Properties

Label 100.6.c.a.49.1
Level 100
Weight 6
Character 100.49
Analytic conductor 16.038
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 100.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.0383819813\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 100.49
Dual form 100.6.c.a.49.2

$q$-expansion

\(f(q)\) \(=\) \(q-22.0000i q^{3} +218.000i q^{7} -241.000 q^{9} +O(q^{10})\) \(q-22.0000i q^{3} +218.000i q^{7} -241.000 q^{9} -480.000 q^{11} +622.000i q^{13} +186.000i q^{17} +1204.00 q^{19} +4796.00 q^{21} +3186.00i q^{23} -44.0000i q^{27} -5526.00 q^{29} +9356.00 q^{31} +10560.0i q^{33} +5618.00i q^{37} +13684.0 q^{39} -14394.0 q^{41} +370.000i q^{43} +16146.0i q^{47} -30717.0 q^{49} +4092.00 q^{51} +4374.00i q^{53} -26488.0i q^{57} +11748.0 q^{59} +13202.0 q^{61} -52538.0i q^{63} -11542.0i q^{67} +70092.0 q^{69} -29532.0 q^{71} -33698.0i q^{73} -104640. i q^{77} -31208.0 q^{79} -59531.0 q^{81} +38466.0i q^{83} +121572. i q^{87} -119514. q^{89} -135596. q^{91} -205832. i q^{93} +94658.0i q^{97} +115680. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 482q^{9} + O(q^{10}) \) \( 2q - 482q^{9} - 960q^{11} + 2408q^{19} + 9592q^{21} - 11052q^{29} + 18712q^{31} + 27368q^{39} - 28788q^{41} - 61434q^{49} + 8184q^{51} + 23496q^{59} + 26404q^{61} + 140184q^{69} - 59064q^{71} - 62416q^{79} - 119062q^{81} - 239028q^{89} - 271192q^{91} + 231360q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) − 22.0000i − 1.41130i −0.708560 0.705650i \(-0.750655\pi\)
0.708560 0.705650i \(-0.249345\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 218.000i 1.68156i 0.541380 + 0.840778i \(0.317902\pi\)
−0.541380 + 0.840778i \(0.682098\pi\)
\(8\) 0 0
\(9\) −241.000 −0.991770
\(10\) 0 0
\(11\) −480.000 −1.19608 −0.598039 0.801467i \(-0.704053\pi\)
−0.598039 + 0.801467i \(0.704053\pi\)
\(12\) 0 0
\(13\) 622.000i 1.02078i 0.859943 + 0.510390i \(0.170499\pi\)
−0.859943 + 0.510390i \(0.829501\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 186.000i 0.156096i 0.996950 + 0.0780478i \(0.0248687\pi\)
−0.996950 + 0.0780478i \(0.975131\pi\)
\(18\) 0 0
\(19\) 1204.00 0.765143 0.382571 0.923926i \(-0.375039\pi\)
0.382571 + 0.923926i \(0.375039\pi\)
\(20\) 0 0
\(21\) 4796.00 2.37318
\(22\) 0 0
\(23\) 3186.00i 1.25582i 0.778287 + 0.627908i \(0.216089\pi\)
−0.778287 + 0.627908i \(0.783911\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 44.0000i − 0.0116156i
\(28\) 0 0
\(29\) −5526.00 −1.22016 −0.610079 0.792341i \(-0.708862\pi\)
−0.610079 + 0.792341i \(0.708862\pi\)
\(30\) 0 0
\(31\) 9356.00 1.74858 0.874291 0.485402i \(-0.161327\pi\)
0.874291 + 0.485402i \(0.161327\pi\)
\(32\) 0 0
\(33\) 10560.0i 1.68803i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5618.00i 0.674648i 0.941389 + 0.337324i \(0.109522\pi\)
−0.941389 + 0.337324i \(0.890478\pi\)
\(38\) 0 0
\(39\) 13684.0 1.44063
\(40\) 0 0
\(41\) −14394.0 −1.33728 −0.668639 0.743587i \(-0.733123\pi\)
−0.668639 + 0.743587i \(0.733123\pi\)
\(42\) 0 0
\(43\) 370.000i 0.0305162i 0.999884 + 0.0152581i \(0.00485699\pi\)
−0.999884 + 0.0152581i \(0.995143\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 16146.0i 1.06615i 0.846066 + 0.533077i \(0.178965\pi\)
−0.846066 + 0.533077i \(0.821035\pi\)
\(48\) 0 0
\(49\) −30717.0 −1.82763
\(50\) 0 0
\(51\) 4092.00 0.220298
\(52\) 0 0
\(53\) 4374.00i 0.213889i 0.994265 + 0.106945i \(0.0341068\pi\)
−0.994265 + 0.106945i \(0.965893\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 26488.0i − 1.07985i
\(58\) 0 0
\(59\) 11748.0 0.439374 0.219687 0.975570i \(-0.429496\pi\)
0.219687 + 0.975570i \(0.429496\pi\)
\(60\) 0 0
\(61\) 13202.0 0.454271 0.227136 0.973863i \(-0.427064\pi\)
0.227136 + 0.973863i \(0.427064\pi\)
\(62\) 0 0
\(63\) − 52538.0i − 1.66772i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 11542.0i − 0.314119i −0.987589 0.157059i \(-0.949799\pi\)
0.987589 0.157059i \(-0.0502014\pi\)
\(68\) 0 0
\(69\) 70092.0 1.77233
\(70\) 0 0
\(71\) −29532.0 −0.695260 −0.347630 0.937632i \(-0.613013\pi\)
−0.347630 + 0.937632i \(0.613013\pi\)
\(72\) 0 0
\(73\) − 33698.0i − 0.740111i −0.929010 0.370056i \(-0.879339\pi\)
0.929010 0.370056i \(-0.120661\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 104640.i − 2.01127i
\(78\) 0 0
\(79\) −31208.0 −0.562598 −0.281299 0.959620i \(-0.590765\pi\)
−0.281299 + 0.959620i \(0.590765\pi\)
\(80\) 0 0
\(81\) −59531.0 −1.00816
\(82\) 0 0
\(83\) 38466.0i 0.612889i 0.951889 + 0.306444i \(0.0991394\pi\)
−0.951889 + 0.306444i \(0.900861\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 121572.i 1.72201i
\(88\) 0 0
\(89\) −119514. −1.59935 −0.799675 0.600432i \(-0.794995\pi\)
−0.799675 + 0.600432i \(0.794995\pi\)
\(90\) 0 0
\(91\) −135596. −1.71650
\(92\) 0 0
\(93\) − 205832.i − 2.46777i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 94658.0i 1.02148i 0.859737 + 0.510738i \(0.170628\pi\)
−0.859737 + 0.510738i \(0.829372\pi\)
\(98\) 0 0
\(99\) 115680. 1.18623
\(100\) 0 0
\(101\) 101046. 0.985634 0.492817 0.870133i \(-0.335967\pi\)
0.492817 + 0.870133i \(0.335967\pi\)
\(102\) 0 0
\(103\) 143434.i 1.33217i 0.745877 + 0.666084i \(0.232031\pi\)
−0.745877 + 0.666084i \(0.767969\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 57054.0i − 0.481755i −0.970555 0.240878i \(-0.922565\pi\)
0.970555 0.240878i \(-0.0774353\pi\)
\(108\) 0 0
\(109\) 3118.00 0.0251368 0.0125684 0.999921i \(-0.495999\pi\)
0.0125684 + 0.999921i \(0.495999\pi\)
\(110\) 0 0
\(111\) 123596. 0.952132
\(112\) 0 0
\(113\) 54534.0i 0.401764i 0.979615 + 0.200882i \(0.0643808\pi\)
−0.979615 + 0.200882i \(0.935619\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 149902.i − 1.01238i
\(118\) 0 0
\(119\) −40548.0 −0.262484
\(120\) 0 0
\(121\) 69349.0 0.430603
\(122\) 0 0
\(123\) 316668.i 1.88730i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 24698.0i 0.135879i 0.997689 + 0.0679395i \(0.0216425\pi\)
−0.997689 + 0.0679395i \(0.978358\pi\)
\(128\) 0 0
\(129\) 8140.00 0.0430675
\(130\) 0 0
\(131\) 236640. 1.20479 0.602393 0.798200i \(-0.294214\pi\)
0.602393 + 0.798200i \(0.294214\pi\)
\(132\) 0 0
\(133\) 262472.i 1.28663i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 22158.0i − 0.100862i −0.998728 0.0504312i \(-0.983940\pi\)
0.998728 0.0504312i \(-0.0160596\pi\)
\(138\) 0 0
\(139\) 193204. 0.848163 0.424081 0.905624i \(-0.360597\pi\)
0.424081 + 0.905624i \(0.360597\pi\)
\(140\) 0 0
\(141\) 355212. 1.50467
\(142\) 0 0
\(143\) − 298560.i − 1.22093i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 675774.i 2.57934i
\(148\) 0 0
\(149\) −448554. −1.65519 −0.827597 0.561322i \(-0.810293\pi\)
−0.827597 + 0.561322i \(0.810293\pi\)
\(150\) 0 0
\(151\) −140860. −0.502742 −0.251371 0.967891i \(-0.580881\pi\)
−0.251371 + 0.967891i \(0.580881\pi\)
\(152\) 0 0
\(153\) − 44826.0i − 0.154811i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 335878.i − 1.08751i −0.839245 0.543754i \(-0.817002\pi\)
0.839245 0.543754i \(-0.182998\pi\)
\(158\) 0 0
\(159\) 96228.0 0.301862
\(160\) 0 0
\(161\) −694548. −2.11173
\(162\) 0 0
\(163\) 101650.i 0.299667i 0.988711 + 0.149833i \(0.0478737\pi\)
−0.988711 + 0.149833i \(0.952126\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 139242.i 0.386348i 0.981164 + 0.193174i \(0.0618782\pi\)
−0.981164 + 0.193174i \(0.938122\pi\)
\(168\) 0 0
\(169\) −15591.0 −0.0419911
\(170\) 0 0
\(171\) −290164. −0.758845
\(172\) 0 0
\(173\) 265014.i 0.673215i 0.941645 + 0.336607i \(0.109279\pi\)
−0.941645 + 0.336607i \(0.890721\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 258456.i − 0.620088i
\(178\) 0 0
\(179\) 142812. 0.333144 0.166572 0.986029i \(-0.446730\pi\)
0.166572 + 0.986029i \(0.446730\pi\)
\(180\) 0 0
\(181\) 109670. 0.248824 0.124412 0.992231i \(-0.460296\pi\)
0.124412 + 0.992231i \(0.460296\pi\)
\(182\) 0 0
\(183\) − 290444.i − 0.641113i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 89280.0i − 0.186703i
\(188\) 0 0
\(189\) 9592.00 0.0195324
\(190\) 0 0
\(191\) 294948. 0.585008 0.292504 0.956264i \(-0.405511\pi\)
0.292504 + 0.956264i \(0.405511\pi\)
\(192\) 0 0
\(193\) − 1.00303e6i − 1.93831i −0.246459 0.969153i \(-0.579267\pi\)
0.246459 0.969153i \(-0.420733\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 823998.i − 1.51273i −0.654151 0.756364i \(-0.726974\pi\)
0.654151 0.756364i \(-0.273026\pi\)
\(198\) 0 0
\(199\) 906712. 1.62307 0.811534 0.584305i \(-0.198633\pi\)
0.811534 + 0.584305i \(0.198633\pi\)
\(200\) 0 0
\(201\) −253924. −0.443316
\(202\) 0 0
\(203\) − 1.20467e6i − 2.05176i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 767826.i − 1.24548i
\(208\) 0 0
\(209\) −577920. −0.915170
\(210\) 0 0
\(211\) 506384. 0.783022 0.391511 0.920173i \(-0.371953\pi\)
0.391511 + 0.920173i \(0.371953\pi\)
\(212\) 0 0
\(213\) 649704.i 0.981220i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.03961e6i 2.94034i
\(218\) 0 0
\(219\) −741356. −1.04452
\(220\) 0 0
\(221\) −115692. −0.159339
\(222\) 0 0
\(223\) 542050.i 0.729923i 0.931023 + 0.364962i \(0.118918\pi\)
−0.931023 + 0.364962i \(0.881082\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 1.44857e6i − 1.86585i −0.360075 0.932924i \(-0.617249\pi\)
0.360075 0.932924i \(-0.382751\pi\)
\(228\) 0 0
\(229\) 478786. 0.603327 0.301663 0.953414i \(-0.402458\pi\)
0.301663 + 0.953414i \(0.402458\pi\)
\(230\) 0 0
\(231\) −2.30208e6 −2.83851
\(232\) 0 0
\(233\) − 374106.i − 0.451445i −0.974192 0.225723i \(-0.927526\pi\)
0.974192 0.225723i \(-0.0724743\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 686576.i 0.793995i
\(238\) 0 0
\(239\) −169416. −0.191849 −0.0959245 0.995389i \(-0.530581\pi\)
−0.0959245 + 0.995389i \(0.530581\pi\)
\(240\) 0 0
\(241\) −353746. −0.392328 −0.196164 0.980571i \(-0.562848\pi\)
−0.196164 + 0.980571i \(0.562848\pi\)
\(242\) 0 0
\(243\) 1.29899e6i 1.41121i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 748888.i 0.781042i
\(248\) 0 0
\(249\) 846252. 0.864971
\(250\) 0 0
\(251\) 1.25520e6 1.25756 0.628780 0.777583i \(-0.283555\pi\)
0.628780 + 0.777583i \(0.283555\pi\)
\(252\) 0 0
\(253\) − 1.52928e6i − 1.50205i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 1.12877e6i − 1.06604i −0.846102 0.533021i \(-0.821057\pi\)
0.846102 0.533021i \(-0.178943\pi\)
\(258\) 0 0
\(259\) −1.22472e6 −1.13446
\(260\) 0 0
\(261\) 1.33177e6 1.21012
\(262\) 0 0
\(263\) 263082.i 0.234532i 0.993101 + 0.117266i \(0.0374130\pi\)
−0.993101 + 0.117266i \(0.962587\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.62931e6i 2.25717i
\(268\) 0 0
\(269\) 1.18774e6 1.00079 0.500393 0.865798i \(-0.333189\pi\)
0.500393 + 0.865798i \(0.333189\pi\)
\(270\) 0 0
\(271\) 431300. 0.356744 0.178372 0.983963i \(-0.442917\pi\)
0.178372 + 0.983963i \(0.442917\pi\)
\(272\) 0 0
\(273\) 2.98311e6i 2.42250i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 743114.i 0.581910i 0.956737 + 0.290955i \(0.0939730\pi\)
−0.956737 + 0.290955i \(0.906027\pi\)
\(278\) 0 0
\(279\) −2.25480e6 −1.73419
\(280\) 0 0
\(281\) 1.92193e6 1.45201 0.726007 0.687687i \(-0.241374\pi\)
0.726007 + 0.687687i \(0.241374\pi\)
\(282\) 0 0
\(283\) 1.63071e6i 1.21035i 0.796092 + 0.605176i \(0.206897\pi\)
−0.796092 + 0.605176i \(0.793103\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 3.13789e6i − 2.24871i
\(288\) 0 0
\(289\) 1.38526e6 0.975634
\(290\) 0 0
\(291\) 2.08248e6 1.44161
\(292\) 0 0
\(293\) − 71250.0i − 0.0484859i −0.999706 0.0242430i \(-0.992282\pi\)
0.999706 0.0242430i \(-0.00771753\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 21120.0i 0.0138932i
\(298\) 0 0
\(299\) −1.98169e6 −1.28191
\(300\) 0 0
\(301\) −80660.0 −0.0513147
\(302\) 0 0
\(303\) − 2.22301e6i − 1.39103i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 1.61762e6i − 0.979560i −0.871846 0.489780i \(-0.837077\pi\)
0.871846 0.489780i \(-0.162923\pi\)
\(308\) 0 0
\(309\) 3.15555e6 1.88009
\(310\) 0 0
\(311\) −682788. −0.400299 −0.200150 0.979765i \(-0.564143\pi\)
−0.200150 + 0.979765i \(0.564143\pi\)
\(312\) 0 0
\(313\) 2.70444e6i 1.56033i 0.625574 + 0.780165i \(0.284865\pi\)
−0.625574 + 0.780165i \(0.715135\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.60347e6i 1.45514i 0.686035 + 0.727568i \(0.259350\pi\)
−0.686035 + 0.727568i \(0.740650\pi\)
\(318\) 0 0
\(319\) 2.65248e6 1.45940
\(320\) 0 0
\(321\) −1.25519e6 −0.679902
\(322\) 0 0
\(323\) 223944.i 0.119435i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 68596.0i − 0.0354756i
\(328\) 0 0
\(329\) −3.51983e6 −1.79280
\(330\) 0 0
\(331\) −661432. −0.331830 −0.165915 0.986140i \(-0.553058\pi\)
−0.165915 + 0.986140i \(0.553058\pi\)
\(332\) 0 0
\(333\) − 1.35394e6i − 0.669096i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.71706e6i 0.823588i 0.911277 + 0.411794i \(0.135098\pi\)
−0.911277 + 0.411794i \(0.864902\pi\)
\(338\) 0 0
\(339\) 1.19975e6 0.567010
\(340\) 0 0
\(341\) −4.49088e6 −2.09144
\(342\) 0 0
\(343\) − 3.03238e6i − 1.39171i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 131370.i 0.0585696i 0.999571 + 0.0292848i \(0.00932298\pi\)
−0.999571 + 0.0292848i \(0.990677\pi\)
\(348\) 0 0
\(349\) −3.50951e6 −1.54235 −0.771175 0.636623i \(-0.780331\pi\)
−0.771175 + 0.636623i \(0.780331\pi\)
\(350\) 0 0
\(351\) 27368.0 0.0118570
\(352\) 0 0
\(353\) − 2.21992e6i − 0.948202i −0.880470 0.474101i \(-0.842773\pi\)
0.880470 0.474101i \(-0.157227\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 892056.i 0.370443i
\(358\) 0 0
\(359\) −4.39730e6 −1.80074 −0.900369 0.435128i \(-0.856703\pi\)
−0.900369 + 0.435128i \(0.856703\pi\)
\(360\) 0 0
\(361\) −1.02648e6 −0.414557
\(362\) 0 0
\(363\) − 1.52568e6i − 0.607710i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 2.29824e6i − 0.890697i −0.895357 0.445348i \(-0.853080\pi\)
0.895357 0.445348i \(-0.146920\pi\)
\(368\) 0 0
\(369\) 3.46895e6 1.32627
\(370\) 0 0
\(371\) −953532. −0.359667
\(372\) 0 0
\(373\) 1.73561e6i 0.645920i 0.946413 + 0.322960i \(0.104678\pi\)
−0.946413 + 0.322960i \(0.895322\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 3.43717e6i − 1.24551i
\(378\) 0 0
\(379\) 5.39115e6 1.92789 0.963947 0.266094i \(-0.0857331\pi\)
0.963947 + 0.266094i \(0.0857331\pi\)
\(380\) 0 0
\(381\) 543356. 0.191766
\(382\) 0 0
\(383\) − 3.27281e6i − 1.14005i −0.821627 0.570026i \(-0.806933\pi\)
0.821627 0.570026i \(-0.193067\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 89170.0i − 0.0302650i
\(388\) 0 0
\(389\) −603114. −0.202081 −0.101040 0.994882i \(-0.532217\pi\)
−0.101040 + 0.994882i \(0.532217\pi\)
\(390\) 0 0
\(391\) −592596. −0.196027
\(392\) 0 0
\(393\) − 5.20608e6i − 1.70032i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 749422.i − 0.238644i −0.992856 0.119322i \(-0.961928\pi\)
0.992856 0.119322i \(-0.0380721\pi\)
\(398\) 0 0
\(399\) 5.77438e6 1.81582
\(400\) 0 0
\(401\) 5.31357e6 1.65016 0.825079 0.565018i \(-0.191131\pi\)
0.825079 + 0.565018i \(0.191131\pi\)
\(402\) 0 0
\(403\) 5.81943e6i 1.78492i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 2.69664e6i − 0.806932i
\(408\) 0 0
\(409\) −999326. −0.295392 −0.147696 0.989033i \(-0.547186\pi\)
−0.147696 + 0.989033i \(0.547186\pi\)
\(410\) 0 0
\(411\) −487476. −0.142347
\(412\) 0 0
\(413\) 2.56106e6i 0.738831i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 4.25049e6i − 1.19701i
\(418\) 0 0
\(419\) −2.03740e6 −0.566944 −0.283472 0.958980i \(-0.591486\pi\)
−0.283472 + 0.958980i \(0.591486\pi\)
\(420\) 0 0
\(421\) −5.11461e6 −1.40640 −0.703198 0.710994i \(-0.748245\pi\)
−0.703198 + 0.710994i \(0.748245\pi\)
\(422\) 0 0
\(423\) − 3.89119e6i − 1.05738i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.87804e6i 0.763882i
\(428\) 0 0
\(429\) −6.56832e6 −1.72310
\(430\) 0 0
\(431\) −3.30404e6 −0.856747 −0.428374 0.903602i \(-0.640913\pi\)
−0.428374 + 0.903602i \(0.640913\pi\)
\(432\) 0 0
\(433\) 2.01638e6i 0.516836i 0.966033 + 0.258418i \(0.0832012\pi\)
−0.966033 + 0.258418i \(0.916799\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.83594e6i 0.960879i
\(438\) 0 0
\(439\) −6.58321e6 −1.63033 −0.815166 0.579227i \(-0.803355\pi\)
−0.815166 + 0.579227i \(0.803355\pi\)
\(440\) 0 0
\(441\) 7.40280e6 1.81259
\(442\) 0 0
\(443\) 4.81783e6i 1.16638i 0.812334 + 0.583192i \(0.198197\pi\)
−0.812334 + 0.583192i \(0.801803\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 9.86819e6i 2.33598i
\(448\) 0 0
\(449\) 6.20399e6 1.45230 0.726149 0.687538i \(-0.241308\pi\)
0.726149 + 0.687538i \(0.241308\pi\)
\(450\) 0 0
\(451\) 6.90912e6 1.59949
\(452\) 0 0
\(453\) 3.09892e6i 0.709520i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.84383e6i 0.636962i 0.947929 + 0.318481i \(0.103173\pi\)
−0.947929 + 0.318481i \(0.896827\pi\)
\(458\) 0 0
\(459\) 8184.00 0.00181315
\(460\) 0 0
\(461\) −1.75605e6 −0.384844 −0.192422 0.981312i \(-0.561634\pi\)
−0.192422 + 0.981312i \(0.561634\pi\)
\(462\) 0 0
\(463\) − 7.66857e6i − 1.66250i −0.555899 0.831250i \(-0.687626\pi\)
0.555899 0.831250i \(-0.312374\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 1.35903e6i − 0.288361i −0.989551 0.144181i \(-0.953945\pi\)
0.989551 0.144181i \(-0.0460546\pi\)
\(468\) 0 0
\(469\) 2.51616e6 0.528209
\(470\) 0 0
\(471\) −7.38932e6 −1.53480
\(472\) 0 0
\(473\) − 177600.i − 0.0364998i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 1.05413e6i − 0.212129i
\(478\) 0 0
\(479\) 2.02706e6 0.403672 0.201836 0.979419i \(-0.435309\pi\)
0.201836 + 0.979419i \(0.435309\pi\)
\(480\) 0 0
\(481\) −3.49440e6 −0.688667
\(482\) 0 0
\(483\) 1.52801e7i 2.98028i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2.46427e6i 0.470833i 0.971895 + 0.235416i \(0.0756454\pi\)
−0.971895 + 0.235416i \(0.924355\pi\)
\(488\) 0 0
\(489\) 2.23630e6 0.422920
\(490\) 0 0
\(491\) 1.03848e7 1.94399 0.971996 0.234998i \(-0.0755084\pi\)
0.971996 + 0.234998i \(0.0755084\pi\)
\(492\) 0 0
\(493\) − 1.02784e6i − 0.190461i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 6.43798e6i − 1.16912i
\(498\) 0 0
\(499\) −6.49416e6 −1.16754 −0.583769 0.811919i \(-0.698423\pi\)
−0.583769 + 0.811919i \(0.698423\pi\)
\(500\) 0 0
\(501\) 3.06332e6 0.545254
\(502\) 0 0
\(503\) 1.03565e7i 1.82513i 0.408931 + 0.912565i \(0.365902\pi\)
−0.408931 + 0.912565i \(0.634098\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 343002.i 0.0592621i
\(508\) 0 0
\(509\) −5.87305e6 −1.00478 −0.502388 0.864643i \(-0.667545\pi\)
−0.502388 + 0.864643i \(0.667545\pi\)
\(510\) 0 0
\(511\) 7.34616e6 1.24454
\(512\) 0 0
\(513\) − 52976.0i − 0.00888763i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 7.75008e6i − 1.27520i
\(518\) 0 0
\(519\) 5.83031e6 0.950108
\(520\) 0 0
\(521\) 2.17295e6 0.350717 0.175358 0.984505i \(-0.443892\pi\)
0.175358 + 0.984505i \(0.443892\pi\)
\(522\) 0 0
\(523\) − 1.07361e6i − 0.171629i −0.996311 0.0858145i \(-0.972651\pi\)
0.996311 0.0858145i \(-0.0273492\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.74022e6i 0.272946i
\(528\) 0 0
\(529\) −3.71425e6 −0.577075
\(530\) 0 0
\(531\) −2.83127e6 −0.435757
\(532\) 0 0
\(533\) − 8.95307e6i − 1.36507i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 3.14186e6i − 0.470166i
\(538\) 0 0
\(539\) 1.47442e7 2.18599
\(540\) 0 0
\(541\) 7.09033e6 1.04153 0.520767 0.853699i \(-0.325646\pi\)
0.520767 + 0.853699i \(0.325646\pi\)
\(542\) 0 0
\(543\) − 2.41274e6i − 0.351165i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 6.69763e6i 0.957091i 0.878063 + 0.478545i \(0.158836\pi\)
−0.878063 + 0.478545i \(0.841164\pi\)
\(548\) 0 0
\(549\) −3.18168e6 −0.450532
\(550\) 0 0
\(551\) −6.65330e6 −0.933595
\(552\) 0 0
\(553\) − 6.80334e6i − 0.946040i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.19008e7i 1.62532i 0.582735 + 0.812662i \(0.301982\pi\)
−0.582735 + 0.812662i \(0.698018\pi\)
\(558\) 0 0
\(559\) −230140. −0.0311503
\(560\) 0 0
\(561\) −1.96416e6 −0.263493
\(562\) 0 0
\(563\) − 8.75636e6i − 1.16427i −0.813093 0.582133i \(-0.802218\pi\)
0.813093 0.582133i \(-0.197782\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 1.29778e7i − 1.69528i
\(568\) 0 0
\(569\) 1.15677e6 0.149784 0.0748922 0.997192i \(-0.476139\pi\)
0.0748922 + 0.997192i \(0.476139\pi\)
\(570\) 0 0
\(571\) −7.07807e6 −0.908500 −0.454250 0.890874i \(-0.650093\pi\)
−0.454250 + 0.890874i \(0.650093\pi\)
\(572\) 0 0
\(573\) − 6.48886e6i − 0.825623i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 3.13404e6i − 0.391890i −0.980615 0.195945i \(-0.937223\pi\)
0.980615 0.195945i \(-0.0627775\pi\)
\(578\) 0 0
\(579\) −2.20667e7 −2.73553
\(580\) 0 0
\(581\) −8.38559e6 −1.03061
\(582\) 0 0
\(583\) − 2.09952e6i − 0.255828i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.13833e7i 1.36355i 0.731561 + 0.681776i \(0.238792\pi\)
−0.731561 + 0.681776i \(0.761208\pi\)
\(588\) 0 0
\(589\) 1.12646e7 1.33791
\(590\) 0 0
\(591\) −1.81280e7 −2.13491
\(592\) 0 0
\(593\) 1.58655e7i 1.85275i 0.376599 + 0.926376i \(0.377094\pi\)
−0.376599 + 0.926376i \(0.622906\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 1.99477e7i − 2.29064i
\(598\) 0 0
\(599\) 1.50998e7 1.71951 0.859756 0.510705i \(-0.170615\pi\)
0.859756 + 0.510705i \(0.170615\pi\)
\(600\) 0 0
\(601\) −8.08705e6 −0.913280 −0.456640 0.889652i \(-0.650947\pi\)
−0.456640 + 0.889652i \(0.650947\pi\)
\(602\) 0 0
\(603\) 2.78162e6i 0.311534i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 710398.i − 0.0782582i −0.999234 0.0391291i \(-0.987542\pi\)
0.999234 0.0391291i \(-0.0124584\pi\)
\(608\) 0 0
\(609\) −2.65027e7 −2.89566
\(610\) 0 0
\(611\) −1.00428e7 −1.08831
\(612\) 0 0
\(613\) − 5.96434e6i − 0.641078i −0.947235 0.320539i \(-0.896136\pi\)
0.947235 0.320539i \(-0.103864\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.48432e7i 1.56970i 0.619689 + 0.784848i \(0.287259\pi\)
−0.619689 + 0.784848i \(0.712741\pi\)
\(618\) 0 0
\(619\) 1.82042e7 1.90961 0.954807 0.297227i \(-0.0960620\pi\)
0.954807 + 0.297227i \(0.0960620\pi\)
\(620\) 0 0
\(621\) 140184. 0.0145871
\(622\) 0 0
\(623\) − 2.60541e7i − 2.68940i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1.27142e7i 1.29158i
\(628\) 0 0
\(629\) −1.04495e6 −0.105310
\(630\) 0 0
\(631\) 1.09461e6 0.109443 0.0547214 0.998502i \(-0.482573\pi\)
0.0547214 + 0.998502i \(0.482573\pi\)
\(632\) 0 0
\(633\) − 1.11404e7i − 1.10508i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 1.91060e7i − 1.86561i
\(638\) 0 0
\(639\) 7.11721e6 0.689537
\(640\) 0 0
\(641\) 7.44046e6 0.715245 0.357622 0.933866i \(-0.383587\pi\)
0.357622 + 0.933866i \(0.383587\pi\)
\(642\) 0 0
\(643\) 1.07915e7i 1.02933i 0.857391 + 0.514665i \(0.172084\pi\)
−0.857391 + 0.514665i \(0.827916\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 9.62998e6i − 0.904409i −0.891914 0.452204i \(-0.850638\pi\)
0.891914 0.452204i \(-0.149362\pi\)
\(648\) 0 0
\(649\) −5.63904e6 −0.525525
\(650\) 0 0
\(651\) 4.48714e7 4.14970
\(652\) 0 0
\(653\) 1.00019e7i 0.917905i 0.888461 + 0.458953i \(0.151775\pi\)
−0.888461 + 0.458953i \(0.848225\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.12122e6i 0.734020i
\(658\) 0 0
\(659\) 4.01060e6 0.359746 0.179873 0.983690i \(-0.442431\pi\)
0.179873 + 0.983690i \(0.442431\pi\)
\(660\) 0 0
\(661\) 1.20338e7 1.07127 0.535636 0.844449i \(-0.320072\pi\)
0.535636 + 0.844449i \(0.320072\pi\)
\(662\) 0 0
\(663\) 2.54522e6i 0.224876i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 1.76058e7i − 1.53229i
\(668\) 0 0
\(669\) 1.19251e7 1.03014
\(670\) 0 0
\(671\) −6.33696e6 −0.543344
\(672\) 0 0
\(673\) − 2.01231e6i − 0.171260i −0.996327 0.0856301i \(-0.972710\pi\)
0.996327 0.0856301i \(-0.0272903\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.62410e7i 1.36188i 0.732337 + 0.680942i \(0.238429\pi\)
−0.732337 + 0.680942i \(0.761571\pi\)
\(678\) 0 0
\(679\) −2.06354e7 −1.71767
\(680\) 0 0
\(681\) −3.18686e7 −2.63327
\(682\) 0 0
\(683\) − 4.62910e6i − 0.379704i −0.981813 0.189852i \(-0.939199\pi\)
0.981813 0.189852i \(-0.0608008\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 1.05333e7i − 0.851476i
\(688\) 0 0
\(689\) −2.72063e6 −0.218334
\(690\) 0 0
\(691\) 1.16794e7 0.930517 0.465258 0.885175i \(-0.345961\pi\)
0.465258 + 0.885175i \(0.345961\pi\)
\(692\) 0 0
\(693\) 2.52182e7i 1.99472i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 2.67728e6i − 0.208743i
\(698\) 0 0
\(699\) −8.23033e6 −0.637125
\(700\) 0 0
\(701\) 1.99543e7 1.53370 0.766851 0.641825i \(-0.221822\pi\)
0.766851 + 0.641825i \(0.221822\pi\)
\(702\) 0 0
\(703\) 6.76407e6i 0.516202i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.20280e7i 1.65740i
\(708\) 0 0
\(709\) 4.88331e6 0.364837 0.182419 0.983221i \(-0.441607\pi\)
0.182419 + 0.983221i \(0.441607\pi\)
\(710\) 0 0
\(711\) 7.52113e6 0.557968
\(712\) 0 0
\(713\) 2.98082e7i 2.19590i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.72715e6i 0.270757i
\(718\) 0 0
\(719\) 1.35778e7 0.979505 0.489753 0.871861i \(-0.337087\pi\)
0.489753 + 0.871861i \(0.337087\pi\)
\(720\) 0 0
\(721\) −3.12686e7 −2.24012
\(722\) 0 0
\(723\) 7.78241e6i 0.553692i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 6.42411e6i 0.450792i 0.974267 + 0.225396i \(0.0723677\pi\)
−0.974267 + 0.225396i \(0.927632\pi\)
\(728\) 0 0
\(729\) 1.41117e7 0.983472
\(730\) 0 0
\(731\) −68820.0 −0.00476345
\(732\) 0 0
\(733\) − 9.08556e6i − 0.624585i −0.949986 0.312293i \(-0.898903\pi\)
0.949986 0.312293i \(-0.101097\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.54016e6i 0.375711i
\(738\) 0 0
\(739\) −2.02457e7 −1.36371 −0.681854 0.731488i \(-0.738826\pi\)
−0.681854 + 0.731488i \(0.738826\pi\)
\(740\) 0 0
\(741\) 1.64755e7 1.10229
\(742\) 0 0
\(743\) 5.44831e6i 0.362067i 0.983477 + 0.181034i \(0.0579443\pi\)
−0.983477 + 0.181034i \(0.942056\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 9.27031e6i − 0.607845i
\(748\) 0 0
\(749\) 1.24378e7 0.810099
\(750\) 0 0
\(751\) −1.14072e6 −0.0738041 −0.0369021 0.999319i \(-0.511749\pi\)
−0.0369021 + 0.999319i \(0.511749\pi\)
\(752\) 0 0
\(753\) − 2.76144e7i − 1.77479i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.90153e7i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(758\) 0 0
\(759\) −3.36442e7 −2.11985
\(760\) 0 0
\(761\) 2.23551e7 1.39931 0.699656 0.714480i \(-0.253337\pi\)
0.699656 + 0.714480i \(0.253337\pi\)
\(762\) 0 0
\(763\) 679724.i 0.0422689i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.30726e6i 0.448504i
\(768\) 0 0
\(769\) 1.00704e7 0.614088 0.307044 0.951695i \(-0.400660\pi\)
0.307044 + 0.951695i \(0.400660\pi\)
\(770\) 0 0
\(771\) −2.48330e7 −1.50451
\(772\) 0 0
\(773\) 4.05963e6i 0.244364i 0.992508 + 0.122182i \(0.0389892\pi\)
−0.992508 + 0.122182i \(0.961011\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.69439e7i 1.60106i
\(778\) 0 0
\(779\) −1.73304e7 −1.02321
\(780\) 0 0
\(781\) 1.41754e7 0.831585
\(782\) 0 0
\(783\) 243144.i 0.0141729i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 1.72256e7i − 0.991372i −0.868502 0.495686i \(-0.834917\pi\)
0.868502 0.495686i \(-0.165083\pi\)
\(788\) 0 0
\(789\) 5.78780e6 0.330995
\(790\) 0 0
\(791\) −1.18884e7 −0.675589
\(792\) 0 0
\(793\) 8.21164e6i 0.463711i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 2.10793e7i − 1.17547i −0.809055 0.587733i \(-0.800020\pi\)
0.809055 0.587733i \(-0.199980\pi\)
\(798\) 0 0
\(799\) −3.00316e6 −0.166422
\(800\) 0 0
\(801\) 2.88029e7 1.58619
\(802\) 0 0
\(803\) 1.61750e7i 0.885231i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 2.61303e7i − 1.41241i
\(808\) 0 0
\(809\) 1.87877e7 1.00926 0.504629 0.863336i \(-0.331629\pi\)
0.504629 + 0.863336i \(0.331629\pi\)
\(810\) 0 0
\(811\) −1.32456e7 −0.707164 −0.353582 0.935404i \(-0.615036\pi\)
−0.353582 + 0.935404i \(0.615036\pi\)
\(812\) 0 0
\(813\) − 9.48860e6i − 0.503473i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 445480.i 0.0233493i
\(818\) 0 0
\(819\) 3.26786e7 1.70237
\(820\) 0 0
\(821\) −7.66925e6 −0.397096 −0.198548 0.980091i \(-0.563623\pi\)
−0.198548 + 0.980091i \(0.563623\pi\)
\(822\) 0 0
\(823\) 8.82786e6i 0.454314i 0.973858 + 0.227157i \(0.0729430\pi\)
−0.973858 + 0.227157i \(0.927057\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 3.06923e7i − 1.56051i −0.625463 0.780254i \(-0.715090\pi\)
0.625463 0.780254i \(-0.284910\pi\)
\(828\) 0 0
\(829\) −3.28414e7 −1.65972 −0.829860 0.557972i \(-0.811580\pi\)
−0.829860 + 0.557972i \(0.811580\pi\)
\(830\) 0 0
\(831\) 1.63485e7 0.821250
\(832\) 0 0
\(833\) − 5.71336e6i − 0.285285i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 411664.i − 0.0203109i
\(838\) 0 0
\(839\) 8.42117e6 0.413017 0.206508 0.978445i \(-0.433790\pi\)
0.206508 + 0.978445i \(0.433790\pi\)
\(840\) 0 0
\(841\) 1.00255e7 0.488784
\(842\) 0 0
\(843\) − 4.22824e7i − 2.04923i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.51181e7i 0.724083i
\(848\) 0 0
\(849\) 3.58757e7 1.70817
\(850\) 0 0
\(851\) −1.78989e7 −0.847234
\(852\) 0 0
\(853\) − 2.35126e7i − 1.10644i −0.833035 0.553221i \(-0.813399\pi\)
0.833035 0.553221i \(-0.186601\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1.13050e7i − 0.525799i −0.964823 0.262900i \(-0.915321\pi\)
0.964823 0.262900i \(-0.0846788\pi\)
\(858\) 0 0
\(859\) −1.00078e7 −0.462758 −0.231379 0.972864i \(-0.574324\pi\)
−0.231379 + 0.972864i \(0.574324\pi\)
\(860\) 0 0
\(861\) −6.90336e7 −3.17360
\(862\) 0 0
\(863\) − 2.61429e7i − 1.19489i −0.801911 0.597443i \(-0.796183\pi\)
0.801911 0.597443i \(-0.203817\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 3.04757e7i − 1.37691i
\(868\) 0 0
\(869\) 1.49798e7 0.672911
\(870\) 0 0
\(871\) 7.17912e6 0.320646
\(872\) 0 0
\(873\) − 2.28126e7i − 1.01307i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 1.92041e6i − 0.0843129i −0.999111 0.0421565i \(-0.986577\pi\)
0.999111 0.0421565i \(-0.0134228\pi\)
\(878\) 0 0
\(879\) −1.56750e6 −0.0684282
\(880\) 0 0
\(881\) −2.56594e7 −1.11380 −0.556899 0.830580i \(-0.688009\pi\)
−0.556899 + 0.830580i \(0.688009\pi\)
\(882\) 0 0
\(883\) 2.05643e7i 0.887590i 0.896128 + 0.443795i \(0.146368\pi\)
−0.896128 + 0.443795i \(0.853632\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.16868e7i 1.35229i 0.736770 + 0.676143i \(0.236350\pi\)
−0.736770 + 0.676143i \(0.763650\pi\)
\(888\) 0 0
\(889\) −5.38416e6 −0.228488
\(890\) 0 0
\(891\) 2.85749e7 1.20584
\(892\) 0 0
\(893\) 1.94398e7i 0.815761i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4.35972e7i 1.80916i
\(898\) 0 0
\(899\) −5.17013e7 −2.13355
\(900\) 0 0
\(901\) −813564. −0.0333872
\(902\) 0 0
\(903\) 1.77452e6i 0.0724205i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 3.96963e6i 0.160225i 0.996786 + 0.0801127i \(0.0255280\pi\)
−0.996786 + 0.0801127i \(0.974472\pi\)
\(908\) 0 0
\(909\) −2.43521e7 −0.977522
\(910\) 0 0
\(911\) −1.37945e7 −0.550692 −0.275346 0.961345i \(-0.588792\pi\)
−0.275346 + 0.961345i \(0.588792\pi\)
\(912\) 0 0
\(913\) − 1.84637e7i − 0.733063i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.15875e7i 2.02592i
\(918\) 0 0
\(919\) −8.08126e6 −0.315639 −0.157819 0.987468i \(-0.550446\pi\)
−0.157819 + 0.987468i \(0.550446\pi\)
\(920\) 0 0
\(921\) −3.55877e7 −1.38245
\(922\) 0 0
\(923\) − 1.83689e7i − 0.709707i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 3.45676e7i − 1.32120i
\(928\) 0 0
\(929\) −2.99956e7 −1.14030 −0.570150 0.821541i \(-0.693115\pi\)
−0.570150 + 0.821541i \(0.693115\pi\)
\(930\) 0 0
\(931\) −3.69833e7 −1.39840
\(932\) 0 0
\(933\) 1.50213e7i 0.564943i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 2.07620e7i − 0.772540i −0.922386 0.386270i \(-0.873763\pi\)
0.922386 0.386270i \(-0.126237\pi\)
\(938\) 0 0
\(939\) 5.94976e7 2.20209
\(940\) 0 0
\(941\) −3.47642e6 −0.127985 −0.0639923 0.997950i \(-0.520383\pi\)
−0.0639923 + 0.997950i \(0.520383\pi\)
\(942\) 0 0
\(943\) − 4.58593e7i − 1.67938i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.86700e6i 0.0676503i 0.999428 + 0.0338252i \(0.0107689\pi\)
−0.999428 + 0.0338252i \(0.989231\pi\)
\(948\) 0 0
\(949\) 2.09602e7 0.755490
\(950\) 0 0
\(951\) 5.72763e7 2.05364
\(952\) 0 0
\(953\) 3.85501e7i 1.37497i 0.726199 + 0.687484i \(0.241285\pi\)
−0.726199 + 0.687484i \(0.758715\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 5.83546e7i − 2.05966i
\(958\) 0 0
\(959\) 4.83044e6 0.169606
\(960\) 0 0
\(961\) 5.89056e7 2.05754
\(962\) 0 0
\(963\) 1.37500e7i 0.477790i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.64875e7i 0.567008i 0.958971 + 0.283504i \(0.0914969\pi\)
−0.958971 + 0.283504i \(0.908503\pi\)
\(968\) 0 0
\(969\) 4.92677e6 0.168559
\(970\) 0 0
\(971\) −2.36976e7 −0.806597 −0.403299 0.915068i \(-0.632136\pi\)
−0.403299 + 0.915068i \(0.632136\pi\)
\(972\) 0 0
\(973\) 4.21185e7i 1.42623i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 5.77590e7i − 1.93590i −0.251143 0.967950i \(-0.580806\pi\)
0.251143 0.967950i \(-0.419194\pi\)
\(978\) 0 0
\(979\) 5.73667e7 1.91295
\(980\) 0 0
\(981\) −751438. −0.0249299
\(982\) 0 0
\(983\) 1.10103e7i 0.363425i 0.983352 + 0.181712i \(0.0581640\pi\)
−0.983352 + 0.181712i \(0.941836\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 7.74362e7i 2.53018i
\(988\) 0 0
\(989\) −1.17882e6 −0.0383228
\(990\) 0 0
\(991\) 3.70807e7 1.19940 0.599700 0.800225i \(-0.295287\pi\)
0.599700 + 0.800225i \(0.295287\pi\)
\(992\) 0 0
\(993\) 1.45515e7i 0.468311i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 4.52935e6i 0.144311i 0.997393 + 0.0721553i \(0.0229877\pi\)
−0.997393 + 0.0721553i \(0.977012\pi\)
\(998\) 0 0
\(999\) 247192. 0.00783647
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 100.6.c.a.49.1 2
3.2 odd 2 900.6.d.h.649.2 2
4.3 odd 2 400.6.c.c.49.2 2
5.2 odd 4 100.6.a.a.1.1 1
5.3 odd 4 20.6.a.a.1.1 1
5.4 even 2 inner 100.6.c.a.49.2 2
15.2 even 4 900.6.a.b.1.1 1
15.8 even 4 180.6.a.e.1.1 1
15.14 odd 2 900.6.d.h.649.1 2
20.3 even 4 80.6.a.b.1.1 1
20.7 even 4 400.6.a.m.1.1 1
20.19 odd 2 400.6.c.c.49.1 2
35.13 even 4 980.6.a.b.1.1 1
40.3 even 4 320.6.a.n.1.1 1
40.13 odd 4 320.6.a.c.1.1 1
60.23 odd 4 720.6.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.6.a.a.1.1 1 5.3 odd 4
80.6.a.b.1.1 1 20.3 even 4
100.6.a.a.1.1 1 5.2 odd 4
100.6.c.a.49.1 2 1.1 even 1 trivial
100.6.c.a.49.2 2 5.4 even 2 inner
180.6.a.e.1.1 1 15.8 even 4
320.6.a.c.1.1 1 40.13 odd 4
320.6.a.n.1.1 1 40.3 even 4
400.6.a.m.1.1 1 20.7 even 4
400.6.c.c.49.1 2 20.19 odd 2
400.6.c.c.49.2 2 4.3 odd 2
720.6.a.l.1.1 1 60.23 odd 4
900.6.a.b.1.1 1 15.2 even 4
900.6.d.h.649.1 2 15.14 odd 2
900.6.d.h.649.2 2 3.2 odd 2
980.6.a.b.1.1 1 35.13 even 4