Properties

Label 200.6.c.d.49.2
Level $200$
Weight $6$
Character 200.49
Analytic conductor $32.077$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [200,6,Mod(49,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 200.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(32.0767639626\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 200.49
Dual form 200.6.c.d.49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{3} -62.0000i q^{7} +239.000 q^{9} +O(q^{10})\) \(q+2.00000i q^{3} -62.0000i q^{7} +239.000 q^{9} -144.000 q^{11} +654.000i q^{13} -1190.00i q^{17} -556.000 q^{19} +124.000 q^{21} -2182.00i q^{23} +964.000i q^{27} +1578.00 q^{29} +9660.00 q^{31} -288.000i q^{33} -3534.00i q^{37} -1308.00 q^{39} +7462.00 q^{41} +7114.00i q^{43} -28294.0i q^{47} +12963.0 q^{49} +2380.00 q^{51} +13046.0i q^{53} -1112.00i q^{57} +37092.0 q^{59} +39570.0 q^{61} -14818.0i q^{63} -56734.0i q^{67} +4364.00 q^{69} +45588.0 q^{71} -11842.0i q^{73} +8928.00i q^{77} -94216.0 q^{79} +56149.0 q^{81} +31482.0i q^{83} +3156.00i q^{87} +94054.0 q^{89} +40548.0 q^{91} +19320.0i q^{93} +23714.0i q^{97} -34416.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 478 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 478 q^{9} - 288 q^{11} - 1112 q^{19} + 248 q^{21} + 3156 q^{29} + 19320 q^{31} - 2616 q^{39} + 14924 q^{41} + 25926 q^{49} + 4760 q^{51} + 74184 q^{59} + 79140 q^{61} + 8728 q^{69} + 91176 q^{71} - 188432 q^{79} + 112298 q^{81} + 188108 q^{89} + 81096 q^{91} - 68832 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.00000i 0.128300i 0.997940 + 0.0641500i \(0.0204336\pi\)
−0.997940 + 0.0641500i \(0.979566\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 62.0000i − 0.478241i −0.970990 0.239120i \(-0.923141\pi\)
0.970990 0.239120i \(-0.0768591\pi\)
\(8\) 0 0
\(9\) 239.000 0.983539
\(10\) 0 0
\(11\) −144.000 −0.358823 −0.179412 0.983774i \(-0.557419\pi\)
−0.179412 + 0.983774i \(0.557419\pi\)
\(12\) 0 0
\(13\) 654.000i 1.07330i 0.843806 + 0.536648i \(0.180310\pi\)
−0.843806 + 0.536648i \(0.819690\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1190.00i − 0.998676i −0.866407 0.499338i \(-0.833577\pi\)
0.866407 0.499338i \(-0.166423\pi\)
\(18\) 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
\(23\) − 2182.00i − 0.860073i −0.902812 0.430036i \(-0.858501\pi\)
0.902812 0.430036i \(-0.141499\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 964.000i 0.254488i
\(28\) 0 0
\(29\) 1578.00 0.348427 0.174214 0.984708i \(-0.444262\pi\)
0.174214 + 0.984708i \(0.444262\pi\)
\(30\) 0 0
\(31\) 9660.00 1.80540 0.902699 0.430273i \(-0.141583\pi\)
0.902699 + 0.430273i \(0.141583\pi\)
\(32\) 0 0
\(33\) − 288.000i − 0.0460371i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 3534.00i − 0.424387i −0.977228 0.212194i \(-0.931939\pi\)
0.977228 0.212194i \(-0.0680607\pi\)
\(38\) 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
\(43\) 7114.00i 0.586736i 0.956000 + 0.293368i \(0.0947761\pi\)
−0.956000 + 0.293368i \(0.905224\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 28294.0i − 1.86831i −0.356863 0.934157i \(-0.616154\pi\)
0.356863 0.934157i \(-0.383846\pi\)
\(48\) 0 0
\(49\) 12963.0 0.771286
\(50\) 0 0
\(51\) 2380.00 0.128130
\(52\) 0 0
\(53\) 13046.0i 0.637952i 0.947763 + 0.318976i \(0.103339\pi\)
−0.947763 + 0.318976i \(0.896661\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1112.00i − 0.0453333i
\(58\) 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
\(63\) − 14818.0i − 0.470368i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 56734.0i − 1.54403i −0.635603 0.772016i \(-0.719248\pi\)
0.635603 0.772016i \(-0.280752\pi\)
\(68\) 0 0
\(69\) 4364.00 0.110347
\(70\) 0 0
\(71\) 45588.0 1.07326 0.536630 0.843818i \(-0.319697\pi\)
0.536630 + 0.843818i \(0.319697\pi\)
\(72\) 0 0
\(73\) − 11842.0i − 0.260087i −0.991508 0.130043i \(-0.958488\pi\)
0.991508 0.130043i \(-0.0415116\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8928.00i 0.171604i
\(78\) 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
\(83\) 31482.0i 0.501611i 0.968037 + 0.250806i \(0.0806955\pi\)
−0.968037 + 0.250806i \(0.919305\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3156.00i 0.0447032i
\(88\) 0 0
\(89\) 94054.0 1.25864 0.629321 0.777145i \(-0.283333\pi\)
0.629321 + 0.777145i \(0.283333\pi\)
\(90\) 0 0
\(91\) 40548.0 0.513294
\(92\) 0 0
\(93\) 19320.0i 0.231633i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 23714.0i 0.255903i 0.991780 + 0.127952i \(0.0408402\pi\)
−0.991780 + 0.127952i \(0.959160\pi\)
\(98\) 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.i − 1.27098i −0.772109 0.635490i \(-0.780798\pi\)
0.772109 0.635490i \(-0.219202\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 193190.i − 1.63127i −0.578569 0.815634i \(-0.696388\pi\)
0.578569 0.815634i \(-0.303612\pi\)
\(108\) 0 0
\(109\) 120046. 0.967791 0.483895 0.875126i \(-0.339222\pi\)
0.483895 + 0.875126i \(0.339222\pi\)
\(110\) 0 0
\(111\) 7068.00 0.0544489
\(112\) 0 0
\(113\) 152646.i 1.12458i 0.826941 + 0.562289i \(0.190079\pi\)
−0.826941 + 0.562289i \(0.809921\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 156306.i 1.05563i
\(118\) 0 0
\(119\) −73780.0 −0.477608
\(120\) 0 0
\(121\) −140315. −0.871246
\(122\) 0 0
\(123\) 14924.0i 0.0889452i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 107906.i 0.593658i 0.954931 + 0.296829i \(0.0959291\pi\)
−0.954931 + 0.296829i \(0.904071\pi\)
\(128\) 0 0
\(129\) −14228.0 −0.0752783
\(130\) 0 0
\(131\) −233072. −1.18662 −0.593310 0.804974i \(-0.702179\pi\)
−0.593310 + 0.804974i \(0.702179\pi\)
\(132\) 0 0
\(133\) 34472.0i 0.168981i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 356082.i 1.62087i 0.585827 + 0.810436i \(0.300770\pi\)
−0.585827 + 0.810436i \(0.699230\pi\)
\(138\) 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
\(143\) − 94176.0i − 0.385124i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 25926.0i 0.0989560i
\(148\) 0 0
\(149\) −27498.0 −0.101469 −0.0507347 0.998712i \(-0.516156\pi\)
−0.0507347 + 0.998712i \(0.516156\pi\)
\(150\) 0 0
\(151\) −136908. −0.488637 −0.244319 0.969695i \(-0.578564\pi\)
−0.244319 + 0.969695i \(0.578564\pi\)
\(152\) 0 0
\(153\) − 284410.i − 0.982237i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 406714.i 1.31686i 0.752641 + 0.658431i \(0.228779\pi\)
−0.752641 + 0.658431i \(0.771221\pi\)
\(158\) 0 0
\(159\) −26092.0 −0.0818492
\(160\) 0 0
\(161\) −135284. −0.411322
\(162\) 0 0
\(163\) 13642.0i 0.0402169i 0.999798 + 0.0201085i \(0.00640116\pi\)
−0.999798 + 0.0201085i \(0.993599\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 203438.i − 0.564470i −0.959345 0.282235i \(-0.908924\pi\)
0.959345 0.282235i \(-0.0910758\pi\)
\(168\) 0 0
\(169\) −56423.0 −0.151964
\(170\) 0 0
\(171\) −132884. −0.347522
\(172\) 0 0
\(173\) − 127242.i − 0.323233i −0.986854 0.161616i \(-0.948329\pi\)
0.986854 0.161616i \(-0.0516707\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 74184.0i 0.177982i
\(178\) 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
\(183\) 79140.0i 0.174690i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 171360.i 0.358348i
\(188\) 0 0
\(189\) 59768.0 0.121707
\(190\) 0 0
\(191\) −412300. −0.817768 −0.408884 0.912586i \(-0.634082\pi\)
−0.408884 + 0.912586i \(0.634082\pi\)
\(192\) 0 0
\(193\) 771654.i 1.49118i 0.666406 + 0.745589i \(0.267832\pi\)
−0.666406 + 0.745589i \(0.732168\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 190238.i − 0.349246i −0.984635 0.174623i \(-0.944129\pi\)
0.984635 0.174623i \(-0.0558707\pi\)
\(198\) 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
\(203\) − 97836.0i − 0.166632i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 521498.i − 0.845915i
\(208\) 0 0
\(209\) 80064.0 0.126786
\(210\) 0 0
\(211\) 928704. 1.43606 0.718028 0.696015i \(-0.245045\pi\)
0.718028 + 0.696015i \(0.245045\pi\)
\(212\) 0 0
\(213\) 91176.0i 0.137699i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 598920.i − 0.863415i
\(218\) 0 0
\(219\) 23684.0 0.0333691
\(220\) 0 0
\(221\) 778260. 1.07187
\(222\) 0 0
\(223\) − 421494.i − 0.567583i −0.958886 0.283791i \(-0.908408\pi\)
0.958886 0.283791i \(-0.0915923\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 991962.i 1.27770i 0.769329 + 0.638852i \(0.220590\pi\)
−0.769329 + 0.638852i \(0.779410\pi\)
\(228\) 0 0
\(229\) 266946. 0.336384 0.168192 0.985754i \(-0.446207\pi\)
0.168192 + 0.985754i \(0.446207\pi\)
\(230\) 0 0
\(231\) −17856.0 −0.0220168
\(232\) 0 0
\(233\) − 960314.i − 1.15884i −0.815029 0.579420i \(-0.803279\pi\)
0.815029 0.579420i \(-0.196721\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 188432.i − 0.217913i
\(238\) 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.i 0.376487i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 363624.i − 0.379237i
\(248\) 0 0
\(249\) −62964.0 −0.0643567
\(250\) 0 0
\(251\) 1.96792e6 1.97162 0.985810 0.167866i \(-0.0536876\pi\)
0.985810 + 0.167866i \(0.0536876\pi\)
\(252\) 0 0
\(253\) 314208.i 0.308614i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 971910.i − 0.917896i −0.888463 0.458948i \(-0.848227\pi\)
0.888463 0.458948i \(-0.151773\pi\)
\(258\) 0 0
\(259\) −219108. −0.202959
\(260\) 0 0
\(261\) 377142. 0.342692
\(262\) 0 0
\(263\) 154770.i 0.137974i 0.997618 + 0.0689870i \(0.0219767\pi\)
−0.997618 + 0.0689870i \(0.978023\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 188108.i 0.161484i
\(268\) 0 0
\(269\) −1.02371e6 −0.862577 −0.431289 0.902214i \(-0.641941\pi\)
−0.431289 + 0.902214i \(0.641941\pi\)
\(270\) 0 0
\(271\) −1.14776e6 −0.949350 −0.474675 0.880161i \(-0.657434\pi\)
−0.474675 + 0.880161i \(0.657434\pi\)
\(272\) 0 0
\(273\) 81096.0i 0.0658556i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.49676e6i − 1.95514i −0.210619 0.977568i \(-0.567548\pi\)
0.210619 0.977568i \(-0.432452\pi\)
\(278\) 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
\(283\) 2.12395e6i 1.57645i 0.615390 + 0.788223i \(0.288999\pi\)
−0.615390 + 0.788223i \(0.711001\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 462644.i − 0.331545i
\(288\) 0 0
\(289\) 3757.00 0.00264604
\(290\) 0 0
\(291\) −47428.0 −0.0328324
\(292\) 0 0
\(293\) − 992722.i − 0.675552i −0.941227 0.337776i \(-0.890325\pi\)
0.941227 0.337776i \(-0.109675\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 138816.i − 0.0913163i
\(298\) 0 0
\(299\) 1.42703e6 0.923112
\(300\) 0 0
\(301\) 441068. 0.280601
\(302\) 0 0
\(303\) − 259348.i − 0.162284i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 487522.i 0.295222i 0.989046 + 0.147611i \(0.0471583\pi\)
−0.989046 + 0.147611i \(0.952842\pi\)
\(308\) 0 0
\(309\) 273692. 0.163067
\(310\) 0 0
\(311\) −444116. −0.260373 −0.130186 0.991490i \(-0.541558\pi\)
−0.130186 + 0.991490i \(0.541558\pi\)
\(312\) 0 0
\(313\) − 47242.0i − 0.0272563i −0.999907 0.0136282i \(-0.995662\pi\)
0.999907 0.0136282i \(-0.00433811\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 694058.i 0.387925i 0.981009 + 0.193962i \(0.0621340\pi\)
−0.981009 + 0.193962i \(0.937866\pi\)
\(318\) 0 0
\(319\) −227232. −0.125024
\(320\) 0 0
\(321\) 386380. 0.209292
\(322\) 0 0
\(323\) 661640.i 0.352871i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 240092.i 0.124168i
\(328\) 0 0
\(329\) −1.75423e6 −0.893504
\(330\) 0 0
\(331\) 82168.0 0.0412223 0.0206112 0.999788i \(-0.493439\pi\)
0.0206112 + 0.999788i \(0.493439\pi\)
\(332\) 0 0
\(333\) − 844626.i − 0.417401i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 727934.i − 0.349154i −0.984644 0.174577i \(-0.944144\pi\)
0.984644 0.174577i \(-0.0558558\pi\)
\(338\) 0 0
\(339\) −305292. −0.144283
\(340\) 0 0
\(341\) −1.39104e6 −0.647819
\(342\) 0 0
\(343\) − 1.84574e6i − 0.847101i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.02298e6i 0.901919i 0.892544 + 0.450959i \(0.148918\pi\)
−0.892544 + 0.450959i \(0.851082\pi\)
\(348\) 0 0
\(349\) −4.40858e6 −1.93747 −0.968736 0.248095i \(-0.920196\pi\)
−0.968736 + 0.248095i \(0.920196\pi\)
\(350\) 0 0
\(351\) −630456. −0.273141
\(352\) 0 0
\(353\) − 1.06965e6i − 0.456883i −0.973558 0.228441i \(-0.926637\pi\)
0.973558 0.228441i \(-0.0733630\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 147560.i − 0.0612771i
\(358\) 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.i − 0.111781i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.64081e6i 1.41102i 0.708700 + 0.705509i \(0.249282\pi\)
−0.708700 + 0.705509i \(0.750718\pi\)
\(368\) 0 0
\(369\) 1.78342e6 0.681847
\(370\) 0 0
\(371\) 808852. 0.305094
\(372\) 0 0
\(373\) 3.17311e6i 1.18090i 0.807074 + 0.590450i \(0.201050\pi\)
−0.807074 + 0.590450i \(0.798950\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.03201e6i 0.373965i
\(378\) 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
\(383\) − 1.98925e6i − 0.692936i −0.938062 0.346468i \(-0.887381\pi\)
0.938062 0.346468i \(-0.112619\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.70025e6i 0.577078i
\(388\) 0 0
\(389\) 5.16495e6 1.73058 0.865291 0.501270i \(-0.167134\pi\)
0.865291 + 0.501270i \(0.167134\pi\)
\(390\) 0 0
\(391\) −2.59658e6 −0.858934
\(392\) 0 0
\(393\) − 466144.i − 0.152243i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 937586.i 0.298562i 0.988795 + 0.149281i \(0.0476959\pi\)
−0.988795 + 0.149281i \(0.952304\pi\)
\(398\) 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
\(403\) 6.31764e6i 1.93773i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 508896.i 0.152280i
\(408\) 0 0
\(409\) −4.06137e6 −1.20051 −0.600254 0.799810i \(-0.704934\pi\)
−0.600254 + 0.799810i \(0.704934\pi\)
\(410\) 0 0
\(411\) −712164. −0.207958
\(412\) 0 0
\(413\) − 2.29970e6i − 0.663433i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 624408.i − 0.175844i
\(418\) 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
\(423\) − 6.76227e6i − 1.83756i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 2.45334e6i − 0.651161i
\(428\) 0 0
\(429\) 188352. 0.0494114
\(430\) 0 0
\(431\) −4.27900e6 −1.10956 −0.554778 0.831998i \(-0.687197\pi\)
−0.554778 + 0.831998i \(0.687197\pi\)
\(432\) 0 0
\(433\) 3.20195e6i 0.820720i 0.911924 + 0.410360i \(0.134597\pi\)
−0.911924 + 0.410360i \(0.865403\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.21319e6i 0.303897i
\(438\) 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
\(443\) 5.43551e6i 1.31593i 0.753050 + 0.657963i \(0.228582\pi\)
−0.753050 + 0.657963i \(0.771418\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 54996.0i − 0.0130185i
\(448\) 0 0
\(449\) 2.99007e6 0.699948 0.349974 0.936759i \(-0.386190\pi\)
0.349974 + 0.936759i \(0.386190\pi\)
\(450\) 0 0
\(451\) −1.07453e6 −0.248758
\(452\) 0 0
\(453\) − 273816.i − 0.0626922i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.01759e6i 1.79578i 0.440218 + 0.897891i \(0.354901\pi\)
−0.440218 + 0.897891i \(0.645099\pi\)
\(458\) 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
\(463\) − 6.14261e6i − 1.33168i −0.746094 0.665840i \(-0.768073\pi\)
0.746094 0.665840i \(-0.231927\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 1.59270e6i − 0.337942i −0.985621 0.168971i \(-0.945956\pi\)
0.985621 0.168971i \(-0.0540444\pi\)
\(468\) 0 0
\(469\) −3.51751e6 −0.738419
\(470\) 0 0
\(471\) −813428. −0.168953
\(472\) 0 0
\(473\) − 1.02442e6i − 0.210535i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.11799e6i 0.627450i
\(478\) 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.i − 0.0527726i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 8.20714e6i − 1.56808i −0.620707 0.784042i \(-0.713154\pi\)
0.620707 0.784042i \(-0.286846\pi\)
\(488\) 0 0
\(489\) −27284.0 −0.00515984
\(490\) 0 0
\(491\) 8.93394e6 1.67240 0.836198 0.548428i \(-0.184773\pi\)
0.836198 + 0.548428i \(0.184773\pi\)
\(492\) 0 0
\(493\) − 1.87782e6i − 0.347966i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 2.82646e6i − 0.513276i
\(498\) 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
\(503\) − 3.68177e6i − 0.648839i −0.945913 0.324420i \(-0.894831\pi\)
0.945913 0.324420i \(-0.105169\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 112846.i − 0.0194969i
\(508\) 0 0
\(509\) 6.73483e6 1.15221 0.576105 0.817375i \(-0.304572\pi\)
0.576105 + 0.817375i \(0.304572\pi\)
\(510\) 0 0
\(511\) −734204. −0.124384
\(512\) 0 0
\(513\) − 535984.i − 0.0899204i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 4.07434e6i 0.670395i
\(518\) 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
\(523\) 1.17300e7i 1.87518i 0.347744 + 0.937589i \(0.386948\pi\)
−0.347744 + 0.937589i \(0.613052\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 1.14954e7i − 1.80301i
\(528\) 0 0
\(529\) 1.67522e6 0.260275
\(530\) 0 0
\(531\) 8.86499e6 1.36440
\(532\) 0 0
\(533\) 4.88015e6i 0.744072i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 189368.i 0.0283381i
\(538\) 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
\(543\) − 1.03404e6i − 0.150500i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 3.24801e6i − 0.464139i −0.972699 0.232070i \(-0.925450\pi\)
0.972699 0.232070i \(-0.0745498\pi\)
\(548\) 0 0
\(549\) 9.45723e6 1.33916
\(550\) 0 0
\(551\) −877368. −0.123113
\(552\) 0 0
\(553\) 5.84139e6i 0.812276i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 9.94446e6i − 1.35814i −0.734075 0.679068i \(-0.762384\pi\)
0.734075 0.679068i \(-0.237616\pi\)
\(558\) 0 0
\(559\) −4.65256e6 −0.629741
\(560\) 0 0
\(561\) −342720. −0.0459761
\(562\) 0 0
\(563\) − 3.89374e6i − 0.517721i −0.965915 0.258861i \(-0.916653\pi\)
0.965915 0.258861i \(-0.0833471\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 3.48124e6i − 0.454754i
\(568\) 0 0
\(569\) 1.11951e7 1.44960 0.724801 0.688958i \(-0.241932\pi\)
0.724801 + 0.688958i \(0.241932\pi\)
\(570\) 0 0
\(571\) −844040. −0.108336 −0.0541680 0.998532i \(-0.517251\pi\)
−0.0541680 + 0.998532i \(0.517251\pi\)
\(572\) 0 0
\(573\) − 824600.i − 0.104920i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 5.13378e6i − 0.641945i −0.947088 0.320973i \(-0.895990\pi\)
0.947088 0.320973i \(-0.104010\pi\)
\(578\) 0 0
\(579\) −1.54331e6 −0.191318
\(580\) 0 0
\(581\) 1.95188e6 0.239891
\(582\) 0 0
\(583\) − 1.87862e6i − 0.228912i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.76156e6i 1.16929i 0.811287 + 0.584647i \(0.198767\pi\)
−0.811287 + 0.584647i \(0.801233\pi\)
\(588\) 0 0
\(589\) −5.37096e6 −0.637916
\(590\) 0 0
\(591\) 380476. 0.0448083
\(592\) 0 0
\(593\) − 966226.i − 0.112835i −0.998407 0.0564173i \(-0.982032\pi\)
0.998407 0.0564173i \(-0.0179677\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 264144.i − 0.0303323i
\(598\) 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
\(603\) − 1.35594e7i − 1.51862i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 9.70767e6i − 1.06941i −0.845040 0.534704i \(-0.820423\pi\)
0.845040 0.534704i \(-0.179577\pi\)
\(608\) 0 0
\(609\) 195672. 0.0213789
\(610\) 0 0
\(611\) 1.85043e7 2.00525
\(612\) 0 0
\(613\) 1.10568e7i 1.18844i 0.804304 + 0.594219i \(0.202539\pi\)
−0.804304 + 0.594219i \(0.797461\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.31174e6i 0.878980i 0.898248 + 0.439490i \(0.144841\pi\)
−0.898248 + 0.439490i \(0.855159\pi\)
\(618\) 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
\(623\) − 5.83135e6i − 0.601934i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 160128.i 0.0162667i
\(628\) 0 0
\(629\) −4.20546e6 −0.423825
\(630\) 0 0
\(631\) −8.20262e6 −0.820123 −0.410062 0.912058i \(-0.634493\pi\)
−0.410062 + 0.912058i \(0.634493\pi\)
\(632\) 0 0
\(633\) 1.85741e6i 0.184246i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 8.47780e6i 0.827818i
\(638\) 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
\(643\) 1.33896e7i 1.27715i 0.769561 + 0.638573i \(0.220475\pi\)
−0.769561 + 0.638573i \(0.779525\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.48254e6i 0.608814i 0.952542 + 0.304407i \(0.0984582\pi\)
−0.952542 + 0.304407i \(0.901542\pi\)
\(648\) 0 0
\(649\) −5.34125e6 −0.497773
\(650\) 0 0
\(651\) 1.19784e6 0.110776
\(652\) 0 0
\(653\) − 1.44907e7i − 1.32986i −0.746904 0.664931i \(-0.768461\pi\)
0.746904 0.664931i \(-0.231539\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 2.83024e6i − 0.255805i
\(658\) 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
\(663\) 1.55652e6i 0.137522i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 3.44320e6i − 0.299673i
\(668\) 0 0
\(669\) 842988. 0.0728209
\(670\) 0 0
\(671\) −5.69808e6 −0.488565
\(672\) 0 0
\(673\) − 3.86655e6i − 0.329068i −0.986371 0.164534i \(-0.947388\pi\)
0.986371 0.164534i \(-0.0526120\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.23856e6i 0.103859i 0.998651 + 0.0519297i \(0.0165372\pi\)
−0.998651 + 0.0519297i \(0.983463\pi\)
\(678\) 0 0
\(679\) 1.47027e6 0.122383
\(680\) 0 0
\(681\) −1.98392e6 −0.163930
\(682\) 0 0
\(683\) 1.31376e7i 1.07762i 0.842427 + 0.538810i \(0.181126\pi\)
−0.842427 + 0.538810i \(0.818874\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 533892.i 0.0431580i
\(688\) 0 0
\(689\) −8.53208e6 −0.684711
\(690\) 0 0
\(691\) −1.23841e7 −0.986664 −0.493332 0.869841i \(-0.664221\pi\)
−0.493332 + 0.869841i \(0.664221\pi\)
\(692\) 0 0
\(693\) 2.13379e6i 0.168779i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 8.87978e6i − 0.692341i
\(698\) 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
\(703\) 1.96490e6i 0.149952i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.03979e6i 0.604917i
\(708\) 0 0
\(709\) −1.22257e7 −0.913397 −0.456699 0.889622i \(-0.650968\pi\)
−0.456699 + 0.889622i \(0.650968\pi\)
\(710\) 0 0
\(711\) −2.25176e7 −1.67051
\(712\) 0 0
\(713\) − 2.10781e7i − 1.55277i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 985392.i 0.0715832i
\(718\) 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.i 0.00797952i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.17271e7i 0.822916i 0.911429 + 0.411458i \(0.134980\pi\)
−0.911429 + 0.411458i \(0.865020\pi\)
\(728\) 0 0
\(729\) 1.29511e7 0.902585
\(730\) 0 0
\(731\) 8.46566e6 0.585959
\(732\) 0 0
\(733\) 1.16512e7i 0.800960i 0.916305 + 0.400480i \(0.131157\pi\)
−0.916305 + 0.400480i \(0.868843\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.16970e6i 0.554035i
\(738\) 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.i 0.0131162i 0.999978 + 0.00655812i \(0.00208753\pi\)
−0.999978 + 0.00655812i \(0.997912\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 7.52420e6i 0.493354i
\(748\) 0 0
\(749\) −1.19778e7 −0.780139
\(750\) 0 0
\(751\) −1.33282e7 −0.862326 −0.431163 0.902274i \(-0.641897\pi\)
−0.431163 + 0.902274i \(0.641897\pi\)
\(752\) 0 0
\(753\) 3.93584e6i 0.252959i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 3.86122e6i − 0.244898i −0.992475 0.122449i \(-0.960925\pi\)
0.992475 0.122449i \(-0.0390748\pi\)
\(758\) 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
\(763\) − 7.44285e6i − 0.462837i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.42582e7i 1.48891i
\(768\) 0 0
\(769\) −2.76358e7 −1.68522 −0.842609 0.538527i \(-0.818981\pi\)
−0.842609 + 0.538527i \(0.818981\pi\)
\(770\) 0 0
\(771\) 1.94382e6 0.117766
\(772\) 0 0
\(773\) − 1.78842e7i − 1.07652i −0.842780 0.538259i \(-0.819082\pi\)
0.842780 0.538259i \(-0.180918\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 438216.i − 0.0260397i
\(778\) 0 0
\(779\) −4.14887e6 −0.244955
\(780\) 0 0
\(781\) −6.56467e6 −0.385111
\(782\) 0 0
\(783\) 1.52119e6i 0.0886706i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 2.15691e7i 1.24135i 0.784067 + 0.620676i \(0.213142\pi\)
−0.784067 + 0.620676i \(0.786858\pi\)
\(788\) 0 0
\(789\) −309540. −0.0177021
\(790\) 0 0
\(791\) 9.46405e6 0.537819
\(792\) 0 0
\(793\) 2.58788e7i 1.46137i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1.03060e7i − 0.574705i −0.957825 0.287353i \(-0.907225\pi\)
0.957825 0.287353i \(-0.0927752\pi\)
\(798\) 0 0
\(799\) −3.36699e7 −1.86584
\(800\) 0 0
\(801\) 2.24789e7 1.23792
\(802\) 0 0
\(803\) 1.70525e6i 0.0933251i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 2.04743e6i − 0.110669i
\(808\) 0 0
\(809\) −372378. −0.0200038 −0.0100019 0.999950i \(-0.503184\pi\)
−0.0100019 + 0.999950i \(0.503184\pi\)
\(810\) 0 0
\(811\) −1.94795e7 −1.03998 −0.519990 0.854173i \(-0.674064\pi\)
−0.519990 + 0.854173i \(0.674064\pi\)
\(812\) 0 0
\(813\) − 2.29551e6i − 0.121802i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 3.95538e6i − 0.207316i
\(818\) 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
\(823\) − 1.78622e7i − 0.919253i −0.888112 0.459626i \(-0.847983\pi\)
0.888112 0.459626i \(-0.152017\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.42560e6i 0.479231i 0.970868 + 0.239616i \(0.0770215\pi\)
−0.970868 + 0.239616i \(0.922979\pi\)
\(828\) 0 0
\(829\) 1.48622e7 0.751098 0.375549 0.926803i \(-0.377454\pi\)
0.375549 + 0.926803i \(0.377454\pi\)
\(830\) 0 0
\(831\) 4.99352e6 0.250844
\(832\) 0 0
\(833\) − 1.54260e7i − 0.770265i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 9.31224e6i 0.459452i
\(838\) 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
\(843\) 3.39080e6i 0.164336i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 8.69953e6i 0.416665i
\(848\) 0 0
\(849\) −4.24791e6 −0.202258
\(850\) 0 0
\(851\) −7.71119e6 −0.365004
\(852\) 0 0
\(853\) − 1.62685e7i − 0.765552i −0.923841 0.382776i \(-0.874968\pi\)
0.923841 0.382776i \(-0.125032\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 2.92667e7i − 1.36120i −0.732656 0.680600i \(-0.761719\pi\)
0.732656 0.680600i \(-0.238281\pi\)
\(858\) 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
\(863\) 1.58052e7i 0.722391i 0.932490 + 0.361196i \(0.117631\pi\)
−0.932490 + 0.361196i \(0.882369\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 7514.00i 0 0.000339487i
\(868\) 0 0
\(869\) 1.35671e7 0.609449
\(870\) 0 0
\(871\) 3.71040e7 1.65720
\(872\) 0 0
\(873\) 5.66765e6i 0.251691i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 4.26834e7i − 1.87396i −0.349384 0.936980i \(-0.613609\pi\)
0.349384 0.936980i \(-0.386391\pi\)
\(878\) 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
\(883\) 1.68471e7i 0.727149i 0.931565 + 0.363574i \(0.118444\pi\)
−0.931565 + 0.363574i \(0.881556\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 8.36792e6i − 0.357115i −0.983929 0.178558i \(-0.942857\pi\)
0.983929 0.178558i \(-0.0571431\pi\)
\(888\) 0 0
\(889\) 6.69017e6 0.283911
\(890\) 0 0
\(891\) −8.08546e6 −0.341201
\(892\) 0 0
\(893\) 1.57315e7i 0.660147i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.85406e6i 0.118435i
\(898\) 0 0
\(899\) 1.52435e7 0.629050
\(900\) 0 0
\(901\) 1.55247e7 0.637107
\(902\) 0 0
\(903\) 882136.i 0.0360011i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 2.57230e7i − 1.03825i −0.854697 0.519127i \(-0.826257\pi\)
0.854697 0.519127i \(-0.173743\pi\)
\(908\) 0 0
\(909\) −3.09921e7 −1.24406
\(910\) 0 0
\(911\) 3.42108e7 1.36574 0.682869 0.730540i \(-0.260732\pi\)
0.682869 + 0.730540i \(0.260732\pi\)
\(912\) 0 0
\(913\) − 4.53341e6i − 0.179990i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.44505e7i 0.567490i
\(918\) 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
\(923\) 2.98146e7i 1.15192i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 3.27062e7i − 1.25006i
\(928\) 0 0
\(929\) −1.34361e7 −0.510781 −0.255390 0.966838i \(-0.582204\pi\)
−0.255390 + 0.966838i \(0.582204\pi\)
\(930\) 0 0
\(931\) −7.20743e6 −0.272525
\(932\) 0 0
\(933\) − 888232.i − 0.0334058i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 7.96529e6i 0.296383i 0.988959 + 0.148191i \(0.0473451\pi\)
−0.988959 + 0.148191i \(0.952655\pi\)
\(938\) 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
\(943\) − 1.62821e7i − 0.596253i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.21769e7i − 1.16592i −0.812500 0.582961i \(-0.801894\pi\)
0.812500 0.582961i \(-0.198106\pi\)
\(948\) 0 0
\(949\) 7.74467e6 0.279150
\(950\) 0 0
\(951\) −1.38812e6 −0.0497708
\(952\) 0 0
\(953\) − 5.33807e6i − 0.190394i −0.995458 0.0951968i \(-0.969652\pi\)
0.995458 0.0951968i \(-0.0303480\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 454464.i − 0.0160406i
\(958\) 0 0
\(959\) 2.20771e7 0.775167
\(960\) 0 0
\(961\) 6.46864e7 2.25946
\(962\) 0 0
\(963\) − 4.61724e7i − 1.60442i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 3.71522e7i 1.27767i 0.769345 + 0.638834i \(0.220583\pi\)
−0.769345 + 0.638834i \(0.779417\pi\)
\(968\) 0 0
\(969\) −1.32328e6 −0.0452733
\(970\) 0 0
\(971\) −1.09865e7 −0.373949 −0.186975 0.982365i \(-0.559868\pi\)
−0.186975 + 0.982365i \(0.559868\pi\)
\(972\) 0 0
\(973\) 1.93566e7i 0.655463i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 2.65054e7i − 0.888379i −0.895933 0.444190i \(-0.853492\pi\)
0.895933 0.444190i \(-0.146508\pi\)
\(978\) 0 0
\(979\) −1.35438e7 −0.451630
\(980\) 0 0
\(981\) 2.86910e7 0.951860
\(982\) 0 0
\(983\) 4.75726e7i 1.57027i 0.619327 + 0.785133i \(0.287406\pi\)
−0.619327 + 0.785133i \(0.712594\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 3.50846e6i − 0.114637i
\(988\) 0 0
\(989\) 1.55227e7 0.504636
\(990\) 0 0
\(991\) 3.22149e7 1.04201 0.521006 0.853553i \(-0.325557\pi\)
0.521006 + 0.853553i \(0.325557\pi\)
\(992\) 0 0
\(993\) 164336.i 0.00528883i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.87072e7i 1.23326i 0.787253 + 0.616630i \(0.211502\pi\)
−0.787253 + 0.616630i \(0.788498\pi\)
\(998\) 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 200.6.c.d.49.2 2
4.3 odd 2 400.6.c.k.49.1 2
5.2 odd 4 200.6.a.b.1.1 1
5.3 odd 4 40.6.a.c.1.1 1
5.4 even 2 inner 200.6.c.d.49.1 2
15.8 even 4 360.6.a.f.1.1 1
20.3 even 4 80.6.a.d.1.1 1
20.7 even 4 400.6.a.h.1.1 1
20.19 odd 2 400.6.c.k.49.2 2
40.3 even 4 320.6.a.h.1.1 1
40.13 odd 4 320.6.a.i.1.1 1
60.23 odd 4 720.6.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.6.a.c.1.1 1 5.3 odd 4
80.6.a.d.1.1 1 20.3 even 4
200.6.a.b.1.1 1 5.2 odd 4
200.6.c.d.49.1 2 5.4 even 2 inner
200.6.c.d.49.2 2 1.1 even 1 trivial
320.6.a.h.1.1 1 40.3 even 4
320.6.a.i.1.1 1 40.13 odd 4
360.6.a.f.1.1 1 15.8 even 4
400.6.a.h.1.1 1 20.7 even 4
400.6.c.k.49.1 2 4.3 odd 2
400.6.c.k.49.2 2 20.19 odd 2
720.6.a.t.1.1 1 60.23 odd 4