Properties

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

$q$-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 \(-0.858521\pi\)
0.902839 0.429979i \(-0.141479\pi\)
\(4\) 0 0
\(5\) 324.492i 1.16094i 0.814283 + 0.580468i \(0.197130\pi\)
−0.814283 + 0.580468i \(0.802870\pi\)
\(6\) 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 \(-0.788128\pi\)
0.786537 0.617544i \(-0.211872\pi\)
\(12\) 0 0
\(13\) − 6289.38i − 0.793973i −0.917824 0.396987i \(-0.870056\pi\)
0.917824 0.396987i \(-0.129944\pi\)
\(14\) 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 \(0.0784903\pi\)
−0.969752 + 0.244093i \(0.921510\pi\)
\(20\) 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 \(0.225830\pi\)
−0.758710 + 0.651429i \(0.774170\pi\)
\(30\) 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 \(-0.946211\pi\)
0.985756 0.168180i \(-0.0537890\pi\)
\(38\) 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 \(0.101992\pi\)
−0.949104 + 0.314962i \(0.898008\pi\)
\(44\) 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 \(-0.840700\pi\)
0.877365 0.479824i \(-0.159300\pi\)
\(54\) 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 \(0.0228185\pi\)
−0.997432 + 0.0716250i \(0.977182\pi\)
\(60\) 0 0
\(61\) 1.55268e6i 0.875843i 0.899013 + 0.437922i \(0.144285\pi\)
−0.899013 + 0.437922i \(0.855715\pi\)
\(62\) 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 \(-0.979533\pi\)
0.997933 0.0642555i \(-0.0204673\pi\)
\(68\) 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 \(0.191445\pi\)
−0.824520 + 0.565833i \(0.808555\pi\)
\(84\) 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 \(-0.903256\pi\)
0.954168 0.299272i \(-0.0967439\pi\)
\(102\) 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 \(-0.993558\pi\)
0.999795 0.0202354i \(-0.00644158\pi\)
\(108\) 0 0
\(109\) − 1.95882e7i − 1.44878i −0.689393 0.724388i \(-0.742123\pi\)
0.689393 0.724388i \(-0.257877\pi\)
\(110\) 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 \(-0.748727\pi\)
0.704274 0.709928i \(-0.251273\pi\)
\(132\) 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 \(0.309325\pi\)
−0.563836 + 0.825886i \(0.690675\pi\)
\(140\) 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 \(-0.928287\pi\)
0.974729 0.223392i \(-0.0717131\pi\)
\(150\) 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 \(0.177170\pi\)
−0.849058 + 0.528300i \(0.822830\pi\)
\(158\) 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 \(0.282504\pi\)
−0.631343 + 0.775504i \(0.717496\pi\)
\(164\) 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 \(0.0464720\pi\)
−0.989361 + 0.145478i \(0.953528\pi\)
\(174\) 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 \(-0.937838\pi\)
0.980992 0.194048i \(-0.0621618\pi\)
\(180\) 0 0
\(181\) 3.96227e6i 0.0496671i 0.999692 + 0.0248335i \(0.00790558\pi\)
−0.999692 + 0.0248335i \(0.992094\pi\)
\(182\) 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 \(0.179798\pi\)
−0.844668 + 0.535290i \(0.820202\pi\)
\(198\) 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 \(0.254776\pi\)
−0.696418 + 0.717636i \(0.745224\pi\)
\(212\) 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 \(-0.848857\pi\)
0.889371 0.457187i \(-0.151143\pi\)
\(228\) 0 0
\(229\) 5.27173e7i 0.290088i 0.989425 + 0.145044i \(0.0463323\pi\)
−0.989425 + 0.145044i \(0.953668\pi\)
\(230\) 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 \(0.0757401\pi\)
−0.971825 + 0.235706i \(0.924260\pi\)
\(252\) 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 \(-0.818450\pi\)
0.841709 0.539931i \(-0.181550\pi\)
\(270\) 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 \(-0.851532\pi\)
0.893181 0.449697i \(-0.148468\pi\)
\(278\) 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 \(-0.824743\pi\)
0.852218 0.523187i \(-0.175257\pi\)
\(284\) 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 \(-0.925026\pi\)
0.972389 0.233366i \(-0.0749741\pi\)
\(294\) 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 \(0.00498916\pi\)
−0.999877 + 0.0156733i \(0.995011\pi\)
\(308\) 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 \(0.384792\pi\)
−0.354086 + 0.935213i \(0.615208\pi\)
\(318\) 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 \(-0.929843\pi\)
0.975809 0.218624i \(-0.0701569\pi\)
\(332\) 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 \(0.251686\pi\)
−0.703352 + 0.710841i \(0.748314\pi\)
\(348\) 0 0
\(349\) − 1.38337e9i − 1.74201i −0.491278 0.871003i \(-0.663470\pi\)
0.491278 0.871003i \(-0.336530\pi\)
\(350\) 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 \(0.267983\pi\)
−0.666051 + 0.745906i \(0.732017\pi\)
\(374\) 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 \(0.0118977\pi\)
−0.999302 + 0.0373689i \(0.988102\pi\)
\(380\) 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 \(0.152676\pi\)
−0.887159 + 0.461464i \(0.847324\pi\)
\(390\) 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 \(-0.696802\pi\)
0.579627 0.814882i \(-0.303198\pi\)
\(398\) 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 \(0.00613114\pi\)
−0.999815 + 0.0192604i \(0.993869\pi\)
\(420\) 0 0
\(421\) − 1.90609e8i − 0.124496i −0.998061 0.0622480i \(-0.980173\pi\)
0.998061 0.0622480i \(-0.0198270\pi\)
\(422\) 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 \(0.0438573\pi\)
−0.990523 + 0.137346i \(0.956143\pi\)
\(444\) 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 \(-0.899660\pi\)
0.950726 0.310033i \(-0.100340\pi\)
\(462\) 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 \(0.0432687\pi\)
−0.990775 + 0.135514i \(0.956731\pi\)
\(468\) 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 \(0.150355\pi\)
−0.890500 + 0.454983i \(0.849645\pi\)
\(492\) 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 \(-0.607251\pi\)
0.330601 0.943771i \(-0.392749\pi\)
\(500\) 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 \(-0.856836\pi\)
0.900550 0.434753i \(-0.143164\pi\)
\(510\) 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 \(0.383157\pi\)
−0.358884 + 0.933382i \(0.616843\pi\)
\(524\) 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 \(-0.976672\pi\)
0.997316 0.0732227i \(-0.0233284\pi\)
\(542\) 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 \(0.00366846\pi\)
−0.999934 + 0.0115246i \(0.996332\pi\)
\(548\) 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 \(0.0217622\pi\)
−0.997664 + 0.0683147i \(0.978238\pi\)
\(558\) 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 \(-0.955877\pi\)
0.990408 0.138172i \(-0.0441227\pi\)
\(564\) 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 \(-0.884492\pi\)
0.934879 0.354966i \(-0.115508\pi\)
\(572\) 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 \(-0.910404\pi\)
0.960647 0.277772i \(-0.0895960\pi\)
\(588\) 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 \(-0.729943\pi\)
0.661178 0.750229i \(-0.270057\pi\)
\(614\) 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 \(0.147237\pi\)
−0.894914 + 0.446239i \(0.852763\pi\)
\(620\) 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 \(-0.611839\pi\)
0.344167 0.938908i \(-0.388161\pi\)
\(644\) 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 \(0.0725569\pi\)
−0.974133 + 0.225975i \(0.927443\pi\)
\(654\) 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 \(0.114449\pi\)
−0.936054 + 0.351856i \(0.885551\pi\)
\(660\) 0 0
\(661\) 1.95604e9i 0.263435i 0.991287 + 0.131717i \(0.0420491\pi\)
−0.991287 + 0.131717i \(0.957951\pi\)
\(662\) 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 \(0.177674\pi\)
−0.848222 + 0.529642i \(0.822326\pi\)
\(678\) 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 \(0.143794\pi\)
−0.899688 + 0.436534i \(0.856206\pi\)
\(684\) 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 \(-0.903108\pi\)
0.954028 0.299717i \(-0.0968922\pi\)
\(692\) 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 \(0.265795\pi\)
−0.671163 + 0.741310i \(0.734205\pi\)
\(702\) 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 \(-0.811581\pi\)
0.829862 0.557969i \(-0.188419\pi\)
\(710\) 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 \(0.122373\pi\)
−0.927007 + 0.375045i \(0.877627\pi\)
\(734\) 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 \(0.156935\pi\)
−0.880905 + 0.473294i \(0.843065\pi\)
\(740\) 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 \(0.0916724\pi\)
−0.958815 + 0.284033i \(0.908328\pi\)
\(758\) 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 \(-0.637350\pi\)
0.418233 0.908340i \(-0.362650\pi\)
\(774\) 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 \(-0.993857\pi\)
0.999814 0.0192965i \(-0.00614264\pi\)
\(788\) 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 \(-0.973074\pi\)
0.996424 0.0844907i \(-0.0269263\pi\)
\(798\) 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 \(-0.842390\pi\)
0.879899 0.475160i \(-0.157610\pi\)
\(812\) 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 \(-0.922573\pi\)
0.970562 0.240853i \(-0.0774271\pi\)
\(822\) 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 \(0.161233\pi\)
−0.874434 + 0.485144i \(0.838767\pi\)
\(828\) 0 0
\(829\) − 2.63296e10i − 1.60510i −0.596582 0.802552i \(-0.703475\pi\)
0.596582 0.802552i \(-0.296525\pi\)
\(830\) 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 \(-0.999449\pi\)
0.999998 0.00173248i \(-0.000551466\pi\)
\(854\) 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 \(0.232769\pi\)
−0.744331 + 0.667811i \(0.767231\pi\)
\(860\) 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 \(0.242397\pi\)
−0.723792 + 0.690018i \(0.757603\pi\)
\(878\) 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 \(-0.828527\pi\)
0.858378 0.513018i \(-0.171473\pi\)
\(884\) 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 \(-0.673328\pi\)
0.518012 0.855373i \(-0.326672\pi\)
\(908\) 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 \(-0.864829\pi\)
0.911183 0.412003i \(-0.135171\pi\)
\(942\) 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 \(0.0832454\pi\)
−0.965997 + 0.258552i \(0.916755\pi\)
\(948\) 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 \(-0.675852\pi\)
0.524780 0.851238i \(-0.324148\pi\)
\(972\) 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 \(-0.814695\pi\)
0.835281 0.549824i \(-0.185305\pi\)
\(998\) 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.17.2 6
3.2 odd 2 288.8.d.b.145.2 6
4.3 odd 2 8.8.b.a.5.1 6
8.3 odd 2 8.8.b.a.5.2 yes 6
8.5 even 2 inner 32.8.b.a.17.5 6
12.11 even 2 72.8.d.b.37.6 6
16.3 odd 4 256.8.a.r.1.2 6
16.5 even 4 256.8.a.q.1.2 6
16.11 odd 4 256.8.a.r.1.5 6
16.13 even 4 256.8.a.q.1.5 6
24.5 odd 2 288.8.d.b.145.5 6
24.11 even 2 72.8.d.b.37.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.b.a.5.1 6 4.3 odd 2
8.8.b.a.5.2 yes 6 8.3 odd 2
32.8.b.a.17.2 6 1.1 even 1 trivial
32.8.b.a.17.5 6 8.5 even 2 inner
72.8.d.b.37.5 6 24.11 even 2
72.8.d.b.37.6 6 12.11 even 2
256.8.a.q.1.2 6 16.5 even 4
256.8.a.q.1.5 6 16.13 even 4
256.8.a.r.1.2 6 16.3 odd 4
256.8.a.r.1.5 6 16.11 odd 4
288.8.d.b.145.2 6 3.2 odd 2
288.8.d.b.145.5 6 24.5 odd 2