Properties

Label 200.6.c.b.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: \(\Z[a_1, \ldots, a_{7}]\)
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.b.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-18.0000i q^{3} -242.000i q^{7} -81.0000 q^{9} +O(q^{10})\) \(q-18.0000i q^{3} -242.000i q^{7} -81.0000 q^{9} +656.000 q^{11} -206.000i q^{13} -1690.00i q^{17} +1364.00 q^{19} -4356.00 q^{21} +2198.00i q^{23} -2916.00i q^{27} +2218.00 q^{29} -1700.00 q^{31} -11808.0i q^{33} +846.000i q^{37} -3708.00 q^{39} -1818.00 q^{41} +10534.0i q^{43} -12074.0i q^{47} -41757.0 q^{49} -30420.0 q^{51} +32586.0i q^{53} -24552.0i q^{57} -8668.00 q^{59} -34670.0 q^{61} +19602.0i q^{63} +47566.0i q^{67} +39564.0 q^{69} +948.000 q^{71} -63102.0i q^{73} -158752. i q^{77} -46536.0 q^{79} -72171.0 q^{81} -88778.0i q^{83} -39924.0i q^{87} +104934. q^{89} -49852.0 q^{91} +30600.0i q^{93} +36254.0i q^{97} -53136.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 162 q^{9} + 1312 q^{11} + 2728 q^{19} - 8712 q^{21} + 4436 q^{29} - 3400 q^{31} - 7416 q^{39} - 3636 q^{41} - 83514 q^{49} - 60840 q^{51} - 17336 q^{59} - 69340 q^{61} + 79128 q^{69} + 1896 q^{71} - 93072 q^{79} - 144342 q^{81} + 209868 q^{89} - 99704 q^{91} - 106272 q^{99}+O(q^{100}) \) 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
\(3\) − 18.0000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 242.000i − 1.86668i −0.358991 0.933341i \(-0.616879\pi\)
0.358991 0.933341i \(-0.383121\pi\)
\(8\) 0 0
\(9\) −81.0000 −0.333333
\(10\) 0 0
\(11\) 656.000 1.63464 0.817320 0.576184i \(-0.195459\pi\)
0.817320 + 0.576184i \(0.195459\pi\)
\(12\) 0 0
\(13\) − 206.000i − 0.338072i −0.985610 0.169036i \(-0.945935\pi\)
0.985610 0.169036i \(-0.0540654\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1690.00i − 1.41829i −0.705064 0.709144i \(-0.749082\pi\)
0.705064 0.709144i \(-0.250918\pi\)
\(18\) 0 0
\(19\) 1364.00 0.866823 0.433411 0.901196i \(-0.357310\pi\)
0.433411 + 0.901196i \(0.357310\pi\)
\(20\) 0 0
\(21\) −4356.00 −2.15546
\(22\) 0 0
\(23\) 2198.00i 0.866379i 0.901303 + 0.433190i \(0.142612\pi\)
−0.901303 + 0.433190i \(0.857388\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 2916.00i − 0.769800i
\(28\) 0 0
\(29\) 2218.00 0.489741 0.244871 0.969556i \(-0.421255\pi\)
0.244871 + 0.969556i \(0.421255\pi\)
\(30\) 0 0
\(31\) −1700.00 −0.317720 −0.158860 0.987301i \(-0.550782\pi\)
−0.158860 + 0.987301i \(0.550782\pi\)
\(32\) 0 0
\(33\) − 11808.0i − 1.88752i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 846.000i 0.101594i 0.998709 + 0.0507968i \(0.0161761\pi\)
−0.998709 + 0.0507968i \(0.983824\pi\)
\(38\) 0 0
\(39\) −3708.00 −0.390372
\(40\) 0 0
\(41\) −1818.00 −0.168902 −0.0844509 0.996428i \(-0.526914\pi\)
−0.0844509 + 0.996428i \(0.526914\pi\)
\(42\) 0 0
\(43\) 10534.0i 0.868805i 0.900719 + 0.434402i \(0.143040\pi\)
−0.900719 + 0.434402i \(0.856960\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 12074.0i − 0.797272i −0.917109 0.398636i \(-0.869484\pi\)
0.917109 0.398636i \(-0.130516\pi\)
\(48\) 0 0
\(49\) −41757.0 −2.48450
\(50\) 0 0
\(51\) −30420.0 −1.63770
\(52\) 0 0
\(53\) 32586.0i 1.59346i 0.604335 + 0.796730i \(0.293439\pi\)
−0.604335 + 0.796730i \(0.706561\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 24552.0i − 1.00092i
\(58\) 0 0
\(59\) −8668.00 −0.324182 −0.162091 0.986776i \(-0.551824\pi\)
−0.162091 + 0.986776i \(0.551824\pi\)
\(60\) 0 0
\(61\) −34670.0 −1.19297 −0.596485 0.802624i \(-0.703436\pi\)
−0.596485 + 0.802624i \(0.703436\pi\)
\(62\) 0 0
\(63\) 19602.0i 0.622227i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 47566.0i 1.29452i 0.762268 + 0.647261i \(0.224086\pi\)
−0.762268 + 0.647261i \(0.775914\pi\)
\(68\) 0 0
\(69\) 39564.0 1.00041
\(70\) 0 0
\(71\) 948.000 0.0223184 0.0111592 0.999938i \(-0.496448\pi\)
0.0111592 + 0.999938i \(0.496448\pi\)
\(72\) 0 0
\(73\) − 63102.0i − 1.38591i −0.720979 0.692957i \(-0.756308\pi\)
0.720979 0.692957i \(-0.243692\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 158752.i − 3.05135i
\(78\) 0 0
\(79\) −46536.0 −0.838921 −0.419461 0.907773i \(-0.637781\pi\)
−0.419461 + 0.907773i \(0.637781\pi\)
\(80\) 0 0
\(81\) −72171.0 −1.22222
\(82\) 0 0
\(83\) − 88778.0i − 1.41452i −0.706952 0.707262i \(-0.749930\pi\)
0.706952 0.707262i \(-0.250070\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 39924.0i − 0.565504i
\(88\) 0 0
\(89\) 104934. 1.40424 0.702120 0.712059i \(-0.252237\pi\)
0.702120 + 0.712059i \(0.252237\pi\)
\(90\) 0 0
\(91\) −49852.0 −0.631072
\(92\) 0 0
\(93\) 30600.0i 0.366872i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 36254.0i 0.391225i 0.980681 + 0.195612i \(0.0626695\pi\)
−0.980681 + 0.195612i \(0.937331\pi\)
\(98\) 0 0
\(99\) −53136.0 −0.544880
\(100\) 0 0
\(101\) 42486.0 0.414422 0.207211 0.978296i \(-0.433561\pi\)
0.207211 + 0.978296i \(0.433561\pi\)
\(102\) 0 0
\(103\) 147934.i 1.37396i 0.726675 + 0.686981i \(0.241065\pi\)
−0.726675 + 0.686981i \(0.758935\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18390.0i 0.155282i 0.996981 + 0.0776412i \(0.0247389\pi\)
−0.996981 + 0.0776412i \(0.975261\pi\)
\(108\) 0 0
\(109\) 145006. 1.16901 0.584507 0.811389i \(-0.301288\pi\)
0.584507 + 0.811389i \(0.301288\pi\)
\(110\) 0 0
\(111\) 15228.0 0.117310
\(112\) 0 0
\(113\) 82746.0i 0.609608i 0.952415 + 0.304804i \(0.0985910\pi\)
−0.952415 + 0.304804i \(0.901409\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 16686.0i 0.112691i
\(118\) 0 0
\(119\) −408980. −2.64749
\(120\) 0 0
\(121\) 269285. 1.67205
\(122\) 0 0
\(123\) 32724.0i 0.195031i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 274446.i 1.50990i 0.655784 + 0.754949i \(0.272338\pi\)
−0.655784 + 0.754949i \(0.727662\pi\)
\(128\) 0 0
\(129\) 189612. 1.00321
\(130\) 0 0
\(131\) 202608. 1.03152 0.515761 0.856733i \(-0.327509\pi\)
0.515761 + 0.856733i \(0.327509\pi\)
\(132\) 0 0
\(133\) − 330088.i − 1.61808i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 48142.0i 0.219141i 0.993979 + 0.109570i \(0.0349475\pi\)
−0.993979 + 0.109570i \(0.965053\pi\)
\(138\) 0 0
\(139\) 111156. 0.487973 0.243987 0.969779i \(-0.421545\pi\)
0.243987 + 0.969779i \(0.421545\pi\)
\(140\) 0 0
\(141\) −217332. −0.920610
\(142\) 0 0
\(143\) − 135136.i − 0.552626i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 751626.i 2.86885i
\(148\) 0 0
\(149\) −243178. −0.897343 −0.448672 0.893697i \(-0.648103\pi\)
−0.448672 + 0.893697i \(0.648103\pi\)
\(150\) 0 0
\(151\) 368852. 1.31647 0.658233 0.752814i \(-0.271304\pi\)
0.658233 + 0.752814i \(0.271304\pi\)
\(152\) 0 0
\(153\) 136890.i 0.472763i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 319546.i − 1.03463i −0.855796 0.517314i \(-0.826932\pi\)
0.855796 0.517314i \(-0.173068\pi\)
\(158\) 0 0
\(159\) 586548. 1.83997
\(160\) 0 0
\(161\) 531916. 1.61725
\(162\) 0 0
\(163\) 69862.0i 0.205955i 0.994684 + 0.102977i \(0.0328369\pi\)
−0.994684 + 0.102977i \(0.967163\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 343422.i 0.952877i 0.879208 + 0.476439i \(0.158073\pi\)
−0.879208 + 0.476439i \(0.841927\pi\)
\(168\) 0 0
\(169\) 328857. 0.885708
\(170\) 0 0
\(171\) −110484. −0.288941
\(172\) 0 0
\(173\) − 1142.00i − 0.00290102i −0.999999 0.00145051i \(-0.999538\pi\)
0.999999 0.00145051i \(-0.000461712\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 156024.i 0.374333i
\(178\) 0 0
\(179\) 86684.0 0.202212 0.101106 0.994876i \(-0.467762\pi\)
0.101106 + 0.994876i \(0.467762\pi\)
\(180\) 0 0
\(181\) −651418. −1.47796 −0.738981 0.673726i \(-0.764693\pi\)
−0.738981 + 0.673726i \(0.764693\pi\)
\(182\) 0 0
\(183\) 624060.i 1.37752i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 1.10864e6i − 2.31839i
\(188\) 0 0
\(189\) −705672. −1.43697
\(190\) 0 0
\(191\) 29140.0 0.0577971 0.0288986 0.999582i \(-0.490800\pi\)
0.0288986 + 0.999582i \(0.490800\pi\)
\(192\) 0 0
\(193\) − 646406.i − 1.24914i −0.780968 0.624571i \(-0.785274\pi\)
0.780968 0.624571i \(-0.214726\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 431138.i − 0.791500i −0.918358 0.395750i \(-0.870485\pi\)
0.918358 0.395750i \(-0.129515\pi\)
\(198\) 0 0
\(199\) 131608. 0.235586 0.117793 0.993038i \(-0.462418\pi\)
0.117793 + 0.993038i \(0.462418\pi\)
\(200\) 0 0
\(201\) 856188. 1.49479
\(202\) 0 0
\(203\) − 536756.i − 0.914191i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 178038.i − 0.288793i
\(208\) 0 0
\(209\) 894784. 1.41694
\(210\) 0 0
\(211\) 1.21078e6 1.87224 0.936118 0.351686i \(-0.114392\pi\)
0.936118 + 0.351686i \(0.114392\pi\)
\(212\) 0 0
\(213\) − 17064.0i − 0.0257710i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 411400.i 0.593082i
\(218\) 0 0
\(219\) −1.13584e6 −1.60031
\(220\) 0 0
\(221\) −348140. −0.479483
\(222\) 0 0
\(223\) 34886.0i 0.0469774i 0.999724 + 0.0234887i \(0.00747737\pi\)
−0.999724 + 0.0234887i \(0.992523\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 124182.i 0.159954i 0.996797 + 0.0799768i \(0.0254846\pi\)
−0.996797 + 0.0799768i \(0.974515\pi\)
\(228\) 0 0
\(229\) 456386. 0.575100 0.287550 0.957766i \(-0.407159\pi\)
0.287550 + 0.957766i \(0.407159\pi\)
\(230\) 0 0
\(231\) −2.85754e6 −3.52340
\(232\) 0 0
\(233\) 252666.i 0.304900i 0.988311 + 0.152450i \(0.0487163\pi\)
−0.988311 + 0.152450i \(0.951284\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 837648.i 0.968703i
\(238\) 0 0
\(239\) −65064.0 −0.0736794 −0.0368397 0.999321i \(-0.511729\pi\)
−0.0368397 + 0.999321i \(0.511729\pi\)
\(240\) 0 0
\(241\) −1.40600e6 −1.55935 −0.779675 0.626185i \(-0.784615\pi\)
−0.779675 + 0.626185i \(0.784615\pi\)
\(242\) 0 0
\(243\) 590490.i 0.641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 280984.i − 0.293048i
\(248\) 0 0
\(249\) −1.59800e6 −1.63335
\(250\) 0 0
\(251\) −548400. −0.549431 −0.274715 0.961526i \(-0.588584\pi\)
−0.274715 + 0.961526i \(0.588584\pi\)
\(252\) 0 0
\(253\) 1.44189e6i 1.41622i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 493830.i 0.466385i 0.972431 + 0.233193i \(0.0749172\pi\)
−0.972431 + 0.233193i \(0.925083\pi\)
\(258\) 0 0
\(259\) 204732. 0.189643
\(260\) 0 0
\(261\) −179658. −0.163247
\(262\) 0 0
\(263\) − 1.07181e6i − 0.955495i −0.878497 0.477748i \(-0.841453\pi\)
0.878497 0.477748i \(-0.158547\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 1.88881e6i − 1.62148i
\(268\) 0 0
\(269\) −999394. −0.842085 −0.421043 0.907041i \(-0.638336\pi\)
−0.421043 + 0.907041i \(0.638336\pi\)
\(270\) 0 0
\(271\) 1.00760e6 0.833425 0.416713 0.909038i \(-0.363182\pi\)
0.416713 + 0.909038i \(0.363182\pi\)
\(272\) 0 0
\(273\) 897336.i 0.728700i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 1.02286e6i − 0.800969i −0.916303 0.400485i \(-0.868842\pi\)
0.916303 0.400485i \(-0.131158\pi\)
\(278\) 0 0
\(279\) 137700. 0.105907
\(280\) 0 0
\(281\) −1.18172e6 −0.892790 −0.446395 0.894836i \(-0.647292\pi\)
−0.446395 + 0.894836i \(0.647292\pi\)
\(282\) 0 0
\(283\) − 917506.i − 0.680993i −0.940246 0.340497i \(-0.889405\pi\)
0.940246 0.340497i \(-0.110595\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 439956.i 0.315286i
\(288\) 0 0
\(289\) −1.43624e6 −1.01154
\(290\) 0 0
\(291\) 652572. 0.451748
\(292\) 0 0
\(293\) − 512302.i − 0.348624i −0.984690 0.174312i \(-0.944230\pi\)
0.984690 0.174312i \(-0.0557701\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 1.91290e6i − 1.25835i
\(298\) 0 0
\(299\) 452788. 0.292898
\(300\) 0 0
\(301\) 2.54923e6 1.62178
\(302\) 0 0
\(303\) − 764748.i − 0.478533i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 1.40946e6i − 0.853505i −0.904368 0.426753i \(-0.859658\pi\)
0.904368 0.426753i \(-0.140342\pi\)
\(308\) 0 0
\(309\) 2.66281e6 1.58652
\(310\) 0 0
\(311\) −2.78604e6 −1.63337 −0.816687 0.577081i \(-0.804192\pi\)
−0.816687 + 0.577081i \(0.804192\pi\)
\(312\) 0 0
\(313\) 1.55086e6i 0.894770i 0.894342 + 0.447385i \(0.147645\pi\)
−0.894342 + 0.447385i \(0.852355\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 377322.i − 0.210894i −0.994425 0.105447i \(-0.966373\pi\)
0.994425 0.105447i \(-0.0336273\pi\)
\(318\) 0 0
\(319\) 1.45501e6 0.800550
\(320\) 0 0
\(321\) 331020. 0.179305
\(322\) 0 0
\(323\) − 2.30516e6i − 1.22940i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 2.61011e6i − 1.34986i
\(328\) 0 0
\(329\) −2.92191e6 −1.48825
\(330\) 0 0
\(331\) −1.63063e6 −0.818062 −0.409031 0.912521i \(-0.634133\pi\)
−0.409031 + 0.912521i \(0.634133\pi\)
\(332\) 0 0
\(333\) − 68526.0i − 0.0338645i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.36717e6i 1.61506i 0.589824 + 0.807532i \(0.299197\pi\)
−0.589824 + 0.807532i \(0.700803\pi\)
\(338\) 0 0
\(339\) 1.48943e6 0.703915
\(340\) 0 0
\(341\) −1.11520e6 −0.519358
\(342\) 0 0
\(343\) 6.03790e6i 2.77109i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 837202.i − 0.373256i −0.982431 0.186628i \(-0.940244\pi\)
0.982431 0.186628i \(-0.0597559\pi\)
\(348\) 0 0
\(349\) 1.51910e6 0.667609 0.333805 0.942642i \(-0.391667\pi\)
0.333805 + 0.942642i \(0.391667\pi\)
\(350\) 0 0
\(351\) −600696. −0.260248
\(352\) 0 0
\(353\) − 3.51851e6i − 1.50287i −0.659806 0.751436i \(-0.729362\pi\)
0.659806 0.751436i \(-0.270638\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 7.36164e6i 3.05706i
\(358\) 0 0
\(359\) 3.57089e6 1.46231 0.731156 0.682210i \(-0.238981\pi\)
0.731156 + 0.682210i \(0.238981\pi\)
\(360\) 0 0
\(361\) −615603. −0.248618
\(362\) 0 0
\(363\) − 4.84713e6i − 1.93071i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.58231e6i 1.38835i 0.719808 + 0.694173i \(0.244230\pi\)
−0.719808 + 0.694173i \(0.755770\pi\)
\(368\) 0 0
\(369\) 147258. 0.0563006
\(370\) 0 0
\(371\) 7.88581e6 2.97448
\(372\) 0 0
\(373\) 635530.i 0.236518i 0.992983 + 0.118259i \(0.0377313\pi\)
−0.992983 + 0.118259i \(0.962269\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 456908.i − 0.165568i
\(378\) 0 0
\(379\) −67060.0 −0.0239809 −0.0119905 0.999928i \(-0.503817\pi\)
−0.0119905 + 0.999928i \(0.503817\pi\)
\(380\) 0 0
\(381\) 4.94003e6 1.74348
\(382\) 0 0
\(383\) 4.45129e6i 1.55056i 0.631618 + 0.775280i \(0.282391\pi\)
−0.631618 + 0.775280i \(0.717609\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 853254.i − 0.289602i
\(388\) 0 0
\(389\) −5.79825e6 −1.94278 −0.971388 0.237496i \(-0.923673\pi\)
−0.971388 + 0.237496i \(0.923673\pi\)
\(390\) 0 0
\(391\) 3.71462e6 1.22878
\(392\) 0 0
\(393\) − 3.64694e6i − 1.19110i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 333874.i − 0.106318i −0.998586 0.0531589i \(-0.983071\pi\)
0.998586 0.0531589i \(-0.0169290\pi\)
\(398\) 0 0
\(399\) −5.94158e6 −1.86840
\(400\) 0 0
\(401\) −2.55689e6 −0.794057 −0.397029 0.917806i \(-0.629959\pi\)
−0.397029 + 0.917806i \(0.629959\pi\)
\(402\) 0 0
\(403\) 350200.i 0.107412i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 554976.i 0.166069i
\(408\) 0 0
\(409\) 3.05511e6 0.903063 0.451531 0.892255i \(-0.350878\pi\)
0.451531 + 0.892255i \(0.350878\pi\)
\(410\) 0 0
\(411\) 866556. 0.253042
\(412\) 0 0
\(413\) 2.09766e6i 0.605145i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 2.00081e6i − 0.563463i
\(418\) 0 0
\(419\) 3.54347e6 0.986038 0.493019 0.870019i \(-0.335893\pi\)
0.493019 + 0.870019i \(0.335893\pi\)
\(420\) 0 0
\(421\) 1.97294e6 0.542511 0.271255 0.962507i \(-0.412561\pi\)
0.271255 + 0.962507i \(0.412561\pi\)
\(422\) 0 0
\(423\) 977994.i 0.265757i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.39014e6i 2.22689i
\(428\) 0 0
\(429\) −2.43245e6 −0.638117
\(430\) 0 0
\(431\) 1.37396e6 0.356270 0.178135 0.984006i \(-0.442994\pi\)
0.178135 + 0.984006i \(0.442994\pi\)
\(432\) 0 0
\(433\) 5.18813e6i 1.32981i 0.746926 + 0.664907i \(0.231529\pi\)
−0.746926 + 0.664907i \(0.768471\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.99807e6i 0.750997i
\(438\) 0 0
\(439\) 2.94082e6 0.728296 0.364148 0.931341i \(-0.381360\pi\)
0.364148 + 0.931341i \(0.381360\pi\)
\(440\) 0 0
\(441\) 3.38232e6 0.828167
\(442\) 0 0
\(443\) − 1.28347e6i − 0.310724i −0.987858 0.155362i \(-0.950346\pi\)
0.987858 0.155362i \(-0.0496544\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4.37720e6i 1.03616i
\(448\) 0 0
\(449\) 4.95263e6 1.15937 0.579683 0.814842i \(-0.303176\pi\)
0.579683 + 0.814842i \(0.303176\pi\)
\(450\) 0 0
\(451\) −1.19261e6 −0.276094
\(452\) 0 0
\(453\) − 6.63934e6i − 1.52012i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 7.91315e6i − 1.77239i −0.463315 0.886194i \(-0.653340\pi\)
0.463315 0.886194i \(-0.346660\pi\)
\(458\) 0 0
\(459\) −4.92804e6 −1.09180
\(460\) 0 0
\(461\) −6.18530e6 −1.35553 −0.677764 0.735280i \(-0.737051\pi\)
−0.677764 + 0.735280i \(0.737051\pi\)
\(462\) 0 0
\(463\) 491934.i 0.106648i 0.998577 + 0.0533242i \(0.0169817\pi\)
−0.998577 + 0.0533242i \(0.983018\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 447442.i − 0.0949390i −0.998873 0.0474695i \(-0.984884\pi\)
0.998873 0.0474695i \(-0.0151157\pi\)
\(468\) 0 0
\(469\) 1.15110e7 2.41646
\(470\) 0 0
\(471\) −5.75183e6 −1.19469
\(472\) 0 0
\(473\) 6.91030e6i 1.42018i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 2.63947e6i − 0.531154i
\(478\) 0 0
\(479\) −8.18487e6 −1.62995 −0.814973 0.579499i \(-0.803248\pi\)
−0.814973 + 0.579499i \(0.803248\pi\)
\(480\) 0 0
\(481\) 174276. 0.0343459
\(482\) 0 0
\(483\) − 9.57449e6i − 1.86744i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 6.21524e6i − 1.18751i −0.804648 0.593753i \(-0.797646\pi\)
0.804648 0.593753i \(-0.202354\pi\)
\(488\) 0 0
\(489\) 1.25752e6 0.237816
\(490\) 0 0
\(491\) 827856. 0.154971 0.0774856 0.996993i \(-0.475311\pi\)
0.0774856 + 0.996993i \(0.475311\pi\)
\(492\) 0 0
\(493\) − 3.74842e6i − 0.694594i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 229416.i − 0.0416613i
\(498\) 0 0
\(499\) 1.04004e7 1.86982 0.934908 0.354890i \(-0.115482\pi\)
0.934908 + 0.354890i \(0.115482\pi\)
\(500\) 0 0
\(501\) 6.18160e6 1.10029
\(502\) 0 0
\(503\) 2.03821e6i 0.359193i 0.983740 + 0.179597i \(0.0574792\pi\)
−0.983740 + 0.179597i \(0.942521\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 5.91943e6i − 1.02273i
\(508\) 0 0
\(509\) −3.66133e6 −0.626390 −0.313195 0.949689i \(-0.601399\pi\)
−0.313195 + 0.949689i \(0.601399\pi\)
\(510\) 0 0
\(511\) −1.52707e7 −2.58706
\(512\) 0 0
\(513\) − 3.97742e6i − 0.667281i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 7.92054e6i − 1.30325i
\(518\) 0 0
\(519\) −20556.0 −0.00334981
\(520\) 0 0
\(521\) 3.24713e6 0.524089 0.262045 0.965056i \(-0.415603\pi\)
0.262045 + 0.965056i \(0.415603\pi\)
\(522\) 0 0
\(523\) − 4.97357e6i − 0.795086i −0.917584 0.397543i \(-0.869863\pi\)
0.917584 0.397543i \(-0.130137\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.87300e6i 0.450619i
\(528\) 0 0
\(529\) 1.60514e6 0.249387
\(530\) 0 0
\(531\) 702108. 0.108061
\(532\) 0 0
\(533\) 374508.i 0.0571009i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 1.56031e6i − 0.233494i
\(538\) 0 0
\(539\) −2.73926e7 −4.06126
\(540\) 0 0
\(541\) 2.42544e6 0.356285 0.178142 0.984005i \(-0.442991\pi\)
0.178142 + 0.984005i \(0.442991\pi\)
\(542\) 0 0
\(543\) 1.17255e7i 1.70660i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 731254.i 0.104496i 0.998634 + 0.0522480i \(0.0166386\pi\)
−0.998634 + 0.0522480i \(0.983361\pi\)
\(548\) 0 0
\(549\) 2.80827e6 0.397656
\(550\) 0 0
\(551\) 3.02535e6 0.424519
\(552\) 0 0
\(553\) 1.12617e7i 1.56600i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 7.71992e6i − 1.05433i −0.849764 0.527163i \(-0.823256\pi\)
0.849764 0.527163i \(-0.176744\pi\)
\(558\) 0 0
\(559\) 2.17000e6 0.293718
\(560\) 0 0
\(561\) −1.99555e7 −2.67705
\(562\) 0 0
\(563\) 3.10576e6i 0.412949i 0.978452 + 0.206475i \(0.0661991\pi\)
−0.978452 + 0.206475i \(0.933801\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.74654e7i 2.28150i
\(568\) 0 0
\(569\) 482498. 0.0624762 0.0312381 0.999512i \(-0.490055\pi\)
0.0312381 + 0.999512i \(0.490055\pi\)
\(570\) 0 0
\(571\) 1.38502e7 1.77773 0.888865 0.458169i \(-0.151495\pi\)
0.888865 + 0.458169i \(0.151495\pi\)
\(572\) 0 0
\(573\) − 524520.i − 0.0667384i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 7.09764e6i − 0.887513i −0.896147 0.443756i \(-0.853646\pi\)
0.896147 0.443756i \(-0.146354\pi\)
\(578\) 0 0
\(579\) −1.16353e7 −1.44239
\(580\) 0 0
\(581\) −2.14843e7 −2.64046
\(582\) 0 0
\(583\) 2.13764e7i 2.60473i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 1.56926e6i − 0.187975i −0.995573 0.0939873i \(-0.970039\pi\)
0.995573 0.0939873i \(-0.0299613\pi\)
\(588\) 0 0
\(589\) −2.31880e6 −0.275407
\(590\) 0 0
\(591\) −7.76048e6 −0.913945
\(592\) 0 0
\(593\) − 1.30477e7i − 1.52370i −0.647756 0.761848i \(-0.724293\pi\)
0.647756 0.761848i \(-0.275707\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 2.36894e6i − 0.272031i
\(598\) 0 0
\(599\) 5.24688e6 0.597495 0.298747 0.954332i \(-0.403431\pi\)
0.298747 + 0.954332i \(0.403431\pi\)
\(600\) 0 0
\(601\) −5.57316e6 −0.629384 −0.314692 0.949194i \(-0.601901\pi\)
−0.314692 + 0.949194i \(0.601901\pi\)
\(602\) 0 0
\(603\) − 3.85285e6i − 0.431508i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 1.98249e6i − 0.218393i −0.994020 0.109197i \(-0.965172\pi\)
0.994020 0.109197i \(-0.0348278\pi\)
\(608\) 0 0
\(609\) −9.66161e6 −1.05562
\(610\) 0 0
\(611\) −2.48724e6 −0.269535
\(612\) 0 0
\(613\) 969810.i 0.104240i 0.998641 + 0.0521201i \(0.0165979\pi\)
−0.998641 + 0.0521201i \(0.983402\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.12946e7i 1.19442i 0.802084 + 0.597211i \(0.203725\pi\)
−0.802084 + 0.597211i \(0.796275\pi\)
\(618\) 0 0
\(619\) 1.80728e7 1.89583 0.947914 0.318528i \(-0.103188\pi\)
0.947914 + 0.318528i \(0.103188\pi\)
\(620\) 0 0
\(621\) 6.40937e6 0.666939
\(622\) 0 0
\(623\) − 2.53940e7i − 2.62127i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 1.61061e7i − 1.63615i
\(628\) 0 0
\(629\) 1.42974e6 0.144089
\(630\) 0 0
\(631\) 4.62634e6 0.462556 0.231278 0.972888i \(-0.425709\pi\)
0.231278 + 0.972888i \(0.425709\pi\)
\(632\) 0 0
\(633\) − 2.17941e7i − 2.16187i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 8.60194e6i 0.839939i
\(638\) 0 0
\(639\) −76788.0 −0.00743946
\(640\) 0 0
\(641\) 1.99058e7 1.91352 0.956762 0.290871i \(-0.0939452\pi\)
0.956762 + 0.290871i \(0.0939452\pi\)
\(642\) 0 0
\(643\) 1.21078e7i 1.15489i 0.816431 + 0.577443i \(0.195949\pi\)
−0.816431 + 0.577443i \(0.804051\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.53124e6i 0.237724i 0.992911 + 0.118862i \(0.0379245\pi\)
−0.992911 + 0.118862i \(0.962075\pi\)
\(648\) 0 0
\(649\) −5.68621e6 −0.529921
\(650\) 0 0
\(651\) 7.40520e6 0.684832
\(652\) 0 0
\(653\) 1.37043e7i 1.25770i 0.777529 + 0.628848i \(0.216473\pi\)
−0.777529 + 0.628848i \(0.783527\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 5.11126e6i 0.461971i
\(658\) 0 0
\(659\) 9.83320e6 0.882026 0.441013 0.897501i \(-0.354619\pi\)
0.441013 + 0.897501i \(0.354619\pi\)
\(660\) 0 0
\(661\) 6.68687e6 0.595278 0.297639 0.954679i \(-0.403801\pi\)
0.297639 + 0.954679i \(0.403801\pi\)
\(662\) 0 0
\(663\) 6.26652e6i 0.553659i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.87516e6i 0.424302i
\(668\) 0 0
\(669\) 627948. 0.0542448
\(670\) 0 0
\(671\) −2.27435e7 −1.95008
\(672\) 0 0
\(673\) − 727566.i − 0.0619205i −0.999521 0.0309603i \(-0.990143\pi\)
0.999521 0.0309603i \(-0.00985654\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 1.86951e7i − 1.56768i −0.620964 0.783839i \(-0.713259\pi\)
0.620964 0.783839i \(-0.286741\pi\)
\(678\) 0 0
\(679\) 8.77347e6 0.730293
\(680\) 0 0
\(681\) 2.23528e6 0.184699
\(682\) 0 0
\(683\) − 1.79850e7i − 1.47523i −0.675223 0.737614i \(-0.735952\pi\)
0.675223 0.737614i \(-0.264048\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 8.21495e6i − 0.664069i
\(688\) 0 0
\(689\) 6.71272e6 0.538704
\(690\) 0 0
\(691\) 1.20006e6 0.0956107 0.0478053 0.998857i \(-0.484777\pi\)
0.0478053 + 0.998857i \(0.484777\pi\)
\(692\) 0 0
\(693\) 1.28589e7i 1.01712i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3.07242e6i 0.239551i
\(698\) 0 0
\(699\) 4.54799e6 0.352068
\(700\) 0 0
\(701\) −1.15194e7 −0.885393 −0.442697 0.896671i \(-0.645978\pi\)
−0.442697 + 0.896671i \(0.645978\pi\)
\(702\) 0 0
\(703\) 1.15394e6i 0.0880636i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.02816e7i − 0.773593i
\(708\) 0 0
\(709\) 610738. 0.0456288 0.0228144 0.999740i \(-0.492737\pi\)
0.0228144 + 0.999740i \(0.492737\pi\)
\(710\) 0 0
\(711\) 3.76942e6 0.279640
\(712\) 0 0
\(713\) − 3.73660e6i − 0.275266i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.17115e6i 0.0850776i
\(718\) 0 0
\(719\) −3.97278e6 −0.286597 −0.143299 0.989680i \(-0.545771\pi\)
−0.143299 + 0.989680i \(0.545771\pi\)
\(720\) 0 0
\(721\) 3.58000e7 2.56475
\(722\) 0 0
\(723\) 2.53080e7i 1.80058i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.23220e7i 0.864658i 0.901716 + 0.432329i \(0.142308\pi\)
−0.901716 + 0.432329i \(0.857692\pi\)
\(728\) 0 0
\(729\) −6.90873e6 −0.481481
\(730\) 0 0
\(731\) 1.78025e7 1.23222
\(732\) 0 0
\(733\) 8.14579e6i 0.559981i 0.960003 + 0.279990i \(0.0903313\pi\)
−0.960003 + 0.279990i \(0.909669\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.12033e7i 2.11608i
\(738\) 0 0
\(739\) −7.16653e6 −0.482723 −0.241361 0.970435i \(-0.577594\pi\)
−0.241361 + 0.970435i \(0.577594\pi\)
\(740\) 0 0
\(741\) −5.05771e6 −0.338383
\(742\) 0 0
\(743\) − 5.65041e6i − 0.375498i −0.982217 0.187749i \(-0.939881\pi\)
0.982217 0.187749i \(-0.0601192\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 7.19102e6i 0.471508i
\(748\) 0 0
\(749\) 4.45038e6 0.289863
\(750\) 0 0
\(751\) 9.09500e6 0.588441 0.294221 0.955738i \(-0.404940\pi\)
0.294221 + 0.955738i \(0.404940\pi\)
\(752\) 0 0
\(753\) 9.87120e6i 0.634428i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.12880e7i 0.715944i 0.933732 + 0.357972i \(0.116532\pi\)
−0.933732 + 0.357972i \(0.883468\pi\)
\(758\) 0 0
\(759\) 2.59540e7 1.63531
\(760\) 0 0
\(761\) 1.52933e7 0.957283 0.478641 0.878011i \(-0.341129\pi\)
0.478641 + 0.878011i \(0.341129\pi\)
\(762\) 0 0
\(763\) − 3.50915e7i − 2.18218i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.78561e6i 0.109597i
\(768\) 0 0
\(769\) −1.77402e6 −0.108179 −0.0540894 0.998536i \(-0.517226\pi\)
−0.0540894 + 0.998536i \(0.517226\pi\)
\(770\) 0 0
\(771\) 8.88894e6 0.538535
\(772\) 0 0
\(773\) − 1.46441e7i − 0.881484i −0.897634 0.440742i \(-0.854715\pi\)
0.897634 0.440742i \(-0.145285\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 3.68518e6i − 0.218981i
\(778\) 0 0
\(779\) −2.47975e6 −0.146408
\(780\) 0 0
\(781\) 621888. 0.0364825
\(782\) 0 0
\(783\) − 6.46769e6i − 0.377003i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 5.97074e6i 0.343630i 0.985129 + 0.171815i \(0.0549632\pi\)
−0.985129 + 0.171815i \(0.945037\pi\)
\(788\) 0 0
\(789\) −1.92926e7 −1.10331
\(790\) 0 0
\(791\) 2.00245e7 1.13794
\(792\) 0 0
\(793\) 7.14202e6i 0.403309i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.40500e7i 1.89876i 0.314125 + 0.949382i \(0.398289\pi\)
−0.314125 + 0.949382i \(0.601711\pi\)
\(798\) 0 0
\(799\) −2.04051e7 −1.13076
\(800\) 0 0
\(801\) −8.49965e6 −0.468080
\(802\) 0 0
\(803\) − 4.13949e7i − 2.26547i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.79891e7i 0.972356i
\(808\) 0 0
\(809\) 2.63540e7 1.41571 0.707857 0.706356i \(-0.249662\pi\)
0.707857 + 0.706356i \(0.249662\pi\)
\(810\) 0 0
\(811\) −9.49658e6 −0.507008 −0.253504 0.967334i \(-0.581583\pi\)
−0.253504 + 0.967334i \(0.581583\pi\)
\(812\) 0 0
\(813\) − 1.81369e7i − 0.962357i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.43684e7i 0.753100i
\(818\) 0 0
\(819\) 4.03801e6 0.210357
\(820\) 0 0
\(821\) −1.59887e7 −0.827856 −0.413928 0.910310i \(-0.635843\pi\)
−0.413928 + 0.910310i \(0.635843\pi\)
\(822\) 0 0
\(823\) 3.18347e7i 1.63833i 0.573559 + 0.819164i \(0.305562\pi\)
−0.573559 + 0.819164i \(0.694438\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1.27575e7i − 0.648635i −0.945948 0.324317i \(-0.894865\pi\)
0.945948 0.324317i \(-0.105135\pi\)
\(828\) 0 0
\(829\) −6.18613e6 −0.312631 −0.156316 0.987707i \(-0.549962\pi\)
−0.156316 + 0.987707i \(0.549962\pi\)
\(830\) 0 0
\(831\) −1.84114e7 −0.924880
\(832\) 0 0
\(833\) 7.05693e7i 3.52374i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 4.95720e6i 0.244581i
\(838\) 0 0
\(839\) 5.66754e6 0.277965 0.138982 0.990295i \(-0.455617\pi\)
0.138982 + 0.990295i \(0.455617\pi\)
\(840\) 0 0
\(841\) −1.55916e7 −0.760154
\(842\) 0 0
\(843\) 2.12710e7i 1.03091i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 6.51670e7i − 3.12118i
\(848\) 0 0
\(849\) −1.65151e7 −0.786343
\(850\) 0 0
\(851\) −1.85951e6 −0.0880185
\(852\) 0 0
\(853\) − 1.76010e7i − 0.828257i −0.910218 0.414129i \(-0.864086\pi\)
0.910218 0.414129i \(-0.135914\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 162162.i − 0.00754218i −0.999993 0.00377109i \(-0.998800\pi\)
0.999993 0.00377109i \(-0.00120038\pi\)
\(858\) 0 0
\(859\) 7.10520e6 0.328544 0.164272 0.986415i \(-0.447473\pi\)
0.164272 + 0.986415i \(0.447473\pi\)
\(860\) 0 0
\(861\) 7.91921e6 0.364061
\(862\) 0 0
\(863\) 4.08956e6i 0.186917i 0.995623 + 0.0934586i \(0.0297923\pi\)
−0.995623 + 0.0934586i \(0.970208\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.58524e7i 1.16803i
\(868\) 0 0
\(869\) −3.05276e7 −1.37133
\(870\) 0 0
\(871\) 9.79860e6 0.437641
\(872\) 0 0
\(873\) − 2.93657e6i − 0.130408i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 809194.i − 0.0355266i −0.999842 0.0177633i \(-0.994345\pi\)
0.999842 0.0177633i \(-0.00565453\pi\)
\(878\) 0 0
\(879\) −9.22144e6 −0.402556
\(880\) 0 0
\(881\) 3.90411e6 0.169466 0.0847329 0.996404i \(-0.472996\pi\)
0.0847329 + 0.996404i \(0.472996\pi\)
\(882\) 0 0
\(883\) 3.58290e7i 1.54644i 0.634138 + 0.773220i \(0.281355\pi\)
−0.634138 + 0.773220i \(0.718645\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 2.77571e7i − 1.18458i −0.805725 0.592290i \(-0.798224\pi\)
0.805725 0.592290i \(-0.201776\pi\)
\(888\) 0 0
\(889\) 6.64159e7 2.81850
\(890\) 0 0
\(891\) −4.73442e7 −1.99789
\(892\) 0 0
\(893\) − 1.64689e7i − 0.691094i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 8.15018e6i − 0.338210i
\(898\) 0 0
\(899\) −3.77060e6 −0.155601
\(900\) 0 0
\(901\) 5.50703e7 2.25999
\(902\) 0 0
\(903\) − 4.58861e7i − 1.87267i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 2.01914e7i − 0.814981i −0.913209 0.407490i \(-0.866404\pi\)
0.913209 0.407490i \(-0.133596\pi\)
\(908\) 0 0
\(909\) −3.44137e6 −0.138141
\(910\) 0 0
\(911\) 2.75179e7 1.09855 0.549274 0.835642i \(-0.314904\pi\)
0.549274 + 0.835642i \(0.314904\pi\)
\(912\) 0 0
\(913\) − 5.82384e7i − 2.31224i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 4.90311e7i − 1.92552i
\(918\) 0 0
\(919\) −1.31786e7 −0.514730 −0.257365 0.966314i \(-0.582854\pi\)
−0.257365 + 0.966314i \(0.582854\pi\)
\(920\) 0 0
\(921\) −2.53702e7 −0.985543
\(922\) 0 0
\(923\) − 195288.i − 0.00754521i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 1.19827e7i − 0.457988i
\(928\) 0 0
\(929\) −4.00688e7 −1.52323 −0.761617 0.648027i \(-0.775594\pi\)
−0.761617 + 0.648027i \(0.775594\pi\)
\(930\) 0 0
\(931\) −5.69565e7 −2.15362
\(932\) 0 0
\(933\) 5.01486e7i 1.88606i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 3.04258e7i − 1.13212i −0.824364 0.566060i \(-0.808467\pi\)
0.824364 0.566060i \(-0.191533\pi\)
\(938\) 0 0
\(939\) 2.79154e7 1.03319
\(940\) 0 0
\(941\) −3.26349e7 −1.20146 −0.600729 0.799452i \(-0.705123\pi\)
−0.600729 + 0.799452i \(0.705123\pi\)
\(942\) 0 0
\(943\) − 3.99596e6i − 0.146333i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.01534e7i − 1.09260i −0.837589 0.546300i \(-0.816036\pi\)
0.837589 0.546300i \(-0.183964\pi\)
\(948\) 0 0
\(949\) −1.29990e7 −0.468538
\(950\) 0 0
\(951\) −6.79180e6 −0.243519
\(952\) 0 0
\(953\) 303066.i 0.0108095i 0.999985 + 0.00540474i \(0.00172039\pi\)
−0.999985 + 0.00540474i \(0.998280\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 2.61901e7i − 0.924396i
\(958\) 0 0
\(959\) 1.16504e7 0.409066
\(960\) 0 0
\(961\) −2.57392e7 −0.899054
\(962\) 0 0
\(963\) − 1.48959e6i − 0.0517608i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 7.39863e6i 0.254440i 0.991875 + 0.127220i \(0.0406054\pi\)
−0.991875 + 0.127220i \(0.959395\pi\)
\(968\) 0 0
\(969\) −4.14929e7 −1.41959
\(970\) 0 0
\(971\) −6.18414e6 −0.210490 −0.105245 0.994446i \(-0.533563\pi\)
−0.105245 + 0.994446i \(0.533563\pi\)
\(972\) 0 0
\(973\) − 2.68998e7i − 0.910891i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1.63928e6i − 0.0549436i −0.999623 0.0274718i \(-0.991254\pi\)
0.999623 0.0274718i \(-0.00874565\pi\)
\(978\) 0 0
\(979\) 6.88367e7 2.29543
\(980\) 0 0
\(981\) −1.17455e7 −0.389671
\(982\) 0 0
\(983\) − 1.13020e7i − 0.373052i −0.982450 0.186526i \(-0.940277\pi\)
0.982450 0.186526i \(-0.0597229\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 5.25943e7i 1.71849i
\(988\) 0 0
\(989\) −2.31537e7 −0.752714
\(990\) 0 0
\(991\) 3.12643e6 0.101126 0.0505632 0.998721i \(-0.483898\pi\)
0.0505632 + 0.998721i \(0.483898\pi\)
\(992\) 0 0
\(993\) 2.93514e7i 0.944616i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.55827e7i 1.13371i 0.823818 + 0.566854i \(0.191840\pi\)
−0.823818 + 0.566854i \(0.808160\pi\)
\(998\) 0 0
\(999\) 2.46694e6 0.0782067
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.b.49.1 2
4.3 odd 2 400.6.c.e.49.2 2
5.2 odd 4 40.6.a.a.1.1 1
5.3 odd 4 200.6.a.d.1.1 1
5.4 even 2 inner 200.6.c.b.49.2 2
15.2 even 4 360.6.a.i.1.1 1
20.3 even 4 400.6.a.b.1.1 1
20.7 even 4 80.6.a.g.1.1 1
20.19 odd 2 400.6.c.e.49.1 2
40.27 even 4 320.6.a.d.1.1 1
40.37 odd 4 320.6.a.m.1.1 1
60.47 odd 4 720.6.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.6.a.a.1.1 1 5.2 odd 4
80.6.a.g.1.1 1 20.7 even 4
200.6.a.d.1.1 1 5.3 odd 4
200.6.c.b.49.1 2 1.1 even 1 trivial
200.6.c.b.49.2 2 5.4 even 2 inner
320.6.a.d.1.1 1 40.27 even 4
320.6.a.m.1.1 1 40.37 odd 4
360.6.a.i.1.1 1 15.2 even 4
400.6.a.b.1.1 1 20.3 even 4
400.6.c.e.49.1 2 20.19 odd 2
400.6.c.e.49.2 2 4.3 odd 2
720.6.a.k.1.1 1 60.47 odd 4