Properties

Label 80.6.a.d.1.1
Level 80
Weight 6
Character 80.1
Self dual yes
Analytic conductor 12.831
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 80.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.8307055850\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 80.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{3} -25.0000 q^{5} +62.0000 q^{7} -239.000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} -25.0000 q^{5} +62.0000 q^{7} -239.000 q^{9} +144.000 q^{11} -654.000 q^{13} -50.0000 q^{15} -1190.00 q^{17} -556.000 q^{19} +124.000 q^{21} -2182.00 q^{23} +625.000 q^{25} -964.000 q^{27} -1578.00 q^{29} -9660.00 q^{31} +288.000 q^{33} -1550.00 q^{35} -3534.00 q^{37} -1308.00 q^{39} +7462.00 q^{41} +7114.00 q^{43} +5975.00 q^{45} +28294.0 q^{47} -12963.0 q^{49} -2380.00 q^{51} -13046.0 q^{53} -3600.00 q^{55} -1112.00 q^{57} +37092.0 q^{59} +39570.0 q^{61} -14818.0 q^{63} +16350.0 q^{65} +56734.0 q^{67} -4364.00 q^{69} -45588.0 q^{71} +11842.0 q^{73} +1250.00 q^{75} +8928.00 q^{77} -94216.0 q^{79} +56149.0 q^{81} +31482.0 q^{83} +29750.0 q^{85} -3156.00 q^{87} -94054.0 q^{89} -40548.0 q^{91} -19320.0 q^{93} +13900.0 q^{95} +23714.0 q^{97} -34416.0 q^{99} +O(q^{100})\)

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
\(3\) 2.00000 0.128300 0.0641500 0.997940i \(-0.479566\pi\)
0.0641500 + 0.997940i \(0.479566\pi\)
\(4\) 0 0
\(5\) −25.0000 −0.447214
\(6\) 0 0
\(7\) 62.0000 0.478241 0.239120 0.970990i \(-0.423141\pi\)
0.239120 + 0.970990i \(0.423141\pi\)
\(8\) 0 0
\(9\) −239.000 −0.983539
\(10\) 0 0
\(11\) 144.000 0.358823 0.179412 0.983774i \(-0.442581\pi\)
0.179412 + 0.983774i \(0.442581\pi\)
\(12\) 0 0
\(13\) −654.000 −1.07330 −0.536648 0.843806i \(-0.680310\pi\)
−0.536648 + 0.843806i \(0.680310\pi\)
\(14\) 0 0
\(15\) −50.0000 −0.0573775
\(16\) 0 0
\(17\) −1190.00 −0.998676 −0.499338 0.866407i \(-0.666423\pi\)
−0.499338 + 0.866407i \(0.666423\pi\)
\(18\) 0 0
\(19\) −556.000 −0.353338 −0.176669 0.984270i \(-0.556532\pi\)
−0.176669 + 0.984270i \(0.556532\pi\)
\(20\) 0 0
\(21\) 124.000 0.0613583
\(22\) 0 0
\(23\) −2182.00 −0.860073 −0.430036 0.902812i \(-0.641499\pi\)
−0.430036 + 0.902812i \(0.641499\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) −964.000 −0.254488
\(28\) 0 0
\(29\) −1578.00 −0.348427 −0.174214 0.984708i \(-0.555738\pi\)
−0.174214 + 0.984708i \(0.555738\pi\)
\(30\) 0 0
\(31\) −9660.00 −1.80540 −0.902699 0.430273i \(-0.858417\pi\)
−0.902699 + 0.430273i \(0.858417\pi\)
\(32\) 0 0
\(33\) 288.000 0.0460371
\(34\) 0 0
\(35\) −1550.00 −0.213876
\(36\) 0 0
\(37\) −3534.00 −0.424387 −0.212194 0.977228i \(-0.568061\pi\)
−0.212194 + 0.977228i \(0.568061\pi\)
\(38\) 0 0
\(39\) −1308.00 −0.137704
\(40\) 0 0
\(41\) 7462.00 0.693259 0.346630 0.938002i \(-0.387326\pi\)
0.346630 + 0.938002i \(0.387326\pi\)
\(42\) 0 0
\(43\) 7114.00 0.586736 0.293368 0.956000i \(-0.405224\pi\)
0.293368 + 0.956000i \(0.405224\pi\)
\(44\) 0 0
\(45\) 5975.00 0.439852
\(46\) 0 0
\(47\) 28294.0 1.86831 0.934157 0.356863i \(-0.116154\pi\)
0.934157 + 0.356863i \(0.116154\pi\)
\(48\) 0 0
\(49\) −12963.0 −0.771286
\(50\) 0 0
\(51\) −2380.00 −0.128130
\(52\) 0 0
\(53\) −13046.0 −0.637952 −0.318976 0.947763i \(-0.603339\pi\)
−0.318976 + 0.947763i \(0.603339\pi\)
\(54\) 0 0
\(55\) −3600.00 −0.160471
\(56\) 0 0
\(57\) −1112.00 −0.0453333
\(58\) 0 0
\(59\) 37092.0 1.38724 0.693618 0.720343i \(-0.256016\pi\)
0.693618 + 0.720343i \(0.256016\pi\)
\(60\) 0 0
\(61\) 39570.0 1.36157 0.680787 0.732481i \(-0.261638\pi\)
0.680787 + 0.732481i \(0.261638\pi\)
\(62\) 0 0
\(63\) −14818.0 −0.470368
\(64\) 0 0
\(65\) 16350.0 0.479992
\(66\) 0 0
\(67\) 56734.0 1.54403 0.772016 0.635603i \(-0.219248\pi\)
0.772016 + 0.635603i \(0.219248\pi\)
\(68\) 0 0
\(69\) −4364.00 −0.110347
\(70\) 0 0
\(71\) −45588.0 −1.07326 −0.536630 0.843818i \(-0.680303\pi\)
−0.536630 + 0.843818i \(0.680303\pi\)
\(72\) 0 0
\(73\) 11842.0 0.260087 0.130043 0.991508i \(-0.458488\pi\)
0.130043 + 0.991508i \(0.458488\pi\)
\(74\) 0 0
\(75\) 1250.00 0.0256600
\(76\) 0 0
\(77\) 8928.00 0.171604
\(78\) 0 0
\(79\) −94216.0 −1.69847 −0.849233 0.528018i \(-0.822935\pi\)
−0.849233 + 0.528018i \(0.822935\pi\)
\(80\) 0 0
\(81\) 56149.0 0.950888
\(82\) 0 0
\(83\) 31482.0 0.501611 0.250806 0.968037i \(-0.419305\pi\)
0.250806 + 0.968037i \(0.419305\pi\)
\(84\) 0 0
\(85\) 29750.0 0.446622
\(86\) 0 0
\(87\) −3156.00 −0.0447032
\(88\) 0 0
\(89\) −94054.0 −1.25864 −0.629321 0.777145i \(-0.716667\pi\)
−0.629321 + 0.777145i \(0.716667\pi\)
\(90\) 0 0
\(91\) −40548.0 −0.513294
\(92\) 0 0
\(93\) −19320.0 −0.231633
\(94\) 0 0
\(95\) 13900.0 0.158018
\(96\) 0 0
\(97\) 23714.0 0.255903 0.127952 0.991780i \(-0.459160\pi\)
0.127952 + 0.991780i \(0.459160\pi\)
\(98\) 0 0
\(99\) −34416.0 −0.352917
\(100\) 0 0
\(101\) −129674. −1.26488 −0.632440 0.774609i \(-0.717947\pi\)
−0.632440 + 0.774609i \(0.717947\pi\)
\(102\) 0 0
\(103\) −136846. −1.27098 −0.635490 0.772109i \(-0.719202\pi\)
−0.635490 + 0.772109i \(0.719202\pi\)
\(104\) 0 0
\(105\) −3100.00 −0.0274403
\(106\) 0 0
\(107\) 193190. 1.63127 0.815634 0.578569i \(-0.196388\pi\)
0.815634 + 0.578569i \(0.196388\pi\)
\(108\) 0 0
\(109\) −120046. −0.967791 −0.483895 0.875126i \(-0.660778\pi\)
−0.483895 + 0.875126i \(0.660778\pi\)
\(110\) 0 0
\(111\) −7068.00 −0.0544489
\(112\) 0 0
\(113\) −152646. −1.12458 −0.562289 0.826941i \(-0.690079\pi\)
−0.562289 + 0.826941i \(0.690079\pi\)
\(114\) 0 0
\(115\) 54550.0 0.384636
\(116\) 0 0
\(117\) 156306. 1.05563
\(118\) 0 0
\(119\) −73780.0 −0.477608
\(120\) 0 0
\(121\) −140315. −0.871246
\(122\) 0 0
\(123\) 14924.0 0.0889452
\(124\) 0 0
\(125\) −15625.0 −0.0894427
\(126\) 0 0
\(127\) −107906. −0.593658 −0.296829 0.954931i \(-0.595929\pi\)
−0.296829 + 0.954931i \(0.595929\pi\)
\(128\) 0 0
\(129\) 14228.0 0.0752783
\(130\) 0 0
\(131\) 233072. 1.18662 0.593310 0.804974i \(-0.297821\pi\)
0.593310 + 0.804974i \(0.297821\pi\)
\(132\) 0 0
\(133\) −34472.0 −0.168981
\(134\) 0 0
\(135\) 24100.0 0.113811
\(136\) 0 0
\(137\) 356082. 1.62087 0.810436 0.585827i \(-0.199230\pi\)
0.810436 + 0.585827i \(0.199230\pi\)
\(138\) 0 0
\(139\) −312204. −1.37057 −0.685285 0.728275i \(-0.740323\pi\)
−0.685285 + 0.728275i \(0.740323\pi\)
\(140\) 0 0
\(141\) 56588.0 0.239705
\(142\) 0 0
\(143\) −94176.0 −0.385124
\(144\) 0 0
\(145\) 39450.0 0.155821
\(146\) 0 0
\(147\) −25926.0 −0.0989560
\(148\) 0 0
\(149\) 27498.0 0.101469 0.0507347 0.998712i \(-0.483844\pi\)
0.0507347 + 0.998712i \(0.483844\pi\)
\(150\) 0 0
\(151\) 136908. 0.488637 0.244319 0.969695i \(-0.421436\pi\)
0.244319 + 0.969695i \(0.421436\pi\)
\(152\) 0 0
\(153\) 284410. 0.982237
\(154\) 0 0
\(155\) 241500. 0.807398
\(156\) 0 0
\(157\) 406714. 1.31686 0.658431 0.752641i \(-0.271221\pi\)
0.658431 + 0.752641i \(0.271221\pi\)
\(158\) 0 0
\(159\) −26092.0 −0.0818492
\(160\) 0 0
\(161\) −135284. −0.411322
\(162\) 0 0
\(163\) 13642.0 0.0402169 0.0201085 0.999798i \(-0.493599\pi\)
0.0201085 + 0.999798i \(0.493599\pi\)
\(164\) 0 0
\(165\) −7200.00 −0.0205884
\(166\) 0 0
\(167\) 203438. 0.564470 0.282235 0.959345i \(-0.408924\pi\)
0.282235 + 0.959345i \(0.408924\pi\)
\(168\) 0 0
\(169\) 56423.0 0.151964
\(170\) 0 0
\(171\) 132884. 0.347522
\(172\) 0 0
\(173\) 127242. 0.323233 0.161616 0.986854i \(-0.448329\pi\)
0.161616 + 0.986854i \(0.448329\pi\)
\(174\) 0 0
\(175\) 38750.0 0.0956482
\(176\) 0 0
\(177\) 74184.0 0.177982
\(178\) 0 0
\(179\) 94684.0 0.220874 0.110437 0.993883i \(-0.464775\pi\)
0.110437 + 0.993883i \(0.464775\pi\)
\(180\) 0 0
\(181\) −517018. −1.17303 −0.586515 0.809938i \(-0.699501\pi\)
−0.586515 + 0.809938i \(0.699501\pi\)
\(182\) 0 0
\(183\) 79140.0 0.174690
\(184\) 0 0
\(185\) 88350.0 0.189792
\(186\) 0 0
\(187\) −171360. −0.358348
\(188\) 0 0
\(189\) −59768.0 −0.121707
\(190\) 0 0
\(191\) 412300. 0.817768 0.408884 0.912586i \(-0.365918\pi\)
0.408884 + 0.912586i \(0.365918\pi\)
\(192\) 0 0
\(193\) −771654. −1.49118 −0.745589 0.666406i \(-0.767832\pi\)
−0.745589 + 0.666406i \(0.767832\pi\)
\(194\) 0 0
\(195\) 32700.0 0.0615831
\(196\) 0 0
\(197\) −190238. −0.349246 −0.174623 0.984635i \(-0.555871\pi\)
−0.174623 + 0.984635i \(0.555871\pi\)
\(198\) 0 0
\(199\) −132072. −0.236417 −0.118208 0.992989i \(-0.537715\pi\)
−0.118208 + 0.992989i \(0.537715\pi\)
\(200\) 0 0
\(201\) 113468. 0.198099
\(202\) 0 0
\(203\) −97836.0 −0.166632
\(204\) 0 0
\(205\) −186550. −0.310035
\(206\) 0 0
\(207\) 521498. 0.845915
\(208\) 0 0
\(209\) −80064.0 −0.126786
\(210\) 0 0
\(211\) −928704. −1.43606 −0.718028 0.696015i \(-0.754955\pi\)
−0.718028 + 0.696015i \(0.754955\pi\)
\(212\) 0 0
\(213\) −91176.0 −0.137699
\(214\) 0 0
\(215\) −177850. −0.262396
\(216\) 0 0
\(217\) −598920. −0.863415
\(218\) 0 0
\(219\) 23684.0 0.0333691
\(220\) 0 0
\(221\) 778260. 1.07187
\(222\) 0 0
\(223\) −421494. −0.567583 −0.283791 0.958886i \(-0.591592\pi\)
−0.283791 + 0.958886i \(0.591592\pi\)
\(224\) 0 0
\(225\) −149375. −0.196708
\(226\) 0 0
\(227\) −991962. −1.27770 −0.638852 0.769329i \(-0.720590\pi\)
−0.638852 + 0.769329i \(0.720590\pi\)
\(228\) 0 0
\(229\) −266946. −0.336384 −0.168192 0.985754i \(-0.553793\pi\)
−0.168192 + 0.985754i \(0.553793\pi\)
\(230\) 0 0
\(231\) 17856.0 0.0220168
\(232\) 0 0
\(233\) 960314. 1.15884 0.579420 0.815029i \(-0.303279\pi\)
0.579420 + 0.815029i \(0.303279\pi\)
\(234\) 0 0
\(235\) −707350. −0.835535
\(236\) 0 0
\(237\) −188432. −0.217913
\(238\) 0 0
\(239\) 492696. 0.557936 0.278968 0.960300i \(-0.410008\pi\)
0.278968 + 0.960300i \(0.410008\pi\)
\(240\) 0 0
\(241\) 56078.0 0.0621942 0.0310971 0.999516i \(-0.490100\pi\)
0.0310971 + 0.999516i \(0.490100\pi\)
\(242\) 0 0
\(243\) 346550. 0.376487
\(244\) 0 0
\(245\) 324075. 0.344929
\(246\) 0 0
\(247\) 363624. 0.379237
\(248\) 0 0
\(249\) 62964.0 0.0643567
\(250\) 0 0
\(251\) −1.96792e6 −1.97162 −0.985810 0.167866i \(-0.946312\pi\)
−0.985810 + 0.167866i \(0.946312\pi\)
\(252\) 0 0
\(253\) −314208. −0.308614
\(254\) 0 0
\(255\) 59500.0 0.0573016
\(256\) 0 0
\(257\) −971910. −0.917896 −0.458948 0.888463i \(-0.651773\pi\)
−0.458948 + 0.888463i \(0.651773\pi\)
\(258\) 0 0
\(259\) −219108. −0.202959
\(260\) 0 0
\(261\) 377142. 0.342692
\(262\) 0 0
\(263\) 154770. 0.137974 0.0689870 0.997618i \(-0.478023\pi\)
0.0689870 + 0.997618i \(0.478023\pi\)
\(264\) 0 0
\(265\) 326150. 0.285301
\(266\) 0 0
\(267\) −188108. −0.161484
\(268\) 0 0
\(269\) 1.02371e6 0.862577 0.431289 0.902214i \(-0.358059\pi\)
0.431289 + 0.902214i \(0.358059\pi\)
\(270\) 0 0
\(271\) 1.14776e6 0.949350 0.474675 0.880161i \(-0.342566\pi\)
0.474675 + 0.880161i \(0.342566\pi\)
\(272\) 0 0
\(273\) −81096.0 −0.0658556
\(274\) 0 0
\(275\) 90000.0 0.0717647
\(276\) 0 0
\(277\) −2.49676e6 −1.95514 −0.977568 0.210619i \(-0.932452\pi\)
−0.977568 + 0.210619i \(0.932452\pi\)
\(278\) 0 0
\(279\) 2.30874e6 1.77568
\(280\) 0 0
\(281\) 1.69540e6 1.28087 0.640436 0.768011i \(-0.278754\pi\)
0.640436 + 0.768011i \(0.278754\pi\)
\(282\) 0 0
\(283\) 2.12395e6 1.57645 0.788223 0.615390i \(-0.211001\pi\)
0.788223 + 0.615390i \(0.211001\pi\)
\(284\) 0 0
\(285\) 27800.0 0.0202737
\(286\) 0 0
\(287\) 462644. 0.331545
\(288\) 0 0
\(289\) −3757.00 −0.00264604
\(290\) 0 0
\(291\) 47428.0 0.0328324
\(292\) 0 0
\(293\) 992722. 0.675552 0.337776 0.941227i \(-0.390325\pi\)
0.337776 + 0.941227i \(0.390325\pi\)
\(294\) 0 0
\(295\) −927300. −0.620391
\(296\) 0 0
\(297\) −138816. −0.0913163
\(298\) 0 0
\(299\) 1.42703e6 0.923112
\(300\) 0 0
\(301\) 441068. 0.280601
\(302\) 0 0
\(303\) −259348. −0.162284
\(304\) 0 0
\(305\) −989250. −0.608915
\(306\) 0 0
\(307\) −487522. −0.295222 −0.147611 0.989046i \(-0.547158\pi\)
−0.147611 + 0.989046i \(0.547158\pi\)
\(308\) 0 0
\(309\) −273692. −0.163067
\(310\) 0 0
\(311\) 444116. 0.260373 0.130186 0.991490i \(-0.458442\pi\)
0.130186 + 0.991490i \(0.458442\pi\)
\(312\) 0 0
\(313\) 47242.0 0.0272563 0.0136282 0.999907i \(-0.495662\pi\)
0.0136282 + 0.999907i \(0.495662\pi\)
\(314\) 0 0
\(315\) 370450. 0.210355
\(316\) 0 0
\(317\) 694058. 0.387925 0.193962 0.981009i \(-0.437866\pi\)
0.193962 + 0.981009i \(0.437866\pi\)
\(318\) 0 0
\(319\) −227232. −0.125024
\(320\) 0 0
\(321\) 386380. 0.209292
\(322\) 0 0
\(323\) 661640. 0.352871
\(324\) 0 0
\(325\) −408750. −0.214659
\(326\) 0 0
\(327\) −240092. −0.124168
\(328\) 0 0
\(329\) 1.75423e6 0.893504
\(330\) 0 0
\(331\) −82168.0 −0.0412223 −0.0206112 0.999788i \(-0.506561\pi\)
−0.0206112 + 0.999788i \(0.506561\pi\)
\(332\) 0 0
\(333\) 844626. 0.417401
\(334\) 0 0
\(335\) −1.41835e6 −0.690512
\(336\) 0 0
\(337\) −727934. −0.349154 −0.174577 0.984644i \(-0.555856\pi\)
−0.174577 + 0.984644i \(0.555856\pi\)
\(338\) 0 0
\(339\) −305292. −0.144283
\(340\) 0 0
\(341\) −1.39104e6 −0.647819
\(342\) 0 0
\(343\) −1.84574e6 −0.847101
\(344\) 0 0
\(345\) 109100. 0.0493488
\(346\) 0 0
\(347\) −2.02298e6 −0.901919 −0.450959 0.892544i \(-0.648918\pi\)
−0.450959 + 0.892544i \(0.648918\pi\)
\(348\) 0 0
\(349\) 4.40858e6 1.93747 0.968736 0.248095i \(-0.0798044\pi\)
0.968736 + 0.248095i \(0.0798044\pi\)
\(350\) 0 0
\(351\) 630456. 0.273141
\(352\) 0 0
\(353\) 1.06965e6 0.456883 0.228441 0.973558i \(-0.426637\pi\)
0.228441 + 0.973558i \(0.426637\pi\)
\(354\) 0 0
\(355\) 1.13970e6 0.479976
\(356\) 0 0
\(357\) −147560. −0.0612771
\(358\) 0 0
\(359\) 32968.0 0.0135007 0.00675035 0.999977i \(-0.497851\pi\)
0.00675035 + 0.999977i \(0.497851\pi\)
\(360\) 0 0
\(361\) −2.16696e6 −0.875152
\(362\) 0 0
\(363\) −280630. −0.111781
\(364\) 0 0
\(365\) −296050. −0.116314
\(366\) 0 0
\(367\) −3.64081e6 −1.41102 −0.705509 0.708700i \(-0.749282\pi\)
−0.705509 + 0.708700i \(0.749282\pi\)
\(368\) 0 0
\(369\) −1.78342e6 −0.681847
\(370\) 0 0
\(371\) −808852. −0.305094
\(372\) 0 0
\(373\) −3.17311e6 −1.18090 −0.590450 0.807074i \(-0.701050\pi\)
−0.590450 + 0.807074i \(0.701050\pi\)
\(374\) 0 0
\(375\) −31250.0 −0.0114755
\(376\) 0 0
\(377\) 1.03201e6 0.373965
\(378\) 0 0
\(379\) −1.60498e6 −0.573947 −0.286973 0.957939i \(-0.592649\pi\)
−0.286973 + 0.957939i \(0.592649\pi\)
\(380\) 0 0
\(381\) −215812. −0.0761664
\(382\) 0 0
\(383\) −1.98925e6 −0.692936 −0.346468 0.938062i \(-0.612619\pi\)
−0.346468 + 0.938062i \(0.612619\pi\)
\(384\) 0 0
\(385\) −223200. −0.0767436
\(386\) 0 0
\(387\) −1.70025e6 −0.577078
\(388\) 0 0
\(389\) −5.16495e6 −1.73058 −0.865291 0.501270i \(-0.832866\pi\)
−0.865291 + 0.501270i \(0.832866\pi\)
\(390\) 0 0
\(391\) 2.59658e6 0.858934
\(392\) 0 0
\(393\) 466144. 0.152243
\(394\) 0 0
\(395\) 2.35540e6 0.759577
\(396\) 0 0
\(397\) 937586. 0.298562 0.149281 0.988795i \(-0.452304\pi\)
0.149281 + 0.988795i \(0.452304\pi\)
\(398\) 0 0
\(399\) −68944.0 −0.0216802
\(400\) 0 0
\(401\) −5.63657e6 −1.75047 −0.875234 0.483699i \(-0.839293\pi\)
−0.875234 + 0.483699i \(0.839293\pi\)
\(402\) 0 0
\(403\) 6.31764e6 1.93773
\(404\) 0 0
\(405\) −1.40373e6 −0.425250
\(406\) 0 0
\(407\) −508896. −0.152280
\(408\) 0 0
\(409\) 4.06137e6 1.20051 0.600254 0.799810i \(-0.295066\pi\)
0.600254 + 0.799810i \(0.295066\pi\)
\(410\) 0 0
\(411\) 712164. 0.207958
\(412\) 0 0
\(413\) 2.29970e6 0.663433
\(414\) 0 0
\(415\) −787050. −0.224327
\(416\) 0 0
\(417\) −624408. −0.175844
\(418\) 0 0
\(419\) 976108. 0.271621 0.135810 0.990735i \(-0.456636\pi\)
0.135810 + 0.990735i \(0.456636\pi\)
\(420\) 0 0
\(421\) −1.62706e6 −0.447403 −0.223701 0.974658i \(-0.571814\pi\)
−0.223701 + 0.974658i \(0.571814\pi\)
\(422\) 0 0
\(423\) −6.76227e6 −1.83756
\(424\) 0 0
\(425\) −743750. −0.199735
\(426\) 0 0
\(427\) 2.45334e6 0.651161
\(428\) 0 0
\(429\) −188352. −0.0494114
\(430\) 0 0
\(431\) 4.27900e6 1.10956 0.554778 0.831998i \(-0.312803\pi\)
0.554778 + 0.831998i \(0.312803\pi\)
\(432\) 0 0
\(433\) −3.20195e6 −0.820720 −0.410360 0.911924i \(-0.634597\pi\)
−0.410360 + 0.911924i \(0.634597\pi\)
\(434\) 0 0
\(435\) 78900.0 0.0199919
\(436\) 0 0
\(437\) 1.21319e6 0.303897
\(438\) 0 0
\(439\) −5.09246e6 −1.26115 −0.630574 0.776129i \(-0.717180\pi\)
−0.630574 + 0.776129i \(0.717180\pi\)
\(440\) 0 0
\(441\) 3.09816e6 0.758590
\(442\) 0 0
\(443\) 5.43551e6 1.31593 0.657963 0.753050i \(-0.271418\pi\)
0.657963 + 0.753050i \(0.271418\pi\)
\(444\) 0 0
\(445\) 2.35135e6 0.562882
\(446\) 0 0
\(447\) 54996.0 0.0130185
\(448\) 0 0
\(449\) −2.99007e6 −0.699948 −0.349974 0.936759i \(-0.613810\pi\)
−0.349974 + 0.936759i \(0.613810\pi\)
\(450\) 0 0
\(451\) 1.07453e6 0.248758
\(452\) 0 0
\(453\) 273816. 0.0626922
\(454\) 0 0
\(455\) 1.01370e6 0.229552
\(456\) 0 0
\(457\) 8.01759e6 1.79578 0.897891 0.440218i \(-0.145099\pi\)
0.897891 + 0.440218i \(0.145099\pi\)
\(458\) 0 0
\(459\) 1.14716e6 0.254151
\(460\) 0 0
\(461\) 2.58462e6 0.566428 0.283214 0.959057i \(-0.408599\pi\)
0.283214 + 0.959057i \(0.408599\pi\)
\(462\) 0 0
\(463\) −6.14261e6 −1.33168 −0.665840 0.746094i \(-0.731927\pi\)
−0.665840 + 0.746094i \(0.731927\pi\)
\(464\) 0 0
\(465\) 483000. 0.103589
\(466\) 0 0
\(467\) 1.59270e6 0.337942 0.168971 0.985621i \(-0.445956\pi\)
0.168971 + 0.985621i \(0.445956\pi\)
\(468\) 0 0
\(469\) 3.51751e6 0.738419
\(470\) 0 0
\(471\) 813428. 0.168953
\(472\) 0 0
\(473\) 1.02442e6 0.210535
\(474\) 0 0
\(475\) −347500. −0.0706677
\(476\) 0 0
\(477\) 3.11799e6 0.627450
\(478\) 0 0
\(479\) −863592. −0.171977 −0.0859884 0.996296i \(-0.527405\pi\)
−0.0859884 + 0.996296i \(0.527405\pi\)
\(480\) 0 0
\(481\) 2.31124e6 0.455493
\(482\) 0 0
\(483\) −270568. −0.0527726
\(484\) 0 0
\(485\) −592850. −0.114443
\(486\) 0 0
\(487\) 8.20714e6 1.56808 0.784042 0.620707i \(-0.213154\pi\)
0.784042 + 0.620707i \(0.213154\pi\)
\(488\) 0 0
\(489\) 27284.0 0.00515984
\(490\) 0 0
\(491\) −8.93394e6 −1.67240 −0.836198 0.548428i \(-0.815227\pi\)
−0.836198 + 0.548428i \(0.815227\pi\)
\(492\) 0 0
\(493\) 1.87782e6 0.347966
\(494\) 0 0
\(495\) 860400. 0.157829
\(496\) 0 0
\(497\) −2.82646e6 −0.513276
\(498\) 0 0
\(499\) −1.11960e6 −0.201284 −0.100642 0.994923i \(-0.532090\pi\)
−0.100642 + 0.994923i \(0.532090\pi\)
\(500\) 0 0
\(501\) 406876. 0.0724215
\(502\) 0 0
\(503\) −3.68177e6 −0.648839 −0.324420 0.945913i \(-0.605169\pi\)
−0.324420 + 0.945913i \(0.605169\pi\)
\(504\) 0 0
\(505\) 3.24185e6 0.565672
\(506\) 0 0
\(507\) 112846. 0.0194969
\(508\) 0 0
\(509\) −6.73483e6 −1.15221 −0.576105 0.817375i \(-0.695428\pi\)
−0.576105 + 0.817375i \(0.695428\pi\)
\(510\) 0 0
\(511\) 734204. 0.124384
\(512\) 0 0
\(513\) 535984. 0.0899204
\(514\) 0 0
\(515\) 3.42115e6 0.568400
\(516\) 0 0
\(517\) 4.07434e6 0.670395
\(518\) 0 0
\(519\) 254484. 0.0414708
\(520\) 0 0
\(521\) 441370. 0.0712375 0.0356187 0.999365i \(-0.488660\pi\)
0.0356187 + 0.999365i \(0.488660\pi\)
\(522\) 0 0
\(523\) 1.17300e7 1.87518 0.937589 0.347744i \(-0.113052\pi\)
0.937589 + 0.347744i \(0.113052\pi\)
\(524\) 0 0
\(525\) 77500.0 0.0122717
\(526\) 0 0
\(527\) 1.14954e7 1.80301
\(528\) 0 0
\(529\) −1.67522e6 −0.260275
\(530\) 0 0
\(531\) −8.86499e6 −1.36440
\(532\) 0 0
\(533\) −4.88015e6 −0.744072
\(534\) 0 0
\(535\) −4.82975e6 −0.729525
\(536\) 0 0
\(537\) 189368. 0.0283381
\(538\) 0 0
\(539\) −1.86667e6 −0.276755
\(540\) 0 0
\(541\) 744158. 0.109313 0.0546565 0.998505i \(-0.482594\pi\)
0.0546565 + 0.998505i \(0.482594\pi\)
\(542\) 0 0
\(543\) −1.03404e6 −0.150500
\(544\) 0 0
\(545\) 3.00115e6 0.432809
\(546\) 0 0
\(547\) 3.24801e6 0.464139 0.232070 0.972699i \(-0.425450\pi\)
0.232070 + 0.972699i \(0.425450\pi\)
\(548\) 0 0
\(549\) −9.45723e6 −1.33916
\(550\) 0 0
\(551\) 877368. 0.123113
\(552\) 0 0
\(553\) −5.84139e6 −0.812276
\(554\) 0 0
\(555\) 176700. 0.0243503
\(556\) 0 0
\(557\) −9.94446e6 −1.35814 −0.679068 0.734075i \(-0.737616\pi\)
−0.679068 + 0.734075i \(0.737616\pi\)
\(558\) 0 0
\(559\) −4.65256e6 −0.629741
\(560\) 0 0
\(561\) −342720. −0.0459761
\(562\) 0 0
\(563\) −3.89374e6 −0.517721 −0.258861 0.965915i \(-0.583347\pi\)
−0.258861 + 0.965915i \(0.583347\pi\)
\(564\) 0 0
\(565\) 3.81615e6 0.502926
\(566\) 0 0
\(567\) 3.48124e6 0.454754
\(568\) 0 0
\(569\) −1.11951e7 −1.44960 −0.724801 0.688958i \(-0.758068\pi\)
−0.724801 + 0.688958i \(0.758068\pi\)
\(570\) 0 0
\(571\) 844040. 0.108336 0.0541680 0.998532i \(-0.482749\pi\)
0.0541680 + 0.998532i \(0.482749\pi\)
\(572\) 0 0
\(573\) 824600. 0.104920
\(574\) 0 0
\(575\) −1.36375e6 −0.172015
\(576\) 0 0
\(577\) −5.13378e6 −0.641945 −0.320973 0.947088i \(-0.604010\pi\)
−0.320973 + 0.947088i \(0.604010\pi\)
\(578\) 0 0
\(579\) −1.54331e6 −0.191318
\(580\) 0 0
\(581\) 1.95188e6 0.239891
\(582\) 0 0
\(583\) −1.87862e6 −0.228912
\(584\) 0 0
\(585\) −3.90765e6 −0.472091
\(586\) 0 0
\(587\) −9.76156e6 −1.16929 −0.584647 0.811287i \(-0.698767\pi\)
−0.584647 + 0.811287i \(0.698767\pi\)
\(588\) 0 0
\(589\) 5.37096e6 0.637916
\(590\) 0 0
\(591\) −380476. −0.0448083
\(592\) 0 0
\(593\) 966226. 0.112835 0.0564173 0.998407i \(-0.482032\pi\)
0.0564173 + 0.998407i \(0.482032\pi\)
\(594\) 0 0
\(595\) 1.84450e6 0.213593
\(596\) 0 0
\(597\) −264144. −0.0303323
\(598\) 0 0
\(599\) 7.90000e6 0.899622 0.449811 0.893124i \(-0.351491\pi\)
0.449811 + 0.893124i \(0.351491\pi\)
\(600\) 0 0
\(601\) 1.03126e7 1.16461 0.582307 0.812969i \(-0.302150\pi\)
0.582307 + 0.812969i \(0.302150\pi\)
\(602\) 0 0
\(603\) −1.35594e7 −1.51862
\(604\) 0 0
\(605\) 3.50788e6 0.389633
\(606\) 0 0
\(607\) 9.70767e6 1.06941 0.534704 0.845040i \(-0.320423\pi\)
0.534704 + 0.845040i \(0.320423\pi\)
\(608\) 0 0
\(609\) −195672. −0.0213789
\(610\) 0 0
\(611\) −1.85043e7 −2.00525
\(612\) 0 0
\(613\) −1.10568e7 −1.18844 −0.594219 0.804304i \(-0.702539\pi\)
−0.594219 + 0.804304i \(0.702539\pi\)
\(614\) 0 0
\(615\) −373100. −0.0397775
\(616\) 0 0
\(617\) 8.31174e6 0.878980 0.439490 0.898248i \(-0.355159\pi\)
0.439490 + 0.898248i \(0.355159\pi\)
\(618\) 0 0
\(619\) −1.15451e7 −1.21108 −0.605539 0.795816i \(-0.707042\pi\)
−0.605539 + 0.795816i \(0.707042\pi\)
\(620\) 0 0
\(621\) 2.10345e6 0.218878
\(622\) 0 0
\(623\) −5.83135e6 −0.601934
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) −160128. −0.0162667
\(628\) 0 0
\(629\) 4.20546e6 0.423825
\(630\) 0 0
\(631\) 8.20262e6 0.820123 0.410062 0.912058i \(-0.365507\pi\)
0.410062 + 0.912058i \(0.365507\pi\)
\(632\) 0 0
\(633\) −1.85741e6 −0.184246
\(634\) 0 0
\(635\) 2.69765e6 0.265492
\(636\) 0 0
\(637\) 8.47780e6 0.827818
\(638\) 0 0
\(639\) 1.08955e7 1.05559
\(640\) 0 0
\(641\) −5.39695e6 −0.518804 −0.259402 0.965769i \(-0.583525\pi\)
−0.259402 + 0.965769i \(0.583525\pi\)
\(642\) 0 0
\(643\) 1.33896e7 1.27715 0.638573 0.769561i \(-0.279525\pi\)
0.638573 + 0.769561i \(0.279525\pi\)
\(644\) 0 0
\(645\) −355700. −0.0336655
\(646\) 0 0
\(647\) −6.48254e6 −0.608814 −0.304407 0.952542i \(-0.598458\pi\)
−0.304407 + 0.952542i \(0.598458\pi\)
\(648\) 0 0
\(649\) 5.34125e6 0.497773
\(650\) 0 0
\(651\) −1.19784e6 −0.110776
\(652\) 0 0
\(653\) 1.44907e7 1.32986 0.664931 0.746904i \(-0.268461\pi\)
0.664931 + 0.746904i \(0.268461\pi\)
\(654\) 0 0
\(655\) −5.82680e6 −0.530673
\(656\) 0 0
\(657\) −2.83024e6 −0.255805
\(658\) 0 0
\(659\) −6.59080e6 −0.591187 −0.295593 0.955314i \(-0.595517\pi\)
−0.295593 + 0.955314i \(0.595517\pi\)
\(660\) 0 0
\(661\) −3.25233e6 −0.289528 −0.144764 0.989466i \(-0.546242\pi\)
−0.144764 + 0.989466i \(0.546242\pi\)
\(662\) 0 0
\(663\) 1.55652e6 0.137522
\(664\) 0 0
\(665\) 861800. 0.0755705
\(666\) 0 0
\(667\) 3.44320e6 0.299673
\(668\) 0 0
\(669\) −842988. −0.0728209
\(670\) 0 0
\(671\) 5.69808e6 0.488565
\(672\) 0 0
\(673\) 3.86655e6 0.329068 0.164534 0.986371i \(-0.447388\pi\)
0.164534 + 0.986371i \(0.447388\pi\)
\(674\) 0 0
\(675\) −602500. −0.0508976
\(676\) 0 0
\(677\) 1.23856e6 0.103859 0.0519297 0.998651i \(-0.483463\pi\)
0.0519297 + 0.998651i \(0.483463\pi\)
\(678\) 0 0
\(679\) 1.47027e6 0.122383
\(680\) 0 0
\(681\) −1.98392e6 −0.163930
\(682\) 0 0
\(683\) 1.31376e7 1.07762 0.538810 0.842427i \(-0.318874\pi\)
0.538810 + 0.842427i \(0.318874\pi\)
\(684\) 0 0
\(685\) −8.90205e6 −0.724876
\(686\) 0 0
\(687\) −533892. −0.0431580
\(688\) 0 0
\(689\) 8.53208e6 0.684711
\(690\) 0 0
\(691\) 1.23841e7 0.986664 0.493332 0.869841i \(-0.335779\pi\)
0.493332 + 0.869841i \(0.335779\pi\)
\(692\) 0 0
\(693\) −2.13379e6 −0.168779
\(694\) 0 0
\(695\) 7.80510e6 0.612938
\(696\) 0 0
\(697\) −8.87978e6 −0.692341
\(698\) 0 0
\(699\) 1.92063e6 0.148679
\(700\) 0 0
\(701\) −9.78952e6 −0.752430 −0.376215 0.926532i \(-0.622775\pi\)
−0.376215 + 0.926532i \(0.622775\pi\)
\(702\) 0 0
\(703\) 1.96490e6 0.149952
\(704\) 0 0
\(705\) −1.41470e6 −0.107199
\(706\) 0 0
\(707\) −8.03979e6 −0.604917
\(708\) 0 0
\(709\) 1.22257e7 0.913397 0.456699 0.889622i \(-0.349032\pi\)
0.456699 + 0.889622i \(0.349032\pi\)
\(710\) 0 0
\(711\) 2.25176e7 1.67051
\(712\) 0 0
\(713\) 2.10781e7 1.55277
\(714\) 0 0
\(715\) 2.35440e6 0.172233
\(716\) 0 0
\(717\) 985392. 0.0715832
\(718\) 0 0
\(719\) 1.35053e7 0.974276 0.487138 0.873325i \(-0.338041\pi\)
0.487138 + 0.873325i \(0.338041\pi\)
\(720\) 0 0
\(721\) −8.48445e6 −0.607835
\(722\) 0 0
\(723\) 112156. 0.00797952
\(724\) 0 0
\(725\) −986250. −0.0696854
\(726\) 0 0
\(727\) −1.17271e7 −0.822916 −0.411458 0.911429i \(-0.634980\pi\)
−0.411458 + 0.911429i \(0.634980\pi\)
\(728\) 0 0
\(729\) −1.29511e7 −0.902585
\(730\) 0 0
\(731\) −8.46566e6 −0.585959
\(732\) 0 0
\(733\) −1.16512e7 −0.800960 −0.400480 0.916305i \(-0.631157\pi\)
−0.400480 + 0.916305i \(0.631157\pi\)
\(734\) 0 0
\(735\) 648150. 0.0442545
\(736\) 0 0
\(737\) 8.16970e6 0.554035
\(738\) 0 0
\(739\) 1.26808e7 0.854155 0.427077 0.904215i \(-0.359543\pi\)
0.427077 + 0.904215i \(0.359543\pi\)
\(740\) 0 0
\(741\) 727248. 0.0486561
\(742\) 0 0
\(743\) 197370. 0.0131162 0.00655812 0.999978i \(-0.497912\pi\)
0.00655812 + 0.999978i \(0.497912\pi\)
\(744\) 0 0
\(745\) −687450. −0.0453785
\(746\) 0 0
\(747\) −7.52420e6 −0.493354
\(748\) 0 0
\(749\) 1.19778e7 0.780139
\(750\) 0 0
\(751\) 1.33282e7 0.862326 0.431163 0.902274i \(-0.358103\pi\)
0.431163 + 0.902274i \(0.358103\pi\)
\(752\) 0 0
\(753\) −3.93584e6 −0.252959
\(754\) 0 0
\(755\) −3.42270e6 −0.218525
\(756\) 0 0
\(757\) −3.86122e6 −0.244898 −0.122449 0.992475i \(-0.539075\pi\)
−0.122449 + 0.992475i \(0.539075\pi\)
\(758\) 0 0
\(759\) −628416. −0.0395952
\(760\) 0 0
\(761\) −8.31756e6 −0.520636 −0.260318 0.965523i \(-0.583827\pi\)
−0.260318 + 0.965523i \(0.583827\pi\)
\(762\) 0 0
\(763\) −7.44285e6 −0.462837
\(764\) 0 0
\(765\) −7.11025e6 −0.439270
\(766\) 0 0
\(767\) −2.42582e7 −1.48891
\(768\) 0 0
\(769\) 2.76358e7 1.68522 0.842609 0.538527i \(-0.181019\pi\)
0.842609 + 0.538527i \(0.181019\pi\)
\(770\) 0 0
\(771\) −1.94382e6 −0.117766
\(772\) 0 0
\(773\) 1.78842e7 1.07652 0.538259 0.842780i \(-0.319082\pi\)
0.538259 + 0.842780i \(0.319082\pi\)
\(774\) 0 0
\(775\) −6.03750e6 −0.361080
\(776\) 0 0
\(777\) −438216. −0.0260397
\(778\) 0 0
\(779\) −4.14887e6 −0.244955
\(780\) 0 0
\(781\) −6.56467e6 −0.385111
\(782\) 0 0
\(783\) 1.52119e6 0.0886706
\(784\) 0 0
\(785\) −1.01679e7 −0.588918
\(786\) 0 0
\(787\) −2.15691e7 −1.24135 −0.620676 0.784067i \(-0.713142\pi\)
−0.620676 + 0.784067i \(0.713142\pi\)
\(788\) 0 0
\(789\) 309540. 0.0177021
\(790\) 0 0
\(791\) −9.46405e6 −0.537819
\(792\) 0 0
\(793\) −2.58788e7 −1.46137
\(794\) 0 0
\(795\) 652300. 0.0366041
\(796\) 0 0
\(797\) −1.03060e7 −0.574705 −0.287353 0.957825i \(-0.592775\pi\)
−0.287353 + 0.957825i \(0.592775\pi\)
\(798\) 0 0
\(799\) −3.36699e7 −1.86584
\(800\) 0 0
\(801\) 2.24789e7 1.23792
\(802\) 0 0
\(803\) 1.70525e6 0.0933251
\(804\) 0 0
\(805\) 3.38210e6 0.183949
\(806\) 0 0
\(807\) 2.04743e6 0.110669
\(808\) 0 0
\(809\) 372378. 0.0200038 0.0100019 0.999950i \(-0.496816\pi\)
0.0100019 + 0.999950i \(0.496816\pi\)
\(810\) 0 0
\(811\) 1.94795e7 1.03998 0.519990 0.854173i \(-0.325936\pi\)
0.519990 + 0.854173i \(0.325936\pi\)
\(812\) 0 0
\(813\) 2.29551e6 0.121802
\(814\) 0 0
\(815\) −341050. −0.0179856
\(816\) 0 0
\(817\) −3.95538e6 −0.207316
\(818\) 0 0
\(819\) 9.69097e6 0.504844
\(820\) 0 0
\(821\) −469318. −0.0243002 −0.0121501 0.999926i \(-0.503868\pi\)
−0.0121501 + 0.999926i \(0.503868\pi\)
\(822\) 0 0
\(823\) −1.78622e7 −0.919253 −0.459626 0.888112i \(-0.652017\pi\)
−0.459626 + 0.888112i \(0.652017\pi\)
\(824\) 0 0
\(825\) 180000. 0.00920741
\(826\) 0 0
\(827\) −9.42560e6 −0.479231 −0.239616 0.970868i \(-0.577021\pi\)
−0.239616 + 0.970868i \(0.577021\pi\)
\(828\) 0 0
\(829\) −1.48622e7 −0.751098 −0.375549 0.926803i \(-0.622546\pi\)
−0.375549 + 0.926803i \(0.622546\pi\)
\(830\) 0 0
\(831\) −4.99352e6 −0.250844
\(832\) 0 0
\(833\) 1.54260e7 0.770265
\(834\) 0 0
\(835\) −5.08595e6 −0.252439
\(836\) 0 0
\(837\) 9.31224e6 0.459452
\(838\) 0 0
\(839\) 4.71170e6 0.231085 0.115543 0.993303i \(-0.463139\pi\)
0.115543 + 0.993303i \(0.463139\pi\)
\(840\) 0 0
\(841\) −1.80211e7 −0.878599
\(842\) 0 0
\(843\) 3.39080e6 0.164336
\(844\) 0 0
\(845\) −1.41058e6 −0.0679602
\(846\) 0 0
\(847\) −8.69953e6 −0.416665
\(848\) 0 0
\(849\) 4.24791e6 0.202258
\(850\) 0 0
\(851\) 7.71119e6 0.365004
\(852\) 0 0
\(853\) 1.62685e7 0.765552 0.382776 0.923841i \(-0.374968\pi\)
0.382776 + 0.923841i \(0.374968\pi\)
\(854\) 0 0
\(855\) −3.32210e6 −0.155417
\(856\) 0 0
\(857\) −2.92667e7 −1.36120 −0.680600 0.732656i \(-0.738281\pi\)
−0.680600 + 0.732656i \(0.738281\pi\)
\(858\) 0 0
\(859\) 3.31062e7 1.53083 0.765413 0.643539i \(-0.222535\pi\)
0.765413 + 0.643539i \(0.222535\pi\)
\(860\) 0 0
\(861\) 925288. 0.0425372
\(862\) 0 0
\(863\) 1.58052e7 0.722391 0.361196 0.932490i \(-0.382369\pi\)
0.361196 + 0.932490i \(0.382369\pi\)
\(864\) 0 0
\(865\) −3.18105e6 −0.144554
\(866\) 0 0
\(867\) −7514.00 −0.000339487 0
\(868\) 0 0
\(869\) −1.35671e7 −0.609449
\(870\) 0 0
\(871\) −3.71040e7 −1.65720
\(872\) 0 0
\(873\) −5.66765e6 −0.251691
\(874\) 0 0
\(875\) −968750. −0.0427752
\(876\) 0 0
\(877\) −4.26834e7 −1.87396 −0.936980 0.349384i \(-0.886391\pi\)
−0.936980 + 0.349384i \(0.886391\pi\)
\(878\) 0 0
\(879\) 1.98544e6 0.0866733
\(880\) 0 0
\(881\) −3.57397e6 −0.155135 −0.0775677 0.996987i \(-0.524715\pi\)
−0.0775677 + 0.996987i \(0.524715\pi\)
\(882\) 0 0
\(883\) 1.68471e7 0.727149 0.363574 0.931565i \(-0.381556\pi\)
0.363574 + 0.931565i \(0.381556\pi\)
\(884\) 0 0
\(885\) −1.85460e6 −0.0795962
\(886\) 0 0
\(887\) 8.36792e6 0.357115 0.178558 0.983929i \(-0.442857\pi\)
0.178558 + 0.983929i \(0.442857\pi\)
\(888\) 0 0
\(889\) −6.69017e6 −0.283911
\(890\) 0 0
\(891\) 8.08546e6 0.341201
\(892\) 0 0
\(893\) −1.57315e7 −0.660147
\(894\) 0 0
\(895\) −2.36710e6 −0.0987777
\(896\) 0 0
\(897\) 2.85406e6 0.118435
\(898\) 0 0
\(899\) 1.52435e7 0.629050
\(900\) 0 0
\(901\) 1.55247e7 0.637107
\(902\) 0 0
\(903\) 882136. 0.0360011
\(904\) 0 0
\(905\) 1.29255e7 0.524595
\(906\) 0 0
\(907\) 2.57230e7 1.03825 0.519127 0.854697i \(-0.326257\pi\)
0.519127 + 0.854697i \(0.326257\pi\)
\(908\) 0 0
\(909\) 3.09921e7 1.24406
\(910\) 0 0
\(911\) −3.42108e7 −1.36574 −0.682869 0.730540i \(-0.739268\pi\)
−0.682869 + 0.730540i \(0.739268\pi\)
\(912\) 0 0
\(913\) 4.53341e6 0.179990
\(914\) 0 0
\(915\) −1.97850e6 −0.0781238
\(916\) 0 0
\(917\) 1.44505e7 0.567490
\(918\) 0 0
\(919\) −2.44034e6 −0.0953149 −0.0476575 0.998864i \(-0.515176\pi\)
−0.0476575 + 0.998864i \(0.515176\pi\)
\(920\) 0 0
\(921\) −975044. −0.0378770
\(922\) 0 0
\(923\) 2.98146e7 1.15192
\(924\) 0 0
\(925\) −2.20875e6 −0.0848774
\(926\) 0 0
\(927\) 3.27062e7 1.25006
\(928\) 0 0
\(929\) 1.34361e7 0.510781 0.255390 0.966838i \(-0.417796\pi\)
0.255390 + 0.966838i \(0.417796\pi\)
\(930\) 0 0
\(931\) 7.20743e6 0.272525
\(932\) 0 0
\(933\) 888232. 0.0334058
\(934\) 0 0
\(935\) 4.28400e6 0.160258
\(936\) 0 0
\(937\) 7.96529e6 0.296383 0.148191 0.988959i \(-0.452655\pi\)
0.148191 + 0.988959i \(0.452655\pi\)
\(938\) 0 0
\(939\) 94484.0 0.00349699
\(940\) 0 0
\(941\) 9.08025e6 0.334290 0.167145 0.985932i \(-0.446545\pi\)
0.167145 + 0.985932i \(0.446545\pi\)
\(942\) 0 0
\(943\) −1.62821e7 −0.596253
\(944\) 0 0
\(945\) 1.49420e6 0.0544289
\(946\) 0 0
\(947\) 3.21769e7 1.16592 0.582961 0.812500i \(-0.301894\pi\)
0.582961 + 0.812500i \(0.301894\pi\)
\(948\) 0 0
\(949\) −7.74467e6 −0.279150
\(950\) 0 0
\(951\) 1.38812e6 0.0497708
\(952\) 0 0
\(953\) 5.33807e6 0.190394 0.0951968 0.995458i \(-0.469652\pi\)
0.0951968 + 0.995458i \(0.469652\pi\)
\(954\) 0 0
\(955\) −1.03075e7 −0.365717
\(956\) 0 0
\(957\) −454464. −0.0160406
\(958\) 0 0
\(959\) 2.20771e7 0.775167
\(960\) 0 0
\(961\) 6.46864e7 2.25946
\(962\) 0 0
\(963\) −4.61724e7 −1.60442
\(964\) 0 0
\(965\) 1.92914e7 0.666875
\(966\) 0 0
\(967\) −3.71522e7 −1.27767 −0.638834 0.769345i \(-0.720583\pi\)
−0.638834 + 0.769345i \(0.720583\pi\)
\(968\) 0 0
\(969\) 1.32328e6 0.0452733
\(970\) 0 0
\(971\) 1.09865e7 0.373949 0.186975 0.982365i \(-0.440132\pi\)
0.186975 + 0.982365i \(0.440132\pi\)
\(972\) 0 0
\(973\) −1.93566e7 −0.655463
\(974\) 0 0
\(975\) −817500. −0.0275408
\(976\) 0 0
\(977\) −2.65054e7 −0.888379 −0.444190 0.895933i \(-0.646508\pi\)
−0.444190 + 0.895933i \(0.646508\pi\)
\(978\) 0 0
\(979\) −1.35438e7 −0.451630
\(980\) 0 0
\(981\) 2.86910e7 0.951860
\(982\) 0 0
\(983\) 4.75726e7 1.57027 0.785133 0.619327i \(-0.212594\pi\)
0.785133 + 0.619327i \(0.212594\pi\)
\(984\) 0 0
\(985\) 4.75595e6 0.156188
\(986\) 0 0
\(987\) 3.50846e6 0.114637
\(988\) 0 0
\(989\) −1.55227e7 −0.504636
\(990\) 0 0
\(991\) −3.22149e7 −1.04201 −0.521006 0.853553i \(-0.674443\pi\)
−0.521006 + 0.853553i \(0.674443\pi\)
\(992\) 0 0
\(993\) −164336. −0.00528883
\(994\) 0 0
\(995\) 3.30180e6 0.105729
\(996\) 0 0
\(997\) 3.87072e7 1.23326 0.616630 0.787253i \(-0.288498\pi\)
0.616630 + 0.787253i \(0.288498\pi\)
\(998\) 0 0
\(999\) 3.40678e6 0.108002
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 80.6.a.d.1.1 1
3.2 odd 2 720.6.a.t.1.1 1
4.3 odd 2 40.6.a.c.1.1 1
5.2 odd 4 400.6.c.k.49.1 2
5.3 odd 4 400.6.c.k.49.2 2
5.4 even 2 400.6.a.h.1.1 1
8.3 odd 2 320.6.a.i.1.1 1
8.5 even 2 320.6.a.h.1.1 1
12.11 even 2 360.6.a.f.1.1 1
20.3 even 4 200.6.c.d.49.1 2
20.7 even 4 200.6.c.d.49.2 2
20.19 odd 2 200.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.6.a.c.1.1 1 4.3 odd 2
80.6.a.d.1.1 1 1.1 even 1 trivial
200.6.a.b.1.1 1 20.19 odd 2
200.6.c.d.49.1 2 20.3 even 4
200.6.c.d.49.2 2 20.7 even 4
320.6.a.h.1.1 1 8.5 even 2
320.6.a.i.1.1 1 8.3 odd 2
360.6.a.f.1.1 1 12.11 even 2
400.6.a.h.1.1 1 5.4 even 2
400.6.c.k.49.1 2 5.2 odd 4
400.6.c.k.49.2 2 5.3 odd 4
720.6.a.t.1.1 1 3.2 odd 2