Properties

Label 200.6.c.d.49.1
Level $200$
Weight $6$
Character 200.49
Analytic conductor $32.077$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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

$q$-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 \(-0.979566\pi\)
0.997940 0.0641500i \(-0.0204336\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 62.0000i 0.478241i 0.970990 + 0.239120i \(0.0768591\pi\)
−0.970990 + 0.239120i \(0.923141\pi\)
\(8\) 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 \(-0.819690\pi\)
0.843806 0.536648i \(-0.180310\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1190.00i 0.998676i 0.866407 + 0.499338i \(0.166423\pi\)
−0.866407 + 0.499338i \(0.833577\pi\)
\(18\) 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 \(0.141499\pi\)
−0.902812 + 0.430036i \(0.858501\pi\)
\(24\) 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 \(0.0680607\pi\)
−0.977228 + 0.212194i \(0.931939\pi\)
\(38\) 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 \(-0.905224\pi\)
0.956000 0.293368i \(-0.0947761\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 28294.0i 1.86831i 0.356863 + 0.934157i \(0.383846\pi\)
−0.356863 + 0.934157i \(0.616154\pi\)
\(48\) 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 \(-0.896661\pi\)
0.947763 0.318976i \(-0.103339\pi\)
\(54\) 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 \(0.280752\pi\)
−0.635603 + 0.772016i \(0.719248\pi\)
\(68\) 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 \(0.0415116\pi\)
−0.991508 + 0.130043i \(0.958488\pi\)
\(74\) 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 \(-0.919305\pi\)
0.968037 0.250806i \(-0.0806955\pi\)
\(84\) 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 \(-0.959160\pi\)
0.991780 0.127952i \(-0.0408402\pi\)
\(98\) 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 \(0.219202\pi\)
−0.772109 + 0.635490i \(0.780798\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 193190.i 1.63127i 0.578569 + 0.815634i \(0.303612\pi\)
−0.578569 + 0.815634i \(0.696388\pi\)
\(108\) 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 \(-0.809921\pi\)
0.826941 0.562289i \(-0.190079\pi\)
\(114\) 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 \(-0.904071\pi\)
0.954931 0.296829i \(-0.0959291\pi\)
\(128\) 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 \(-0.699230\pi\)
0.585827 0.810436i \(-0.300770\pi\)
\(138\) 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 \(-0.771221\pi\)
0.752641 0.658431i \(-0.228779\pi\)
\(158\) 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 \(-0.993599\pi\)
0.999798 0.0201085i \(-0.00640116\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 203438.i 0.564470i 0.959345 + 0.282235i \(0.0910758\pi\)
−0.959345 + 0.282235i \(0.908924\pi\)
\(168\) 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 \(0.0516707\pi\)
−0.986854 + 0.161616i \(0.948329\pi\)
\(174\) 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 \(-0.732168\pi\)
0.666406 0.745589i \(-0.267832\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 190238.i 0.349246i 0.984635 + 0.174623i \(0.0558707\pi\)
−0.984635 + 0.174623i \(0.944129\pi\)
\(198\) 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 \(0.0915923\pi\)
−0.958886 + 0.283791i \(0.908408\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 991962.i − 1.27770i −0.769329 0.638852i \(-0.779410\pi\)
0.769329 0.638852i \(-0.220590\pi\)
\(228\) 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 \(0.196721\pi\)
−0.815029 + 0.579420i \(0.803279\pi\)
\(234\) 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 \(0.151773\pi\)
−0.888463 + 0.458948i \(0.848227\pi\)
\(258\) 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 \(-0.978023\pi\)
0.997618 0.0689870i \(-0.0219767\pi\)
\(264\) 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 \(0.432452\pi\)
−0.210619 + 0.977568i \(0.567548\pi\)
\(278\) 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 \(-0.711001\pi\)
0.615390 0.788223i \(-0.288999\pi\)
\(284\) 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 \(0.109675\pi\)
−0.941227 + 0.337776i \(0.890325\pi\)
\(294\) 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 \(-0.952842\pi\)
0.989046 0.147611i \(-0.0471583\pi\)
\(308\) 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 \(0.00433811\pi\)
−0.999907 + 0.0136282i \(0.995662\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 694058.i − 0.387925i −0.981009 0.193962i \(-0.937866\pi\)
0.981009 0.193962i \(-0.0621340\pi\)
\(318\) 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 \(0.0558558\pi\)
−0.984644 + 0.174577i \(0.944144\pi\)
\(338\) 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 \(-0.851082\pi\)
0.892544 0.450959i \(-0.148918\pi\)
\(348\) 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 \(0.0733630\pi\)
−0.973558 + 0.228441i \(0.926637\pi\)
\(354\) 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 \(-0.750718\pi\)
0.708700 0.705509i \(-0.249282\pi\)
\(368\) 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 \(-0.798950\pi\)
0.807074 0.590450i \(-0.201050\pi\)
\(374\) 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 \(0.112619\pi\)
−0.938062 + 0.346468i \(0.887381\pi\)
\(384\) 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 \(-0.952304\pi\)
0.988795 0.149281i \(-0.0476959\pi\)
\(398\) 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 \(-0.865403\pi\)
0.911924 0.410360i \(-0.134597\pi\)
\(434\) 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 \(-0.771418\pi\)
0.753050 0.657963i \(-0.228582\pi\)
\(444\) 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 \(-0.645099\pi\)
0.440218 0.897891i \(-0.354901\pi\)
\(458\) 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 \(0.231927\pi\)
−0.746094 + 0.665840i \(0.768073\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.59270e6i 0.337942i 0.985621 + 0.168971i \(0.0540444\pi\)
−0.985621 + 0.168971i \(0.945956\pi\)
\(468\) 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 \(0.286846\pi\)
−0.620707 + 0.784042i \(0.713154\pi\)
\(488\) 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 \(0.105169\pi\)
−0.945913 + 0.324420i \(0.894831\pi\)
\(504\) 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 \(-0.613052\pi\)
0.347744 0.937589i \(-0.386948\pi\)
\(524\) 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 \(0.0745498\pi\)
−0.972699 + 0.232070i \(0.925450\pi\)
\(548\) 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 \(0.237616\pi\)
−0.734075 + 0.679068i \(0.762384\pi\)
\(558\) 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 \(0.0833471\pi\)
−0.965915 + 0.258861i \(0.916653\pi\)
\(564\) 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 \(0.104010\pi\)
−0.947088 + 0.320973i \(0.895990\pi\)
\(578\) 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 \(-0.801233\pi\)
0.811287 0.584647i \(-0.198767\pi\)
\(588\) 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 \(0.0179677\pi\)
−0.998407 + 0.0564173i \(0.982032\pi\)
\(594\) 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 \(0.179577\pi\)
−0.845040 + 0.534704i \(0.820423\pi\)
\(608\) 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 \(-0.797461\pi\)
0.804304 0.594219i \(-0.202539\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 8.31174e6i − 0.878980i −0.898248 0.439490i \(-0.855159\pi\)
0.898248 0.439490i \(-0.144841\pi\)
\(618\) 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 \(-0.779525\pi\)
0.769561 0.638573i \(-0.220475\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 6.48254e6i − 0.608814i −0.952542 0.304407i \(-0.901542\pi\)
0.952542 0.304407i \(-0.0984582\pi\)
\(648\) 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 \(0.231539\pi\)
−0.746904 + 0.664931i \(0.768461\pi\)
\(654\) 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 \(0.0526120\pi\)
−0.986371 + 0.164534i \(0.947388\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 1.23856e6i − 0.103859i −0.998651 0.0519297i \(-0.983463\pi\)
0.998651 0.0519297i \(-0.0165372\pi\)
\(678\) 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 \(-0.818874\pi\)
0.842427 0.538810i \(-0.181126\pi\)
\(684\) 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 \(-0.865020\pi\)
0.911429 0.411458i \(-0.134980\pi\)
\(728\) 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 \(-0.868843\pi\)
0.916305 0.400480i \(-0.131157\pi\)
\(734\) 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 \(-0.997912\pi\)
0.999978 0.00655812i \(-0.00208753\pi\)
\(744\) 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 \(0.0390748\pi\)
−0.992475 + 0.122449i \(0.960925\pi\)
\(758\) 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 \(0.180918\pi\)
−0.842780 + 0.538259i \(0.819082\pi\)
\(774\) 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 \(-0.786858\pi\)
0.784067 0.620676i \(-0.213142\pi\)
\(788\) 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 \(0.0927752\pi\)
−0.957825 + 0.287353i \(0.907225\pi\)
\(798\) 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 \(0.152017\pi\)
−0.888112 + 0.459626i \(0.847983\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 9.42560e6i − 0.479231i −0.970868 0.239616i \(-0.922979\pi\)
0.970868 0.239616i \(-0.0770215\pi\)
\(828\) 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 \(0.125032\pi\)
−0.923841 + 0.382776i \(0.874968\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.92667e7i 1.36120i 0.732656 + 0.680600i \(0.238281\pi\)
−0.732656 + 0.680600i \(0.761719\pi\)
\(858\) 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 \(-0.882369\pi\)
0.932490 0.361196i \(-0.117631\pi\)
\(864\) 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 \(0.386391\pi\)
−0.349384 + 0.936980i \(0.613609\pi\)
\(878\) 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 \(-0.881556\pi\)
0.931565 0.363574i \(-0.118444\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.36792e6i 0.357115i 0.983929 + 0.178558i \(0.0571431\pi\)
−0.983929 + 0.178558i \(0.942857\pi\)
\(888\) 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 \(0.173743\pi\)
−0.854697 + 0.519127i \(0.826257\pi\)
\(908\) 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 \(-0.952655\pi\)
0.988959 0.148191i \(-0.0473451\pi\)
\(938\) 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 \(0.198106\pi\)
−0.812500 + 0.582961i \(0.801894\pi\)
\(948\) 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 \(0.0303480\pi\)
−0.995458 + 0.0951968i \(0.969652\pi\)
\(954\) 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 \(-0.779417\pi\)
0.769345 0.638834i \(-0.220583\pi\)
\(968\) 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 \(0.146508\pi\)
−0.895933 + 0.444190i \(0.853492\pi\)
\(978\) 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 \(-0.712594\pi\)
0.619327 0.785133i \(-0.287406\pi\)
\(984\) 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 \(-0.788498\pi\)
0.787253 0.616630i \(-0.211502\pi\)
\(998\) 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.49.1 2
4.3 odd 2 400.6.c.k.49.2 2
5.2 odd 4 40.6.a.c.1.1 1
5.3 odd 4 200.6.a.b.1.1 1
5.4 even 2 inner 200.6.c.d.49.2 2
15.2 even 4 360.6.a.f.1.1 1
20.3 even 4 400.6.a.h.1.1 1
20.7 even 4 80.6.a.d.1.1 1
20.19 odd 2 400.6.c.k.49.1 2
40.27 even 4 320.6.a.h.1.1 1
40.37 odd 4 320.6.a.i.1.1 1
60.47 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 5.2 odd 4
80.6.a.d.1.1 1 20.7 even 4
200.6.a.b.1.1 1 5.3 odd 4
200.6.c.d.49.1 2 1.1 even 1 trivial
200.6.c.d.49.2 2 5.4 even 2 inner
320.6.a.h.1.1 1 40.27 even 4
320.6.a.i.1.1 1 40.37 odd 4
360.6.a.f.1.1 1 15.2 even 4
400.6.a.h.1.1 1 20.3 even 4
400.6.c.k.49.1 2 20.19 odd 2
400.6.c.k.49.2 2 4.3 odd 2
720.6.a.t.1.1 1 60.47 odd 4