Properties

Label 2760.3.g.a.2161.11
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 2161.11
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.38

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} -1.58351i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} -1.58351i q^{7} +3.00000 q^{9} -7.96580i q^{11} +2.01096 q^{13} +3.87298i q^{15} -2.85301i q^{17} +15.7647i q^{19} +2.74273i q^{21} +(-22.6256 - 4.13331i) q^{23} -5.00000 q^{25} -5.19615 q^{27} -37.4744 q^{29} +17.2247 q^{31} +13.7972i q^{33} -3.54085 q^{35} +60.3907i q^{37} -3.48309 q^{39} -21.0549 q^{41} -15.3496i q^{43} -6.70820i q^{45} +7.43748 q^{47} +46.4925 q^{49} +4.94155i q^{51} +30.4235i q^{53} -17.8121 q^{55} -27.3053i q^{57} +36.6568 q^{59} -24.7355i q^{61} -4.75054i q^{63} -4.49665i q^{65} -124.907i q^{67} +(39.1886 + 7.15910i) q^{69} -119.986 q^{71} -66.2872 q^{73} +8.66025 q^{75} -12.6140 q^{77} +47.4317i q^{79} +9.00000 q^{81} -74.7878i q^{83} -6.37952 q^{85} +64.9076 q^{87} +78.4768i q^{89} -3.18439i q^{91} -29.8340 q^{93} +35.2509 q^{95} +107.673i q^{97} -23.8974i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 1.58351i 0.226216i −0.993583 0.113108i \(-0.963919\pi\)
0.993583 0.113108i \(-0.0360807\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 7.96580i 0.724163i −0.932146 0.362082i \(-0.882066\pi\)
0.932146 0.362082i \(-0.117934\pi\)
\(12\) 0 0
\(13\) 2.01096 0.154689 0.0773447 0.997004i \(-0.475356\pi\)
0.0773447 + 0.997004i \(0.475356\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 2.85301i 0.167824i −0.996473 0.0839120i \(-0.973259\pi\)
0.996473 0.0839120i \(-0.0267415\pi\)
\(18\) 0 0
\(19\) 15.7647i 0.829721i 0.909885 + 0.414861i \(0.136170\pi\)
−0.909885 + 0.414861i \(0.863830\pi\)
\(20\) 0 0
\(21\) 2.74273i 0.130606i
\(22\) 0 0
\(23\) −22.6256 4.13331i −0.983720 0.179709i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) −37.4744 −1.29222 −0.646111 0.763244i \(-0.723606\pi\)
−0.646111 + 0.763244i \(0.723606\pi\)
\(30\) 0 0
\(31\) 17.2247 0.555635 0.277818 0.960634i \(-0.410389\pi\)
0.277818 + 0.960634i \(0.410389\pi\)
\(32\) 0 0
\(33\) 13.7972i 0.418096i
\(34\) 0 0
\(35\) −3.54085 −0.101167
\(36\) 0 0
\(37\) 60.3907i 1.63218i 0.577925 + 0.816090i \(0.303863\pi\)
−0.577925 + 0.816090i \(0.696137\pi\)
\(38\) 0 0
\(39\) −3.48309 −0.0893099
\(40\) 0 0
\(41\) −21.0549 −0.513534 −0.256767 0.966473i \(-0.582657\pi\)
−0.256767 + 0.966473i \(0.582657\pi\)
\(42\) 0 0
\(43\) 15.3496i 0.356967i −0.983943 0.178483i \(-0.942881\pi\)
0.983943 0.178483i \(-0.0571191\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) 7.43748 0.158244 0.0791221 0.996865i \(-0.474788\pi\)
0.0791221 + 0.996865i \(0.474788\pi\)
\(48\) 0 0
\(49\) 46.4925 0.948826
\(50\) 0 0
\(51\) 4.94155i 0.0968932i
\(52\) 0 0
\(53\) 30.4235i 0.574028i 0.957926 + 0.287014i \(0.0926625\pi\)
−0.957926 + 0.287014i \(0.907337\pi\)
\(54\) 0 0
\(55\) −17.8121 −0.323856
\(56\) 0 0
\(57\) 27.3053i 0.479040i
\(58\) 0 0
\(59\) 36.6568 0.621302 0.310651 0.950524i \(-0.399453\pi\)
0.310651 + 0.950524i \(0.399453\pi\)
\(60\) 0 0
\(61\) 24.7355i 0.405500i −0.979231 0.202750i \(-0.935012\pi\)
0.979231 0.202750i \(-0.0649878\pi\)
\(62\) 0 0
\(63\) 4.75054i 0.0754054i
\(64\) 0 0
\(65\) 4.49665i 0.0691792i
\(66\) 0 0
\(67\) 124.907i 1.86428i −0.362095 0.932141i \(-0.617938\pi\)
0.362095 0.932141i \(-0.382062\pi\)
\(68\) 0 0
\(69\) 39.1886 + 7.15910i 0.567951 + 0.103755i
\(70\) 0 0
\(71\) −119.986 −1.68994 −0.844970 0.534814i \(-0.820382\pi\)
−0.844970 + 0.534814i \(0.820382\pi\)
\(72\) 0 0
\(73\) −66.2872 −0.908044 −0.454022 0.890991i \(-0.650011\pi\)
−0.454022 + 0.890991i \(0.650011\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) −12.6140 −0.163818
\(78\) 0 0
\(79\) 47.4317i 0.600402i 0.953876 + 0.300201i \(0.0970537\pi\)
−0.953876 + 0.300201i \(0.902946\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 74.7878i 0.901057i −0.892762 0.450529i \(-0.851236\pi\)
0.892762 0.450529i \(-0.148764\pi\)
\(84\) 0 0
\(85\) −6.37952 −0.0750532
\(86\) 0 0
\(87\) 64.9076 0.746064
\(88\) 0 0
\(89\) 78.4768i 0.881762i 0.897566 + 0.440881i \(0.145334\pi\)
−0.897566 + 0.440881i \(0.854666\pi\)
\(90\) 0 0
\(91\) 3.18439i 0.0349933i
\(92\) 0 0
\(93\) −29.8340 −0.320796
\(94\) 0 0
\(95\) 35.2509 0.371063
\(96\) 0 0
\(97\) 107.673i 1.11003i 0.831840 + 0.555016i \(0.187288\pi\)
−0.831840 + 0.555016i \(0.812712\pi\)
\(98\) 0 0
\(99\) 23.8974i 0.241388i
\(100\) 0 0
\(101\) 188.466 1.86600 0.933000 0.359877i \(-0.117181\pi\)
0.933000 + 0.359877i \(0.117181\pi\)
\(102\) 0 0
\(103\) 101.012i 0.980698i 0.871526 + 0.490349i \(0.163131\pi\)
−0.871526 + 0.490349i \(0.836869\pi\)
\(104\) 0 0
\(105\) 6.13292 0.0584088
\(106\) 0 0
\(107\) 139.082i 1.29983i 0.760007 + 0.649914i \(0.225195\pi\)
−0.760007 + 0.649914i \(0.774805\pi\)
\(108\) 0 0
\(109\) 127.811i 1.17258i −0.810103 0.586288i \(-0.800589\pi\)
0.810103 0.586288i \(-0.199411\pi\)
\(110\) 0 0
\(111\) 104.600i 0.942340i
\(112\) 0 0
\(113\) 52.5032i 0.464630i −0.972641 0.232315i \(-0.925370\pi\)
0.972641 0.232315i \(-0.0746300\pi\)
\(114\) 0 0
\(115\) −9.24236 + 50.5923i −0.0803684 + 0.439933i
\(116\) 0 0
\(117\) 6.03288 0.0515631
\(118\) 0 0
\(119\) −4.51778 −0.0379645
\(120\) 0 0
\(121\) 57.5461 0.475587
\(122\) 0 0
\(123\) 36.4681 0.296489
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −19.6899 −0.155039 −0.0775195 0.996991i \(-0.524700\pi\)
−0.0775195 + 0.996991i \(0.524700\pi\)
\(128\) 0 0
\(129\) 26.5862i 0.206095i
\(130\) 0 0
\(131\) −74.2188 −0.566556 −0.283278 0.959038i \(-0.591422\pi\)
−0.283278 + 0.959038i \(0.591422\pi\)
\(132\) 0 0
\(133\) 24.9636 0.187696
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 101.870i 0.743576i 0.928318 + 0.371788i \(0.121255\pi\)
−0.928318 + 0.371788i \(0.878745\pi\)
\(138\) 0 0
\(139\) −34.3389 −0.247042 −0.123521 0.992342i \(-0.539419\pi\)
−0.123521 + 0.992342i \(0.539419\pi\)
\(140\) 0 0
\(141\) −12.8821 −0.0913623
\(142\) 0 0
\(143\) 16.0189i 0.112020i
\(144\) 0 0
\(145\) 83.7954i 0.577899i
\(146\) 0 0
\(147\) −80.5273 −0.547805
\(148\) 0 0
\(149\) 134.475i 0.902520i 0.892393 + 0.451260i \(0.149025\pi\)
−0.892393 + 0.451260i \(0.850975\pi\)
\(150\) 0 0
\(151\) 264.466 1.75143 0.875715 0.482828i \(-0.160390\pi\)
0.875715 + 0.482828i \(0.160390\pi\)
\(152\) 0 0
\(153\) 8.55902i 0.0559413i
\(154\) 0 0
\(155\) 38.5156i 0.248488i
\(156\) 0 0
\(157\) 234.029i 1.49063i 0.666711 + 0.745317i \(0.267702\pi\)
−0.666711 + 0.745317i \(0.732298\pi\)
\(158\) 0 0
\(159\) 52.6950i 0.331415i
\(160\) 0 0
\(161\) −6.54515 + 35.8279i −0.0406531 + 0.222533i
\(162\) 0 0
\(163\) 84.8179 0.520355 0.260177 0.965561i \(-0.416219\pi\)
0.260177 + 0.965561i \(0.416219\pi\)
\(164\) 0 0
\(165\) 30.8514 0.186978
\(166\) 0 0
\(167\) 199.963 1.19739 0.598693 0.800979i \(-0.295687\pi\)
0.598693 + 0.800979i \(0.295687\pi\)
\(168\) 0 0
\(169\) −164.956 −0.976071
\(170\) 0 0
\(171\) 47.2941i 0.276574i
\(172\) 0 0
\(173\) −213.353 −1.23325 −0.616627 0.787255i \(-0.711501\pi\)
−0.616627 + 0.787255i \(0.711501\pi\)
\(174\) 0 0
\(175\) 7.91757i 0.0452433i
\(176\) 0 0
\(177\) −63.4914 −0.358709
\(178\) 0 0
\(179\) −78.1533 −0.436611 −0.218305 0.975881i \(-0.570053\pi\)
−0.218305 + 0.975881i \(0.570053\pi\)
\(180\) 0 0
\(181\) 206.052i 1.13841i 0.822196 + 0.569205i \(0.192749\pi\)
−0.822196 + 0.569205i \(0.807251\pi\)
\(182\) 0 0
\(183\) 42.8431i 0.234115i
\(184\) 0 0
\(185\) 135.038 0.729933
\(186\) 0 0
\(187\) −22.7265 −0.121532
\(188\) 0 0
\(189\) 8.22818i 0.0435354i
\(190\) 0 0
\(191\) 149.720i 0.783874i −0.919992 0.391937i \(-0.871805\pi\)
0.919992 0.391937i \(-0.128195\pi\)
\(192\) 0 0
\(193\) 96.4435 0.499707 0.249854 0.968284i \(-0.419617\pi\)
0.249854 + 0.968284i \(0.419617\pi\)
\(194\) 0 0
\(195\) 7.78842i 0.0399406i
\(196\) 0 0
\(197\) −354.345 −1.79871 −0.899353 0.437224i \(-0.855962\pi\)
−0.899353 + 0.437224i \(0.855962\pi\)
\(198\) 0 0
\(199\) 51.5428i 0.259009i 0.991579 + 0.129504i \(0.0413386\pi\)
−0.991579 + 0.129504i \(0.958661\pi\)
\(200\) 0 0
\(201\) 216.345i 1.07634i
\(202\) 0 0
\(203\) 59.3413i 0.292322i
\(204\) 0 0
\(205\) 47.0801i 0.229659i
\(206\) 0 0
\(207\) −67.8767 12.3999i −0.327907 0.0599030i
\(208\) 0 0
\(209\) 125.578 0.600854
\(210\) 0 0
\(211\) 22.0709 0.104602 0.0523008 0.998631i \(-0.483345\pi\)
0.0523008 + 0.998631i \(0.483345\pi\)
\(212\) 0 0
\(213\) 207.821 0.975687
\(214\) 0 0
\(215\) −34.3227 −0.159640
\(216\) 0 0
\(217\) 27.2755i 0.125694i
\(218\) 0 0
\(219\) 114.813 0.524259
\(220\) 0 0
\(221\) 5.73729i 0.0259606i
\(222\) 0 0
\(223\) −359.616 −1.61263 −0.806314 0.591488i \(-0.798541\pi\)
−0.806314 + 0.591488i \(0.798541\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 249.973i 1.10120i −0.834769 0.550601i \(-0.814399\pi\)
0.834769 0.550601i \(-0.185601\pi\)
\(228\) 0 0
\(229\) 436.784i 1.90735i 0.300833 + 0.953677i \(0.402735\pi\)
−0.300833 + 0.953677i \(0.597265\pi\)
\(230\) 0 0
\(231\) 21.8480 0.0945801
\(232\) 0 0
\(233\) −8.36522 −0.0359022 −0.0179511 0.999839i \(-0.505714\pi\)
−0.0179511 + 0.999839i \(0.505714\pi\)
\(234\) 0 0
\(235\) 16.6307i 0.0707690i
\(236\) 0 0
\(237\) 82.1542i 0.346642i
\(238\) 0 0
\(239\) 415.870 1.74004 0.870020 0.493016i \(-0.164106\pi\)
0.870020 + 0.493016i \(0.164106\pi\)
\(240\) 0 0
\(241\) 208.769i 0.866263i 0.901331 + 0.433132i \(0.142592\pi\)
−0.901331 + 0.433132i \(0.857408\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 103.960i 0.424328i
\(246\) 0 0
\(247\) 31.7022i 0.128349i
\(248\) 0 0
\(249\) 129.536i 0.520226i
\(250\) 0 0
\(251\) 387.080i 1.54215i 0.636744 + 0.771076i \(0.280281\pi\)
−0.636744 + 0.771076i \(0.719719\pi\)
\(252\) 0 0
\(253\) −32.9251 + 180.231i −0.130139 + 0.712374i
\(254\) 0 0
\(255\) 11.0497 0.0433320
\(256\) 0 0
\(257\) 304.210 1.18370 0.591848 0.806049i \(-0.298399\pi\)
0.591848 + 0.806049i \(0.298399\pi\)
\(258\) 0 0
\(259\) 95.6295 0.369226
\(260\) 0 0
\(261\) −112.423 −0.430740
\(262\) 0 0
\(263\) 286.363i 1.08883i 0.838816 + 0.544416i \(0.183249\pi\)
−0.838816 + 0.544416i \(0.816751\pi\)
\(264\) 0 0
\(265\) 68.0289 0.256713
\(266\) 0 0
\(267\) 135.926i 0.509085i
\(268\) 0 0
\(269\) −51.0877 −0.189917 −0.0949585 0.995481i \(-0.530272\pi\)
−0.0949585 + 0.995481i \(0.530272\pi\)
\(270\) 0 0
\(271\) −59.6073 −0.219953 −0.109977 0.993934i \(-0.535078\pi\)
−0.109977 + 0.993934i \(0.535078\pi\)
\(272\) 0 0
\(273\) 5.51552i 0.0202034i
\(274\) 0 0
\(275\) 39.8290i 0.144833i
\(276\) 0 0
\(277\) 462.638 1.67017 0.835086 0.550120i \(-0.185418\pi\)
0.835086 + 0.550120i \(0.185418\pi\)
\(278\) 0 0
\(279\) 51.6741 0.185212
\(280\) 0 0
\(281\) 345.916i 1.23102i 0.788130 + 0.615509i \(0.211049\pi\)
−0.788130 + 0.615509i \(0.788951\pi\)
\(282\) 0 0
\(283\) 129.005i 0.455848i 0.973679 + 0.227924i \(0.0731937\pi\)
−0.973679 + 0.227924i \(0.926806\pi\)
\(284\) 0 0
\(285\) −61.0564 −0.214233
\(286\) 0 0
\(287\) 33.3407i 0.116170i
\(288\) 0 0
\(289\) 280.860 0.971835
\(290\) 0 0
\(291\) 186.495i 0.640877i
\(292\) 0 0
\(293\) 217.485i 0.742270i 0.928579 + 0.371135i \(0.121031\pi\)
−0.928579 + 0.371135i \(0.878969\pi\)
\(294\) 0 0
\(295\) 81.9671i 0.277855i
\(296\) 0 0
\(297\) 41.3915i 0.139365i
\(298\) 0 0
\(299\) −45.4991 8.31193i −0.152171 0.0277991i
\(300\) 0 0
\(301\) −24.3063 −0.0807517
\(302\) 0 0
\(303\) −326.433 −1.07734
\(304\) 0 0
\(305\) −55.3102 −0.181345
\(306\) 0 0
\(307\) 94.1784 0.306770 0.153385 0.988167i \(-0.450983\pi\)
0.153385 + 0.988167i \(0.450983\pi\)
\(308\) 0 0
\(309\) 174.958i 0.566206i
\(310\) 0 0
\(311\) 255.801 0.822511 0.411255 0.911520i \(-0.365090\pi\)
0.411255 + 0.911520i \(0.365090\pi\)
\(312\) 0 0
\(313\) 315.756i 1.00881i 0.863469 + 0.504403i \(0.168287\pi\)
−0.863469 + 0.504403i \(0.831713\pi\)
\(314\) 0 0
\(315\) −10.6225 −0.0337223
\(316\) 0 0
\(317\) 153.344 0.483735 0.241867 0.970309i \(-0.422240\pi\)
0.241867 + 0.970309i \(0.422240\pi\)
\(318\) 0 0
\(319\) 298.514i 0.935779i
\(320\) 0 0
\(321\) 240.897i 0.750457i
\(322\) 0 0
\(323\) 44.9768 0.139247
\(324\) 0 0
\(325\) −10.0548 −0.0309379
\(326\) 0 0
\(327\) 221.375i 0.676987i
\(328\) 0 0
\(329\) 11.7774i 0.0357974i
\(330\) 0 0
\(331\) −337.990 −1.02112 −0.510558 0.859843i \(-0.670561\pi\)
−0.510558 + 0.859843i \(0.670561\pi\)
\(332\) 0 0
\(333\) 181.172i 0.544060i
\(334\) 0 0
\(335\) −279.300 −0.833733
\(336\) 0 0
\(337\) 438.136i 1.30011i 0.759888 + 0.650054i \(0.225254\pi\)
−0.759888 + 0.650054i \(0.774746\pi\)
\(338\) 0 0
\(339\) 90.9382i 0.268254i
\(340\) 0 0
\(341\) 137.208i 0.402371i
\(342\) 0 0
\(343\) 151.214i 0.440856i
\(344\) 0 0
\(345\) 16.0082 87.6284i 0.0464007 0.253995i
\(346\) 0 0
\(347\) −424.211 −1.22251 −0.611255 0.791433i \(-0.709335\pi\)
−0.611255 + 0.791433i \(0.709335\pi\)
\(348\) 0 0
\(349\) 75.1527 0.215337 0.107669 0.994187i \(-0.465661\pi\)
0.107669 + 0.994187i \(0.465661\pi\)
\(350\) 0 0
\(351\) −10.4493 −0.0297700
\(352\) 0 0
\(353\) −590.245 −1.67208 −0.836040 0.548668i \(-0.815135\pi\)
−0.836040 + 0.548668i \(0.815135\pi\)
\(354\) 0 0
\(355\) 268.296i 0.755764i
\(356\) 0 0
\(357\) 7.82502 0.0219188
\(358\) 0 0
\(359\) 317.245i 0.883691i −0.897091 0.441846i \(-0.854324\pi\)
0.897091 0.441846i \(-0.145676\pi\)
\(360\) 0 0
\(361\) 112.474 0.311563
\(362\) 0 0
\(363\) −99.6727 −0.274580
\(364\) 0 0
\(365\) 148.223i 0.406089i
\(366\) 0 0
\(367\) 495.495i 1.35012i 0.737762 + 0.675061i \(0.235883\pi\)
−0.737762 + 0.675061i \(0.764117\pi\)
\(368\) 0 0
\(369\) −63.1646 −0.171178
\(370\) 0 0
\(371\) 48.1760 0.129854
\(372\) 0 0
\(373\) 114.793i 0.307756i −0.988090 0.153878i \(-0.950824\pi\)
0.988090 0.153878i \(-0.0491762\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) −75.3596 −0.199893
\(378\) 0 0
\(379\) 587.936i 1.55128i −0.631175 0.775641i \(-0.717427\pi\)
0.631175 0.775641i \(-0.282573\pi\)
\(380\) 0 0
\(381\) 34.1040 0.0895118
\(382\) 0 0
\(383\) 218.823i 0.571340i −0.958328 0.285670i \(-0.907784\pi\)
0.958328 0.285670i \(-0.0922161\pi\)
\(384\) 0 0
\(385\) 28.2057i 0.0732615i
\(386\) 0 0
\(387\) 46.0487i 0.118989i
\(388\) 0 0
\(389\) 96.7351i 0.248676i 0.992240 + 0.124338i \(0.0396808\pi\)
−0.992240 + 0.124338i \(0.960319\pi\)
\(390\) 0 0
\(391\) −11.7924 + 64.5509i −0.0301595 + 0.165092i
\(392\) 0 0
\(393\) 128.551 0.327101
\(394\) 0 0
\(395\) 106.061 0.268508
\(396\) 0 0
\(397\) 2.57843 0.00649478 0.00324739 0.999995i \(-0.498966\pi\)
0.00324739 + 0.999995i \(0.498966\pi\)
\(398\) 0 0
\(399\) −43.2383 −0.108367
\(400\) 0 0
\(401\) 728.763i 1.81736i 0.417490 + 0.908682i \(0.362910\pi\)
−0.417490 + 0.908682i \(0.637090\pi\)
\(402\) 0 0
\(403\) 34.6382 0.0859508
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) 481.060 1.18197
\(408\) 0 0
\(409\) 246.149 0.601832 0.300916 0.953651i \(-0.402708\pi\)
0.300916 + 0.953651i \(0.402708\pi\)
\(410\) 0 0
\(411\) 176.444i 0.429304i
\(412\) 0 0
\(413\) 58.0466i 0.140549i
\(414\) 0 0
\(415\) −167.231 −0.402965
\(416\) 0 0
\(417\) 59.4767 0.142630
\(418\) 0 0
\(419\) 370.230i 0.883605i −0.897112 0.441802i \(-0.854339\pi\)
0.897112 0.441802i \(-0.145661\pi\)
\(420\) 0 0
\(421\) 57.1171i 0.135670i 0.997697 + 0.0678350i \(0.0216092\pi\)
−0.997697 + 0.0678350i \(0.978391\pi\)
\(422\) 0 0
\(423\) 22.3124 0.0527481
\(424\) 0 0
\(425\) 14.2650i 0.0335648i
\(426\) 0 0
\(427\) −39.1690 −0.0917306
\(428\) 0 0
\(429\) 27.7456i 0.0646750i
\(430\) 0 0
\(431\) 4.71597i 0.0109419i −0.999985 0.00547096i \(-0.998259\pi\)
0.999985 0.00547096i \(-0.00174147\pi\)
\(432\) 0 0
\(433\) 500.927i 1.15687i 0.815727 + 0.578437i \(0.196337\pi\)
−0.815727 + 0.578437i \(0.803663\pi\)
\(434\) 0 0
\(435\) 145.138i 0.333650i
\(436\) 0 0
\(437\) 65.1604 356.685i 0.149108 0.816213i
\(438\) 0 0
\(439\) 403.703 0.919597 0.459798 0.888023i \(-0.347922\pi\)
0.459798 + 0.888023i \(0.347922\pi\)
\(440\) 0 0
\(441\) 139.477 0.316275
\(442\) 0 0
\(443\) −761.733 −1.71949 −0.859744 0.510725i \(-0.829377\pi\)
−0.859744 + 0.510725i \(0.829377\pi\)
\(444\) 0 0
\(445\) 175.479 0.394336
\(446\) 0 0
\(447\) 232.918i 0.521070i
\(448\) 0 0
\(449\) 466.529 1.03904 0.519519 0.854459i \(-0.326111\pi\)
0.519519 + 0.854459i \(0.326111\pi\)
\(450\) 0 0
\(451\) 167.719i 0.371882i
\(452\) 0 0
\(453\) −458.069 −1.01119
\(454\) 0 0
\(455\) −7.12050 −0.0156495
\(456\) 0 0
\(457\) 776.202i 1.69847i 0.528013 + 0.849236i \(0.322937\pi\)
−0.528013 + 0.849236i \(0.677063\pi\)
\(458\) 0 0
\(459\) 14.8247i 0.0322977i
\(460\) 0 0
\(461\) 414.512 0.899158 0.449579 0.893241i \(-0.351574\pi\)
0.449579 + 0.893241i \(0.351574\pi\)
\(462\) 0 0
\(463\) −631.558 −1.36406 −0.682028 0.731326i \(-0.738902\pi\)
−0.682028 + 0.731326i \(0.738902\pi\)
\(464\) 0 0
\(465\) 66.7109i 0.143464i
\(466\) 0 0
\(467\) 150.487i 0.322241i −0.986935 0.161120i \(-0.948489\pi\)
0.986935 0.161120i \(-0.0515108\pi\)
\(468\) 0 0
\(469\) −197.792 −0.421731
\(470\) 0 0
\(471\) 405.351i 0.860618i
\(472\) 0 0
\(473\) −122.272 −0.258502
\(474\) 0 0
\(475\) 78.8235i 0.165944i
\(476\) 0 0
\(477\) 91.2704i 0.191343i
\(478\) 0 0
\(479\) 422.507i 0.882060i 0.897492 + 0.441030i \(0.145387\pi\)
−0.897492 + 0.441030i \(0.854613\pi\)
\(480\) 0 0
\(481\) 121.443i 0.252481i
\(482\) 0 0
\(483\) 11.3365 62.0557i 0.0234711 0.128480i
\(484\) 0 0
\(485\) 240.764 0.496421
\(486\) 0 0
\(487\) −480.779 −0.987226 −0.493613 0.869682i \(-0.664324\pi\)
−0.493613 + 0.869682i \(0.664324\pi\)
\(488\) 0 0
\(489\) −146.909 −0.300427
\(490\) 0 0
\(491\) −284.622 −0.579678 −0.289839 0.957075i \(-0.593602\pi\)
−0.289839 + 0.957075i \(0.593602\pi\)
\(492\) 0 0
\(493\) 106.915i 0.216866i
\(494\) 0 0
\(495\) −53.4362 −0.107952
\(496\) 0 0
\(497\) 189.999i 0.382292i
\(498\) 0 0
\(499\) 310.548 0.622341 0.311171 0.950354i \(-0.399279\pi\)
0.311171 + 0.950354i \(0.399279\pi\)
\(500\) 0 0
\(501\) −346.347 −0.691311
\(502\) 0 0
\(503\) 483.143i 0.960523i 0.877125 + 0.480262i \(0.159458\pi\)
−0.877125 + 0.480262i \(0.840542\pi\)
\(504\) 0 0
\(505\) 421.423i 0.834500i
\(506\) 0 0
\(507\) 285.712 0.563535
\(508\) 0 0
\(509\) −326.052 −0.640574 −0.320287 0.947321i \(-0.603779\pi\)
−0.320287 + 0.947321i \(0.603779\pi\)
\(510\) 0 0
\(511\) 104.967i 0.205414i
\(512\) 0 0
\(513\) 81.9158i 0.159680i
\(514\) 0 0
\(515\) 225.869 0.438581
\(516\) 0 0
\(517\) 59.2454i 0.114595i
\(518\) 0 0
\(519\) 369.538 0.712020
\(520\) 0 0
\(521\) 544.754i 1.04559i −0.852458 0.522796i \(-0.824889\pi\)
0.852458 0.522796i \(-0.175111\pi\)
\(522\) 0 0
\(523\) 656.345i 1.25496i 0.778632 + 0.627481i \(0.215914\pi\)
−0.778632 + 0.627481i \(0.784086\pi\)
\(524\) 0 0
\(525\) 13.7136i 0.0261212i
\(526\) 0 0
\(527\) 49.1422i 0.0932489i
\(528\) 0 0
\(529\) 494.832 + 187.037i 0.935409 + 0.353567i
\(530\) 0 0
\(531\) 109.970 0.207101
\(532\) 0 0
\(533\) −42.3406 −0.0794382
\(534\) 0 0
\(535\) 310.996 0.581301
\(536\) 0 0
\(537\) 135.365 0.252077
\(538\) 0 0
\(539\) 370.350i 0.687105i
\(540\) 0 0
\(541\) 223.806 0.413690 0.206845 0.978374i \(-0.433680\pi\)
0.206845 + 0.978374i \(0.433680\pi\)
\(542\) 0 0
\(543\) 356.893i 0.657262i
\(544\) 0 0
\(545\) −285.794 −0.524392
\(546\) 0 0
\(547\) −359.674 −0.657539 −0.328769 0.944410i \(-0.606634\pi\)
−0.328769 + 0.944410i \(0.606634\pi\)
\(548\) 0 0
\(549\) 74.2064i 0.135167i
\(550\) 0 0
\(551\) 590.773i 1.07218i
\(552\) 0 0
\(553\) 75.1088 0.135821
\(554\) 0 0
\(555\) −233.892 −0.421427
\(556\) 0 0
\(557\) 815.315i 1.46376i −0.681433 0.731880i \(-0.738643\pi\)
0.681433 0.731880i \(-0.261357\pi\)
\(558\) 0 0
\(559\) 30.8674i 0.0552189i
\(560\) 0 0
\(561\) 39.3634 0.0701665
\(562\) 0 0
\(563\) 623.361i 1.10721i −0.832779 0.553606i \(-0.813251\pi\)
0.832779 0.553606i \(-0.186749\pi\)
\(564\) 0 0
\(565\) −117.401 −0.207789
\(566\) 0 0
\(567\) 14.2516i 0.0251351i
\(568\) 0 0
\(569\) 544.770i 0.957417i −0.877974 0.478708i \(-0.841105\pi\)
0.877974 0.478708i \(-0.158895\pi\)
\(570\) 0 0
\(571\) 769.116i 1.34696i −0.739204 0.673482i \(-0.764798\pi\)
0.739204 0.673482i \(-0.235202\pi\)
\(572\) 0 0
\(573\) 259.323i 0.452570i
\(574\) 0 0
\(575\) 113.128 + 20.6665i 0.196744 + 0.0359418i
\(576\) 0 0
\(577\) −950.293 −1.64696 −0.823478 0.567349i \(-0.807969\pi\)
−0.823478 + 0.567349i \(0.807969\pi\)
\(578\) 0 0
\(579\) −167.045 −0.288506
\(580\) 0 0
\(581\) −118.427 −0.203834
\(582\) 0 0
\(583\) 242.347 0.415690
\(584\) 0 0
\(585\) 13.4899i 0.0230597i
\(586\) 0 0
\(587\) 665.247 1.13330 0.566650 0.823958i \(-0.308239\pi\)
0.566650 + 0.823958i \(0.308239\pi\)
\(588\) 0 0
\(589\) 271.542i 0.461022i
\(590\) 0 0
\(591\) 613.743 1.03848
\(592\) 0 0
\(593\) 183.058 0.308698 0.154349 0.988016i \(-0.450672\pi\)
0.154349 + 0.988016i \(0.450672\pi\)
\(594\) 0 0
\(595\) 10.1021i 0.0169783i
\(596\) 0 0
\(597\) 89.2747i 0.149539i
\(598\) 0 0
\(599\) −226.175 −0.377588 −0.188794 0.982017i \(-0.560458\pi\)
−0.188794 + 0.982017i \(0.560458\pi\)
\(600\) 0 0
\(601\) −450.282 −0.749221 −0.374610 0.927182i \(-0.622223\pi\)
−0.374610 + 0.927182i \(0.622223\pi\)
\(602\) 0 0
\(603\) 374.721i 0.621428i
\(604\) 0 0
\(605\) 128.677i 0.212689i
\(606\) 0 0
\(607\) 904.860 1.49071 0.745354 0.666669i \(-0.232281\pi\)
0.745354 + 0.666669i \(0.232281\pi\)
\(608\) 0 0
\(609\) 102.782i 0.168772i
\(610\) 0 0
\(611\) 14.9565 0.0244787
\(612\) 0 0
\(613\) 1078.10i 1.75873i 0.476146 + 0.879366i \(0.342033\pi\)
−0.476146 + 0.879366i \(0.657967\pi\)
\(614\) 0 0
\(615\) 81.5452i 0.132594i
\(616\) 0 0
\(617\) 106.109i 0.171975i 0.996296 + 0.0859875i \(0.0274045\pi\)
−0.996296 + 0.0859875i \(0.972595\pi\)
\(618\) 0 0
\(619\) 32.4229i 0.0523794i −0.999657 0.0261897i \(-0.991663\pi\)
0.999657 0.0261897i \(-0.00833740\pi\)
\(620\) 0 0
\(621\) 117.566 + 21.4773i 0.189317 + 0.0345850i
\(622\) 0 0
\(623\) 124.269 0.199469
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −217.508 −0.346903
\(628\) 0 0
\(629\) 172.295 0.273919
\(630\) 0 0
\(631\) 927.414i 1.46975i 0.678201 + 0.734876i \(0.262760\pi\)
−0.678201 + 0.734876i \(0.737240\pi\)
\(632\) 0 0
\(633\) −38.2280 −0.0603918
\(634\) 0 0
\(635\) 44.0281i 0.0693355i
\(636\) 0 0
\(637\) 93.4946 0.146773
\(638\) 0 0
\(639\) −359.957 −0.563313
\(640\) 0 0
\(641\) 1071.73i 1.67196i −0.548760 0.835980i \(-0.684900\pi\)
0.548760 0.835980i \(-0.315100\pi\)
\(642\) 0 0
\(643\) 1048.35i 1.63040i −0.579177 0.815202i \(-0.696626\pi\)
0.579177 0.815202i \(-0.303374\pi\)
\(644\) 0 0
\(645\) 59.4486 0.0921684
\(646\) 0 0
\(647\) −334.285 −0.516670 −0.258335 0.966055i \(-0.583174\pi\)
−0.258335 + 0.966055i \(0.583174\pi\)
\(648\) 0 0
\(649\) 292.001i 0.449924i
\(650\) 0 0
\(651\) 47.2426i 0.0725693i
\(652\) 0 0
\(653\) 597.700 0.915314 0.457657 0.889129i \(-0.348689\pi\)
0.457657 + 0.889129i \(0.348689\pi\)
\(654\) 0 0
\(655\) 165.958i 0.253371i
\(656\) 0 0
\(657\) −198.862 −0.302681
\(658\) 0 0
\(659\) 475.799i 0.722002i 0.932565 + 0.361001i \(0.117565\pi\)
−0.932565 + 0.361001i \(0.882435\pi\)
\(660\) 0 0
\(661\) 43.0152i 0.0650759i 0.999471 + 0.0325380i \(0.0103590\pi\)
−0.999471 + 0.0325380i \(0.989641\pi\)
\(662\) 0 0
\(663\) 9.93727i 0.0149883i
\(664\) 0 0
\(665\) 55.8204i 0.0839404i
\(666\) 0 0
\(667\) 847.880 + 154.893i 1.27118 + 0.232224i
\(668\) 0 0
\(669\) 622.873 0.931051
\(670\) 0 0
\(671\) −197.038 −0.293648
\(672\) 0 0
\(673\) −543.120 −0.807013 −0.403507 0.914977i \(-0.632209\pi\)
−0.403507 + 0.914977i \(0.632209\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 652.373i 0.963624i −0.876275 0.481812i \(-0.839979\pi\)
0.876275 0.481812i \(-0.160021\pi\)
\(678\) 0 0
\(679\) 170.502 0.251107
\(680\) 0 0
\(681\) 432.965i 0.635779i
\(682\) 0 0
\(683\) −596.324 −0.873096 −0.436548 0.899681i \(-0.643799\pi\)
−0.436548 + 0.899681i \(0.643799\pi\)
\(684\) 0 0
\(685\) 227.788 0.332537
\(686\) 0 0
\(687\) 756.532i 1.10121i
\(688\) 0 0
\(689\) 61.1804i 0.0887959i
\(690\) 0 0
\(691\) −722.267 −1.04525 −0.522624 0.852563i \(-0.675047\pi\)
−0.522624 + 0.852563i \(0.675047\pi\)
\(692\) 0 0
\(693\) −37.8419 −0.0546059
\(694\) 0 0
\(695\) 76.7840i 0.110481i
\(696\) 0 0
\(697\) 60.0697i 0.0861833i
\(698\) 0 0
\(699\) 14.4890 0.0207282
\(700\) 0 0
\(701\) 374.410i 0.534108i −0.963682 0.267054i \(-0.913950\pi\)
0.963682 0.267054i \(-0.0860502\pi\)
\(702\) 0 0
\(703\) −952.041 −1.35425
\(704\) 0 0
\(705\) 28.8052i 0.0408585i
\(706\) 0 0
\(707\) 298.439i 0.422120i
\(708\) 0 0
\(709\) 682.760i 0.962991i −0.876449 0.481495i \(-0.840094\pi\)
0.876449 0.481495i \(-0.159906\pi\)
\(710\) 0 0
\(711\) 142.295i 0.200134i
\(712\) 0 0
\(713\) −389.718 71.1950i −0.546589 0.0998527i
\(714\) 0 0
\(715\) −35.8194 −0.0500970
\(716\) 0 0
\(717\) −720.307 −1.00461
\(718\) 0 0
\(719\) 1199.29 1.66799 0.833997 0.551769i \(-0.186047\pi\)
0.833997 + 0.551769i \(0.186047\pi\)
\(720\) 0 0
\(721\) 159.954 0.221850
\(722\) 0 0
\(723\) 361.599i 0.500137i
\(724\) 0 0
\(725\) 187.372 0.258444
\(726\) 0 0
\(727\) 957.231i 1.31669i −0.752718 0.658343i \(-0.771258\pi\)
0.752718 0.658343i \(-0.228742\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) −43.7924 −0.0599076
\(732\) 0 0
\(733\) 1126.29i 1.53655i −0.640118 0.768277i \(-0.721115\pi\)
0.640118 0.768277i \(-0.278885\pi\)
\(734\) 0 0
\(735\) 180.065i 0.244986i
\(736\) 0 0
\(737\) −994.983 −1.35005
\(738\) 0 0
\(739\) 845.704 1.14439 0.572195 0.820118i \(-0.306092\pi\)
0.572195 + 0.820118i \(0.306092\pi\)
\(740\) 0 0
\(741\) 54.9098i 0.0741023i
\(742\) 0 0
\(743\) 38.6258i 0.0519862i 0.999662 + 0.0259931i \(0.00827480\pi\)
−0.999662 + 0.0259931i \(0.991725\pi\)
\(744\) 0 0
\(745\) 300.696 0.403619
\(746\) 0 0
\(747\) 224.363i 0.300352i
\(748\) 0 0
\(749\) 220.238 0.294043
\(750\) 0 0
\(751\) 24.1195i 0.0321165i −0.999871 0.0160582i \(-0.994888\pi\)
0.999871 0.0160582i \(-0.00511172\pi\)
\(752\) 0 0
\(753\) 670.442i 0.890361i
\(754\) 0 0
\(755\) 591.364i 0.783264i
\(756\) 0 0
\(757\) 136.738i 0.180632i −0.995913 0.0903158i \(-0.971212\pi\)
0.995913 0.0903158i \(-0.0287876\pi\)
\(758\) 0 0
\(759\) 57.0280 312.169i 0.0751356 0.411289i
\(760\) 0 0
\(761\) −907.651 −1.19271 −0.596354 0.802721i \(-0.703385\pi\)
−0.596354 + 0.802721i \(0.703385\pi\)
\(762\) 0 0
\(763\) −202.390 −0.265256
\(764\) 0 0
\(765\) −19.1386 −0.0250177
\(766\) 0 0
\(767\) 73.7154 0.0961088
\(768\) 0 0
\(769\) 1142.39i 1.48555i 0.669540 + 0.742776i \(0.266491\pi\)
−0.669540 + 0.742776i \(0.733509\pi\)
\(770\) 0 0
\(771\) −526.907 −0.683407
\(772\) 0 0
\(773\) 586.323i 0.758504i −0.925294 0.379252i \(-0.876181\pi\)
0.925294 0.379252i \(-0.123819\pi\)
\(774\) 0 0
\(775\) −86.1234 −0.111127
\(776\) 0 0
\(777\) −165.635 −0.213173
\(778\) 0 0
\(779\) 331.924i 0.426090i
\(780\) 0 0
\(781\) 955.782i 1.22379i
\(782\) 0 0
\(783\) 194.723 0.248688
\(784\) 0 0
\(785\) 523.306 0.666632
\(786\) 0 0
\(787\) 12.7384i 0.0161860i −0.999967 0.00809301i \(-0.997424\pi\)
0.999967 0.00809301i \(-0.00257611\pi\)
\(788\) 0 0
\(789\) 495.995i 0.628637i
\(790\) 0 0
\(791\) −83.1395 −0.105107
\(792\) 0 0
\(793\) 49.7421i 0.0627265i
\(794\) 0 0
\(795\) −117.830 −0.148213
\(796\) 0 0
\(797\) 334.922i 0.420228i 0.977677 + 0.210114i \(0.0673835\pi\)
−0.977677 + 0.210114i \(0.932617\pi\)
\(798\) 0 0