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Properties

Label 900.6.a.b.1.1
Level $900$
Weight $6$
Character 900.1
Self dual yes
Analytic conductor $144.345$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,6,Mod(1,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 900.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(144.345437832\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 900.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-218.000 q^{7} +O(q^{10})\) \(q-218.000 q^{7} +480.000 q^{11} +622.000 q^{13} +186.000 q^{17} -1204.00 q^{19} -3186.00 q^{23} -5526.00 q^{29} +9356.00 q^{31} -5618.00 q^{37} +14394.0 q^{41} +370.000 q^{43} +16146.0 q^{47} +30717.0 q^{49} -4374.00 q^{53} +11748.0 q^{59} +13202.0 q^{61} +11542.0 q^{67} +29532.0 q^{71} -33698.0 q^{73} -104640. q^{77} +31208.0 q^{79} -38466.0 q^{83} -119514. q^{89} -135596. q^{91} -94658.0 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −218.000 −1.68156 −0.840778 − 0.541380i \(-0.817902\pi\)
−0.840778 + 0.541380i \(0.817902\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 480.000 1.19608 0.598039 − 0.801467i \(-0.295947\pi\)
0.598039 + 0.801467i \(0.295947\pi\)
\(12\) 0 0
\(13\) 622.000 1.02078 0.510390 − 0.859943i \(-0.329501\pi\)
0.510390 + 0.859943i \(0.329501\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 186.000 0.156096 0.0780478 − 0.996950i \(-0.475131\pi\)
0.0780478 + 0.996950i \(0.475131\pi\)
\(18\) 0 0
\(19\) −1204.00 −0.765143 −0.382571 − 0.923926i \(-0.624961\pi\)
−0.382571 + 0.923926i \(0.624961\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3186.00 −1.25582 −0.627908 − 0.778287i \(-0.716089\pi\)
−0.627908 + 0.778287i \(0.716089\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 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
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5618.00 −0.674648 −0.337324 − 0.941389i \(-0.609522\pi\)
−0.337324 + 0.941389i \(0.609522\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 14394.0 1.33728 0.668639 − 0.743587i \(-0.266877\pi\)
0.668639 + 0.743587i \(0.266877\pi\)
\(42\) 0 0
\(43\) 370.000 0.0305162 0.0152581 − 0.999884i \(-0.495143\pi\)
0.0152581 + 0.999884i \(0.495143\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 16146.0 1.06615 0.533077 − 0.846066i \(-0.321035\pi\)
0.533077 + 0.846066i \(0.321035\pi\)
\(48\) 0 0
\(49\) 30717.0 1.82763
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4374.00 −0.213889 −0.106945 − 0.994265i \(-0.534107\pi\)
−0.106945 + 0.994265i \(0.534107\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 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
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11542.0 0.314119 0.157059 − 0.987589i \(-0.449799\pi\)
0.157059 + 0.987589i \(0.449799\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 29532.0 0.695260 0.347630 − 0.937632i \(-0.386987\pi\)
0.347630 + 0.937632i \(0.386987\pi\)
\(72\) 0 0
\(73\) −33698.0 −0.740111 −0.370056 − 0.929010i \(-0.620661\pi\)
−0.370056 + 0.929010i \(0.620661\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −104640. −2.01127
\(78\) 0 0
\(79\) 31208.0 0.562598 0.281299 − 0.959620i \(-0.409235\pi\)
0.281299 + 0.959620i \(0.409235\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −38466.0 −0.612889 −0.306444 − 0.951889i \(-0.599139\pi\)
−0.306444 + 0.951889i \(0.599139\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 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
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −94658.0 −1.02148 −0.510738 − 0.859737i \(-0.670628\pi\)
−0.510738 + 0.859737i \(0.670628\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −101046. −0.985634 −0.492817 − 0.870133i \(-0.664033\pi\)
−0.492817 + 0.870133i \(0.664033\pi\)
\(102\) 0 0
\(103\) 143434. 1.33217 0.666084 − 0.745877i \(-0.267969\pi\)
0.666084 + 0.745877i \(0.267969\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −57054.0 −0.481755 −0.240878 − 0.970555i \(-0.577435\pi\)
−0.240878 + 0.970555i \(0.577435\pi\)
\(108\) 0 0
\(109\) −3118.00 −0.0251368 −0.0125684 − 0.999921i \(-0.504001\pi\)
−0.0125684 + 0.999921i \(0.504001\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −54534.0 −0.401764 −0.200882 − 0.979615i \(-0.564381\pi\)
−0.200882 + 0.979615i \(0.564381\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −40548.0 −0.262484
\(120\) 0 0
\(121\) 69349.0 0.430603
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −24698.0 −0.135879 −0.0679395 − 0.997689i \(-0.521642\pi\)
−0.0679395 + 0.997689i \(0.521642\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −236640. −1.20479 −0.602393 − 0.798200i \(-0.705786\pi\)
−0.602393 + 0.798200i \(0.705786\pi\)
\(132\) 0 0
\(133\) 262472. 1.28663
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −22158.0 −0.100862 −0.0504312 − 0.998728i \(-0.516060\pi\)
−0.0504312 + 0.998728i \(0.516060\pi\)
\(138\) 0 0
\(139\) −193204. −0.848163 −0.424081 − 0.905624i \(-0.639403\pi\)
−0.424081 + 0.905624i \(0.639403\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 298560. 1.22093
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 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
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 335878. 1.08751 0.543754 − 0.839245i \(-0.317002\pi\)
0.543754 + 0.839245i \(0.317002\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 694548. 2.11173
\(162\) 0 0
\(163\) 101650. 0.299667 0.149833 − 0.988711i \(-0.452126\pi\)
0.149833 + 0.988711i \(0.452126\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 139242. 0.386348 0.193174 − 0.981164i \(-0.438122\pi\)
0.193174 + 0.981164i \(0.438122\pi\)
\(168\) 0 0
\(169\) 15591.0 0.0419911
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −265014. −0.673215 −0.336607 − 0.941645i \(-0.609279\pi\)
−0.336607 + 0.941645i \(0.609279\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 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
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 89280.0 0.186703
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −294948. −0.585008 −0.292504 − 0.956264i \(-0.594489\pi\)
−0.292504 + 0.956264i \(0.594489\pi\)
\(192\) 0 0
\(193\) −1.00303e6 −1.93831 −0.969153 − 0.246459i \(-0.920733\pi\)
−0.969153 + 0.246459i \(0.920733\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −823998. −1.51273 −0.756364 − 0.654151i \(-0.773026\pi\)
−0.756364 + 0.654151i \(0.773026\pi\)
\(198\) 0 0
\(199\) −906712. −1.62307 −0.811534 − 0.584305i \(-0.801367\pi\)
−0.811534 + 0.584305i \(0.801367\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.20467e6 2.05176
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.03961e6 −2.94034
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 115692. 0.159339
\(222\) 0 0
\(223\) 542050. 0.729923 0.364962 − 0.931023i \(-0.381082\pi\)
0.364962 + 0.931023i \(0.381082\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.44857e6 −1.86585 −0.932924 − 0.360075i \(-0.882751\pi\)
−0.932924 + 0.360075i \(0.882751\pi\)
\(228\) 0 0
\(229\) −478786. −0.603327 −0.301663 − 0.953414i \(-0.597542\pi\)
−0.301663 + 0.953414i \(0.597542\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 374106. 0.451445 0.225723 − 0.974192i \(-0.427526\pi\)
0.225723 + 0.974192i \(0.427526\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 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
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −748888. −0.781042
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.25520e6 −1.25756 −0.628780 − 0.777583i \(-0.716445\pi\)
−0.628780 + 0.777583i \(0.716445\pi\)
\(252\) 0 0
\(253\) −1.52928e6 −1.50205
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.12877e6 −1.06604 −0.533021 − 0.846102i \(-0.678943\pi\)
−0.533021 + 0.846102i \(0.678943\pi\)
\(258\) 0 0
\(259\) 1.22472e6 1.13446
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −263082. −0.234532 −0.117266 − 0.993101i \(-0.537413\pi\)
−0.117266 + 0.993101i \(0.537413\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 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
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −743114. −0.581910 −0.290955 − 0.956737i \(-0.593973\pi\)
−0.290955 + 0.956737i \(0.593973\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.92193e6 −1.45201 −0.726007 − 0.687687i \(-0.758626\pi\)
−0.726007 + 0.687687i \(0.758626\pi\)
\(282\) 0 0
\(283\) 1.63071e6 1.21035 0.605176 − 0.796092i \(-0.293103\pi\)
0.605176 + 0.796092i \(0.293103\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.13789e6 −2.24871
\(288\) 0 0
\(289\) −1.38526e6 −0.975634
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 71250.0 0.0484859 0.0242430 − 0.999706i \(-0.492282\pi\)
0.0242430 + 0.999706i \(0.492282\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.98169e6 −1.28191
\(300\) 0 0
\(301\) −80660.0 −0.0513147
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.61762e6 0.979560 0.489780 − 0.871846i \(-0.337077\pi\)
0.489780 + 0.871846i \(0.337077\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 682788. 0.400299 0.200150 − 0.979765i \(-0.435857\pi\)
0.200150 + 0.979765i \(0.435857\pi\)
\(312\) 0 0
\(313\) 2.70444e6 1.56033 0.780165 − 0.625574i \(-0.215135\pi\)
0.780165 + 0.625574i \(0.215135\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.60347e6 1.45514 0.727568 − 0.686035i \(-0.240650\pi\)
0.727568 + 0.686035i \(0.240650\pi\)
\(318\) 0 0
\(319\) −2.65248e6 −1.45940
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −223944. −0.119435
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 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
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.71706e6 −0.823588 −0.411794 − 0.911277i \(-0.635098\pi\)
−0.411794 + 0.911277i \(0.635098\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.49088e6 2.09144
\(342\) 0 0
\(343\) −3.03238e6 −1.39171
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 131370. 0.0585696 0.0292848 − 0.999571i \(-0.490677\pi\)
0.0292848 + 0.999571i \(0.490677\pi\)
\(348\) 0 0
\(349\) 3.50951e6 1.54235 0.771175 − 0.636623i \(-0.219669\pi\)
0.771175 + 0.636623i \(0.219669\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.21992e6 0.948202 0.474101 − 0.880470i \(-0.342773\pi\)
0.474101 + 0.880470i \(0.342773\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 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
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2.29824e6 0.890697 0.445348 − 0.895357i \(-0.353080\pi\)
0.445348 + 0.895357i \(0.353080\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 953532. 0.359667
\(372\) 0 0
\(373\) 1.73561e6 0.645920 0.322960 − 0.946413i \(-0.395322\pi\)
0.322960 + 0.946413i \(0.395322\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.43717e6 −1.24551
\(378\) 0 0
\(379\) −5.39115e6 −1.92789 −0.963947 − 0.266094i \(-0.914267\pi\)
−0.963947 + 0.266094i \(0.914267\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.27281e6 1.14005 0.570026 − 0.821627i \(-0.306933\pi\)
0.570026 + 0.821627i \(0.306933\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 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
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 749422. 0.238644 0.119322 − 0.992856i \(-0.461928\pi\)
0.119322 + 0.992856i \(0.461928\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5.31357e6 −1.65016 −0.825079 − 0.565018i \(-0.808869\pi\)
−0.825079 + 0.565018i \(0.808869\pi\)
\(402\) 0 0
\(403\) 5.81943e6 1.78492
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.69664e6 −0.806932
\(408\) 0 0
\(409\) 999326. 0.295392 0.147696 − 0.989033i \(-0.452814\pi\)
0.147696 + 0.989033i \(0.452814\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.56106e6 −0.738831
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 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
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.87804e6 −0.763882
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.30404e6 0.856747 0.428374 − 0.903602i \(-0.359087\pi\)
0.428374 + 0.903602i \(0.359087\pi\)
\(432\) 0 0
\(433\) 2.01638e6 0.516836 0.258418 − 0.966033i \(-0.416799\pi\)
0.258418 + 0.966033i \(0.416799\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.83594e6 0.960879
\(438\) 0 0
\(439\) 6.58321e6 1.63033 0.815166 − 0.579227i \(-0.196645\pi\)
0.815166 + 0.579227i \(0.196645\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.81783e6 −1.16638 −0.583192 − 0.812334i \(-0.698197\pi\)
−0.583192 + 0.812334i \(0.698197\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 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
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2.84383e6 −0.636962 −0.318481 − 0.947929i \(-0.603173\pi\)
−0.318481 + 0.947929i \(0.603173\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.75605e6 0.384844 0.192422 − 0.981312i \(-0.438366\pi\)
0.192422 + 0.981312i \(0.438366\pi\)
\(462\) 0 0
\(463\) −7.66857e6 −1.66250 −0.831250 − 0.555899i \(-0.812374\pi\)
−0.831250 + 0.555899i \(0.812374\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.35903e6 −0.288361 −0.144181 − 0.989551i \(-0.546055\pi\)
−0.144181 + 0.989551i \(0.546055\pi\)
\(468\) 0 0
\(469\) −2.51616e6 −0.528209
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 177600. 0.0364998
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 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
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.46427e6 −0.470833 −0.235416 − 0.971895i \(-0.575645\pi\)
−0.235416 + 0.971895i \(0.575645\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.03848e7 −1.94399 −0.971996 − 0.234998i \(-0.924492\pi\)
−0.971996 + 0.234998i \(0.924492\pi\)
\(492\) 0 0
\(493\) −1.02784e6 −0.190461
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.43798e6 −1.16912
\(498\) 0 0
\(499\) 6.49416e6 1.16754 0.583769 − 0.811919i \(-0.301577\pi\)
0.583769 + 0.811919i \(0.301577\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1.03565e7 −1.82513 −0.912565 − 0.408931i \(-0.865902\pi\)
−0.912565 + 0.408931i \(0.865902\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 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
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 7.75008e6 1.27520
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2.17295e6 −0.350717 −0.175358 − 0.984505i \(-0.556108\pi\)
−0.175358 + 0.984505i \(0.556108\pi\)
\(522\) 0 0
\(523\) −1.07361e6 −0.171629 −0.0858145 − 0.996311i \(-0.527349\pi\)
−0.0858145 + 0.996311i \(0.527349\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.74022e6 0.272946
\(528\) 0 0
\(529\) 3.71425e6 0.577075
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.95307e6 1.36507
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 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
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −6.69763e6 −0.957091 −0.478545 − 0.878063i \(-0.658836\pi\)
−0.478545 + 0.878063i \(0.658836\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.65330e6 0.933595
\(552\) 0 0
\(553\) −6.80334e6 −0.946040
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.19008e7 1.62532 0.812662 − 0.582735i \(-0.198018\pi\)
0.812662 + 0.582735i \(0.198018\pi\)
\(558\) 0 0
\(559\) 230140. 0.0311503
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.75636e6 1.16427 0.582133 − 0.813093i \(-0.302218\pi\)
0.582133 + 0.813093i \(0.302218\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 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
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 3.13404e6 0.391890 0.195945 − 0.980615i \(-0.437223\pi\)
0.195945 + 0.980615i \(0.437223\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.38559e6 1.03061
\(582\) 0 0
\(583\) −2.09952e6 −0.255828
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.13833e7 1.36355 0.681776 − 0.731561i \(-0.261208\pi\)
0.681776 + 0.731561i \(0.261208\pi\)
\(588\) 0 0
\(589\) −1.12646e7 −1.33791
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.58655e7 −1.85275 −0.926376 − 0.376599i \(-0.877094\pi\)
−0.926376 + 0.376599i \(0.877094\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 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
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 710398. 0.0782582 0.0391291 − 0.999234i \(-0.487542\pi\)
0.0391291 + 0.999234i \(0.487542\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.00428e7 1.08831
\(612\) 0 0
\(613\) −5.96434e6 −0.641078 −0.320539 − 0.947235i \(-0.603864\pi\)
−0.320539 + 0.947235i \(0.603864\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.48432e7 1.56970 0.784848 − 0.619689i \(-0.212741\pi\)
0.784848 + 0.619689i \(0.212741\pi\)
\(618\) 0 0
\(619\) −1.82042e7 −1.90961 −0.954807 − 0.297227i \(-0.903938\pi\)
−0.954807 + 0.297227i \(0.903938\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.60541e7 2.68940
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.91060e7 1.86561
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −7.44046e6 −0.715245 −0.357622 − 0.933866i \(-0.616413\pi\)
−0.357622 + 0.933866i \(0.616413\pi\)
\(642\) 0 0
\(643\) 1.07915e7 1.02933 0.514665 − 0.857391i \(-0.327916\pi\)
0.514665 + 0.857391i \(0.327916\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.62998e6 −0.904409 −0.452204 − 0.891914i \(-0.649362\pi\)
−0.452204 + 0.891914i \(0.649362\pi\)
\(648\) 0 0
\(649\) 5.63904e6 0.525525
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.00019e7 −0.917905 −0.458953 − 0.888461i \(-0.651775\pi\)
−0.458953 + 0.888461i \(0.651775\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 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
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.76058e7 1.53229
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6.33696e6 0.543344
\(672\) 0 0
\(673\) −2.01231e6 −0.171260 −0.0856301 − 0.996327i \(-0.527290\pi\)
−0.0856301 + 0.996327i \(0.527290\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.62410e7 1.36188 0.680942 − 0.732337i \(-0.261571\pi\)
0.680942 + 0.732337i \(0.261571\pi\)
\(678\) 0 0
\(679\) 2.06354e7 1.71767
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4.62910e6 0.379704 0.189852 − 0.981813i \(-0.439199\pi\)
0.189852 + 0.981813i \(0.439199\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 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
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.67728e6 0.208743
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.99543e7 −1.53370 −0.766851 − 0.641825i \(-0.778178\pi\)
−0.766851 + 0.641825i \(0.778178\pi\)
\(702\) 0 0
\(703\) 6.76407e6 0.516202
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.20280e7 1.65740
\(708\) 0 0
\(709\) −4.88331e6 −0.364837 −0.182419 − 0.983221i \(-0.558393\pi\)
−0.182419 + 0.983221i \(0.558393\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.98082e7 −2.19590
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 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
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −6.42411e6 −0.450792 −0.225396 − 0.974267i \(-0.572368\pi\)
−0.225396 + 0.974267i \(0.572368\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 68820.0 0.00476345
\(732\) 0 0
\(733\) −9.08556e6 −0.624585 −0.312293 − 0.949986i \(-0.601097\pi\)
−0.312293 + 0.949986i \(0.601097\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.54016e6 0.375711
\(738\) 0 0
\(739\) 2.02457e7 1.36371 0.681854 − 0.731488i \(-0.261174\pi\)
0.681854 + 0.731488i \(0.261174\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5.44831e6 −0.362067 −0.181034 − 0.983477i \(-0.557944\pi\)
−0.181034 + 0.983477i \(0.557944\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 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
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.90153e7 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.23551e7 −1.39931 −0.699656 − 0.714480i \(-0.746663\pi\)
−0.699656 + 0.714480i \(0.746663\pi\)
\(762\) 0 0
\(763\) 679724. 0.0422689
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.30726e6 0.448504
\(768\) 0 0
\(769\) −1.00704e7 −0.614088 −0.307044 − 0.951695i \(-0.599340\pi\)
−0.307044 + 0.951695i \(0.599340\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −4.05963e6 −0.244364 −0.122182 − 0.992508i \(-0.538989\pi\)
−0.122182 + 0.992508i \(0.538989\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.73304e7 −1.02321
\(780\) 0 0
\(781\) 1.41754e7 0.831585
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1.72256e7 0.991372 0.495686 − 0.868502i \(-0.334917\pi\)
0.495686 + 0.868502i \(0.334917\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.18884e7 0.675589
\(792\) 0 0
\(793\) 8.21164e6 0.463711
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2.10793e7 −1.17547 −0.587733 − 0.809055i \(-0.699980\pi\)
−0.587733 + 0.809055i \(0.699980\pi\)
\(798\) 0 0
\(799\) 3.00316e6 0.166422
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.61750e7 −0.885231
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 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
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −445480. −0.0233493
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 7.66925e6 0.397096 0.198548 − 0.980091i \(-0.436377\pi\)
0.198548 + 0.980091i \(0.436377\pi\)
\(822\) 0 0
\(823\) 8.82786e6 0.454314 0.227157 − 0.973858i \(-0.427057\pi\)
0.227157 + 0.973858i \(0.427057\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3.06923e7 −1.56051 −0.780254 − 0.625463i \(-0.784910\pi\)
−0.780254 + 0.625463i \(0.784910\pi\)
\(828\) 0 0
\(829\) 3.28414e7 1.65972 0.829860 − 0.557972i \(-0.188420\pi\)
0.829860 + 0.557972i \(0.188420\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.71336e6 0.285285
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 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
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.51181e7 −0.724083
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.78989e7 0.847234
\(852\) 0 0
\(853\) −2.35126e7 −1.10644 −0.553221 − 0.833035i \(-0.686601\pi\)
−0.553221 + 0.833035i \(0.686601\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.13050e7 −0.525799 −0.262900 − 0.964823i \(-0.584679\pi\)
−0.262900 + 0.964823i \(0.584679\pi\)
\(858\) 0 0
\(859\) 1.00078e7 0.462758 0.231379 − 0.972864i \(-0.425676\pi\)
0.231379 + 0.972864i \(0.425676\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.61429e7 1.19489 0.597443 − 0.801911i \(-0.296183\pi\)
0.597443 + 0.801911i \(0.296183\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.49798e7 0.672911
\(870\) 0 0
\(871\) 7.17912e6 0.320646
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.92041e6 0.0843129 0.0421565 − 0.999111i \(-0.486577\pi\)
0.0421565 + 0.999111i \(0.486577\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2.56594e7 1.11380 0.556899 − 0.830580i \(-0.311991\pi\)
0.556899 + 0.830580i \(0.311991\pi\)
\(882\) 0 0
\(883\) 2.05643e7 0.887590 0.443795 − 0.896128i \(-0.353632\pi\)
0.443795 + 0.896128i \(0.353632\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.16868e7 1.35229 0.676143 − 0.736770i \(-0.263650\pi\)
0.676143 + 0.736770i \(0.263650\pi\)
\(888\) 0 0
\(889\) 5.38416e6 0.228488
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.94398e7 −0.815761
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −5.17013e7 −2.13355
\(900\) 0 0
\(901\) −813564. −0.0333872
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −3.96963e6 −0.160225 −0.0801127 − 0.996786i \(-0.525528\pi\)
−0.0801127 + 0.996786i \(0.525528\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.37945e7 0.550692 0.275346 − 0.961345i \(-0.411208\pi\)
0.275346 + 0.961345i \(0.411208\pi\)
\(912\) 0 0
\(913\) −1.84637e7 −0.733063
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.15875e7 2.02592
\(918\) 0 0
\(919\) 8.08126e6 0.315639 0.157819 − 0.987468i \(-0.449554\pi\)
0.157819 + 0.987468i \(0.449554\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.83689e7 0.709707
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 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
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.07620e7 0.772540 0.386270 − 0.922386i \(-0.373763\pi\)
0.386270 + 0.922386i \(0.373763\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 3.47642e6 0.127985 0.0639923 − 0.997950i \(-0.479617\pi\)
0.0639923 + 0.997950i \(0.479617\pi\)
\(942\) 0 0
\(943\) −4.58593e7 −1.67938
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.86700e6 0.0676503 0.0338252 − 0.999428i \(-0.489231\pi\)
0.0338252 + 0.999428i \(0.489231\pi\)
\(948\) 0 0
\(949\) −2.09602e7 −0.755490
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −3.85501e7 −1.37497 −0.687484 − 0.726199i \(-0.741285\pi\)
−0.687484 + 0.726199i \(0.741285\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.83044e6 0.169606
\(960\) 0 0
\(961\) 5.89056e7 2.05754
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.64875e7 −0.567008 −0.283504 − 0.958971i \(-0.591497\pi\)
−0.283504 + 0.958971i \(0.591497\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2.36976e7 0.806597 0.403299 − 0.915068i \(-0.367864\pi\)
0.403299 + 0.915068i \(0.367864\pi\)
\(972\) 0 0
\(973\) 4.21185e7 1.42623
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −5.77590e7 −1.93590 −0.967950 − 0.251143i \(-0.919194\pi\)
−0.967950 + 0.251143i \(0.919194\pi\)
\(978\) 0 0
\(979\) −5.73667e7 −1.91295
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.10103e7 −0.363425 −0.181712 − 0.983352i \(-0.558164\pi\)
−0.181712 + 0.983352i \(0.558164\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 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
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −4.52935e6 −0.144311 −0.0721553 − 0.997393i \(-0.522988\pi\)
−0.0721553 + 0.997393i \(0.522988\pi\)
\(998\) 0 0
\(999\) 0 0
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 900.6.a.b.1.1 1
3.2 odd 2 100.6.a.a.1.1 1
5.2 odd 4 900.6.d.h.649.1 2
5.3 odd 4 900.6.d.h.649.2 2
5.4 even 2 180.6.a.e.1.1 1
12.11 even 2 400.6.a.m.1.1 1
15.2 even 4 100.6.c.a.49.2 2
15.8 even 4 100.6.c.a.49.1 2
15.14 odd 2 20.6.a.a.1.1 ✓ 1
20.19 odd 2 720.6.a.l.1.1 1
60.23 odd 4 400.6.c.c.49.2 2
60.47 odd 4 400.6.c.c.49.1 2
60.59 even 2 80.6.a.b.1.1 1
105.104 even 2 980.6.a.b.1.1 1
120.29 odd 2 320.6.a.c.1.1 1
120.59 even 2 320.6.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.6.a.a.1.1 ✓ 1 15.14 odd 2
80.6.a.b.1.1 1 60.59 even 2
100.6.a.a.1.1 1 3.2 odd 2
100.6.c.a.49.1 2 15.8 even 4
100.6.c.a.49.2 2 15.2 even 4
180.6.a.e.1.1 1 5.4 even 2
320.6.a.c.1.1 1 120.29 odd 2
320.6.a.n.1.1 1 120.59 even 2
400.6.a.m.1.1 1 12.11 even 2
400.6.c.c.49.1 2 60.47 odd 4
400.6.c.c.49.2 2 60.23 odd 4
720.6.a.l.1.1 1 20.19 odd 2
900.6.a.b.1.1 1 1.1 even 1 trivial
900.6.d.h.649.1 2 5.2 odd 4
900.6.d.h.649.2 2 5.3 odd 4
980.6.a.b.1.1 1 105.104 even 2