Properties

Label 784.6.a.c.1.1
Level $784$
Weight $6$
Character 784.1
Self dual yes
Analytic conductor $125.741$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,6,Mod(1,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, 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("784.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.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: \(125.740914733\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 784.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-14.0000 q^{3} +56.0000 q^{5} -47.0000 q^{9} +O(q^{10})\) \(q-14.0000 q^{3} +56.0000 q^{5} -47.0000 q^{9} -232.000 q^{11} +140.000 q^{13} -784.000 q^{15} +1722.00 q^{17} -98.0000 q^{19} -1824.00 q^{23} +11.0000 q^{25} +4060.00 q^{27} +3418.00 q^{29} -7644.00 q^{31} +3248.00 q^{33} -10398.0 q^{37} -1960.00 q^{39} +17962.0 q^{41} -10880.0 q^{43} -2632.00 q^{45} +9324.00 q^{47} -24108.0 q^{51} +2262.00 q^{53} -12992.0 q^{55} +1372.00 q^{57} -2730.00 q^{59} -25648.0 q^{61} +7840.00 q^{65} +48404.0 q^{67} +25536.0 q^{69} +58560.0 q^{71} -68082.0 q^{73} -154.000 q^{75} -31784.0 q^{79} -45419.0 q^{81} -20538.0 q^{83} +96432.0 q^{85} -47852.0 q^{87} +50582.0 q^{89} +107016. q^{93} -5488.00 q^{95} +58506.0 q^{97} +10904.0 q^{99} +O(q^{100})\)

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
\(3\) −14.0000 −0.898100 −0.449050 0.893507i \(-0.648238\pi\)
−0.449050 + 0.893507i \(0.648238\pi\)
\(4\) 0 0
\(5\) 56.0000 1.00176 0.500879 0.865517i \(-0.333010\pi\)
0.500879 + 0.865517i \(0.333010\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −47.0000 −0.193416
\(10\) 0 0
\(11\) −232.000 −0.578104 −0.289052 0.957313i \(-0.593340\pi\)
−0.289052 + 0.957313i \(0.593340\pi\)
\(12\) 0 0
\(13\) 140.000 0.229757 0.114879 0.993380i \(-0.463352\pi\)
0.114879 + 0.993380i \(0.463352\pi\)
\(14\) 0 0
\(15\) −784.000 −0.899680
\(16\) 0 0
\(17\) 1722.00 1.44514 0.722572 0.691296i \(-0.242960\pi\)
0.722572 + 0.691296i \(0.242960\pi\)
\(18\) 0 0
\(19\) −98.0000 −0.0622791 −0.0311395 0.999515i \(-0.509914\pi\)
−0.0311395 + 0.999515i \(0.509914\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1824.00 −0.718961 −0.359480 0.933153i \(-0.617046\pi\)
−0.359480 + 0.933153i \(0.617046\pi\)
\(24\) 0 0
\(25\) 11.0000 0.00352000
\(26\) 0 0
\(27\) 4060.00 1.07181
\(28\) 0 0
\(29\) 3418.00 0.754705 0.377352 0.926070i \(-0.376835\pi\)
0.377352 + 0.926070i \(0.376835\pi\)
\(30\) 0 0
\(31\) −7644.00 −1.42862 −0.714310 0.699830i \(-0.753259\pi\)
−0.714310 + 0.699830i \(0.753259\pi\)
\(32\) 0 0
\(33\) 3248.00 0.519196
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10398.0 −1.24866 −0.624332 0.781159i \(-0.714629\pi\)
−0.624332 + 0.781159i \(0.714629\pi\)
\(38\) 0 0
\(39\) −1960.00 −0.206345
\(40\) 0 0
\(41\) 17962.0 1.66876 0.834382 0.551186i \(-0.185825\pi\)
0.834382 + 0.551186i \(0.185825\pi\)
\(42\) 0 0
\(43\) −10880.0 −0.897342 −0.448671 0.893697i \(-0.648102\pi\)
−0.448671 + 0.893697i \(0.648102\pi\)
\(44\) 0 0
\(45\) −2632.00 −0.193756
\(46\) 0 0
\(47\) 9324.00 0.615684 0.307842 0.951438i \(-0.400393\pi\)
0.307842 + 0.951438i \(0.400393\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −24108.0 −1.29788
\(52\) 0 0
\(53\) 2262.00 0.110612 0.0553061 0.998469i \(-0.482387\pi\)
0.0553061 + 0.998469i \(0.482387\pi\)
\(54\) 0 0
\(55\) −12992.0 −0.579121
\(56\) 0 0
\(57\) 1372.00 0.0559329
\(58\) 0 0
\(59\) −2730.00 −0.102102 −0.0510508 0.998696i \(-0.516257\pi\)
−0.0510508 + 0.998696i \(0.516257\pi\)
\(60\) 0 0
\(61\) −25648.0 −0.882529 −0.441264 0.897377i \(-0.645470\pi\)
−0.441264 + 0.897377i \(0.645470\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7840.00 0.230161
\(66\) 0 0
\(67\) 48404.0 1.31733 0.658664 0.752437i \(-0.271122\pi\)
0.658664 + 0.752437i \(0.271122\pi\)
\(68\) 0 0
\(69\) 25536.0 0.645699
\(70\) 0 0
\(71\) 58560.0 1.37865 0.689327 0.724450i \(-0.257906\pi\)
0.689327 + 0.724450i \(0.257906\pi\)
\(72\) 0 0
\(73\) −68082.0 −1.49529 −0.747645 0.664099i \(-0.768815\pi\)
−0.747645 + 0.664099i \(0.768815\pi\)
\(74\) 0 0
\(75\) −154.000 −0.00316131
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −31784.0 −0.572982 −0.286491 0.958083i \(-0.592489\pi\)
−0.286491 + 0.958083i \(0.592489\pi\)
\(80\) 0 0
\(81\) −45419.0 −0.769175
\(82\) 0 0
\(83\) −20538.0 −0.327237 −0.163619 0.986524i \(-0.552317\pi\)
−0.163619 + 0.986524i \(0.552317\pi\)
\(84\) 0 0
\(85\) 96432.0 1.44768
\(86\) 0 0
\(87\) −47852.0 −0.677801
\(88\) 0 0
\(89\) 50582.0 0.676894 0.338447 0.940985i \(-0.390098\pi\)
0.338447 + 0.940985i \(0.390098\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 107016. 1.28304
\(94\) 0 0
\(95\) −5488.00 −0.0623886
\(96\) 0 0
\(97\) 58506.0 0.631351 0.315676 0.948867i \(-0.397769\pi\)
0.315676 + 0.948867i \(0.397769\pi\)
\(98\) 0 0
\(99\) 10904.0 0.111814
\(100\) 0 0
\(101\) −38696.0 −0.377453 −0.188726 0.982030i \(-0.560436\pi\)
−0.188726 + 0.982030i \(0.560436\pi\)
\(102\) 0 0
\(103\) 53060.0 0.492804 0.246402 0.969168i \(-0.420752\pi\)
0.246402 + 0.969168i \(0.420752\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 146324. 1.23554 0.617769 0.786360i \(-0.288037\pi\)
0.617769 + 0.786360i \(0.288037\pi\)
\(108\) 0 0
\(109\) 92898.0 0.748928 0.374464 0.927241i \(-0.377827\pi\)
0.374464 + 0.927241i \(0.377827\pi\)
\(110\) 0 0
\(111\) 145572. 1.12143
\(112\) 0 0
\(113\) −83354.0 −0.614088 −0.307044 0.951695i \(-0.599340\pi\)
−0.307044 + 0.951695i \(0.599340\pi\)
\(114\) 0 0
\(115\) −102144. −0.720225
\(116\) 0 0
\(117\) −6580.00 −0.0444387
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −107227. −0.665795
\(122\) 0 0
\(123\) −251468. −1.49872
\(124\) 0 0
\(125\) −174384. −0.998232
\(126\) 0 0
\(127\) −60384.0 −0.332210 −0.166105 0.986108i \(-0.553119\pi\)
−0.166105 + 0.986108i \(0.553119\pi\)
\(128\) 0 0
\(129\) 152320. 0.805903
\(130\) 0 0
\(131\) −61586.0 −0.313548 −0.156774 0.987635i \(-0.550109\pi\)
−0.156774 + 0.987635i \(0.550109\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 227360. 1.07369
\(136\) 0 0
\(137\) −204462. −0.930703 −0.465352 0.885126i \(-0.654072\pi\)
−0.465352 + 0.885126i \(0.654072\pi\)
\(138\) 0 0
\(139\) −35406.0 −0.155432 −0.0777159 0.996976i \(-0.524763\pi\)
−0.0777159 + 0.996976i \(0.524763\pi\)
\(140\) 0 0
\(141\) −130536. −0.552946
\(142\) 0 0
\(143\) −32480.0 −0.132824
\(144\) 0 0
\(145\) 191408. 0.756032
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −20226.0 −0.0746353 −0.0373177 0.999303i \(-0.511881\pi\)
−0.0373177 + 0.999303i \(0.511881\pi\)
\(150\) 0 0
\(151\) −70904.0 −0.253063 −0.126531 0.991963i \(-0.540384\pi\)
−0.126531 + 0.991963i \(0.540384\pi\)
\(152\) 0 0
\(153\) −80934.0 −0.279513
\(154\) 0 0
\(155\) −428064. −1.43113
\(156\) 0 0
\(157\) −293524. −0.950374 −0.475187 0.879885i \(-0.657620\pi\)
−0.475187 + 0.879885i \(0.657620\pi\)
\(158\) 0 0
\(159\) −31668.0 −0.0993408
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −13192.0 −0.0388903 −0.0194452 0.999811i \(-0.506190\pi\)
−0.0194452 + 0.999811i \(0.506190\pi\)
\(164\) 0 0
\(165\) 181888. 0.520109
\(166\) 0 0
\(167\) 493612. 1.36960 0.684801 0.728730i \(-0.259889\pi\)
0.684801 + 0.728730i \(0.259889\pi\)
\(168\) 0 0
\(169\) −351693. −0.947212
\(170\) 0 0
\(171\) 4606.00 0.0120457
\(172\) 0 0
\(173\) −240716. −0.611490 −0.305745 0.952113i \(-0.598906\pi\)
−0.305745 + 0.952113i \(0.598906\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 38220.0 0.0916975
\(178\) 0 0
\(179\) −294932. −0.688001 −0.344001 0.938969i \(-0.611782\pi\)
−0.344001 + 0.938969i \(0.611782\pi\)
\(180\) 0 0
\(181\) 336980. 0.764553 0.382277 0.924048i \(-0.375140\pi\)
0.382277 + 0.924048i \(0.375140\pi\)
\(182\) 0 0
\(183\) 359072. 0.792600
\(184\) 0 0
\(185\) −582288. −1.25086
\(186\) 0 0
\(187\) −399504. −0.835444
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −358264. −0.710591 −0.355296 0.934754i \(-0.615620\pi\)
−0.355296 + 0.934754i \(0.615620\pi\)
\(192\) 0 0
\(193\) −989554. −1.91226 −0.956128 0.292948i \(-0.905364\pi\)
−0.956128 + 0.292948i \(0.905364\pi\)
\(194\) 0 0
\(195\) −109760. −0.206708
\(196\) 0 0
\(197\) −990050. −1.81757 −0.908786 0.417263i \(-0.862989\pi\)
−0.908786 + 0.417263i \(0.862989\pi\)
\(198\) 0 0
\(199\) −840756. −1.50500 −0.752501 0.658591i \(-0.771153\pi\)
−0.752501 + 0.658591i \(0.771153\pi\)
\(200\) 0 0
\(201\) −677656. −1.18309
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1.00587e6 1.67170
\(206\) 0 0
\(207\) 85728.0 0.139058
\(208\) 0 0
\(209\) 22736.0 0.0360038
\(210\) 0 0
\(211\) −1.15073e6 −1.77938 −0.889689 0.456568i \(-0.849079\pi\)
−0.889689 + 0.456568i \(0.849079\pi\)
\(212\) 0 0
\(213\) −819840. −1.23817
\(214\) 0 0
\(215\) −609280. −0.898919
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 953148. 1.34292
\(220\) 0 0
\(221\) 241080. 0.332032
\(222\) 0 0
\(223\) −824264. −1.10995 −0.554976 0.831866i \(-0.687273\pi\)
−0.554976 + 0.831866i \(0.687273\pi\)
\(224\) 0 0
\(225\) −517.000 −0.000680823 0
\(226\) 0 0
\(227\) 74382.0 0.0958083 0.0479042 0.998852i \(-0.484746\pi\)
0.0479042 + 0.998852i \(0.484746\pi\)
\(228\) 0 0
\(229\) −1.13196e6 −1.42640 −0.713199 0.700961i \(-0.752755\pi\)
−0.713199 + 0.700961i \(0.752755\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −198726. −0.239809 −0.119904 0.992785i \(-0.538259\pi\)
−0.119904 + 0.992785i \(0.538259\pi\)
\(234\) 0 0
\(235\) 522144. 0.616766
\(236\) 0 0
\(237\) 444976. 0.514595
\(238\) 0 0
\(239\) −482904. −0.546847 −0.273424 0.961894i \(-0.588156\pi\)
−0.273424 + 0.961894i \(0.588156\pi\)
\(240\) 0 0
\(241\) −805910. −0.893807 −0.446904 0.894582i \(-0.647473\pi\)
−0.446904 + 0.894582i \(0.647473\pi\)
\(242\) 0 0
\(243\) −350714. −0.381011
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −13720.0 −0.0143091
\(248\) 0 0
\(249\) 287532. 0.293892
\(250\) 0 0
\(251\) 430738. 0.431548 0.215774 0.976443i \(-0.430773\pi\)
0.215774 + 0.976443i \(0.430773\pi\)
\(252\) 0 0
\(253\) 423168. 0.415634
\(254\) 0 0
\(255\) −1.35005e6 −1.30017
\(256\) 0 0
\(257\) 1.17691e6 1.11150 0.555751 0.831349i \(-0.312431\pi\)
0.555751 + 0.831349i \(0.312431\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −160646. −0.145972
\(262\) 0 0
\(263\) −1.29098e6 −1.15088 −0.575438 0.817845i \(-0.695169\pi\)
−0.575438 + 0.817845i \(0.695169\pi\)
\(264\) 0 0
\(265\) 126672. 0.110807
\(266\) 0 0
\(267\) −708148. −0.607919
\(268\) 0 0
\(269\) 1.27756e6 1.07646 0.538232 0.842797i \(-0.319093\pi\)
0.538232 + 0.842797i \(0.319093\pi\)
\(270\) 0 0
\(271\) 1.65054e6 1.36522 0.682612 0.730781i \(-0.260844\pi\)
0.682612 + 0.730781i \(0.260844\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2552.00 −0.00203493
\(276\) 0 0
\(277\) −1.06409e6 −0.833257 −0.416628 0.909077i \(-0.636788\pi\)
−0.416628 + 0.909077i \(0.636788\pi\)
\(278\) 0 0
\(279\) 359268. 0.276317
\(280\) 0 0
\(281\) −22342.0 −0.0168794 −0.00843969 0.999964i \(-0.502686\pi\)
−0.00843969 + 0.999964i \(0.502686\pi\)
\(282\) 0 0
\(283\) −2.49574e6 −1.85239 −0.926196 0.377042i \(-0.876941\pi\)
−0.926196 + 0.377042i \(0.876941\pi\)
\(284\) 0 0
\(285\) 76832.0 0.0560312
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.54543e6 1.08844
\(290\) 0 0
\(291\) −819084. −0.567017
\(292\) 0 0
\(293\) 1.93178e6 1.31458 0.657291 0.753637i \(-0.271702\pi\)
0.657291 + 0.753637i \(0.271702\pi\)
\(294\) 0 0
\(295\) −152880. −0.102281
\(296\) 0 0
\(297\) −941920. −0.619616
\(298\) 0 0
\(299\) −255360. −0.165187
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 541744. 0.338991
\(304\) 0 0
\(305\) −1.43629e6 −0.884081
\(306\) 0 0
\(307\) −459074. −0.277995 −0.138997 0.990293i \(-0.544388\pi\)
−0.138997 + 0.990293i \(0.544388\pi\)
\(308\) 0 0
\(309\) −742840. −0.442587
\(310\) 0 0
\(311\) 667128. 0.391118 0.195559 0.980692i \(-0.437348\pi\)
0.195559 + 0.980692i \(0.437348\pi\)
\(312\) 0 0
\(313\) 111034. 0.0640612 0.0320306 0.999487i \(-0.489803\pi\)
0.0320306 + 0.999487i \(0.489803\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −68778.0 −0.0384416 −0.0192208 0.999815i \(-0.506119\pi\)
−0.0192208 + 0.999815i \(0.506119\pi\)
\(318\) 0 0
\(319\) −792976. −0.436298
\(320\) 0 0
\(321\) −2.04854e6 −1.10964
\(322\) 0 0
\(323\) −168756. −0.0900022
\(324\) 0 0
\(325\) 1540.00 0.000808746 0
\(326\) 0 0
\(327\) −1.30057e6 −0.672613
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 564448. 0.283174 0.141587 0.989926i \(-0.454779\pi\)
0.141587 + 0.989926i \(0.454779\pi\)
\(332\) 0 0
\(333\) 488706. 0.241511
\(334\) 0 0
\(335\) 2.71062e6 1.31965
\(336\) 0 0
\(337\) 2.07729e6 0.996376 0.498188 0.867069i \(-0.333999\pi\)
0.498188 + 0.867069i \(0.333999\pi\)
\(338\) 0 0
\(339\) 1.16696e6 0.551512
\(340\) 0 0
\(341\) 1.77341e6 0.825891
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1.43002e6 0.646834
\(346\) 0 0
\(347\) 53248.0 0.0237399 0.0118700 0.999930i \(-0.496222\pi\)
0.0118700 + 0.999930i \(0.496222\pi\)
\(348\) 0 0
\(349\) 2.27200e6 0.998494 0.499247 0.866460i \(-0.333610\pi\)
0.499247 + 0.866460i \(0.333610\pi\)
\(350\) 0 0
\(351\) 568400. 0.246256
\(352\) 0 0
\(353\) −4.00645e6 −1.71129 −0.855644 0.517565i \(-0.826838\pi\)
−0.855644 + 0.517565i \(0.826838\pi\)
\(354\) 0 0
\(355\) 3.27936e6 1.38108
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −73784.0 −0.0302152 −0.0151076 0.999886i \(-0.504809\pi\)
−0.0151076 + 0.999886i \(0.504809\pi\)
\(360\) 0 0
\(361\) −2.46650e6 −0.996121
\(362\) 0 0
\(363\) 1.50118e6 0.597951
\(364\) 0 0
\(365\) −3.81259e6 −1.49792
\(366\) 0 0
\(367\) 1.40431e6 0.544250 0.272125 0.962262i \(-0.412274\pi\)
0.272125 + 0.962262i \(0.412274\pi\)
\(368\) 0 0
\(369\) −844214. −0.322765
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.60323e6 −0.596657 −0.298329 0.954463i \(-0.596429\pi\)
−0.298329 + 0.954463i \(0.596429\pi\)
\(374\) 0 0
\(375\) 2.44138e6 0.896513
\(376\) 0 0
\(377\) 478520. 0.173399
\(378\) 0 0
\(379\) 4.77012e6 1.70581 0.852906 0.522064i \(-0.174838\pi\)
0.852906 + 0.522064i \(0.174838\pi\)
\(380\) 0 0
\(381\) 845376. 0.298358
\(382\) 0 0
\(383\) −2.23079e6 −0.777072 −0.388536 0.921434i \(-0.627019\pi\)
−0.388536 + 0.921434i \(0.627019\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 511360. 0.173560
\(388\) 0 0
\(389\) 4.84024e6 1.62178 0.810892 0.585196i \(-0.198982\pi\)
0.810892 + 0.585196i \(0.198982\pi\)
\(390\) 0 0
\(391\) −3.14093e6 −1.03900
\(392\) 0 0
\(393\) 862204. 0.281597
\(394\) 0 0
\(395\) −1.77990e6 −0.573989
\(396\) 0 0
\(397\) −995820. −0.317106 −0.158553 0.987350i \(-0.550683\pi\)
−0.158553 + 0.987350i \(0.550683\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.31605e6 −1.02982 −0.514909 0.857245i \(-0.672174\pi\)
−0.514909 + 0.857245i \(0.672174\pi\)
\(402\) 0 0
\(403\) −1.07016e6 −0.328236
\(404\) 0 0
\(405\) −2.54346e6 −0.770527
\(406\) 0 0
\(407\) 2.41234e6 0.721858
\(408\) 0 0
\(409\) −3.07273e6 −0.908274 −0.454137 0.890932i \(-0.650052\pi\)
−0.454137 + 0.890932i \(0.650052\pi\)
\(410\) 0 0
\(411\) 2.86247e6 0.835865
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1.15013e6 −0.327813
\(416\) 0 0
\(417\) 495684. 0.139593
\(418\) 0 0
\(419\) 2.81438e6 0.783154 0.391577 0.920145i \(-0.371930\pi\)
0.391577 + 0.920145i \(0.371930\pi\)
\(420\) 0 0
\(421\) 3.05802e6 0.840883 0.420441 0.907320i \(-0.361875\pi\)
0.420441 + 0.907320i \(0.361875\pi\)
\(422\) 0 0
\(423\) −438228. −0.119083
\(424\) 0 0
\(425\) 18942.0 0.00508690
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 454720. 0.119289
\(430\) 0 0
\(431\) −1.93750e6 −0.502398 −0.251199 0.967936i \(-0.580825\pi\)
−0.251199 + 0.967936i \(0.580825\pi\)
\(432\) 0 0
\(433\) −3.94790e6 −1.01192 −0.505961 0.862557i \(-0.668862\pi\)
−0.505961 + 0.862557i \(0.668862\pi\)
\(434\) 0 0
\(435\) −2.67971e6 −0.678993
\(436\) 0 0
\(437\) 178752. 0.0447762
\(438\) 0 0
\(439\) −7.41770e6 −1.83700 −0.918498 0.395426i \(-0.870597\pi\)
−0.918498 + 0.395426i \(0.870597\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.40269e6 −0.339589 −0.169794 0.985480i \(-0.554310\pi\)
−0.169794 + 0.985480i \(0.554310\pi\)
\(444\) 0 0
\(445\) 2.83259e6 0.678085
\(446\) 0 0
\(447\) 283164. 0.0670300
\(448\) 0 0
\(449\) −590574. −0.138248 −0.0691239 0.997608i \(-0.522020\pi\)
−0.0691239 + 0.997608i \(0.522020\pi\)
\(450\) 0 0
\(451\) −4.16718e6 −0.964720
\(452\) 0 0
\(453\) 992656. 0.227276
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2.90484e6 −0.650627 −0.325313 0.945606i \(-0.605470\pi\)
−0.325313 + 0.945606i \(0.605470\pi\)
\(458\) 0 0
\(459\) 6.99132e6 1.54891
\(460\) 0 0
\(461\) 922684. 0.202209 0.101105 0.994876i \(-0.467762\pi\)
0.101105 + 0.994876i \(0.467762\pi\)
\(462\) 0 0
\(463\) −7.18235e6 −1.55709 −0.778546 0.627588i \(-0.784042\pi\)
−0.778546 + 0.627588i \(0.784042\pi\)
\(464\) 0 0
\(465\) 5.99290e6 1.28530
\(466\) 0 0
\(467\) −612570. −0.129976 −0.0649881 0.997886i \(-0.520701\pi\)
−0.0649881 + 0.997886i \(0.520701\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 4.10934e6 0.853531
\(472\) 0 0
\(473\) 2.52416e6 0.518757
\(474\) 0 0
\(475\) −1078.00 −0.000219222 0
\(476\) 0 0
\(477\) −106314. −0.0213941
\(478\) 0 0
\(479\) 2.60330e6 0.518424 0.259212 0.965820i \(-0.416537\pi\)
0.259212 + 0.965820i \(0.416537\pi\)
\(480\) 0 0
\(481\) −1.45572e6 −0.286890
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.27634e6 0.632461
\(486\) 0 0
\(487\) −5.46309e6 −1.04380 −0.521898 0.853008i \(-0.674776\pi\)
−0.521898 + 0.853008i \(0.674776\pi\)
\(488\) 0 0
\(489\) 184688. 0.0349274
\(490\) 0 0
\(491\) −1.64090e6 −0.307170 −0.153585 0.988135i \(-0.549082\pi\)
−0.153585 + 0.988135i \(0.549082\pi\)
\(492\) 0 0
\(493\) 5.88580e6 1.09066
\(494\) 0 0
\(495\) 610624. 0.112011
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −2.99796e6 −0.538983 −0.269491 0.963003i \(-0.586856\pi\)
−0.269491 + 0.963003i \(0.586856\pi\)
\(500\) 0 0
\(501\) −6.91057e6 −1.23004
\(502\) 0 0
\(503\) −6.89405e6 −1.21494 −0.607469 0.794343i \(-0.707815\pi\)
−0.607469 + 0.794343i \(0.707815\pi\)
\(504\) 0 0
\(505\) −2.16698e6 −0.378117
\(506\) 0 0
\(507\) 4.92370e6 0.850691
\(508\) 0 0
\(509\) −2.30476e6 −0.394305 −0.197152 0.980373i \(-0.563169\pi\)
−0.197152 + 0.980373i \(0.563169\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −397880. −0.0667511
\(514\) 0 0
\(515\) 2.97136e6 0.493671
\(516\) 0 0
\(517\) −2.16317e6 −0.355929
\(518\) 0 0
\(519\) 3.37002e6 0.549180
\(520\) 0 0
\(521\) 1.20960e7 1.95231 0.976155 0.217073i \(-0.0696509\pi\)
0.976155 + 0.217073i \(0.0696509\pi\)
\(522\) 0 0
\(523\) 5.48443e6 0.876753 0.438377 0.898791i \(-0.355554\pi\)
0.438377 + 0.898791i \(0.355554\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.31630e7 −2.06456
\(528\) 0 0
\(529\) −3.10937e6 −0.483095
\(530\) 0 0
\(531\) 128310. 0.0197480
\(532\) 0 0
\(533\) 2.51468e6 0.383411
\(534\) 0 0
\(535\) 8.19414e6 1.23771
\(536\) 0 0
\(537\) 4.12905e6 0.617894
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −6.71799e6 −0.986839 −0.493420 0.869791i \(-0.664253\pi\)
−0.493420 + 0.869791i \(0.664253\pi\)
\(542\) 0 0
\(543\) −4.71772e6 −0.686646
\(544\) 0 0
\(545\) 5.20229e6 0.750245
\(546\) 0 0
\(547\) 5.00235e6 0.714835 0.357418 0.933945i \(-0.383657\pi\)
0.357418 + 0.933945i \(0.383657\pi\)
\(548\) 0 0
\(549\) 1.20546e6 0.170695
\(550\) 0 0
\(551\) −334964. −0.0470023
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 8.15203e6 1.12340
\(556\) 0 0
\(557\) 9.01961e6 1.23183 0.615913 0.787814i \(-0.288787\pi\)
0.615913 + 0.787814i \(0.288787\pi\)
\(558\) 0 0
\(559\) −1.52320e6 −0.206171
\(560\) 0 0
\(561\) 5.59306e6 0.750312
\(562\) 0 0
\(563\) 1.24051e7 1.64941 0.824707 0.565561i \(-0.191340\pi\)
0.824707 + 0.565561i \(0.191340\pi\)
\(564\) 0 0
\(565\) −4.66782e6 −0.615167
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.48804e6 0.840103 0.420052 0.907500i \(-0.362012\pi\)
0.420052 + 0.907500i \(0.362012\pi\)
\(570\) 0 0
\(571\) 1.02285e7 1.31287 0.656435 0.754382i \(-0.272064\pi\)
0.656435 + 0.754382i \(0.272064\pi\)
\(572\) 0 0
\(573\) 5.01570e6 0.638182
\(574\) 0 0
\(575\) −20064.0 −0.00253074
\(576\) 0 0
\(577\) −2.65338e6 −0.331787 −0.165894 0.986144i \(-0.553051\pi\)
−0.165894 + 0.986144i \(0.553051\pi\)
\(578\) 0 0
\(579\) 1.38538e7 1.71740
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −524784. −0.0639454
\(584\) 0 0
\(585\) −368480. −0.0445168
\(586\) 0 0
\(587\) −1.43044e7 −1.71346 −0.856729 0.515766i \(-0.827507\pi\)
−0.856729 + 0.515766i \(0.827507\pi\)
\(588\) 0 0
\(589\) 749112. 0.0889731
\(590\) 0 0
\(591\) 1.38607e7 1.63236
\(592\) 0 0
\(593\) 1.00265e7 1.17088 0.585442 0.810714i \(-0.300921\pi\)
0.585442 + 0.810714i \(0.300921\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.17706e7 1.35164
\(598\) 0 0
\(599\) 7.52292e6 0.856681 0.428341 0.903617i \(-0.359098\pi\)
0.428341 + 0.903617i \(0.359098\pi\)
\(600\) 0 0
\(601\) −3.38625e6 −0.382413 −0.191207 0.981550i \(-0.561240\pi\)
−0.191207 + 0.981550i \(0.561240\pi\)
\(602\) 0 0
\(603\) −2.27499e6 −0.254792
\(604\) 0 0
\(605\) −6.00471e6 −0.666966
\(606\) 0 0
\(607\) −6.90861e6 −0.761060 −0.380530 0.924769i \(-0.624258\pi\)
−0.380530 + 0.924769i \(0.624258\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.30536e6 0.141458
\(612\) 0 0
\(613\) −9.68896e6 −1.04142 −0.520710 0.853734i \(-0.674333\pi\)
−0.520710 + 0.853734i \(0.674333\pi\)
\(614\) 0 0
\(615\) −1.40822e7 −1.50135
\(616\) 0 0
\(617\) −7.84742e6 −0.829877 −0.414939 0.909849i \(-0.636197\pi\)
−0.414939 + 0.909849i \(0.636197\pi\)
\(618\) 0 0
\(619\) −1.01972e7 −1.06968 −0.534840 0.844953i \(-0.679628\pi\)
−0.534840 + 0.844953i \(0.679628\pi\)
\(620\) 0 0
\(621\) −7.40544e6 −0.770587
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −9.79988e6 −1.00351
\(626\) 0 0
\(627\) −318304. −0.0323350
\(628\) 0 0
\(629\) −1.79054e7 −1.80450
\(630\) 0 0
\(631\) 8.36258e6 0.836116 0.418058 0.908420i \(-0.362711\pi\)
0.418058 + 0.908420i \(0.362711\pi\)
\(632\) 0 0
\(633\) 1.61102e7 1.59806
\(634\) 0 0
\(635\) −3.38150e6 −0.332794
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −2.75232e6 −0.266653
\(640\) 0 0
\(641\) 1.10283e6 0.106014 0.0530070 0.998594i \(-0.483119\pi\)
0.0530070 + 0.998594i \(0.483119\pi\)
\(642\) 0 0
\(643\) 1.71354e7 1.63443 0.817217 0.576330i \(-0.195516\pi\)
0.817217 + 0.576330i \(0.195516\pi\)
\(644\) 0 0
\(645\) 8.52992e6 0.807320
\(646\) 0 0
\(647\) −54964.0 −0.00516200 −0.00258100 0.999997i \(-0.500822\pi\)
−0.00258100 + 0.999997i \(0.500822\pi\)
\(648\) 0 0
\(649\) 633360. 0.0590254
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −485166. −0.0445254 −0.0222627 0.999752i \(-0.507087\pi\)
−0.0222627 + 0.999752i \(0.507087\pi\)
\(654\) 0 0
\(655\) −3.44882e6 −0.314099
\(656\) 0 0
\(657\) 3.19985e6 0.289212
\(658\) 0 0
\(659\) 2.72136e6 0.244103 0.122051 0.992524i \(-0.461053\pi\)
0.122051 + 0.992524i \(0.461053\pi\)
\(660\) 0 0
\(661\) 2.14525e6 0.190974 0.0954869 0.995431i \(-0.469559\pi\)
0.0954869 + 0.995431i \(0.469559\pi\)
\(662\) 0 0
\(663\) −3.37512e6 −0.298198
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −6.23443e6 −0.542603
\(668\) 0 0
\(669\) 1.15397e7 0.996848
\(670\) 0 0
\(671\) 5.95034e6 0.510194
\(672\) 0 0
\(673\) 2.92796e6 0.249188 0.124594 0.992208i \(-0.460237\pi\)
0.124594 + 0.992208i \(0.460237\pi\)
\(674\) 0 0
\(675\) 44660.0 0.00377276
\(676\) 0 0
\(677\) 1.34992e7 1.13198 0.565988 0.824414i \(-0.308495\pi\)
0.565988 + 0.824414i \(0.308495\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −1.04135e6 −0.0860455
\(682\) 0 0
\(683\) 5.42972e6 0.445375 0.222688 0.974890i \(-0.428517\pi\)
0.222688 + 0.974890i \(0.428517\pi\)
\(684\) 0 0
\(685\) −1.14499e7 −0.932340
\(686\) 0 0
\(687\) 1.58474e7 1.28105
\(688\) 0 0
\(689\) 316680. 0.0254140
\(690\) 0 0
\(691\) 2.08280e7 1.65940 0.829702 0.558207i \(-0.188510\pi\)
0.829702 + 0.558207i \(0.188510\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.98274e6 −0.155705
\(696\) 0 0
\(697\) 3.09306e7 2.41160
\(698\) 0 0
\(699\) 2.78216e6 0.215372
\(700\) 0 0
\(701\) 2.35141e7 1.80731 0.903655 0.428261i \(-0.140874\pi\)
0.903655 + 0.428261i \(0.140874\pi\)
\(702\) 0 0
\(703\) 1.01900e6 0.0777656
\(704\) 0 0
\(705\) −7.31002e6 −0.553918
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.95747e7 −1.46244 −0.731221 0.682140i \(-0.761049\pi\)
−0.731221 + 0.682140i \(0.761049\pi\)
\(710\) 0 0
\(711\) 1.49385e6 0.110824
\(712\) 0 0
\(713\) 1.39427e7 1.02712
\(714\) 0 0
\(715\) −1.81888e6 −0.133057
\(716\) 0 0
\(717\) 6.76066e6 0.491124
\(718\) 0 0
\(719\) −2.61152e7 −1.88396 −0.941978 0.335674i \(-0.891036\pi\)
−0.941978 + 0.335674i \(0.891036\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1.12827e7 0.802729
\(724\) 0 0
\(725\) 37598.0 0.00265656
\(726\) 0 0
\(727\) 1.54126e7 1.08154 0.540768 0.841172i \(-0.318134\pi\)
0.540768 + 0.841172i \(0.318134\pi\)
\(728\) 0 0
\(729\) 1.59468e7 1.11136
\(730\) 0 0
\(731\) −1.87354e7 −1.29679
\(732\) 0 0
\(733\) 1.69868e7 1.16776 0.583878 0.811841i \(-0.301535\pi\)
0.583878 + 0.811841i \(0.301535\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.12297e7 −0.761554
\(738\) 0 0
\(739\) −2.01511e6 −0.135734 −0.0678669 0.997694i \(-0.521619\pi\)
−0.0678669 + 0.997694i \(0.521619\pi\)
\(740\) 0 0
\(741\) 192080. 0.0128510
\(742\) 0 0
\(743\) 1.51381e7 1.00600 0.503001 0.864286i \(-0.332229\pi\)
0.503001 + 0.864286i \(0.332229\pi\)
\(744\) 0 0
\(745\) −1.13266e6 −0.0747666
\(746\) 0 0
\(747\) 965286. 0.0632928
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −7.21401e6 −0.466742 −0.233371 0.972388i \(-0.574976\pi\)
−0.233371 + 0.972388i \(0.574976\pi\)
\(752\) 0 0
\(753\) −6.03033e6 −0.387573
\(754\) 0 0
\(755\) −3.97062e6 −0.253508
\(756\) 0 0
\(757\) −1.09697e7 −0.695755 −0.347877 0.937540i \(-0.613097\pi\)
−0.347877 + 0.937540i \(0.613097\pi\)
\(758\) 0 0
\(759\) −5.92435e6 −0.373281
\(760\) 0 0
\(761\) −1.92442e7 −1.20459 −0.602293 0.798275i \(-0.705746\pi\)
−0.602293 + 0.798275i \(0.705746\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −4.53230e6 −0.280005
\(766\) 0 0
\(767\) −382200. −0.0234586
\(768\) 0 0
\(769\) −8.21185e6 −0.500755 −0.250378 0.968148i \(-0.580555\pi\)
−0.250378 + 0.968148i \(0.580555\pi\)
\(770\) 0 0
\(771\) −1.64767e7 −0.998241
\(772\) 0 0
\(773\) −1.86187e7 −1.12073 −0.560363 0.828247i \(-0.689338\pi\)
−0.560363 + 0.828247i \(0.689338\pi\)
\(774\) 0 0
\(775\) −84084.0 −0.00502874
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.76028e6 −0.103929
\(780\) 0 0
\(781\) −1.35859e7 −0.797006
\(782\) 0 0
\(783\) 1.38771e7 0.808898
\(784\) 0 0
\(785\) −1.64373e7 −0.952045
\(786\) 0 0
\(787\) 2.62501e7 1.51075 0.755377 0.655291i \(-0.227454\pi\)
0.755377 + 0.655291i \(0.227454\pi\)
\(788\) 0 0
\(789\) 1.80737e7 1.03360
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −3.59072e6 −0.202768
\(794\) 0 0
\(795\) −1.77341e6 −0.0995155
\(796\) 0 0
\(797\) 1.00373e7 0.559720 0.279860 0.960041i \(-0.409712\pi\)
0.279860 + 0.960041i \(0.409712\pi\)
\(798\) 0 0
\(799\) 1.60559e7 0.889751
\(800\) 0 0
\(801\) −2.37735e6 −0.130922
\(802\) 0 0
\(803\) 1.57950e7 0.864433
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1.78858e7 −0.966772
\(808\) 0 0
\(809\) 1.40884e7 0.756816 0.378408 0.925639i \(-0.376472\pi\)
0.378408 + 0.925639i \(0.376472\pi\)
\(810\) 0 0
\(811\) 1.81433e7 0.968646 0.484323 0.874889i \(-0.339066\pi\)
0.484323 + 0.874889i \(0.339066\pi\)
\(812\) 0 0
\(813\) −2.31076e7 −1.22611
\(814\) 0 0
\(815\) −738752. −0.0389587
\(816\) 0 0
\(817\) 1.06624e6 0.0558856
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.13669e7 −1.10633 −0.553164 0.833072i \(-0.686580\pi\)
−0.553164 + 0.833072i \(0.686580\pi\)
\(822\) 0 0
\(823\) −1.78017e7 −0.916142 −0.458071 0.888916i \(-0.651459\pi\)
−0.458071 + 0.888916i \(0.651459\pi\)
\(824\) 0 0
\(825\) 35728.0 0.00182757
\(826\) 0 0
\(827\) −1.62921e7 −0.828350 −0.414175 0.910197i \(-0.635930\pi\)
−0.414175 + 0.910197i \(0.635930\pi\)
\(828\) 0 0
\(829\) 2.08499e6 0.105370 0.0526851 0.998611i \(-0.483222\pi\)
0.0526851 + 0.998611i \(0.483222\pi\)
\(830\) 0 0
\(831\) 1.48973e7 0.748348
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 2.76423e7 1.37201
\(836\) 0 0
\(837\) −3.10346e7 −1.53120
\(838\) 0 0
\(839\) −2.27850e7 −1.11749 −0.558745 0.829340i \(-0.688717\pi\)
−0.558745 + 0.829340i \(0.688717\pi\)
\(840\) 0 0
\(841\) −8.82842e6 −0.430421
\(842\) 0 0
\(843\) 312788. 0.0151594
\(844\) 0 0
\(845\) −1.96948e7 −0.948877
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 3.49403e7 1.66363
\(850\) 0 0
\(851\) 1.89660e7 0.897740
\(852\) 0 0
\(853\) 2.26975e7 1.06808 0.534042 0.845458i \(-0.320672\pi\)
0.534042 + 0.845458i \(0.320672\pi\)
\(854\) 0 0
\(855\) 257936. 0.0120669
\(856\) 0 0
\(857\) −2.52900e7 −1.17624 −0.588120 0.808774i \(-0.700132\pi\)
−0.588120 + 0.808774i \(0.700132\pi\)
\(858\) 0 0
\(859\) −1.03947e7 −0.480652 −0.240326 0.970692i \(-0.577254\pi\)
−0.240326 + 0.970692i \(0.577254\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −4.33399e7 −1.98089 −0.990447 0.137892i \(-0.955967\pi\)
−0.990447 + 0.137892i \(0.955967\pi\)
\(864\) 0 0
\(865\) −1.34801e7 −0.612566
\(866\) 0 0
\(867\) −2.16360e7 −0.977527
\(868\) 0 0
\(869\) 7.37389e6 0.331243
\(870\) 0 0
\(871\) 6.77656e6 0.302666
\(872\) 0 0
\(873\) −2.74978e6 −0.122113
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.71659e7 1.63172 0.815861 0.578248i \(-0.196264\pi\)
0.815861 + 0.578248i \(0.196264\pi\)
\(878\) 0 0
\(879\) −2.70449e7 −1.18063
\(880\) 0 0
\(881\) −9.04785e6 −0.392740 −0.196370 0.980530i \(-0.562915\pi\)
−0.196370 + 0.980530i \(0.562915\pi\)
\(882\) 0 0
\(883\) −3.29679e7 −1.42295 −0.711474 0.702712i \(-0.751972\pi\)
−0.711474 + 0.702712i \(0.751972\pi\)
\(884\) 0 0
\(885\) 2.14032e6 0.0918588
\(886\) 0 0
\(887\) −1.61099e7 −0.687517 −0.343758 0.939058i \(-0.611700\pi\)
−0.343758 + 0.939058i \(0.611700\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.05372e7 0.444663
\(892\) 0 0
\(893\) −913752. −0.0383442
\(894\) 0 0
\(895\) −1.65162e7 −0.689211
\(896\) 0 0
\(897\) 3.57504e6 0.148354
\(898\) 0 0
\(899\) −2.61272e7 −1.07819
\(900\) 0 0
\(901\) 3.89516e6 0.159850
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.88709e7 0.765898
\(906\) 0 0
\(907\) 4.47286e7 1.80537 0.902686 0.430300i \(-0.141592\pi\)
0.902686 + 0.430300i \(0.141592\pi\)
\(908\) 0 0
\(909\) 1.81871e6 0.0730053
\(910\) 0 0
\(911\) 6.60518e6 0.263687 0.131844 0.991271i \(-0.457910\pi\)
0.131844 + 0.991271i \(0.457910\pi\)
\(912\) 0 0
\(913\) 4.76482e6 0.189177
\(914\) 0 0
\(915\) 2.01080e7 0.793993
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 3.08930e7 1.20662 0.603311 0.797506i \(-0.293848\pi\)
0.603311 + 0.797506i \(0.293848\pi\)
\(920\) 0 0
\(921\) 6.42704e6 0.249667
\(922\) 0 0
\(923\) 8.19840e6 0.316756
\(924\) 0 0
\(925\) −114378. −0.00439530
\(926\) 0 0
\(927\) −2.49382e6 −0.0953160
\(928\) 0 0
\(929\) 4.87215e6 0.185217 0.0926087 0.995703i \(-0.470479\pi\)
0.0926087 + 0.995703i \(0.470479\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −9.33979e6 −0.351264
\(934\) 0 0
\(935\) −2.23722e7 −0.836913
\(936\) 0 0
\(937\) 3.25004e7 1.20932 0.604658 0.796485i \(-0.293310\pi\)
0.604658 + 0.796485i \(0.293310\pi\)
\(938\) 0 0
\(939\) −1.55448e6 −0.0575334
\(940\) 0 0
\(941\) 2.64040e6 0.0972066 0.0486033 0.998818i \(-0.484523\pi\)
0.0486033 + 0.998818i \(0.484523\pi\)
\(942\) 0 0
\(943\) −3.27627e7 −1.19978
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.08179e7 1.47903 0.739513 0.673142i \(-0.235056\pi\)
0.739513 + 0.673142i \(0.235056\pi\)
\(948\) 0 0
\(949\) −9.53148e6 −0.343554
\(950\) 0 0
\(951\) 962892. 0.0345244
\(952\) 0 0
\(953\) −6.71983e6 −0.239677 −0.119838 0.992793i \(-0.538238\pi\)
−0.119838 + 0.992793i \(0.538238\pi\)
\(954\) 0 0
\(955\) −2.00628e7 −0.711841
\(956\) 0 0
\(957\) 1.11017e7 0.391840
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2.98016e7 1.04095
\(962\) 0 0
\(963\) −6.87723e6 −0.238972
\(964\) 0 0
\(965\) −5.54150e7 −1.91562
\(966\) 0 0
\(967\) 2.78979e6 0.0959413 0.0479707 0.998849i \(-0.484725\pi\)
0.0479707 + 0.998849i \(0.484725\pi\)
\(968\) 0 0
\(969\) 2.36258e6 0.0808310
\(970\) 0 0
\(971\) 3.33594e7 1.13545 0.567727 0.823217i \(-0.307823\pi\)
0.567727 + 0.823217i \(0.307823\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −21560.0 −0.000726335 0
\(976\) 0 0
\(977\) −7.60033e6 −0.254739 −0.127370 0.991855i \(-0.540653\pi\)
−0.127370 + 0.991855i \(0.540653\pi\)
\(978\) 0 0
\(979\) −1.17350e7 −0.391316
\(980\) 0 0
\(981\) −4.36621e6 −0.144854
\(982\) 0 0
\(983\) −5.79760e6 −0.191366 −0.0956829 0.995412i \(-0.530503\pi\)
−0.0956829 + 0.995412i \(0.530503\pi\)
\(984\) 0 0
\(985\) −5.54428e7 −1.82077
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.98451e7 0.645153
\(990\) 0 0
\(991\) −1.26825e7 −0.410224 −0.205112 0.978739i \(-0.565756\pi\)
−0.205112 + 0.978739i \(0.565756\pi\)
\(992\) 0 0
\(993\) −7.90227e6 −0.254319
\(994\) 0 0
\(995\) −4.70823e7 −1.50765
\(996\) 0 0
\(997\) −1.44400e7 −0.460077 −0.230039 0.973182i \(-0.573885\pi\)
−0.230039 + 0.973182i \(0.573885\pi\)
\(998\) 0 0
\(999\) −4.22159e7 −1.33833
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 784.6.a.c.1.1 1
4.3 odd 2 49.6.a.a.1.1 1
7.6 odd 2 112.6.a.g.1.1 1
12.11 even 2 441.6.a.k.1.1 1
21.20 even 2 1008.6.a.y.1.1 1
28.3 even 6 49.6.c.c.30.1 2
28.11 odd 6 49.6.c.b.30.1 2
28.19 even 6 49.6.c.c.18.1 2
28.23 odd 6 49.6.c.b.18.1 2
28.27 even 2 7.6.a.a.1.1 1
56.13 odd 2 448.6.a.c.1.1 1
56.27 even 2 448.6.a.m.1.1 1
84.83 odd 2 63.6.a.e.1.1 1
140.27 odd 4 175.6.b.a.99.1 2
140.83 odd 4 175.6.b.a.99.2 2
140.139 even 2 175.6.a.b.1.1 1
308.307 odd 2 847.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 1 28.27 even 2
49.6.a.a.1.1 1 4.3 odd 2
49.6.c.b.18.1 2 28.23 odd 6
49.6.c.b.30.1 2 28.11 odd 6
49.6.c.c.18.1 2 28.19 even 6
49.6.c.c.30.1 2 28.3 even 6
63.6.a.e.1.1 1 84.83 odd 2
112.6.a.g.1.1 1 7.6 odd 2
175.6.a.b.1.1 1 140.139 even 2
175.6.b.a.99.1 2 140.27 odd 4
175.6.b.a.99.2 2 140.83 odd 4
441.6.a.k.1.1 1 12.11 even 2
448.6.a.c.1.1 1 56.13 odd 2
448.6.a.m.1.1 1 56.27 even 2
784.6.a.c.1.1 1 1.1 even 1 trivial
847.6.a.b.1.1 1 308.307 odd 2
1008.6.a.y.1.1 1 21.20 even 2