Properties

Label 256.8.a.r.1.2
Level $256$
Weight $8$
Character 256.1
Self dual yes
Analytic conductor $79.971$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

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

Embedding invariants

Embedding label 1.2
Root \(-5.81430\) of defining polynomial
Character \(\chi\) \(=\) 256.1

$q$-expansion

\(f(q)\) \(=\) \(q-40.2163 q^{3} +324.492 q^{5} +956.960 q^{7} -569.651 q^{9} +O(q^{10})\) \(q-40.2163 q^{3} +324.492 q^{5} +956.960 q^{7} -569.651 q^{9} +5452.20 q^{11} +6289.38 q^{13} -13049.8 q^{15} +34587.3 q^{17} +14595.6 q^{19} -38485.4 q^{21} +24667.5 q^{23} +27169.8 q^{25} +110862. q^{27} -171116. q^{29} +111688. q^{31} -219267. q^{33} +310526. q^{35} -103636. q^{37} -252935. q^{39} -71691.3 q^{41} -328419. q^{43} -184847. q^{45} +119043. q^{47} +92230.3 q^{49} -1.39097e6 q^{51} -1.04011e6 q^{53} +1.76919e6 q^{55} -586982. q^{57} -225984. q^{59} -1.55268e6 q^{61} -545133. q^{63} +2.04085e6 q^{65} -316375. q^{67} -992033. q^{69} -538965. q^{71} +2.68512e6 q^{73} -1.09267e6 q^{75} +5.21754e6 q^{77} +8.22632e6 q^{79} -3.21264e6 q^{81} +5.89510e6 q^{83} +1.12233e7 q^{85} +6.88164e6 q^{87} -437005. q^{89} +6.01868e6 q^{91} -4.49169e6 q^{93} +4.73616e6 q^{95} -7.84322e6 q^{97} -3.10585e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 688 q^{7} + 2918 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 688 q^{7} + 2918 q^{9} + 17872 q^{15} + 1452 q^{17} + 1296 q^{23} + 39314 q^{25} - 89280 q^{31} + 53880 q^{33} + 328208 q^{39} - 521244 q^{41} + 1566432 q^{47} - 511050 q^{49} + 3270256 q^{55} + 1889896 q^{57} + 5776816 q^{63} + 1416480 q^{65} + 7597104 q^{71} - 2089564 q^{73} + 16015904 q^{79} - 723058 q^{81} + 37453776 q^{87} - 2169084 q^{89} + 48537936 q^{95} - 1088308 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −40.2163 −0.859959 −0.429979 0.902839i \(-0.641479\pi\)
−0.429979 + 0.902839i \(0.641479\pi\)
\(4\) 0 0
\(5\) 324.492 1.16094 0.580468 0.814283i \(-0.302870\pi\)
0.580468 + 0.814283i \(0.302870\pi\)
\(6\) 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
\(11\) 5452.20 1.23509 0.617544 0.786537i \(-0.288128\pi\)
0.617544 + 0.786537i \(0.288128\pi\)
\(12\) 0 0
\(13\) 6289.38 0.793973 0.396987 0.917824i \(-0.370056\pi\)
0.396987 + 0.917824i \(0.370056\pi\)
\(14\) 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
\(19\) 14595.6 0.488186 0.244093 0.969752i \(-0.421510\pi\)
0.244093 + 0.969752i \(0.421510\pi\)
\(20\) 0 0
\(21\) −38485.4 −0.906835
\(22\) 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. 1.08395
\(28\) 0 0
\(29\) −171116. −1.30286 −0.651429 0.758710i \(-0.725830\pi\)
−0.651429 + 0.758710i \(0.725830\pi\)
\(30\) 0 0
\(31\) 111688. 0.673352 0.336676 0.941620i \(-0.390697\pi\)
0.336676 + 0.941620i \(0.390697\pi\)
\(32\) 0 0
\(33\) −219267. −1.06212
\(34\) 0 0
\(35\) 310526. 1.22422
\(36\) 0 0
\(37\) −103636. −0.336360 −0.168180 0.985756i \(-0.553789\pi\)
−0.168180 + 0.985756i \(0.553789\pi\)
\(38\) 0 0
\(39\) −252935. −0.682784
\(40\) 0 0
\(41\) −71691.3 −0.162451 −0.0812256 0.996696i \(-0.525883\pi\)
−0.0812256 + 0.996696i \(0.525883\pi\)
\(42\) 0 0
\(43\) −328419. −0.629925 −0.314962 0.949104i \(-0.601992\pi\)
−0.314962 + 0.949104i \(0.601992\pi\)
\(44\) 0 0
\(45\) −184847. −0.302391
\(46\) 0 0
\(47\) 119043. 0.167248 0.0836241 0.996497i \(-0.473350\pi\)
0.0836241 + 0.996497i \(0.473350\pi\)
\(48\) 0 0
\(49\) 92230.3 0.111992
\(50\) 0 0
\(51\) −1.39097e6 −1.46833
\(52\) 0 0
\(53\) −1.04011e6 −0.959648 −0.479824 0.877365i \(-0.659300\pi\)
−0.479824 + 0.877365i \(0.659300\pi\)
\(54\) 0 0
\(55\) 1.76919e6 1.43386
\(56\) 0 0
\(57\) −586982. −0.419820
\(58\) 0 0
\(59\) −225984. −0.143250 −0.0716250 0.997432i \(-0.522818\pi\)
−0.0716250 + 0.997432i \(0.522818\pi\)
\(60\) 0 0
\(61\) −1.55268e6 −0.875843 −0.437922 0.899013i \(-0.644285\pi\)
−0.437922 + 0.899013i \(0.644285\pi\)
\(62\) 0 0
\(63\) −545133. −0.274670
\(64\) 0 0
\(65\) 2.04085e6 0.921753
\(66\) 0 0
\(67\) −316375. −0.128511 −0.0642555 0.997933i \(-0.520467\pi\)
−0.0642555 + 0.997933i \(0.520467\pi\)
\(68\) 0 0
\(69\) −992033. −0.363542
\(70\) 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 \(-0.367645\pi\)
0.403928 + 0.914791i \(0.367645\pi\)
\(74\) 0 0
\(75\) −1.09267e6 −0.299071
\(76\) 0 0
\(77\) 5.21754e6 1.30241
\(78\) 0 0
\(79\) 8.22632e6 1.87720 0.938600 0.345007i \(-0.112123\pi\)
0.938600 + 0.345007i \(0.112123\pi\)
\(80\) 0 0
\(81\) −3.21264e6 −0.671684
\(82\) 0 0
\(83\) 5.89510e6 1.13167 0.565833 0.824520i \(-0.308555\pi\)
0.565833 + 0.824520i \(0.308555\pi\)
\(84\) 0 0
\(85\) 1.12233e7 1.98223
\(86\) 0 0
\(87\) 6.88164e6 1.12040
\(88\) 0 0
\(89\) −437005. −0.0657085 −0.0328542 0.999460i \(-0.510460\pi\)
−0.0328542 + 0.999460i \(0.510460\pi\)
\(90\) 0 0
\(91\) 6.01868e6 0.837253
\(92\) 0 0
\(93\) −4.49169e6 −0.579055
\(94\) 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
\(99\) −3.10585e6 −0.321705
\(100\) 0 0
\(101\) −6.19757e6 −0.598545 −0.299272 0.954168i \(-0.596744\pi\)
−0.299272 + 0.954168i \(0.596744\pi\)
\(102\) 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. 0.0404709 0.0202354 0.999795i \(-0.493558\pi\)
0.0202354 + 0.999795i \(0.493558\pi\)
\(108\) 0 0
\(109\) 1.95882e7 1.44878 0.724388 0.689393i \(-0.242123\pi\)
0.724388 + 0.689393i \(0.242123\pi\)
\(110\) 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
\(115\) 8.00438e6 0.490778
\(116\) 0 0
\(117\) −3.58275e6 −0.206807
\(118\) 0 0
\(119\) 3.30987e7 1.80051
\(120\) 0 0
\(121\) 1.02394e7 0.525441
\(122\) 0 0
\(123\) 2.88316e6 0.139701
\(124\) 0 0
\(125\) −1.65345e7 −0.757193
\(126\) 0 0
\(127\) 3.96314e7 1.71683 0.858413 0.512959i \(-0.171451\pi\)
0.858413 + 0.512959i \(0.171451\pi\)
\(128\) 0 0
\(129\) 1.32078e7 0.541709
\(130\) 0 0
\(131\) −3.65337e7 −1.41986 −0.709928 0.704274i \(-0.751273\pi\)
−0.709928 + 0.704274i \(0.751273\pi\)
\(132\) 0 0
\(133\) 1.39675e7 0.514797
\(134\) 0 0
\(135\) 3.59739e7 1.25840
\(136\) 0 0
\(137\) 2.56967e7 0.853799 0.426899 0.904299i \(-0.359606\pi\)
0.426899 + 0.904299i \(0.359606\pi\)
\(138\) 0 0
\(139\) −5.23001e7 −1.65177 −0.825886 0.563836i \(-0.809325\pi\)
−0.825886 + 0.563836i \(0.809325\pi\)
\(140\) 0 0
\(141\) −4.78747e6 −0.143827
\(142\) 0 0
\(143\) 3.42910e7 0.980626
\(144\) 0 0
\(145\) −5.55256e7 −1.51254
\(146\) 0 0
\(147\) −3.70916e6 −0.0963085
\(148\) 0 0
\(149\) −1.80406e7 −0.446785 −0.223392 0.974729i \(-0.571713\pi\)
−0.223392 + 0.974729i \(0.571713\pi\)
\(150\) 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
\(155\) 3.62420e7 0.781719
\(156\) 0 0
\(157\) −5.12341e7 −1.05660 −0.528300 0.849058i \(-0.677170\pi\)
−0.528300 + 0.849058i \(0.677170\pi\)
\(158\) 0 0
\(159\) 4.18292e7 0.825258
\(160\) 0 0
\(161\) 2.36058e7 0.445787
\(162\) 0 0
\(163\) 8.57572e7 1.55101 0.775504 0.631343i \(-0.217496\pi\)
0.775504 + 0.631343i \(0.217496\pi\)
\(164\) 0 0
\(165\) −7.11504e7 −1.23306
\(166\) 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
\(171\) −8.31441e6 −0.127158
\(172\) 0 0
\(173\) −1.98148e7 −0.290956 −0.145478 0.989361i \(-0.546472\pi\)
−0.145478 + 0.989361i \(0.546472\pi\)
\(174\) 0 0
\(175\) 2.60005e7 0.366731
\(176\) 0 0
\(177\) 9.08822e6 0.123189
\(178\) 0 0
\(179\) −2.97800e7 −0.388096 −0.194048 0.980992i \(-0.562162\pi\)
−0.194048 + 0.980992i \(0.562162\pi\)
\(180\) 0 0
\(181\) 3.96227e6 0.0496671 0.0248335 0.999692i \(-0.492094\pi\)
0.0248335 + 0.999692i \(0.492094\pi\)
\(182\) 0 0
\(183\) 6.24429e7 0.753189
\(184\) 0 0
\(185\) −3.36290e7 −0.390493
\(186\) 0 0
\(187\) 1.88577e8 2.10884
\(188\) 0 0
\(189\) 1.06091e8 1.14304
\(190\) 0 0
\(191\) −4.80105e7 −0.498562 −0.249281 0.968431i \(-0.580194\pi\)
−0.249281 + 0.968431i \(0.580194\pi\)
\(192\) 0 0
\(193\) −4.72502e6 −0.0473100 −0.0236550 0.999720i \(-0.507530\pi\)
−0.0236550 + 0.999720i \(0.507530\pi\)
\(194\) 0 0
\(195\) −8.20754e7 −0.792669
\(196\) 0 0
\(197\) 1.14882e8 1.07058 0.535290 0.844668i \(-0.320202\pi\)
0.535290 + 0.844668i \(0.320202\pi\)
\(198\) 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
\(203\) −1.63751e8 −1.37388
\(204\) 0 0
\(205\) −2.32632e7 −0.188596
\(206\) 0 0
\(207\) −1.40518e7 −0.110112
\(208\) 0 0
\(209\) 7.95784e7 0.602953
\(210\) 0 0
\(211\) 1.95850e8 1.43527 0.717636 0.696418i \(-0.245224\pi\)
0.717636 + 0.696418i \(0.245224\pi\)
\(212\) 0 0
\(213\) 2.16752e7 0.153686
\(214\) 0 0
\(215\) −1.06569e8 −0.731302
\(216\) 0 0
\(217\) 1.06881e8 0.710057
\(218\) 0 0
\(219\) −1.07986e8 −0.694723
\(220\) 0 0
\(221\) 2.17532e8 1.35566
\(222\) 0 0
\(223\) 1.08024e8 0.652311 0.326156 0.945316i \(-0.394247\pi\)
0.326156 + 0.945316i \(0.394247\pi\)
\(224\) 0 0
\(225\) −1.54773e7 −0.0905851
\(226\) 0 0
\(227\) −1.61144e8 −0.914374 −0.457187 0.889371i \(-0.651143\pi\)
−0.457187 + 0.889371i \(0.651143\pi\)
\(228\) 0 0
\(229\) 5.27173e7 0.290088 0.145044 0.989425i \(-0.453668\pi\)
0.145044 + 0.989425i \(0.453668\pi\)
\(230\) 0 0
\(231\) −2.09830e8 −1.12002
\(232\) 0 0
\(233\) 1.79423e8 0.929249 0.464625 0.885508i \(-0.346189\pi\)
0.464625 + 0.885508i \(0.346189\pi\)
\(234\) 0 0
\(235\) 3.86285e7 0.194165
\(236\) 0 0
\(237\) −3.30832e8 −1.61431
\(238\) 0 0
\(239\) −8.42441e7 −0.399160 −0.199580 0.979882i \(-0.563958\pi\)
−0.199580 + 0.979882i \(0.563958\pi\)
\(240\) 0 0
\(241\) −2.12302e8 −0.977000 −0.488500 0.872564i \(-0.662456\pi\)
−0.488500 + 0.872564i \(0.662456\pi\)
\(242\) 0 0
\(243\) −1.13255e8 −0.506333
\(244\) 0 0
\(245\) 2.99280e7 0.130016
\(246\) 0 0
\(247\) 9.17975e7 0.387607
\(248\) 0 0
\(249\) −2.37079e8 −0.973186
\(250\) 0 0
\(251\) −1.18102e8 −0.471411 −0.235706 0.971825i \(-0.575740\pi\)
−0.235706 + 0.971825i \(0.575740\pi\)
\(252\) 0 0
\(253\) 1.34492e8 0.522125
\(254\) 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
\(259\) −9.91755e7 −0.354695
\(260\) 0 0
\(261\) 9.74762e7 0.339357
\(262\) 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
\(267\) 1.75747e7 0.0565066
\(268\) 0 0
\(269\) 3.44748e8 1.07986 0.539931 0.841709i \(-0.318450\pi\)
0.539931 + 0.841709i \(0.318450\pi\)
\(270\) 0 0
\(271\) −4.42513e8 −1.35062 −0.675311 0.737533i \(-0.735990\pi\)
−0.675311 + 0.737533i \(0.735990\pi\)
\(272\) 0 0
\(273\) −2.42049e8 −0.720003
\(274\) 0 0
\(275\) 1.48135e8 0.429531
\(276\) 0 0
\(277\) −3.18148e8 −0.899395 −0.449697 0.893181i \(-0.648468\pi\)
−0.449697 + 0.893181i \(0.648468\pi\)
\(278\) 0 0
\(279\) −6.36234e7 −0.175389
\(280\) 0 0
\(281\) −1.28497e8 −0.345478 −0.172739 0.984968i \(-0.555262\pi\)
−0.172739 + 0.984968i \(0.555262\pi\)
\(282\) 0 0
\(283\) 3.98970e8 1.04637 0.523187 0.852218i \(-0.324743\pi\)
0.523187 + 0.852218i \(0.324743\pi\)
\(284\) 0 0
\(285\) −1.90471e8 −0.487384
\(286\) 0 0
\(287\) −6.86057e7 −0.171306
\(288\) 0 0
\(289\) 7.85942e8 1.91535
\(290\) 0 0
\(291\) 3.15425e8 0.750362
\(292\) 0 0
\(293\) −2.00958e8 −0.466732 −0.233366 0.972389i \(-0.574974\pi\)
−0.233366 + 0.972389i \(0.574974\pi\)
\(294\) 0 0
\(295\) −7.33298e7 −0.166304
\(296\) 0 0
\(297\) 6.04444e8 1.33878
\(298\) 0 0
\(299\) 1.55143e8 0.335647
\(300\) 0 0
\(301\) −3.14284e8 −0.664262
\(302\) 0 0
\(303\) 2.49243e8 0.514724
\(304\) 0 0
\(305\) −5.03830e8 −1.01680
\(306\) 0 0
\(307\) 1.58918e7 0.0313465 0.0156733 0.999877i \(-0.495011\pi\)
0.0156733 + 0.999877i \(0.495011\pi\)
\(308\) 0 0
\(309\) −2.65353e8 −0.511646
\(310\) 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 \(-0.403251\pi\)
0.299288 + 0.954163i \(0.403251\pi\)
\(314\) 0 0
\(315\) −1.76891e8 −0.318874
\(316\) 0 0
\(317\) −1.06084e9 −1.87043 −0.935213 0.354086i \(-0.884792\pi\)
−0.935213 + 0.354086i \(0.884792\pi\)
\(318\) 0 0
\(319\) −9.32958e8 −1.60914
\(320\) 0 0
\(321\) −2.06247e7 −0.0348033
\(322\) 0 0
\(323\) 5.04824e8 0.833548
\(324\) 0 0
\(325\) 1.70881e8 0.276123
\(326\) 0 0
\(327\) −7.87763e8 −1.24589
\(328\) 0 0
\(329\) 1.13919e8 0.176365
\(330\) 0 0
\(331\) 2.88487e8 0.437249 0.218624 0.975809i \(-0.429843\pi\)
0.218624 + 0.975809i \(0.429843\pi\)
\(332\) 0 0
\(333\) 5.90363e7 0.0876121
\(334\) 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
\(339\) −7.59590e8 −1.05896
\(340\) 0 0
\(341\) 6.08948e8 0.831649
\(342\) 0 0
\(343\) −6.99837e8 −0.936414
\(344\) 0 0
\(345\) −3.21907e8 −0.422049
\(346\) 0 0
\(347\) −1.10651e9 −1.42168 −0.710841 0.703352i \(-0.751686\pi\)
−0.710841 + 0.703352i \(0.751686\pi\)
\(348\) 0 0
\(349\) 1.38337e9 1.74201 0.871003 0.491278i \(-0.163470\pi\)
0.871003 + 0.491278i \(0.163470\pi\)
\(350\) 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
\(355\) −1.74890e8 −0.207475
\(356\) 0 0
\(357\) −1.33111e9 −1.54837
\(358\) 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
\(363\) −4.11789e8 −0.451858
\(364\) 0 0
\(365\) 8.71299e8 0.937869
\(366\) 0 0
\(367\) −7.49367e8 −0.791341 −0.395670 0.918393i \(-0.629488\pi\)
−0.395670 + 0.918393i \(0.629488\pi\)
\(368\) 0 0
\(369\) 4.08390e7 0.0423139
\(370\) 0 0
\(371\) −9.95340e8 −1.01196
\(372\) 0 0
\(373\) 1.49519e9 1.49181 0.745906 0.666051i \(-0.232017\pi\)
0.745906 + 0.666051i \(0.232017\pi\)
\(374\) 0 0
\(375\) 6.64957e8 0.651155
\(376\) 0 0
\(377\) −1.07621e9 −1.03443
\(378\) 0 0
\(379\) −7.92096e7 −0.0747379 −0.0373689 0.999302i \(-0.511898\pi\)
−0.0373689 + 0.999302i \(0.511898\pi\)
\(380\) 0 0
\(381\) −1.59383e9 −1.47640
\(382\) 0 0
\(383\) 4.80285e8 0.436820 0.218410 0.975857i \(-0.429913\pi\)
0.218410 + 0.975857i \(0.429913\pi\)
\(384\) 0 0
\(385\) 1.69305e9 1.51202
\(386\) 0 0
\(387\) 1.87084e8 0.164077
\(388\) 0 0
\(389\) 1.07150e9 0.922928 0.461464 0.887159i \(-0.347324\pi\)
0.461464 + 0.887159i \(0.347324\pi\)
\(390\) 0 0
\(391\) 8.53180e8 0.721809
\(392\) 0 0
\(393\) 1.46925e9 1.22102
\(394\) 0 0
\(395\) 2.66937e9 2.17931
\(396\) 0 0
\(397\) 2.03185e9 1.62976 0.814882 0.579627i \(-0.196802\pi\)
0.814882 + 0.579627i \(0.196802\pi\)
\(398\) 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
\(403\) 7.02451e8 0.534624
\(404\) 0 0
\(405\) −1.04248e9 −0.779782
\(406\) 0 0
\(407\) −5.65044e8 −0.415434
\(408\) 0 0
\(409\) −3.30242e8 −0.238672 −0.119336 0.992854i \(-0.538077\pi\)
−0.119336 + 0.992854i \(0.538077\pi\)
\(410\) 0 0
\(411\) −1.03343e9 −0.734232
\(412\) 0 0
\(413\) −2.16257e8 −0.151059
\(414\) 0 0
\(415\) 1.91291e9 1.31379
\(416\) 0 0
\(417\) 2.10331e9 1.42046
\(418\) 0 0
\(419\) 5.80021e7 0.0385207 0.0192604 0.999815i \(-0.493869\pi\)
0.0192604 + 0.999815i \(0.493869\pi\)
\(420\) 0 0
\(421\) −1.90609e8 −0.124496 −0.0622480 0.998061i \(-0.519827\pi\)
−0.0622480 + 0.998061i \(0.519827\pi\)
\(422\) 0 0
\(423\) −6.78129e7 −0.0435633
\(424\) 0 0
\(425\) 9.39731e8 0.593803
\(426\) 0 0
\(427\) −1.48585e9 −0.923586
\(428\) 0 0
\(429\) −1.37906e9 −0.843298
\(430\) 0 0
\(431\) 2.42923e9 1.46150 0.730749 0.682646i \(-0.239171\pi\)
0.730749 + 0.682646i \(0.239171\pi\)
\(432\) 0 0
\(433\) −2.37902e9 −1.40828 −0.704141 0.710060i \(-0.748668\pi\)
−0.704141 + 0.710060i \(0.748668\pi\)
\(434\) 0 0
\(435\) 2.23303e9 1.30072
\(436\) 0 0
\(437\) 3.60037e8 0.206378
\(438\) 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
\(443\) −5.02643e8 −0.274692 −0.137346 0.990523i \(-0.543857\pi\)
−0.137346 + 0.990523i \(0.543857\pi\)
\(444\) 0 0
\(445\) −1.41805e8 −0.0762834
\(446\) 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
\(451\) −3.90876e8 −0.200641
\(452\) 0 0
\(453\) 1.55792e9 0.787410
\(454\) 0 0
\(455\) 1.95301e9 0.971998
\(456\) 0 0
\(457\) 2.68422e9 1.31556 0.657782 0.753209i \(-0.271495\pi\)
0.657782 + 0.753209i \(0.271495\pi\)
\(458\) 0 0
\(459\) 3.83442e9 1.85078
\(460\) 0 0
\(461\) 1.30434e9 0.620065 0.310033 0.950726i \(-0.399660\pi\)
0.310033 + 0.950726i \(0.399660\pi\)
\(462\) 0 0
\(463\) 2.86853e9 1.34315 0.671577 0.740934i \(-0.265617\pi\)
0.671577 + 0.740934i \(0.265617\pi\)
\(464\) 0 0
\(465\) −1.45752e9 −0.672246
\(466\) 0 0
\(467\) 5.96519e8 0.271029 0.135514 0.990775i \(-0.456731\pi\)
0.135514 + 0.990775i \(0.456731\pi\)
\(468\) 0 0
\(469\) −3.02758e8 −0.135516
\(470\) 0 0
\(471\) 2.06045e9 0.908632
\(472\) 0 0
\(473\) −1.79061e9 −0.778012
\(474\) 0 0
\(475\) 3.96561e8 0.169778
\(476\) 0 0
\(477\) 5.92497e8 0.249961
\(478\) 0 0
\(479\) −2.16068e9 −0.898289 −0.449144 0.893459i \(-0.648271\pi\)
−0.449144 + 0.893459i \(0.648271\pi\)
\(480\) 0 0
\(481\) −6.51805e8 −0.267061
\(482\) 0 0
\(483\) −9.49337e8 −0.383359
\(484\) 0 0
\(485\) −2.54506e9 −1.01298
\(486\) 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
\(491\) −2.38677e9 −0.909966 −0.454983 0.890500i \(-0.650355\pi\)
−0.454983 + 0.890500i \(0.650355\pi\)
\(492\) 0 0
\(493\) −5.91843e9 −2.22455
\(494\) 0 0
\(495\) −1.00782e9 −0.373479
\(496\) 0 0
\(497\) −5.15769e8 −0.188455
\(498\) 0 0
\(499\) −5.23900e9 −1.88754 −0.943771 0.330601i \(-0.892749\pi\)
−0.943771 + 0.330601i \(0.892749\pi\)
\(500\) 0 0
\(501\) −4.25772e9 −1.51267
\(502\) 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
\(507\) 9.32706e8 0.317846
\(508\) 0 0
\(509\) 2.58693e9 0.869505 0.434753 0.900550i \(-0.356836\pi\)
0.434753 + 0.900550i \(0.356836\pi\)
\(510\) 0 0
\(511\) 2.56955e9 0.851892
\(512\) 0 0
\(513\) 1.61811e9 0.529171
\(514\) 0 0
\(515\) 2.14105e9 0.690718
\(516\) 0 0
\(517\) 6.49047e8 0.206566
\(518\) 0 0
\(519\) 7.96876e8 0.250210
\(520\) 0 0
\(521\) −1.08542e8 −0.0336253 −0.0168127 0.999859i \(-0.505352\pi\)
−0.0168127 + 0.999859i \(0.505352\pi\)
\(522\) 0 0
\(523\) −6.10725e9 −1.86676 −0.933382 0.358884i \(-0.883157\pi\)
−0.933382 + 0.358884i \(0.883157\pi\)
\(524\) 0 0
\(525\) −1.04564e9 −0.315374
\(526\) 0 0
\(527\) 3.86300e9 1.14971
\(528\) 0 0
\(529\) −2.79634e9 −0.821288
\(530\) 0 0
\(531\) 1.28732e8 0.0373125
\(532\) 0 0
\(533\) −4.50894e8 −0.128982
\(534\) 0 0
\(535\) 1.66414e8 0.0469841
\(536\) 0 0
\(537\) 1.19764e9 0.333747
\(538\) 0 0
\(539\) 5.02858e8 0.138320
\(540\) 0 0
\(541\) 5.39345e8 0.146445 0.0732227 0.997316i \(-0.476672\pi\)
0.0732227 + 0.997316i \(0.476672\pi\)
\(542\) 0 0
\(543\) −1.59348e8 −0.0427116
\(544\) 0 0
\(545\) 6.35620e9 1.68194
\(546\) 0 0
\(547\) 8.82287e7 0.0230491 0.0115246 0.999934i \(-0.496332\pi\)
0.0115246 + 0.999934i \(0.496332\pi\)
\(548\) 0 0
\(549\) 8.84483e8 0.228132
\(550\) 0 0
\(551\) −2.49754e9 −0.636037
\(552\) 0 0
\(553\) 7.87226e9 1.97953
\(554\) 0 0
\(555\) 1.35243e9 0.335807
\(556\) 0 0
\(557\) −5.57233e8 −0.136629 −0.0683147 0.997664i \(-0.521762\pi\)
−0.0683147 + 0.997664i \(0.521762\pi\)
\(558\) 0 0
\(559\) −2.06555e9 −0.500143
\(560\) 0 0
\(561\) −7.58386e9 −1.81351
\(562\) 0 0
\(563\) −1.17012e9 −0.276344 −0.138172 0.990408i \(-0.544123\pi\)
−0.138172 + 0.990408i \(0.544123\pi\)
\(564\) 0 0
\(565\) 6.12887e9 1.42959
\(566\) 0 0
\(567\) −3.07437e9 −0.708297
\(568\) 0 0
\(569\) 2.39181e9 0.544295 0.272147 0.962256i \(-0.412266\pi\)
0.272147 + 0.962256i \(0.412266\pi\)
\(570\) 0 0
\(571\) 3.15823e9 0.709933 0.354966 0.934879i \(-0.384492\pi\)
0.354966 + 0.934879i \(0.384492\pi\)
\(572\) 0 0
\(573\) 1.93080e9 0.428743
\(574\) 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
\(579\) 1.90023e8 0.0406846
\(580\) 0 0
\(581\) 5.64138e9 1.19335
\(582\) 0 0
\(583\) −5.67087e9 −1.18525
\(584\) 0 0
\(585\) −1.16257e9 −0.240090
\(586\) 0 0
\(587\) 2.72240e9 0.555544 0.277772 0.960647i \(-0.410404\pi\)
0.277772 + 0.960647i \(0.410404\pi\)
\(588\) 0 0
\(589\) 1.63016e9 0.328721
\(590\) 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
\(595\) 1.07402e10 2.09028
\(596\) 0 0
\(597\) 4.86346e8 0.0935482
\(598\) 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 \(-0.780123\pi\)
−0.770759 + 0.637127i \(0.780123\pi\)
\(602\) 0 0
\(603\) 1.80223e8 0.0334734
\(604\) 0 0
\(605\) 3.32259e9 0.610004
\(606\) 0 0
\(607\) −4.60087e9 −0.834986 −0.417493 0.908680i \(-0.637091\pi\)
−0.417493 + 0.908680i \(0.637091\pi\)
\(608\) 0 0
\(609\) 6.58546e9 1.18148
\(610\) 0 0
\(611\) 7.48707e8 0.132791
\(612\) 0 0
\(613\) −8.55728e9 −1.50046 −0.750229 0.661178i \(-0.770057\pi\)
−0.750229 + 0.661178i \(0.770057\pi\)
\(614\) 0 0
\(615\) 9.35561e8 0.162184
\(616\) 0 0
\(617\) 2.58089e9 0.442355 0.221178 0.975234i \(-0.429010\pi\)
0.221178 + 0.975234i \(0.429010\pi\)
\(618\) 0 0
\(619\) −5.26641e9 −0.892478 −0.446239 0.894914i \(-0.647237\pi\)
−0.446239 + 0.894914i \(0.647237\pi\)
\(620\) 0 0
\(621\) 2.73469e9 0.458234
\(622\) 0 0
\(623\) −4.18197e8 −0.0692903
\(624\) 0 0
\(625\) −7.48796e9 −1.22683
\(626\) 0 0
\(627\) −3.20035e9 −0.518514
\(628\) 0 0
\(629\) −3.58449e9 −0.574314
\(630\) 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
\(635\) 1.28601e10 1.99313
\(636\) 0 0
\(637\) 5.80071e8 0.0889187
\(638\) 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
\(643\) −1.26588e10 −1.87782 −0.938908 0.344167i \(-0.888161\pi\)
−0.938908 + 0.344167i \(0.888161\pi\)
\(644\) 0 0
\(645\) 4.28582e9 0.628890
\(646\) 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
\(651\) −4.29837e9 −0.610620
\(652\) 0 0
\(653\) −3.21579e9 −0.451951 −0.225975 0.974133i \(-0.572557\pi\)
−0.225975 + 0.974133i \(0.572557\pi\)
\(654\) 0 0
\(655\) −1.18549e10 −1.64836
\(656\) 0 0
\(657\) −1.52958e9 −0.210423
\(658\) 0 0
\(659\) 5.17004e9 0.703711 0.351856 0.936054i \(-0.385551\pi\)
0.351856 + 0.936054i \(0.385551\pi\)
\(660\) 0 0
\(661\) 1.95604e9 0.263435 0.131717 0.991287i \(-0.457951\pi\)
0.131717 + 0.991287i \(0.457951\pi\)
\(662\) 0 0
\(663\) −8.74835e9 −1.16581
\(664\) 0 0
\(665\) 4.53232e9 0.597647
\(666\) 0 0
\(667\) −4.22099e9 −0.550775
\(668\) 0 0
\(669\) −4.34434e9 −0.560961
\(670\) 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
\(675\) 3.01211e9 0.376971
\(676\) 0 0
\(677\) 8.55209e9 1.05928 0.529642 0.848222i \(-0.322326\pi\)
0.529642 + 0.848222i \(0.322326\pi\)
\(678\) 0 0
\(679\) −7.50565e9 −0.920119
\(680\) 0 0
\(681\) 6.48061e9 0.786323
\(682\) 0 0
\(683\) −7.26976e9 −0.873067 −0.436534 0.899688i \(-0.643794\pi\)
−0.436534 + 0.899688i \(0.643794\pi\)
\(684\) 0 0
\(685\) 8.33837e9 0.991206
\(686\) 0 0
\(687\) −2.12010e9 −0.249463
\(688\) 0 0
\(689\) −6.54162e9 −0.761935
\(690\) 0 0
\(691\) −5.19893e9 −0.599434 −0.299717 0.954028i \(-0.596892\pi\)
−0.299717 + 0.954028i \(0.596892\pi\)
\(692\) 0 0
\(693\) −2.97218e9 −0.339241
\(694\) 0 0
\(695\) −1.69709e10 −1.91760
\(696\) 0 0
\(697\) −2.47961e9 −0.277376
\(698\) 0 0
\(699\) −7.21572e9 −0.799116
\(700\) 0 0
\(701\) −1.35221e10 −1.48262 −0.741310 0.671163i \(-0.765795\pi\)
−0.741310 + 0.671163i \(0.765795\pi\)
\(702\) 0 0
\(703\) −1.51263e9 −0.164206
\(704\) 0 0
\(705\) −1.55349e9 −0.166974
\(706\) 0 0
\(707\) −5.93083e9 −0.631172
\(708\) 0 0
\(709\) −1.05901e10 −1.11594 −0.557969 0.829862i \(-0.688419\pi\)
−0.557969 + 0.829862i \(0.688419\pi\)
\(710\) 0 0
\(711\) −4.68613e9 −0.488957
\(712\) 0 0
\(713\) 2.75507e9 0.284655
\(714\) 0 0
\(715\) 1.11271e10 1.13845
\(716\) 0 0
\(717\) 3.38799e9 0.343261
\(718\) 0 0
\(719\) 1.49161e10 1.49659 0.748297 0.663363i \(-0.230872\pi\)
0.748297 + 0.663363i \(0.230872\pi\)
\(720\) 0 0
\(721\) 6.31417e9 0.627398
\(722\) 0 0
\(723\) 8.53800e9 0.840180
\(724\) 0 0
\(725\) −4.64919e9 −0.453100
\(726\) 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
\(731\) −1.13591e10 −1.07556
\(732\) 0 0
\(733\) −7.99792e9 −0.750090 −0.375045 0.927007i \(-0.622373\pi\)
−0.375045 + 0.927007i \(0.622373\pi\)
\(734\) 0 0
\(735\) −1.20359e9 −0.111808
\(736\) 0 0
\(737\) −1.72494e9 −0.158722
\(738\) 0 0
\(739\) 1.03852e10 0.946588 0.473294 0.880905i \(-0.343065\pi\)
0.473294 + 0.880905i \(0.343065\pi\)
\(740\) 0 0
\(741\) −3.69175e9 −0.333326
\(742\) 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
\(747\) −3.35815e9 −0.294766
\(748\) 0 0
\(749\) 4.90772e8 0.0426770
\(750\) 0 0
\(751\) −1.34330e10 −1.15726 −0.578631 0.815589i \(-0.696413\pi\)
−0.578631 + 0.815589i \(0.696413\pi\)
\(752\) 0 0
\(753\) 4.74963e9 0.405394
\(754\) 0 0
\(755\) −1.25703e10 −1.06300
\(756\) 0 0
\(757\) 6.78007e9 0.568065 0.284033 0.958815i \(-0.408328\pi\)
0.284033 + 0.958815i \(0.408328\pi\)
\(758\) 0 0
\(759\) −5.40877e9 −0.449006
\(760\) 0 0
\(761\) −8.01137e9 −0.658962 −0.329481 0.944162i \(-0.606874\pi\)
−0.329481 + 0.944162i \(0.606874\pi\)
\(762\) 0 0
\(763\) 1.87451e10 1.52775
\(764\) 0 0
\(765\) −6.39335e9 −0.516314
\(766\) 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
\(771\) −5.12609e9 −0.402806
\(772\) 0 0
\(773\) −2.33296e10 −1.81668 −0.908340 0.418233i \(-0.862650\pi\)
−0.908340 + 0.418233i \(0.862650\pi\)
\(774\) 0 0
\(775\) 3.03456e9 0.234174
\(776\) 0 0
\(777\) 3.98847e9 0.305023
\(778\) 0 0
\(779\) −1.04638e9 −0.0793064
\(780\) 0 0
\(781\) −2.93855e9 −0.220726
\(782\) 0 0
\(783\) −1.89703e10 −1.41224
\(784\) 0 0
\(785\) −1.66250e10 −1.22664
\(786\) 0 0
\(787\) −5.27740e8 −0.0385930 −0.0192965 0.999814i \(-0.506143\pi\)
−0.0192965 + 0.999814i \(0.506143\pi\)
\(788\) 0 0
\(789\) 1.74187e10 1.26254
\(790\) 0 0
\(791\) 1.80747e10 1.29853
\(792\) 0 0
\(793\) −9.76536e9 −0.695396
\(794\) 0 0
\(795\) 1.35732e10 0.958072
\(796\) 0 0
\(797\) 2.41514e9 0.168981 0.0844907 0.996424i \(-0.473074\pi\)
0.0844907 + 0.996424i \(0.473074\pi\)
\(798\) 0 0
\(799\) 4.11738e9 0.285566
\(800\) 0 0
\(801\) 2.48940e8 0.0171152
\(802\) 0 0
\(803\) 1.46398e10 0.997773
\(804\) 0 0
\(805\) 7.65988e9 0.517531
\(806\) 0 0
\(807\) −1.38645e10 −0.928636
\(808\) 0 0
\(809\) −1.16364e10 −0.772679 −0.386340 0.922357i \(-0.626261\pi\)
−0.386340 + 0.922357i \(0.626261\pi\)
\(810\) 0 0
\(811\) 1.44359e10 0.950320 0.475160 0.879899i \(-0.342390\pi\)
0.475160 + 0.879899i \(0.342390\pi\)
\(812\) 0 0
\(813\) 1.77962e10 1.16148
\(814\) 0 0
\(815\) 2.78275e10 1.80062
\(816\) 0 0
\(817\) −4.79348e9 −0.307521
\(818\) 0 0
\(819\) −3.42855e9 −0.218080
\(820\) 0 0
\(821\) −7.63805e9 −0.481706 −0.240853 0.970562i \(-0.577427\pi\)
−0.240853 + 0.970562i \(0.577427\pi\)
\(822\) 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
\(827\) −1.57823e10 −0.970288 −0.485144 0.874434i \(-0.661233\pi\)
−0.485144 + 0.874434i \(0.661233\pi\)
\(828\) 0 0
\(829\) 2.63296e10 1.60510 0.802552 0.596582i \(-0.203475\pi\)
0.802552 + 0.596582i \(0.203475\pi\)
\(830\) 0 0
\(831\) 1.27947e10 0.773442
\(832\) 0 0
\(833\) 3.19000e9 0.191220
\(834\) 0 0
\(835\) 3.43541e10 2.04210
\(836\) 0 0
\(837\) 1.23820e10 0.729882
\(838\) 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
\(843\) 5.16767e9 0.297097
\(844\) 0 0
\(845\) −7.52569e9 −0.429090
\(846\) 0 0
\(847\) 9.79866e9 0.554083
\(848\) 0 0
\(849\) −1.60451e10 −0.899839
\(850\) 0 0
\(851\) −2.55643e9 −0.142194
\(852\) 0 0
\(853\) −6.28088e7 −0.00346496 −0.00173248 0.999998i \(-0.500551\pi\)
−0.00173248 + 0.999998i \(0.500551\pi\)
\(854\) 0 0
\(855\) −2.69796e9 −0.147623
\(856\) 0 0
\(857\) 6.25595e9 0.339516 0.169758 0.985486i \(-0.445701\pi\)
0.169758 + 0.985486i \(0.445701\pi\)
\(858\) 0 0
\(859\) −2.48119e10 −1.33562 −0.667811 0.744331i \(-0.732769\pi\)
−0.667811 + 0.744331i \(0.732769\pi\)
\(860\) 0 0
\(861\) 2.75907e9 0.147316
\(862\) 0 0
\(863\) 1.28944e10 0.682909 0.341455 0.939898i \(-0.389080\pi\)
0.341455 + 0.939898i \(0.389080\pi\)
\(864\) 0 0
\(865\) −6.42973e9 −0.337782
\(866\) 0 0
\(867\) −3.16077e10 −1.64712
\(868\) 0 0
\(869\) 4.48516e10 2.31851
\(870\) 0 0
\(871\) −1.98980e9 −0.102034
\(872\) 0 0
\(873\) 4.46789e9 0.227276
\(874\) 0 0
\(875\) −1.58229e10 −0.798468
\(876\) 0 0
\(877\) −2.75670e10 −1.38004 −0.690018 0.723792i \(-0.742397\pi\)
−0.690018 + 0.723792i \(0.742397\pi\)
\(878\) 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
\(883\) −2.09906e10 −1.02604 −0.513018 0.858378i \(-0.671473\pi\)
−0.513018 + 0.858378i \(0.671473\pi\)
\(884\) 0 0
\(885\) 2.94905e9 0.143015
\(886\) 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
\(891\) −1.75160e10 −0.829588
\(892\) 0 0
\(893\) 1.73751e9 0.0816483
\(894\) 0 0
\(895\) −9.66337e9 −0.450555
\(896\) 0 0
\(897\) −6.23927e9 −0.288643
\(898\) 0 0
\(899\) −1.91117e10 −0.877282
\(900\) 0 0
\(901\) −3.59744e10 −1.63854
\(902\) 0 0
\(903\) 1.26393e10 0.571238
\(904\) 0 0
\(905\) 1.28572e9 0.0576603
\(906\) 0 0
\(907\) 3.84425e10 1.71075 0.855373 0.518012i \(-0.173328\pi\)
0.855373 + 0.518012i \(0.173328\pi\)
\(908\) 0 0
\(909\) 3.53045e9 0.155904
\(910\) 0 0
\(911\) 3.64507e10 1.59732 0.798660 0.601783i \(-0.205543\pi\)
0.798660 + 0.601783i \(0.205543\pi\)
\(912\) 0 0
\(913\) 3.21413e10 1.39771
\(914\) 0 0
\(915\) 2.02622e10 0.874405
\(916\) 0 0
\(917\) −3.49613e10 −1.49725
\(918\) 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
\(923\) −3.38976e9 −0.141894
\(924\) 0 0
\(925\) −2.81577e9 −0.116977
\(926\) 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
\(931\) 1.34616e9 0.0546730
\(932\) 0 0
\(933\) 1.96139e10 0.790638
\(934\) 0 0
\(935\) 6.11916e10 2.44823
\(936\) 0 0
\(937\) −3.70014e10 −1.46937 −0.734683 0.678411i \(-0.762669\pi\)
−0.734683 + 0.678411i \(0.762669\pi\)
\(938\) 0 0
\(939\) −1.30595e10 −0.514750
\(940\) 0 0
\(941\) 2.10617e10 0.824005 0.412003 0.911183i \(-0.364829\pi\)
0.412003 + 0.911183i \(0.364829\pi\)
\(942\) 0 0
\(943\) −1.76844e9 −0.0686752
\(944\) 0 0
\(945\) 3.44256e10 1.32700
\(946\) 0 0
\(947\) 1.35146e10 0.517104 0.258552 0.965997i \(-0.416755\pi\)
0.258552 + 0.965997i \(0.416755\pi\)
\(948\) 0 0
\(949\) 1.68877e10 0.641416
\(950\) 0 0
\(951\) 4.26628e10 1.60849
\(952\) 0 0
\(953\) −4.26831e10 −1.59746 −0.798731 0.601688i \(-0.794495\pi\)
−0.798731 + 0.601688i \(0.794495\pi\)
\(954\) 0 0
\(955\) −1.55790e10 −0.578799
\(956\) 0 0
\(957\) 3.75201e10 1.38380
\(958\) 0 0
\(959\) 2.45907e10 0.900340
\(960\) 0 0
\(961\) −1.50383e10 −0.546597
\(962\) 0 0
\(963\) −2.92142e8 −0.0105415
\(964\) 0 0
\(965\) −1.53323e9 −0.0549239
\(966\) 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
\(971\) 4.85678e10 1.70248 0.851238 0.524780i \(-0.175852\pi\)
0.851238 + 0.524780i \(0.175852\pi\)
\(972\) 0 0
\(973\) −5.00491e10 −1.74181
\(974\) 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
\(979\) −2.38264e9 −0.0811557
\(980\) 0 0
\(981\) −1.11584e10 −0.377364
\(982\) 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
\(987\) −4.58142e9 −0.151667
\(988\) 0 0
\(989\) −8.10126e9 −0.266296
\(990\) 0 0
\(991\) 1.49883e10 0.489208 0.244604 0.969623i \(-0.421342\pi\)
0.244604 + 0.969623i \(0.421342\pi\)
\(992\) 0 0
\(993\) −1.16019e10 −0.376016
\(994\) 0 0
\(995\) −3.92416e9 −0.126289
\(996\) 0 0
\(997\) −3.44101e10 −1.09965 −0.549824 0.835281i \(-0.685305\pi\)
−0.549824 + 0.835281i \(0.685305\pi\)
\(998\) 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 256.8.a.r.1.2 6
4.3 odd 2 256.8.a.q.1.5 6
8.3 odd 2 256.8.a.q.1.2 6
8.5 even 2 inner 256.8.a.r.1.5 6
16.3 odd 4 32.8.b.a.17.5 6
16.5 even 4 8.8.b.a.5.1 6
16.11 odd 4 32.8.b.a.17.2 6
16.13 even 4 8.8.b.a.5.2 yes 6
48.5 odd 4 72.8.d.b.37.6 6
48.11 even 4 288.8.d.b.145.2 6
48.29 odd 4 72.8.d.b.37.5 6
48.35 even 4 288.8.d.b.145.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.b.a.5.1 6 16.5 even 4
8.8.b.a.5.2 yes 6 16.13 even 4
32.8.b.a.17.2 6 16.11 odd 4
32.8.b.a.17.5 6 16.3 odd 4
72.8.d.b.37.5 6 48.29 odd 4
72.8.d.b.37.6 6 48.5 odd 4
256.8.a.q.1.2 6 8.3 odd 2
256.8.a.q.1.5 6 4.3 odd 2
256.8.a.r.1.2 6 1.1 even 1 trivial
256.8.a.r.1.5 6 8.5 even 2 inner
288.8.d.b.145.2 6 48.11 even 4
288.8.d.b.145.5 6 48.35 even 4