e9i − 0.892478i −0.894914 0.446239i \(-0.852763\pi\)
0.894914 0.446239i \(-0.147237\pi\)
\(620\) 0 0
\(621\) 2.73469e9i 0.458234i
\(622\) 0 0
\(623\) 4.18197e8 0.0692903
\(624\) 0 0
\(625\) −7.48796e9 −1.22683
\(626\) 0 0
\(627\) 3.20035e9i 0.518514i
\(628\) 0 0
\(629\) 3.58449e9i 0.574314i
\(630\) 0 0
\(631\) −8.32515e9 −1.31914 −0.659568 0.751645i \(-0.729261\pi\)
−0.659568 + 0.751645i \(0.729261\pi\)
\(632\) 0 0
\(633\) 7.87635e9 1.23428
\(634\) 0 0
\(635\) 1.28601e10i 1.99313i
\(636\) 0 0
\(637\) 5.80071e8i 0.0889187i
\(638\) 0 0
\(639\) −3.07022e8 −0.0465496
\(640\) 0 0
\(641\) 4.26190e9 0.639146 0.319573 0.947562i \(-0.396460\pi\)
0.319573 + 0.947562i \(0.396460\pi\)
\(642\) 0 0
\(643\) 1.26588e10i 1.87782i 0.344167 + 0.938908i \(0.388161\pi\)
−0.344167 + 0.938908i \(0.611839\pi\)
\(644\) 0 0
\(645\) − 4.28582e9i − 0.628890i
\(646\) 0 0
\(647\) 7.38061e9 1.07134 0.535670 0.844427i \(-0.320059\pi\)
0.535670 + 0.844427i \(0.320059\pi\)
\(648\) 0 0
\(649\) 1.23211e9 0.176926
\(650\) 0 0
\(651\) − 4.29837e9i − 0.610620i
\(652\) 0 0
\(653\) − 3.21579e9i − 0.451951i −0.974133 0.225975i \(-0.927443\pi\)
0.974133 0.225975i \(-0.0725569\pi\)
\(654\) 0 0
\(655\) 1.18549e10 1.64836
\(656\) 0 0
\(657\) −1.52958e9 −0.210423
\(658\) 0 0
\(659\) − 5.17004e9i − 0.703711i −0.936054 0.351856i \(-0.885551\pi\)
0.936054 0.351856i \(-0.114449\pi\)
\(660\) 0 0
\(661\) − 1.95604e9i − 0.263435i −0.991287 0.131717i \(-0.957951\pi\)
0.991287 0.131717i \(-0.0420491\pi\)
\(662\) 0 0
\(663\) −8.74835e9 −1.16581
\(664\) 0 0
\(665\) −4.53232e9 −0.597647
\(666\) 0 0
\(667\) − 4.22099e9i − 0.550775i
\(668\) 0 0
\(669\) − 4.34434e9i − 0.560961i
\(670\) 0 0
\(671\) 8.46551e9 1.08174
\(672\) 0 0
\(673\) 1.54679e9 0.195605 0.0978024 0.995206i \(-0.468819\pi\)
0.0978024 + 0.995206i \(0.468819\pi\)
\(674\) 0 0
\(675\) − 3.01211e9i − 0.376971i
\(676\) 0 0
\(677\) − 8.55209e9i − 1.05928i −0.848222 0.529642i \(-0.822326\pi\)
0.848222 0.529642i \(-0.177674\pi\)
\(678\) 0 0
\(679\) −7.50565e9 −0.920119
\(680\) 0 0
\(681\) −6.48061e9 −0.786323
\(682\) 0 0
\(683\) − 7.26976e9i − 0.873067i −0.899688 0.436534i \(-0.856206\pi\)
0.899688 0.436534i \(-0.143794\pi\)
\(684\) 0 0
\(685\) 8.33837e9i 0.991206i
\(686\) 0 0
\(687\) 2.12010e9 0.249463
\(688\) 0 0
\(689\) −6.54162e9 −0.761935
\(690\) 0 0
\(691\) 5.19893e9i 0.599434i 0.954028 + 0.299717i \(0.0968922\pi\)
−0.954028 + 0.299717i \(0.903108\pi\)
\(692\) 0 0
\(693\) 2.97218e9i 0.339241i
\(694\) 0 0
\(695\) −1.69709e10 −1.91760
\(696\) 0 0
\(697\) 2.47961e9 0.277376
\(698\) 0 0
\(699\) − 7.21572e9i − 0.799116i
\(700\) 0 0
\(701\) − 1.35221e10i − 1.48262i −0.671163 0.741310i \(-0.734205\pi\)
0.671163 0.741310i \(-0.265795\pi\)
\(702\) 0 0
\(703\) 1.51263e9 0.164206
\(704\) 0 0
\(705\) −1.55349e9 −0.166974
\(706\) 0 0
\(707\) 5.93083e9i 0.631172i
\(708\) 0 0
\(709\) 1.05901e10i 1.11594i 0.829862 + 0.557969i \(0.188419\pi\)
−0.829862 + 0.557969i \(0.811581\pi\)
\(710\) 0 0
\(711\) −4.68613e9 −0.488957
\(712\) 0 0
\(713\) −2.75507e9 −0.284655
\(714\) 0 0
\(715\) 1.11271e10i 1.13845i
\(716\) 0 0
\(717\) 3.38799e9i 0.343261i
\(718\) 0 0
\(719\) −1.49161e10 −1.49659 −0.748297 0.663363i \(-0.769128\pi\)
−0.748297 + 0.663363i \(0.769128\pi\)
\(720\) 0 0
\(721\) 6.31417e9 0.627398
\(722\) 0 0
\(723\) − 8.53800e9i − 0.840180i
\(724\) 0 0
\(725\) 4.64919e9i 0.453100i
\(726\) 0 0
\(727\) 8.90159e9 0.859206 0.429603 0.903018i \(-0.358654\pi\)
0.429603 + 0.903018i \(0.358654\pi\)
\(728\) 0 0
\(729\) −1.15808e10 −1.10711
\(730\) 0 0
\(731\) − 1.13591e10i − 1.07556i
\(732\) 0 0
\(733\) − 7.99792e9i − 0.750090i −0.927007 0.375045i \(-0.877627\pi\)
0.927007 0.375045i \(-0.122373\pi\)
\(734\) 0 0
\(735\) 1.20359e9 0.111808
\(736\) 0 0
\(737\) −1.72494e9 −0.158722
\(738\) 0 0
\(739\) − 1.03852e10i − 0.946588i −0.880905 0.473294i \(-0.843065\pi\)
0.880905 0.473294i \(-0.156935\pi\)
\(740\) 0 0
\(741\) 3.69175e9i 0.333326i
\(742\) 0 0
\(743\) 3.73477e9 0.334044 0.167022 0.985953i \(-0.446585\pi\)
0.167022 + 0.985953i \(0.446585\pi\)
\(744\) 0 0
\(745\) 5.85402e9 0.518689
\(746\) 0 0
\(747\) − 3.35815e9i − 0.294766i
\(748\) 0 0
\(749\) 4.90772e8i 0.0426770i
\(750\) 0 0
\(751\) 1.34330e10 1.15726 0.578631 0.815589i \(-0.303587\pi\)
0.578631 + 0.815589i \(0.303587\pi\)
\(752\) 0 0
\(753\) 4.74963e9 0.405394
\(754\) 0 0
\(755\) 1.25703e10i 1.06300i
\(756\) 0 0
\(757\) − 6.78007e9i − 0.568065i −0.958815 0.284033i \(-0.908328\pi\)
0.958815 0.284033i \(-0.0916724\pi\)
\(758\) 0 0
\(759\) −5.40877e9 −0.449006
\(760\) 0 0
\(761\) 8.01137e9 0.658962 0.329481 0.944162i \(-0.393126\pi\)
0.329481 + 0.944162i \(0.393126\pi\)
\(762\) 0 0
\(763\) 1.87451e10i 1.52775i
\(764\) 0 0
\(765\) − 6.39335e9i − 0.516314i
\(766\) 0 0
\(767\) 1.42130e9 0.113737
\(768\) 0 0
\(769\) 1.46553e10 1.16213 0.581063 0.813858i \(-0.302637\pi\)
0.581063 + 0.813858i \(0.302637\pi\)
\(770\) 0 0
\(771\) 5.12609e9i 0.402806i
\(772\) 0 0
\(773\) 2.33296e10i 1.81668i 0.418233 + 0.908340i \(0.362650\pi\)
−0.418233 + 0.908340i \(0.637350\pi\)
\(774\) 0 0
\(775\) 3.03456e9 0.234174
\(776\) 0 0
\(777\) −3.98847e9 −0.305023
\(778\) 0 0
\(779\) − 1.04638e9i − 0.0793064i
\(780\) 0 0
\(781\) − 2.93855e9i − 0.220726i
\(782\) 0 0
\(783\) 1.89703e10 1.41224
\(784\) 0 0
\(785\) −1.66250e10 −1.22664
\(786\) 0 0
\(787\) 5.27740e8i 0.0385930i 0.999814 + 0.0192965i \(0.00614264\pi\)
−0.999814 + 0.0192965i \(0.993857\pi\)
\(788\) 0 0
\(789\) − 1.74187e10i − 1.26254i
\(790\) 0 0
\(791\) 1.80747e10 1.29853
\(792\) 0 0
\(793\) 9.76536e9 0.695396
\(794\) 0 0
\(795\) 1.35732e10i 0.958072i
\(796\) 0 0
\(797\) 2.41514e9i 0.168981i 0.996424 + 0.0844907i \(0.0269263\pi\)
−0.996424 + 0.0844907i \(0.973074\pi\)
\(798\) 0 0
\(799\) −4.11738e9 −0.285566
\(800\) 0 0
\(801\) 2.48940e8 0.0171152
\(802\) 0 0
\(803\) − 1.46398e10i − 0.997773i
\(804\) 0 0
\(805\) − 7.65988e9i − 0.517531i
\(806\) 0 0
\(807\) −1.38645e10 −0.928636
\(808\) 0 0
\(809\) 1.16364e10 0.772679 0.386340 0.922357i \(-0.373739\pi\)
0.386340 + 0.922357i \(0.373739\pi\)
\(810\) 0 0
\(811\) 1.44359e10i 0.950320i 0.879899 + 0.475160i \(0.157610\pi\)
−0.879899 + 0.475160i \(0.842390\pi\)
\(812\) 0 0
\(813\) 1.77962e10i 1.16148i
\(814\) 0 0
\(815\) −2.78275e10 −1.80062
\(816\) 0 0
\(817\) −4.79348e9 −0.307521
\(818\) 0 0
\(819\) 3.42855e9i 0.218080i
\(820\) 0 0
\(821\) 7.63805e9i 0.481706i 0.970562 + 0.240853i \(0.0774271\pi\)
−0.970562 + 0.240853i \(0.922573\pi\)
\(822\) 0 0
\(823\) 2.16446e10 1.35348 0.676738 0.736224i \(-0.263393\pi\)
0.676738 + 0.736224i \(0.263393\pi\)
\(824\) 0 0
\(825\) 5.95746e9 0.369379
\(826\) 0 0
\(827\) − 1.57823e10i − 0.970288i −0.874434 0.485144i \(-0.838767\pi\)
0.874434 0.485144i \(-0.161233\pi\)
\(828\) 0 0
\(829\) 2.63296e10i 1.60510i 0.596582 + 0.802552i \(0.296525\pi\)
−0.596582 + 0.802552i \(0.703475\pi\)
\(830\) 0 0
\(831\) −1.27947e10 −0.773442
\(832\) 0 0
\(833\) 3.19000e9 0.191220
\(834\) 0 0
\(835\) − 3.43541e10i − 2.04210i
\(836\) 0 0
\(837\) − 1.23820e10i − 0.729882i
\(838\) 0 0
\(839\) 1.84995e9 0.108142 0.0540710 0.998537i \(-0.482780\pi\)
0.0540710 + 0.998537i \(0.482780\pi\)
\(840\) 0 0
\(841\) −1.20307e10 −0.697438
\(842\) 0 0
\(843\) 5.16767e9i 0.297097i
\(844\) 0 0
\(845\) − 7.52569e9i − 0.429090i
\(846\) 0 0
\(847\) −9.79866e9 −0.554083
\(848\) 0 0
\(849\) −1.60451e10 −0.899839
\(850\) 0 0
\(851\) 2.55643e9i 0.142194i
\(852\) 0 0
\(853\) 6.28088e7i 0.00346496i 0.999998 + 0.00173248i \(0.000551466\pi\)
−0.999998 + 0.00173248i \(0.999449\pi\)
\(854\) 0 0
\(855\) −2.69796e9 −0.147623
\(856\) 0 0
\(857\) −6.25595e9 −0.339516 −0.169758 0.985486i \(-0.554299\pi\)
−0.169758 + 0.985486i \(0.554299\pi\)
\(858\) 0 0
\(859\) − 2.48119e10i − 1.33562i −0.744331 0.667811i \(-0.767231\pi\)
0.744331 0.667811i \(-0.232769\pi\)
\(860\) 0 0
\(861\) 2.75907e9i 0.147316i
\(862\) 0 0
\(863\) −1.28944e10 −0.682909 −0.341455 0.939898i \(-0.610920\pi\)
−0.341455 + 0.939898i \(0.610920\pi\)
\(864\) 0 0
\(865\) −6.42973e9 −0.337782
\(866\) 0 0
\(867\) 3.16077e10i 1.64712i
\(868\) 0 0
\(869\) − 4.48516e10i − 2.31851i
\(870\) 0 0
\(871\) −1.98980e9 −0.102034
\(872\) 0 0
\(873\) −4.46789e9 −0.227276
\(874\) 0 0
\(875\) − 1.58229e10i − 0.798468i
\(876\) 0 0
\(877\) − 2.75670e10i − 1.38004i −0.723792 0.690018i \(-0.757603\pi\)
0.723792 0.690018i \(-0.242397\pi\)
\(878\) 0 0
\(879\) −8.08177e9 −0.401371
\(880\) 0 0
\(881\) 2.74343e10 1.35169 0.675847 0.737042i \(-0.263778\pi\)
0.675847 + 0.737042i \(0.263778\pi\)
\(882\) 0 0
\(883\) 2.09906e10i 1.02604i 0.858378 + 0.513018i \(0.171473\pi\)
−0.858378 + 0.513018i \(0.828527\pi\)
\(884\) 0 0
\(885\) − 2.94905e9i − 0.143015i
\(886\) 0 0
\(887\) 2.39134e10 1.15056 0.575280 0.817957i \(-0.304893\pi\)
0.575280 + 0.817957i \(0.304893\pi\)
\(888\) 0 0
\(889\) −3.79257e10 −1.81041
\(890\) 0 0
\(891\) − 1.75160e10i − 0.829588i
\(892\) 0 0
\(893\) 1.73751e9i 0.0816483i
\(894\) 0 0
\(895\) 9.66337e9 0.450555
\(896\) 0 0
\(897\) −6.23927e9 −0.288643
\(898\) 0 0
\(899\) 1.91117e10i 0.877282i
\(900\) 0 0
\(901\) 3.59744e10i 1.63854i
\(902\) 0 0
\(903\) 1.26393e10 0.571238
\(904\) 0 0
\(905\) −1.28572e9 −0.0576603
\(906\) 0 0
\(907\) 3.84425e10i 1.71075i 0.518012 + 0.855373i \(0.326672\pi\)
−0.518012 + 0.855373i \(0.673328\pi\)
\(908\) 0 0
\(909\) 3.53045e9i 0.155904i
\(910\) 0 0
\(911\) −3.64507e10 −1.59732 −0.798660 0.601783i \(-0.794457\pi\)
−0.798660 + 0.601783i \(0.794457\pi\)
\(912\) 0 0
\(913\) 3.21413e10 1.39771
\(914\) 0 0
\(915\) − 2.02622e10i − 0.874405i
\(916\) 0 0
\(917\) 3.49613e10i 1.49725i
\(918\) 0 0
\(919\) −2.37343e10 −1.00872 −0.504361 0.863493i \(-0.668272\pi\)
−0.504361 + 0.863493i \(0.668272\pi\)
\(920\) 0 0
\(921\) 6.39111e8 0.0269567
\(922\) 0 0
\(923\) − 3.38976e9i − 0.141894i
\(924\) 0 0
\(925\) − 2.81577e9i − 0.116977i
\(926\) 0 0
\(927\) 3.75864e9 0.154972
\(928\) 0 0
\(929\) −4.23952e10 −1.73485 −0.867424 0.497570i \(-0.834226\pi\)
−0.867424 + 0.497570i \(0.834226\pi\)
\(930\) 0 0
\(931\) − 1.34616e9i − 0.0546730i
\(932\) 0 0
\(933\) − 1.96139e10i − 0.790638i
\(934\) 0 0
\(935\) 6.11916e10 2.44823
\(936\) 0 0
\(937\) 3.70014e10 1.46937 0.734683 0.678411i \(-0.237331\pi\)
0.734683 + 0.678411i \(0.237331\pi\)
\(938\) 0 0
\(939\) − 1.30595e10i − 0.514750i
\(940\) 0 0
\(941\) 2.10617e10i 0.824005i 0.911183 + 0.412003i \(0.135171\pi\)
−0.911183 + 0.412003i \(0.864829\pi\)
\(942\) 0 0
\(943\) 1.76844e9 0.0686752
\(944\) 0 0
\(945\) 3.44256e10 1.32700
\(946\) 0 0
\(947\) − 1.35146e10i − 0.517104i −0.965997 0.258552i \(-0.916755\pi\)
0.965997 0.258552i \(-0.0832454\pi\)
\(948\) 0 0
\(949\) − 1.68877e10i − 0.641416i
\(950\) 0 0
\(951\) 4.26628e10 1.60849
\(952\) 0 0
\(953\) 4.26831e10 1.59746 0.798731 0.601688i \(-0.205505\pi\)
0.798731 + 0.601688i \(0.205505\pi\)
\(954\) 0 0
\(955\) − 1.55790e10i − 0.578799i
\(956\) 0 0
\(957\) 3.75201e10i 1.38380i
\(958\) 0 0
\(959\) −2.45907e10 −0.900340
\(960\) 0 0
\(961\) −1.50383e10 −0.546597
\(962\) 0 0
\(963\) 2.92142e8i 0.0105415i
\(964\) 0 0
\(965\) 1.53323e9i 0.0549239i
\(966\) 0 0
\(967\) −2.32274e10 −0.826053 −0.413026 0.910719i \(-0.635528\pi\)
−0.413026 + 0.910719i \(0.635528\pi\)
\(968\) 0 0
\(969\) 2.03021e10 0.716817
\(970\) 0 0
\(971\) 4.85678e10i 1.70248i 0.524780 + 0.851238i \(0.324148\pi\)
−0.524780 + 0.851238i \(0.675852\pi\)
\(972\) 0 0
\(973\) − 5.00491e10i − 1.74181i
\(974\) 0 0
\(975\) 6.87221e9 0.237454
\(976\) 0 0
\(977\) 1.54549e10 0.530193 0.265096 0.964222i \(-0.414596\pi\)
0.265096 + 0.964222i \(0.414596\pi\)
\(978\) 0 0
\(979\) 2.38264e9i 0.0811557i
\(980\) 0 0
\(981\) 1.11584e10i 0.377364i
\(982\) 0 0
\(983\) 1.59192e10 0.534546 0.267273 0.963621i \(-0.413877\pi\)
0.267273 + 0.963621i \(0.413877\pi\)
\(984\) 0 0
\(985\) −3.72782e10 −1.24288
\(986\) 0 0
\(987\) − 4.58142e9i − 0.151667i
\(988\) 0 0
\(989\) − 8.10126e9i − 0.266296i
\(990\) 0 0
\(991\) −1.49883e10 −0.489208 −0.244604 0.969623i \(-0.578658\pi\)
−0.244604 + 0.969623i \(0.578658\pi\)
\(992\) 0 0
\(993\) −1.16019e10 −0.376016
\(994\) 0 0
\(995\) 3.92416e9i 0.126289i
\(996\) 0 0
\(997\) 3.44101e10i 1.09965i 0.835281 + 0.549824i \(0.185305\pi\)
−0.835281 + 0.549824i \(0.814695\pi\)
\(998\) 0 0
\(999\) −1.14893e10 −0.364598
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.17.5 6
3.2 odd 2 288.8.d.b.145.5 6
4.3 odd 2 8.8.b.a.5.2 yes 6
8.3 odd 2 8.8.b.a.5.1 6
8.5 even 2 inner 32.8.b.a.17.2 6
12.11 even 2 72.8.d.b.37.5 6
16.3 odd 4 256.8.a.r.1.5 6
16.5 even 4 256.8.a.q.1.5 6
16.11 odd 4 256.8.a.r.1.2 6
16.13 even 4 256.8.a.q.1.2 6
24.5 odd 2 288.8.d.b.145.2 6
24.11 even 2 72.8.d.b.37.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.b.a.5.1 6 8.3 odd 2
8.8.b.a.5.2 yes 6 4.3 odd 2
32.8.b.a.17.2 6 8.5 even 2 inner
32.8.b.a.17.5 6 1.1 even 1 trivial
72.8.d.b.37.5 6 12.11 even 2
72.8.d.b.37.6 6 24.11 even 2
256.8.a.q.1.2 6 16.13 even 4
256.8.a.q.1.5 6 16.5 even 4
256.8.a.r.1.2 6 16.11 odd 4
256.8.a.r.1.5 6 16.3 odd 4
288.8.d.b.145.2 6 24.5 odd 2
288.8.d.b.145.5 6 3.2 odd 2