Properties

Label 2760.3.g.a.2161.7
Level $2760$
Weight $3$
Character 2760.2161
Analytic conductor $75.205$
Analytic rank $0$
Dimension $96$
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] = [2760,3,Mod(2161,2760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2760, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2760.2161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2760.g (of order \(2\), degree \(1\), 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: \(75.2045529634\)
Analytic rank: \(0\)
Dimension: \(96\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2161.7
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.42

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} -5.50203i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} -5.50203i q^{7} +3.00000 q^{9} +11.8537i q^{11} +2.34701 q^{13} +3.87298i q^{15} -0.110434i q^{17} -8.77566i q^{19} +9.52980i q^{21} +(17.5067 - 14.9169i) q^{23} -5.00000 q^{25} -5.19615 q^{27} -28.8089 q^{29} +18.1945 q^{31} -20.5312i q^{33} -12.3029 q^{35} +22.1105i q^{37} -4.06514 q^{39} +63.8105 q^{41} +46.9599i q^{43} -6.70820i q^{45} +2.76124 q^{47} +18.7276 q^{49} +0.191278i q^{51} -18.3207i q^{53} +26.5057 q^{55} +15.1999i q^{57} -68.2845 q^{59} -31.7306i q^{61} -16.5061i q^{63} -5.24807i q^{65} +49.2646i q^{67} +(-30.3226 + 25.8368i) q^{69} -65.3549 q^{71} +144.408 q^{73} +8.66025 q^{75} +65.2195 q^{77} +111.198i q^{79} +9.00000 q^{81} +19.2611i q^{83} -0.246939 q^{85} +49.8985 q^{87} +142.315i q^{89} -12.9133i q^{91} -31.5137 q^{93} -19.6230 q^{95} -90.9019i q^{97} +35.5611i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 96 q + 288 q^{9} - 16 q^{23} - 480 q^{25} - 80 q^{31} + 80 q^{35} + 48 q^{39} + 112 q^{41} + 32 q^{47} - 688 q^{49} - 80 q^{55} - 496 q^{59} - 96 q^{69} - 416 q^{71} - 320 q^{73} + 864 q^{81} + 192 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1657\) \(1841\) \(2071\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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
\(3\) −1.73205 −0.577350
\(4\) 0 0
\(5\) 2.23607i 0.447214i
\(6\) 0 0
\(7\) 5.50203i 0.786005i −0.919537 0.393002i \(-0.871436\pi\)
0.919537 0.393002i \(-0.128564\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 11.8537i 1.07761i 0.842431 + 0.538805i \(0.181124\pi\)
−0.842431 + 0.538805i \(0.818876\pi\)
\(12\) 0 0
\(13\) 2.34701 0.180539 0.0902696 0.995917i \(-0.471227\pi\)
0.0902696 + 0.995917i \(0.471227\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 0.110434i 0.00649615i −0.999995 0.00324807i \(-0.998966\pi\)
0.999995 0.00324807i \(-0.00103390\pi\)
\(18\) 0 0
\(19\) 8.77566i 0.461877i −0.972968 0.230938i \(-0.925820\pi\)
0.972968 0.230938i \(-0.0741796\pi\)
\(20\) 0 0
\(21\) 9.52980i 0.453800i
\(22\) 0 0
\(23\) 17.5067 14.9169i 0.761163 0.648561i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) −28.8089 −0.993411 −0.496706 0.867919i \(-0.665457\pi\)
−0.496706 + 0.867919i \(0.665457\pi\)
\(30\) 0 0
\(31\) 18.1945 0.586918 0.293459 0.955972i \(-0.405194\pi\)
0.293459 + 0.955972i \(0.405194\pi\)
\(32\) 0 0
\(33\) 20.5312i 0.622158i
\(34\) 0 0
\(35\) −12.3029 −0.351512
\(36\) 0 0
\(37\) 22.1105i 0.597582i 0.954319 + 0.298791i \(0.0965833\pi\)
−0.954319 + 0.298791i \(0.903417\pi\)
\(38\) 0 0
\(39\) −4.06514 −0.104234
\(40\) 0 0
\(41\) 63.8105 1.55635 0.778177 0.628045i \(-0.216145\pi\)
0.778177 + 0.628045i \(0.216145\pi\)
\(42\) 0 0
\(43\) 46.9599i 1.09209i 0.837756 + 0.546045i \(0.183867\pi\)
−0.837756 + 0.546045i \(0.816133\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) 2.76124 0.0587497 0.0293749 0.999568i \(-0.490648\pi\)
0.0293749 + 0.999568i \(0.490648\pi\)
\(48\) 0 0
\(49\) 18.7276 0.382196
\(50\) 0 0
\(51\) 0.191278i 0.00375055i
\(52\) 0 0
\(53\) 18.3207i 0.345674i −0.984950 0.172837i \(-0.944707\pi\)
0.984950 0.172837i \(-0.0552934\pi\)
\(54\) 0 0
\(55\) 26.5057 0.481922
\(56\) 0 0
\(57\) 15.1999i 0.266665i
\(58\) 0 0
\(59\) −68.2845 −1.15736 −0.578682 0.815553i \(-0.696433\pi\)
−0.578682 + 0.815553i \(0.696433\pi\)
\(60\) 0 0
\(61\) 31.7306i 0.520174i −0.965585 0.260087i \(-0.916249\pi\)
0.965585 0.260087i \(-0.0837512\pi\)
\(62\) 0 0
\(63\) 16.5061i 0.262002i
\(64\) 0 0
\(65\) 5.24807i 0.0807395i
\(66\) 0 0
\(67\) 49.2646i 0.735292i 0.929966 + 0.367646i \(0.119836\pi\)
−0.929966 + 0.367646i \(0.880164\pi\)
\(68\) 0 0
\(69\) −30.3226 + 25.8368i −0.439458 + 0.374447i
\(70\) 0 0
\(71\) −65.3549 −0.920492 −0.460246 0.887791i \(-0.652239\pi\)
−0.460246 + 0.887791i \(0.652239\pi\)
\(72\) 0 0
\(73\) 144.408 1.97819 0.989097 0.147268i \(-0.0470479\pi\)
0.989097 + 0.147268i \(0.0470479\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) 65.2195 0.847006
\(78\) 0 0
\(79\) 111.198i 1.40757i 0.710415 + 0.703783i \(0.248507\pi\)
−0.710415 + 0.703783i \(0.751493\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 19.2611i 0.232061i 0.993246 + 0.116031i \(0.0370171\pi\)
−0.993246 + 0.116031i \(0.962983\pi\)
\(84\) 0 0
\(85\) −0.246939 −0.00290517
\(86\) 0 0
\(87\) 49.8985 0.573546
\(88\) 0 0
\(89\) 142.315i 1.59905i 0.600635 + 0.799523i \(0.294915\pi\)
−0.600635 + 0.799523i \(0.705085\pi\)
\(90\) 0 0
\(91\) 12.9133i 0.141905i
\(92\) 0 0
\(93\) −31.5137 −0.338857
\(94\) 0 0
\(95\) −19.6230 −0.206558
\(96\) 0 0
\(97\) 90.9019i 0.937133i −0.883428 0.468566i \(-0.844771\pi\)
0.883428 0.468566i \(-0.155229\pi\)
\(98\) 0 0
\(99\) 35.5611i 0.359203i
\(100\) 0 0
\(101\) −1.74122 −0.0172398 −0.00861989 0.999963i \(-0.502744\pi\)
−0.00861989 + 0.999963i \(0.502744\pi\)
\(102\) 0 0
\(103\) 15.9065i 0.154432i −0.997014 0.0772160i \(-0.975397\pi\)
0.997014 0.0772160i \(-0.0246031\pi\)
\(104\) 0 0
\(105\) 21.3093 0.202946
\(106\) 0 0
\(107\) 140.119i 1.30953i −0.755834 0.654764i \(-0.772768\pi\)
0.755834 0.654764i \(-0.227232\pi\)
\(108\) 0 0
\(109\) 125.287i 1.14942i −0.818355 0.574712i \(-0.805114\pi\)
0.818355 0.574712i \(-0.194886\pi\)
\(110\) 0 0
\(111\) 38.2965i 0.345014i
\(112\) 0 0
\(113\) 182.184i 1.61225i −0.591745 0.806125i \(-0.701561\pi\)
0.591745 0.806125i \(-0.298439\pi\)
\(114\) 0 0
\(115\) −33.3552 39.1463i −0.290045 0.340402i
\(116\) 0 0
\(117\) 7.04103 0.0601797
\(118\) 0 0
\(119\) −0.607614 −0.00510600
\(120\) 0 0
\(121\) −19.5103 −0.161242
\(122\) 0 0
\(123\) −110.523 −0.898562
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) 46.3228 0.364747 0.182373 0.983229i \(-0.441622\pi\)
0.182373 + 0.983229i \(0.441622\pi\)
\(128\) 0 0
\(129\) 81.3369i 0.630519i
\(130\) 0 0
\(131\) 53.0627 0.405059 0.202529 0.979276i \(-0.435084\pi\)
0.202529 + 0.979276i \(0.435084\pi\)
\(132\) 0 0
\(133\) −48.2840 −0.363037
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 0.310275i 0.00226478i −0.999999 0.00113239i \(-0.999640\pi\)
0.999999 0.00113239i \(-0.000360451\pi\)
\(138\) 0 0
\(139\) 114.371 0.822817 0.411408 0.911451i \(-0.365037\pi\)
0.411408 + 0.911451i \(0.365037\pi\)
\(140\) 0 0
\(141\) −4.78260 −0.0339192
\(142\) 0 0
\(143\) 27.8207i 0.194551i
\(144\) 0 0
\(145\) 64.4187i 0.444267i
\(146\) 0 0
\(147\) −32.4372 −0.220661
\(148\) 0 0
\(149\) 59.2372i 0.397565i 0.980044 + 0.198783i \(0.0636987\pi\)
−0.980044 + 0.198783i \(0.936301\pi\)
\(150\) 0 0
\(151\) −132.709 −0.878869 −0.439434 0.898275i \(-0.644821\pi\)
−0.439434 + 0.898275i \(0.644821\pi\)
\(152\) 0 0
\(153\) 0.331303i 0.00216538i
\(154\) 0 0
\(155\) 40.6841i 0.262478i
\(156\) 0 0
\(157\) 193.687i 1.23368i −0.787090 0.616838i \(-0.788413\pi\)
0.787090 0.616838i \(-0.211587\pi\)
\(158\) 0 0
\(159\) 31.7325i 0.199575i
\(160\) 0 0
\(161\) −82.0733 96.3227i −0.509772 0.598278i
\(162\) 0 0
\(163\) 75.8510 0.465343 0.232672 0.972555i \(-0.425253\pi\)
0.232672 + 0.972555i \(0.425253\pi\)
\(164\) 0 0
\(165\) −45.9092 −0.278238
\(166\) 0 0
\(167\) 1.21238 0.00725975 0.00362988 0.999993i \(-0.498845\pi\)
0.00362988 + 0.999993i \(0.498845\pi\)
\(168\) 0 0
\(169\) −163.492 −0.967406
\(170\) 0 0
\(171\) 26.3270i 0.153959i
\(172\) 0 0
\(173\) 16.8561 0.0974343 0.0487171 0.998813i \(-0.484487\pi\)
0.0487171 + 0.998813i \(0.484487\pi\)
\(174\) 0 0
\(175\) 27.5102i 0.157201i
\(176\) 0 0
\(177\) 118.272 0.668204
\(178\) 0 0
\(179\) 19.5765 0.109366 0.0546829 0.998504i \(-0.482585\pi\)
0.0546829 + 0.998504i \(0.482585\pi\)
\(180\) 0 0
\(181\) 238.939i 1.32011i −0.751219 0.660053i \(-0.770534\pi\)
0.751219 0.660053i \(-0.229466\pi\)
\(182\) 0 0
\(183\) 54.9590i 0.300322i
\(184\) 0 0
\(185\) 49.4406 0.267247
\(186\) 0 0
\(187\) 1.30906 0.00700031
\(188\) 0 0
\(189\) 28.5894i 0.151267i
\(190\) 0 0
\(191\) 258.841i 1.35519i −0.735435 0.677595i \(-0.763022\pi\)
0.735435 0.677595i \(-0.236978\pi\)
\(192\) 0 0
\(193\) 144.039 0.746316 0.373158 0.927768i \(-0.378275\pi\)
0.373158 + 0.927768i \(0.378275\pi\)
\(194\) 0 0
\(195\) 9.08992i 0.0466150i
\(196\) 0 0
\(197\) −19.8192 −0.100605 −0.0503026 0.998734i \(-0.516019\pi\)
−0.0503026 + 0.998734i \(0.516019\pi\)
\(198\) 0 0
\(199\) 258.512i 1.29905i −0.760339 0.649527i \(-0.774967\pi\)
0.760339 0.649527i \(-0.225033\pi\)
\(200\) 0 0
\(201\) 85.3287i 0.424521i
\(202\) 0 0
\(203\) 158.508i 0.780826i
\(204\) 0 0
\(205\) 142.685i 0.696023i
\(206\) 0 0
\(207\) 52.5202 44.7507i 0.253721 0.216187i
\(208\) 0 0
\(209\) 104.024 0.497723
\(210\) 0 0
\(211\) 381.861 1.80977 0.904884 0.425658i \(-0.139957\pi\)
0.904884 + 0.425658i \(0.139957\pi\)
\(212\) 0 0
\(213\) 113.198 0.531446
\(214\) 0 0
\(215\) 105.005 0.488398
\(216\) 0 0
\(217\) 100.107i 0.461321i
\(218\) 0 0
\(219\) −250.122 −1.14211
\(220\) 0 0
\(221\) 0.259191i 0.00117281i
\(222\) 0 0
\(223\) 83.4115 0.374043 0.187021 0.982356i \(-0.440117\pi\)
0.187021 + 0.982356i \(0.440117\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 150.009i 0.660834i 0.943835 + 0.330417i \(0.107189\pi\)
−0.943835 + 0.330417i \(0.892811\pi\)
\(228\) 0 0
\(229\) 389.190i 1.69952i −0.527171 0.849759i \(-0.676747\pi\)
0.527171 0.849759i \(-0.323253\pi\)
\(230\) 0 0
\(231\) −112.963 −0.489019
\(232\) 0 0
\(233\) 449.403 1.92877 0.964384 0.264508i \(-0.0852094\pi\)
0.964384 + 0.264508i \(0.0852094\pi\)
\(234\) 0 0
\(235\) 6.17432i 0.0262737i
\(236\) 0 0
\(237\) 192.600i 0.812658i
\(238\) 0 0
\(239\) −362.830 −1.51812 −0.759059 0.651022i \(-0.774340\pi\)
−0.759059 + 0.651022i \(0.774340\pi\)
\(240\) 0 0
\(241\) 50.9144i 0.211263i −0.994405 0.105632i \(-0.966314\pi\)
0.994405 0.105632i \(-0.0336864\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 41.8762i 0.170923i
\(246\) 0 0
\(247\) 20.5965i 0.0833868i
\(248\) 0 0
\(249\) 33.3612i 0.133981i
\(250\) 0 0
\(251\) 34.2170i 0.136323i 0.997674 + 0.0681614i \(0.0217133\pi\)
−0.997674 + 0.0681614i \(0.978287\pi\)
\(252\) 0 0
\(253\) 176.820 + 207.520i 0.698895 + 0.820236i
\(254\) 0 0
\(255\) 0.427711 0.00167730
\(256\) 0 0
\(257\) 442.180 1.72055 0.860273 0.509834i \(-0.170293\pi\)
0.860273 + 0.509834i \(0.170293\pi\)
\(258\) 0 0
\(259\) 121.653 0.469702
\(260\) 0 0
\(261\) −86.4268 −0.331137
\(262\) 0 0
\(263\) 388.400i 1.47681i 0.674360 + 0.738403i \(0.264420\pi\)
−0.674360 + 0.738403i \(0.735580\pi\)
\(264\) 0 0
\(265\) −40.9664 −0.154590
\(266\) 0 0
\(267\) 246.497i 0.923210i
\(268\) 0 0
\(269\) 366.196 1.36132 0.680662 0.732598i \(-0.261692\pi\)
0.680662 + 0.732598i \(0.261692\pi\)
\(270\) 0 0
\(271\) −215.006 −0.793381 −0.396691 0.917952i \(-0.629841\pi\)
−0.396691 + 0.917952i \(0.629841\pi\)
\(272\) 0 0
\(273\) 22.3665i 0.0819287i
\(274\) 0 0
\(275\) 59.2685i 0.215522i
\(276\) 0 0
\(277\) 326.108 1.17729 0.588643 0.808393i \(-0.299663\pi\)
0.588643 + 0.808393i \(0.299663\pi\)
\(278\) 0 0
\(279\) 54.5834 0.195639
\(280\) 0 0
\(281\) 126.370i 0.449715i 0.974392 + 0.224857i \(0.0721916\pi\)
−0.974392 + 0.224857i \(0.927808\pi\)
\(282\) 0 0
\(283\) 182.918i 0.646352i 0.946339 + 0.323176i \(0.104751\pi\)
−0.946339 + 0.323176i \(0.895249\pi\)
\(284\) 0 0
\(285\) 33.9880 0.119256
\(286\) 0 0
\(287\) 351.088i 1.22330i
\(288\) 0 0
\(289\) 288.988 0.999958
\(290\) 0 0
\(291\) 157.447i 0.541054i
\(292\) 0 0
\(293\) 206.613i 0.705165i −0.935781 0.352583i \(-0.885304\pi\)
0.935781 0.352583i \(-0.114696\pi\)
\(294\) 0 0
\(295\) 152.689i 0.517589i
\(296\) 0 0
\(297\) 61.5936i 0.207386i
\(298\) 0 0
\(299\) 41.0885 35.0101i 0.137420 0.117091i
\(300\) 0 0
\(301\) 258.375 0.858388
\(302\) 0 0
\(303\) 3.01588 0.00995339
\(304\) 0 0
\(305\) −70.9517 −0.232629
\(306\) 0 0
\(307\) 122.490 0.398990 0.199495 0.979899i \(-0.436070\pi\)
0.199495 + 0.979899i \(0.436070\pi\)
\(308\) 0 0
\(309\) 27.5509i 0.0891613i
\(310\) 0 0
\(311\) 518.339 1.66669 0.833343 0.552757i \(-0.186424\pi\)
0.833343 + 0.552757i \(0.186424\pi\)
\(312\) 0 0
\(313\) 262.353i 0.838189i −0.907943 0.419094i \(-0.862348\pi\)
0.907943 0.419094i \(-0.137652\pi\)
\(314\) 0 0
\(315\) −36.9088 −0.117171
\(316\) 0 0
\(317\) −344.774 −1.08762 −0.543808 0.839209i \(-0.683018\pi\)
−0.543808 + 0.839209i \(0.683018\pi\)
\(318\) 0 0
\(319\) 341.492i 1.07051i
\(320\) 0 0
\(321\) 242.694i 0.756056i
\(322\) 0 0
\(323\) −0.969135 −0.00300042
\(324\) 0 0
\(325\) −11.7350 −0.0361078
\(326\) 0 0
\(327\) 217.004i 0.663621i
\(328\) 0 0
\(329\) 15.1924i 0.0461776i
\(330\) 0 0
\(331\) −248.689 −0.751328 −0.375664 0.926756i \(-0.622585\pi\)
−0.375664 + 0.926756i \(0.622585\pi\)
\(332\) 0 0
\(333\) 66.3316i 0.199194i
\(334\) 0 0
\(335\) 110.159 0.328833
\(336\) 0 0
\(337\) 95.6156i 0.283726i 0.989886 + 0.141863i \(0.0453092\pi\)
−0.989886 + 0.141863i \(0.954691\pi\)
\(338\) 0 0
\(339\) 315.552i 0.930833i
\(340\) 0 0
\(341\) 215.672i 0.632469i
\(342\) 0 0
\(343\) 372.640i 1.08641i
\(344\) 0 0
\(345\) 57.7729 + 67.8034i 0.167458 + 0.196531i
\(346\) 0 0
\(347\) −198.267 −0.571374 −0.285687 0.958323i \(-0.592222\pi\)
−0.285687 + 0.958323i \(0.592222\pi\)
\(348\) 0 0
\(349\) −525.363 −1.50534 −0.752669 0.658399i \(-0.771234\pi\)
−0.752669 + 0.658399i \(0.771234\pi\)
\(350\) 0 0
\(351\) −12.1954 −0.0347448
\(352\) 0 0
\(353\) 564.683 1.59967 0.799834 0.600221i \(-0.204921\pi\)
0.799834 + 0.600221i \(0.204921\pi\)
\(354\) 0 0
\(355\) 146.138i 0.411656i
\(356\) 0 0
\(357\) 1.05242 0.00294795
\(358\) 0 0
\(359\) 635.765i 1.77093i −0.464704 0.885466i \(-0.653839\pi\)
0.464704 0.885466i \(-0.346161\pi\)
\(360\) 0 0
\(361\) 283.988 0.786670
\(362\) 0 0
\(363\) 33.7928 0.0930930
\(364\) 0 0
\(365\) 322.906i 0.884675i
\(366\) 0 0
\(367\) 55.1742i 0.150338i 0.997171 + 0.0751692i \(0.0239497\pi\)
−0.997171 + 0.0751692i \(0.976050\pi\)
\(368\) 0 0
\(369\) 191.432 0.518785
\(370\) 0 0
\(371\) −100.801 −0.271702
\(372\) 0 0
\(373\) 345.272i 0.925662i −0.886447 0.462831i \(-0.846834\pi\)
0.886447 0.462831i \(-0.153166\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) −67.6148 −0.179350
\(378\) 0 0
\(379\) 577.228i 1.52303i −0.648148 0.761514i \(-0.724456\pi\)
0.648148 0.761514i \(-0.275544\pi\)
\(380\) 0 0
\(381\) −80.2335 −0.210587
\(382\) 0 0
\(383\) 510.264i 1.33228i 0.745826 + 0.666141i \(0.232055\pi\)
−0.745826 + 0.666141i \(0.767945\pi\)
\(384\) 0 0
\(385\) 145.835i 0.378793i
\(386\) 0 0
\(387\) 140.880i 0.364030i
\(388\) 0 0
\(389\) 122.802i 0.315686i 0.987464 + 0.157843i \(0.0504539\pi\)
−0.987464 + 0.157843i \(0.949546\pi\)
\(390\) 0 0
\(391\) −1.64734 1.93335i −0.00421314 0.00494463i
\(392\) 0 0
\(393\) −91.9073 −0.233861
\(394\) 0 0
\(395\) 248.646 0.629482
\(396\) 0 0
\(397\) 93.2998 0.235012 0.117506 0.993072i \(-0.462510\pi\)
0.117506 + 0.993072i \(0.462510\pi\)
\(398\) 0 0
\(399\) 83.6303 0.209600
\(400\) 0 0
\(401\) 414.895i 1.03465i −0.855789 0.517326i \(-0.826928\pi\)
0.855789 0.517326i \(-0.173072\pi\)
\(402\) 0 0
\(403\) 42.7026 0.105962
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) −262.091 −0.643959
\(408\) 0 0
\(409\) 651.580 1.59311 0.796553 0.604569i \(-0.206655\pi\)
0.796553 + 0.604569i \(0.206655\pi\)
\(410\) 0 0
\(411\) 0.537411i 0.00130757i
\(412\) 0 0
\(413\) 375.704i 0.909694i
\(414\) 0 0
\(415\) 43.0691 0.103781
\(416\) 0 0
\(417\) −198.097 −0.475053
\(418\) 0 0
\(419\) 161.426i 0.385265i 0.981271 + 0.192632i \(0.0617025\pi\)
−0.981271 + 0.192632i \(0.938298\pi\)
\(420\) 0 0
\(421\) 644.943i 1.53193i −0.642882 0.765965i \(-0.722262\pi\)
0.642882 0.765965i \(-0.277738\pi\)
\(422\) 0 0
\(423\) 8.28371 0.0195832
\(424\) 0 0
\(425\) 0.552172i 0.00129923i
\(426\) 0 0
\(427\) −174.583 −0.408859
\(428\) 0 0
\(429\) 48.1869i 0.112324i
\(430\) 0 0
\(431\) 129.874i 0.301331i 0.988585 + 0.150666i \(0.0481417\pi\)
−0.988585 + 0.150666i \(0.951858\pi\)
\(432\) 0 0
\(433\) 659.250i 1.52252i −0.648449 0.761258i \(-0.724582\pi\)
0.648449 0.761258i \(-0.275418\pi\)
\(434\) 0 0
\(435\) 111.576i 0.256498i
\(436\) 0 0
\(437\) −130.906 153.633i −0.299555 0.351563i
\(438\) 0 0
\(439\) −367.167 −0.836371 −0.418185 0.908362i \(-0.637334\pi\)
−0.418185 + 0.908362i \(0.637334\pi\)
\(440\) 0 0
\(441\) 56.1828 0.127399
\(442\) 0 0
\(443\) −335.706 −0.757801 −0.378901 0.925437i \(-0.623698\pi\)
−0.378901 + 0.925437i \(0.623698\pi\)
\(444\) 0 0
\(445\) 318.226 0.715115
\(446\) 0 0
\(447\) 102.602i 0.229534i
\(448\) 0 0
\(449\) −842.623 −1.87667 −0.938333 0.345732i \(-0.887630\pi\)
−0.938333 + 0.345732i \(0.887630\pi\)
\(450\) 0 0
\(451\) 756.391i 1.67714i
\(452\) 0 0
\(453\) 229.859 0.507415
\(454\) 0 0
\(455\) −28.8751 −0.0634617
\(456\) 0 0
\(457\) 267.223i 0.584732i −0.956307 0.292366i \(-0.905557\pi\)
0.956307 0.292366i \(-0.0944425\pi\)
\(458\) 0 0
\(459\) 0.573834i 0.00125018i
\(460\) 0 0
\(461\) 243.817 0.528887 0.264444 0.964401i \(-0.414812\pi\)
0.264444 + 0.964401i \(0.414812\pi\)
\(462\) 0 0
\(463\) 580.692 1.25419 0.627097 0.778941i \(-0.284243\pi\)
0.627097 + 0.778941i \(0.284243\pi\)
\(464\) 0 0
\(465\) 70.4669i 0.151542i
\(466\) 0 0
\(467\) 157.556i 0.337379i 0.985669 + 0.168690i \(0.0539536\pi\)
−0.985669 + 0.168690i \(0.946046\pi\)
\(468\) 0 0
\(469\) 271.055 0.577943
\(470\) 0 0
\(471\) 335.476i 0.712263i
\(472\) 0 0
\(473\) −556.648 −1.17685
\(474\) 0 0
\(475\) 43.8783i 0.0923753i
\(476\) 0 0
\(477\) 54.9622i 0.115225i
\(478\) 0 0
\(479\) 454.526i 0.948907i −0.880281 0.474453i \(-0.842646\pi\)
0.880281 0.474453i \(-0.157354\pi\)
\(480\) 0 0
\(481\) 51.8936i 0.107887i
\(482\) 0 0
\(483\) 142.155 + 166.836i 0.294317 + 0.345416i
\(484\) 0 0
\(485\) −203.263 −0.419099
\(486\) 0 0
\(487\) 237.227 0.487118 0.243559 0.969886i \(-0.421685\pi\)
0.243559 + 0.969886i \(0.421685\pi\)
\(488\) 0 0
\(489\) −131.378 −0.268666
\(490\) 0 0
\(491\) −561.357 −1.14329 −0.571646 0.820500i \(-0.693695\pi\)
−0.571646 + 0.820500i \(0.693695\pi\)
\(492\) 0 0
\(493\) 3.18150i 0.00645334i
\(494\) 0 0
\(495\) 79.5171 0.160641
\(496\) 0 0
\(497\) 359.585i 0.723511i
\(498\) 0 0
\(499\) 239.009 0.478977 0.239488 0.970899i \(-0.423020\pi\)
0.239488 + 0.970899i \(0.423020\pi\)
\(500\) 0 0
\(501\) −2.09990 −0.00419142
\(502\) 0 0
\(503\) 556.669i 1.10670i 0.832950 + 0.553348i \(0.186650\pi\)
−0.832950 + 0.553348i \(0.813350\pi\)
\(504\) 0 0
\(505\) 3.89348i 0.00770986i
\(506\) 0 0
\(507\) 283.176 0.558532
\(508\) 0 0
\(509\) −938.758 −1.84432 −0.922160 0.386810i \(-0.873577\pi\)
−0.922160 + 0.386810i \(0.873577\pi\)
\(510\) 0 0
\(511\) 794.538i 1.55487i
\(512\) 0 0
\(513\) 45.5997i 0.0888882i
\(514\) 0 0
\(515\) −35.5680 −0.0690641
\(516\) 0 0
\(517\) 32.7309i 0.0633093i
\(518\) 0 0
\(519\) −29.1957 −0.0562537
\(520\) 0 0
\(521\) 698.305i 1.34032i −0.742218 0.670158i \(-0.766226\pi\)
0.742218 0.670158i \(-0.233774\pi\)
\(522\) 0 0
\(523\) 640.010i 1.22373i 0.790963 + 0.611864i \(0.209580\pi\)
−0.790963 + 0.611864i \(0.790420\pi\)
\(524\) 0 0
\(525\) 47.6490i 0.0907600i
\(526\) 0 0
\(527\) 2.00930i 0.00381271i
\(528\) 0 0
\(529\) 83.9726 522.293i 0.158738 0.987321i
\(530\) 0 0
\(531\) −204.853 −0.385788
\(532\) 0 0
\(533\) 149.764 0.280983
\(534\) 0 0
\(535\) −313.317 −0.585639
\(536\) 0 0
\(537\) −33.9074 −0.0631423
\(538\) 0 0
\(539\) 221.992i 0.411858i
\(540\) 0 0
\(541\) −370.932 −0.685641 −0.342820 0.939401i \(-0.611382\pi\)
−0.342820 + 0.939401i \(0.611382\pi\)
\(542\) 0 0
\(543\) 413.855i 0.762163i
\(544\) 0 0
\(545\) −280.151 −0.514038
\(546\) 0 0
\(547\) −312.903 −0.572035 −0.286018 0.958224i \(-0.592332\pi\)
−0.286018 + 0.958224i \(0.592332\pi\)
\(548\) 0 0
\(549\) 95.1918i 0.173391i
\(550\) 0 0
\(551\) 252.817i 0.458833i
\(552\) 0 0
\(553\) 611.813 1.10635
\(554\) 0 0
\(555\) −85.6337 −0.154295
\(556\) 0 0
\(557\) 100.417i 0.180281i −0.995929 0.0901406i \(-0.971268\pi\)
0.995929 0.0901406i \(-0.0287316\pi\)
\(558\) 0 0
\(559\) 110.215i 0.197165i
\(560\) 0 0
\(561\) −2.26735 −0.00404163
\(562\) 0 0
\(563\) 37.5847i 0.0667578i −0.999443 0.0333789i \(-0.989373\pi\)
0.999443 0.0333789i \(-0.0106268\pi\)
\(564\) 0 0
\(565\) −407.376 −0.721020
\(566\) 0 0
\(567\) 49.5183i 0.0873339i
\(568\) 0 0
\(569\) 53.0312i 0.0932008i 0.998914 + 0.0466004i \(0.0148387\pi\)
−0.998914 + 0.0466004i \(0.985161\pi\)
\(570\) 0 0
\(571\) 66.0754i 0.115719i −0.998325 0.0578593i \(-0.981573\pi\)
0.998325 0.0578593i \(-0.0184275\pi\)
\(572\) 0 0
\(573\) 448.326i 0.782420i
\(574\) 0 0
\(575\) −87.5337 + 74.5845i −0.152233 + 0.129712i
\(576\) 0 0
\(577\) 374.008 0.648194 0.324097 0.946024i \(-0.394940\pi\)
0.324097 + 0.946024i \(0.394940\pi\)
\(578\) 0 0
\(579\) −249.483 −0.430886
\(580\) 0 0
\(581\) 105.975 0.182401
\(582\) 0 0
\(583\) 217.169 0.372502
\(584\) 0 0
\(585\) 15.7442i 0.0269132i
\(586\) 0 0
\(587\) −78.7875 −0.134221 −0.0671103 0.997746i \(-0.521378\pi\)
−0.0671103 + 0.997746i \(0.521378\pi\)
\(588\) 0 0
\(589\) 159.668i 0.271084i
\(590\) 0 0
\(591\) 34.3279 0.0580845
\(592\) 0 0
\(593\) −444.050 −0.748820 −0.374410 0.927263i \(-0.622155\pi\)
−0.374410 + 0.927263i \(0.622155\pi\)
\(594\) 0 0
\(595\) 1.35867i 0.00228347i
\(596\) 0 0
\(597\) 447.755i 0.750009i
\(598\) 0 0
\(599\) 604.141 1.00858 0.504291 0.863534i \(-0.331754\pi\)
0.504291 + 0.863534i \(0.331754\pi\)
\(600\) 0 0
\(601\) 362.483 0.603133 0.301566 0.953445i \(-0.402490\pi\)
0.301566 + 0.953445i \(0.402490\pi\)
\(602\) 0 0
\(603\) 147.794i 0.245097i
\(604\) 0 0
\(605\) 43.6263i 0.0721095i
\(606\) 0 0
\(607\) −589.187 −0.970655 −0.485327 0.874333i \(-0.661300\pi\)
−0.485327 + 0.874333i \(0.661300\pi\)
\(608\) 0 0
\(609\) 274.543i 0.450810i
\(610\) 0 0
\(611\) 6.48065 0.0106066
\(612\) 0 0
\(613\) 729.517i 1.19008i 0.803697 + 0.595038i \(0.202863\pi\)
−0.803697 + 0.595038i \(0.797137\pi\)
\(614\) 0 0
\(615\) 247.137i 0.401849i
\(616\) 0 0
\(617\) 750.992i 1.21717i 0.793490 + 0.608584i \(0.208262\pi\)
−0.793490 + 0.608584i \(0.791738\pi\)
\(618\) 0 0
\(619\) 203.872i 0.329357i −0.986347 0.164679i \(-0.947341\pi\)
0.986347 0.164679i \(-0.0526587\pi\)
\(620\) 0 0
\(621\) −90.9677 + 77.5105i −0.146486 + 0.124816i
\(622\) 0 0
\(623\) 783.023 1.25686
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −180.175 −0.287360
\(628\) 0 0
\(629\) 2.44176 0.00388198
\(630\) 0 0
\(631\) 43.2711i 0.0685754i 0.999412 + 0.0342877i \(0.0109163\pi\)
−0.999412 + 0.0342877i \(0.989084\pi\)
\(632\) 0 0
\(633\) −661.403 −1.04487
\(634\) 0 0
\(635\) 103.581i 0.163120i
\(636\) 0 0
\(637\) 43.9539 0.0690014
\(638\) 0 0
\(639\) −196.065 −0.306831
\(640\) 0 0
\(641\) 359.821i 0.561343i 0.959804 + 0.280671i \(0.0905571\pi\)
−0.959804 + 0.280671i \(0.909443\pi\)
\(642\) 0 0
\(643\) 8.11907i 0.0126269i −0.999980 0.00631343i \(-0.997990\pi\)
0.999980 0.00631343i \(-0.00200964\pi\)
\(644\) 0 0
\(645\) −181.875 −0.281976
\(646\) 0 0
\(647\) −883.871 −1.36611 −0.683053 0.730368i \(-0.739348\pi\)
−0.683053 + 0.730368i \(0.739348\pi\)
\(648\) 0 0
\(649\) 809.424i 1.24719i
\(650\) 0 0
\(651\) 173.390i 0.266344i
\(652\) 0 0
\(653\) 139.393 0.213466 0.106733 0.994288i \(-0.465961\pi\)
0.106733 + 0.994288i \(0.465961\pi\)
\(654\) 0 0
\(655\) 118.652i 0.181148i
\(656\) 0 0
\(657\) 433.224 0.659398
\(658\) 0 0
\(659\) 359.157i 0.545003i 0.962155 + 0.272502i \(0.0878510\pi\)
−0.962155 + 0.272502i \(0.912149\pi\)
\(660\) 0 0
\(661\) 890.126i 1.34664i 0.739353 + 0.673318i \(0.235131\pi\)
−0.739353 + 0.673318i \(0.764869\pi\)
\(662\) 0 0
\(663\) 0.448931i 0.000677121i
\(664\) 0 0
\(665\) 107.966i 0.162355i
\(666\) 0 0
\(667\) −504.351 + 429.740i −0.756148 + 0.644287i
\(668\) 0 0
\(669\) −144.473 −0.215954
\(670\) 0 0
\(671\) 376.125 0.560544
\(672\) 0 0
\(673\) 326.453 0.485072 0.242536 0.970142i \(-0.422021\pi\)
0.242536 + 0.970142i \(0.422021\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 122.684i 0.181217i 0.995887 + 0.0906086i \(0.0288812\pi\)
−0.995887 + 0.0906086i \(0.971119\pi\)
\(678\) 0 0
\(679\) −500.145 −0.736591
\(680\) 0 0
\(681\) 259.824i 0.381533i
\(682\) 0 0
\(683\) 391.600 0.573352 0.286676 0.958028i \(-0.407450\pi\)
0.286676 + 0.958028i \(0.407450\pi\)
\(684\) 0 0
\(685\) −0.693795 −0.00101284
\(686\) 0 0
\(687\) 674.096i 0.981217i
\(688\) 0 0
\(689\) 42.9989i 0.0624077i
\(690\) 0 0
\(691\) 1105.57 1.59996 0.799979 0.600028i \(-0.204844\pi\)
0.799979 + 0.600028i \(0.204844\pi\)
\(692\) 0 0
\(693\) 195.658 0.282335
\(694\) 0 0
\(695\) 255.742i 0.367975i
\(696\) 0 0
\(697\) 7.04688i 0.0101103i
\(698\) 0 0
\(699\) −778.388 −1.11357
\(700\) 0 0
\(701\) 82.7851i 0.118096i −0.998255 0.0590479i \(-0.981194\pi\)
0.998255 0.0590479i \(-0.0188065\pi\)
\(702\) 0 0
\(703\) 194.034 0.276009
\(704\) 0 0
\(705\) 10.6942i 0.0151691i
\(706\) 0 0
\(707\) 9.58024i 0.0135505i
\(708\) 0 0
\(709\) 853.003i 1.20311i 0.798833 + 0.601553i \(0.205451\pi\)
−0.798833 + 0.601553i \(0.794549\pi\)
\(710\) 0 0
\(711\) 333.593i 0.469188i
\(712\) 0 0
\(713\) 318.526 271.405i 0.446740 0.380652i
\(714\) 0 0
\(715\) 62.2091 0.0870057
\(716\) 0 0
\(717\) 628.440 0.876485
\(718\) 0 0
\(719\) −633.465 −0.881037 −0.440518 0.897744i \(-0.645205\pi\)
−0.440518 + 0.897744i \(0.645205\pi\)
\(720\) 0 0
\(721\) −87.5181 −0.121384
\(722\) 0 0
\(723\) 88.1864i 0.121973i
\(724\) 0 0
\(725\) 144.045 0.198682
\(726\) 0 0
\(727\) 353.114i 0.485713i −0.970062 0.242857i \(-0.921916\pi\)
0.970062 0.242857i \(-0.0780845\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 5.18599 0.00709438
\(732\) 0 0
\(733\) 399.217i 0.544634i 0.962208 + 0.272317i \(0.0877899\pi\)
−0.962208 + 0.272317i \(0.912210\pi\)
\(734\) 0 0
\(735\) 72.5317i 0.0986826i
\(736\) 0 0
\(737\) −583.967 −0.792357
\(738\) 0 0
\(739\) 233.005 0.315298 0.157649 0.987495i \(-0.449609\pi\)
0.157649 + 0.987495i \(0.449609\pi\)
\(740\) 0 0
\(741\) 35.6743i 0.0481434i
\(742\) 0 0
\(743\) 861.438i 1.15940i −0.814828 0.579702i \(-0.803169\pi\)
0.814828 0.579702i \(-0.196831\pi\)
\(744\) 0 0
\(745\) 132.458 0.177796
\(746\) 0 0
\(747\) 57.7833i 0.0773538i
\(748\) 0 0
\(749\) −770.942 −1.02930
\(750\) 0 0
\(751\) 850.083i 1.13194i 0.824427 + 0.565968i \(0.191497\pi\)
−0.824427 + 0.565968i \(0.808503\pi\)
\(752\) 0 0
\(753\) 59.2657i 0.0787060i
\(754\) 0 0
\(755\) 296.747i 0.393042i
\(756\) 0 0
\(757\) 51.8570i 0.0685032i 0.999413 + 0.0342516i \(0.0109048\pi\)
−0.999413 + 0.0342516i \(0.989095\pi\)
\(758\) 0 0
\(759\) −306.262 359.435i −0.403507 0.473564i
\(760\) 0 0
\(761\) −423.114 −0.555998 −0.277999 0.960581i \(-0.589671\pi\)
−0.277999 + 0.960581i \(0.589671\pi\)
\(762\) 0 0
\(763\) −689.335 −0.903454
\(764\) 0 0
\(765\) −0.740817 −0.000968388
\(766\) 0 0
\(767\) −160.264 −0.208949
\(768\) 0 0
\(769\) 14.3387i 0.0186459i 0.999957 + 0.00932293i \(0.00296762\pi\)
−0.999957 + 0.00932293i \(0.997032\pi\)
\(770\) 0 0
\(771\) −765.879 −0.993358
\(772\) 0 0
\(773\) 1273.43i 1.64739i 0.567037 + 0.823693i \(0.308090\pi\)
−0.567037 + 0.823693i \(0.691910\pi\)
\(774\) 0 0
\(775\) −90.9723 −0.117384
\(776\) 0 0
\(777\) −210.709 −0.271183
\(778\) 0 0
\(779\) 559.979i 0.718844i
\(780\) 0 0
\(781\) 774.698i 0.991931i
\(782\) 0 0
\(783\) 149.696 0.191182
\(784\) 0 0
\(785\) −433.098 −0.551717
\(786\) 0 0
\(787\) 1.32875i 0.00168838i 1.00000 0.000844190i \(0.000268714\pi\)
−1.00000 0.000844190i \(0.999731\pi\)
\(788\) 0 0
\(789\) 672.729i 0.852634i
\(790\) 0 0
\(791\) −1002.38 −1.26724
\(792\) 0 0
\(793\) 74.4720i 0.0939117i
\(794\) 0 0
\(795\) 70.9559 0.0892527
\(796\) 0 0
\(797\) 367.697i 0.461351i 0.973031 + 0.230675i \(0.0740936\pi\)
−0.973031 + 0.230675i \(0.925906\pi\)
\(798\) 0 0
\(799\)