Properties

Label 2760.3.g.a.2161.5
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 2161.5
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.44

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} -7.15300i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} -7.15300i q^{7} +3.00000 q^{9} +7.91141i q^{11} -4.18560 q^{13} +3.87298i q^{15} +13.5574i q^{17} +14.2010i q^{19} +12.3894i q^{21} +(-9.50929 - 20.9421i) q^{23} -5.00000 q^{25} -5.19615 q^{27} +19.4273 q^{29} +26.3262 q^{31} -13.7030i q^{33} -15.9946 q^{35} -55.4949i q^{37} +7.24967 q^{39} +27.8249 q^{41} -28.2189i q^{43} -6.70820i q^{45} -17.0401 q^{47} -2.16538 q^{49} -23.4821i q^{51} -56.7971i q^{53} +17.6904 q^{55} -24.5969i q^{57} +3.87747 q^{59} +13.8252i q^{61} -21.4590i q^{63} +9.35928i q^{65} -37.0559i q^{67} +(16.4706 + 36.2729i) q^{69} -58.0398 q^{71} -12.9561 q^{73} +8.66025 q^{75} +56.5903 q^{77} -55.9230i q^{79} +9.00000 q^{81} +22.1867i q^{83} +30.3152 q^{85} -33.6490 q^{87} +60.7868i q^{89} +29.9396i q^{91} -45.5983 q^{93} +31.7544 q^{95} +139.101i q^{97} +23.7342i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 7.15300i 1.02186i −0.859623 0.510928i \(-0.829302\pi\)
0.859623 0.510928i \(-0.170698\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 7.91141i 0.719219i 0.933103 + 0.359610i \(0.117090\pi\)
−0.933103 + 0.359610i \(0.882910\pi\)
\(12\) 0 0
\(13\) −4.18560 −0.321969 −0.160985 0.986957i \(-0.551467\pi\)
−0.160985 + 0.986957i \(0.551467\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 13.5574i 0.797492i 0.917061 + 0.398746i \(0.130555\pi\)
−0.917061 + 0.398746i \(0.869445\pi\)
\(18\) 0 0
\(19\) 14.2010i 0.747421i 0.927545 + 0.373711i \(0.121915\pi\)
−0.927545 + 0.373711i \(0.878085\pi\)
\(20\) 0 0
\(21\) 12.3894i 0.589969i
\(22\) 0 0
\(23\) −9.50929 20.9421i −0.413448 0.910528i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 19.4273 0.669905 0.334953 0.942235i \(-0.391280\pi\)
0.334953 + 0.942235i \(0.391280\pi\)
\(30\) 0 0
\(31\) 26.3262 0.849232 0.424616 0.905373i \(-0.360409\pi\)
0.424616 + 0.905373i \(0.360409\pi\)
\(32\) 0 0
\(33\) 13.7030i 0.415241i
\(34\) 0 0
\(35\) −15.9946 −0.456988
\(36\) 0 0
\(37\) 55.4949i 1.49986i −0.661515 0.749932i \(-0.730086\pi\)
0.661515 0.749932i \(-0.269914\pi\)
\(38\) 0 0
\(39\) 7.24967 0.185889
\(40\) 0 0
\(41\) 27.8249 0.678657 0.339328 0.940668i \(-0.389800\pi\)
0.339328 + 0.940668i \(0.389800\pi\)
\(42\) 0 0
\(43\) 28.2189i 0.656254i −0.944634 0.328127i \(-0.893583\pi\)
0.944634 0.328127i \(-0.106417\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) −17.0401 −0.362555 −0.181278 0.983432i \(-0.558023\pi\)
−0.181278 + 0.983432i \(0.558023\pi\)
\(48\) 0 0
\(49\) −2.16538 −0.0441914
\(50\) 0 0
\(51\) 23.4821i 0.460432i
\(52\) 0 0
\(53\) 56.7971i 1.07164i −0.844331 0.535822i \(-0.820002\pi\)
0.844331 0.535822i \(-0.179998\pi\)
\(54\) 0 0
\(55\) 17.6904 0.321645
\(56\) 0 0
\(57\) 24.5969i 0.431524i
\(58\) 0 0
\(59\) 3.87747 0.0657198 0.0328599 0.999460i \(-0.489538\pi\)
0.0328599 + 0.999460i \(0.489538\pi\)
\(60\) 0 0
\(61\) 13.8252i 0.226643i 0.993558 + 0.113321i \(0.0361490\pi\)
−0.993558 + 0.113321i \(0.963851\pi\)
\(62\) 0 0
\(63\) 21.4590i 0.340619i
\(64\) 0 0
\(65\) 9.35928i 0.143989i
\(66\) 0 0
\(67\) 37.0559i 0.553074i −0.961003 0.276537i \(-0.910813\pi\)
0.961003 0.276537i \(-0.0891868\pi\)
\(68\) 0 0
\(69\) 16.4706 + 36.2729i 0.238704 + 0.525694i
\(70\) 0 0
\(71\) −58.0398 −0.817462 −0.408731 0.912655i \(-0.634029\pi\)
−0.408731 + 0.912655i \(0.634029\pi\)
\(72\) 0 0
\(73\) −12.9561 −0.177480 −0.0887401 0.996055i \(-0.528284\pi\)
−0.0887401 + 0.996055i \(0.528284\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) 56.5903 0.734939
\(78\) 0 0
\(79\) 55.9230i 0.707887i −0.935267 0.353943i \(-0.884841\pi\)
0.935267 0.353943i \(-0.115159\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 22.1867i 0.267310i 0.991028 + 0.133655i \(0.0426714\pi\)
−0.991028 + 0.133655i \(0.957329\pi\)
\(84\) 0 0
\(85\) 30.3152 0.356649
\(86\) 0 0
\(87\) −33.6490 −0.386770
\(88\) 0 0
\(89\) 60.7868i 0.682998i 0.939882 + 0.341499i \(0.110935\pi\)
−0.939882 + 0.341499i \(0.889065\pi\)
\(90\) 0 0
\(91\) 29.9396i 0.329006i
\(92\) 0 0
\(93\) −45.5983 −0.490304
\(94\) 0 0
\(95\) 31.7544 0.334257
\(96\) 0 0
\(97\) 139.101i 1.43403i 0.697060 + 0.717013i \(0.254491\pi\)
−0.697060 + 0.717013i \(0.745509\pi\)
\(98\) 0 0
\(99\) 23.7342i 0.239740i
\(100\) 0 0
\(101\) −100.688 −0.996915 −0.498457 0.866914i \(-0.666100\pi\)
−0.498457 + 0.866914i \(0.666100\pi\)
\(102\) 0 0
\(103\) 64.1195i 0.622519i −0.950325 0.311260i \(-0.899249\pi\)
0.950325 0.311260i \(-0.100751\pi\)
\(104\) 0 0
\(105\) 27.7034 0.263842
\(106\) 0 0
\(107\) 1.84859i 0.0172766i 0.999963 + 0.00863829i \(0.00274969\pi\)
−0.999963 + 0.00863829i \(0.997250\pi\)
\(108\) 0 0
\(109\) 110.026i 1.00941i −0.863291 0.504706i \(-0.831601\pi\)
0.863291 0.504706i \(-0.168399\pi\)
\(110\) 0 0
\(111\) 96.1201i 0.865947i
\(112\) 0 0
\(113\) 92.3802i 0.817524i 0.912641 + 0.408762i \(0.134039\pi\)
−0.912641 + 0.408762i \(0.865961\pi\)
\(114\) 0 0
\(115\) −46.8281 + 21.2634i −0.407200 + 0.184899i
\(116\) 0 0
\(117\) −12.5568 −0.107323
\(118\) 0 0
\(119\) 96.9759 0.814923
\(120\) 0 0
\(121\) 58.4096 0.482724
\(122\) 0 0
\(123\) −48.1942 −0.391823
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −210.660 −1.65874 −0.829369 0.558701i \(-0.811300\pi\)
−0.829369 + 0.558701i \(0.811300\pi\)
\(128\) 0 0
\(129\) 48.8766i 0.378888i
\(130\) 0 0
\(131\) −214.706 −1.63897 −0.819487 0.573098i \(-0.805741\pi\)
−0.819487 + 0.573098i \(0.805741\pi\)
\(132\) 0 0
\(133\) 101.580 0.763757
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 69.6618i 0.508480i 0.967141 + 0.254240i \(0.0818254\pi\)
−0.967141 + 0.254240i \(0.918175\pi\)
\(138\) 0 0
\(139\) −118.596 −0.853209 −0.426605 0.904438i \(-0.640290\pi\)
−0.426605 + 0.904438i \(0.640290\pi\)
\(140\) 0 0
\(141\) 29.5143 0.209321
\(142\) 0 0
\(143\) 33.1140i 0.231566i
\(144\) 0 0
\(145\) 43.4407i 0.299591i
\(146\) 0 0
\(147\) 3.75055 0.0255139
\(148\) 0 0
\(149\) 4.73942i 0.0318082i −0.999874 0.0159041i \(-0.994937\pi\)
0.999874 0.0159041i \(-0.00506265\pi\)
\(150\) 0 0
\(151\) 138.319 0.916017 0.458009 0.888948i \(-0.348563\pi\)
0.458009 + 0.888948i \(0.348563\pi\)
\(152\) 0 0
\(153\) 40.6721i 0.265831i
\(154\) 0 0
\(155\) 58.8672i 0.379788i
\(156\) 0 0
\(157\) 148.254i 0.944295i −0.881520 0.472148i \(-0.843479\pi\)
0.881520 0.472148i \(-0.156521\pi\)
\(158\) 0 0
\(159\) 98.3755i 0.618714i
\(160\) 0 0
\(161\) −149.799 + 68.0200i −0.930429 + 0.422484i
\(162\) 0 0
\(163\) −245.350 −1.50522 −0.752608 0.658468i \(-0.771205\pi\)
−0.752608 + 0.658468i \(0.771205\pi\)
\(164\) 0 0
\(165\) −30.6408 −0.185702
\(166\) 0 0
\(167\) −233.825 −1.40015 −0.700076 0.714068i \(-0.746851\pi\)
−0.700076 + 0.714068i \(0.746851\pi\)
\(168\) 0 0
\(169\) −151.481 −0.896336
\(170\) 0 0
\(171\) 42.6030i 0.249140i
\(172\) 0 0
\(173\) 183.932 1.06319 0.531595 0.846999i \(-0.321593\pi\)
0.531595 + 0.846999i \(0.321593\pi\)
\(174\) 0 0
\(175\) 35.7650i 0.204371i
\(176\) 0 0
\(177\) −6.71597 −0.0379434
\(178\) 0 0
\(179\) 238.744 1.33377 0.666884 0.745162i \(-0.267628\pi\)
0.666884 + 0.745162i \(0.267628\pi\)
\(180\) 0 0
\(181\) 173.294i 0.957424i −0.877972 0.478712i \(-0.841104\pi\)
0.877972 0.478712i \(-0.158896\pi\)
\(182\) 0 0
\(183\) 23.9460i 0.130852i
\(184\) 0 0
\(185\) −124.090 −0.670759
\(186\) 0 0
\(187\) −107.258 −0.573572
\(188\) 0 0
\(189\) 37.1681i 0.196656i
\(190\) 0 0
\(191\) 161.864i 0.847454i −0.905790 0.423727i \(-0.860722\pi\)
0.905790 0.423727i \(-0.139278\pi\)
\(192\) 0 0
\(193\) 8.54469 0.0442730 0.0221365 0.999755i \(-0.492953\pi\)
0.0221365 + 0.999755i \(0.492953\pi\)
\(194\) 0 0
\(195\) 16.2108i 0.0831321i
\(196\) 0 0
\(197\) −30.8682 −0.156691 −0.0783457 0.996926i \(-0.524964\pi\)
−0.0783457 + 0.996926i \(0.524964\pi\)
\(198\) 0 0
\(199\) 65.5687i 0.329491i 0.986336 + 0.164746i \(0.0526803\pi\)
−0.986336 + 0.164746i \(0.947320\pi\)
\(200\) 0 0
\(201\) 64.1827i 0.319317i
\(202\) 0 0
\(203\) 138.963i 0.684548i
\(204\) 0 0
\(205\) 62.2184i 0.303504i
\(206\) 0 0
\(207\) −28.5279 62.8264i −0.137816 0.303509i
\(208\) 0 0
\(209\) −112.350 −0.537559
\(210\) 0 0
\(211\) −235.552 −1.11636 −0.558180 0.829720i \(-0.688500\pi\)
−0.558180 + 0.829720i \(0.688500\pi\)
\(212\) 0 0
\(213\) 100.528 0.471962
\(214\) 0 0
\(215\) −63.0994 −0.293486
\(216\) 0 0
\(217\) 188.311i 0.867794i
\(218\) 0 0
\(219\) 22.4405 0.102468
\(220\) 0 0
\(221\) 56.7457i 0.256768i
\(222\) 0 0
\(223\) −290.398 −1.30223 −0.651117 0.758977i \(-0.725699\pi\)
−0.651117 + 0.758977i \(0.725699\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 133.737i 0.589150i −0.955628 0.294575i \(-0.904822\pi\)
0.955628 0.294575i \(-0.0951781\pi\)
\(228\) 0 0
\(229\) 105.255i 0.459628i 0.973235 + 0.229814i \(0.0738117\pi\)
−0.973235 + 0.229814i \(0.926188\pi\)
\(230\) 0 0
\(231\) −98.0173 −0.424317
\(232\) 0 0
\(233\) 214.886 0.922260 0.461130 0.887333i \(-0.347444\pi\)
0.461130 + 0.887333i \(0.347444\pi\)
\(234\) 0 0
\(235\) 38.1028i 0.162140i
\(236\) 0 0
\(237\) 96.8615i 0.408699i
\(238\) 0 0
\(239\) 134.459 0.562589 0.281295 0.959621i \(-0.409236\pi\)
0.281295 + 0.959621i \(0.409236\pi\)
\(240\) 0 0
\(241\) 283.577i 1.17667i 0.808619 + 0.588333i \(0.200216\pi\)
−0.808619 + 0.588333i \(0.799784\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 4.84194i 0.0197630i
\(246\) 0 0
\(247\) 59.4397i 0.240647i
\(248\) 0 0
\(249\) 38.4285i 0.154331i
\(250\) 0 0
\(251\) 215.034i 0.856709i −0.903611 0.428354i \(-0.859094\pi\)
0.903611 0.428354i \(-0.140906\pi\)
\(252\) 0 0
\(253\) 165.682 75.2319i 0.654869 0.297359i
\(254\) 0 0
\(255\) −52.5075 −0.205912
\(256\) 0 0
\(257\) −138.767 −0.539951 −0.269975 0.962867i \(-0.587016\pi\)
−0.269975 + 0.962867i \(0.587016\pi\)
\(258\) 0 0
\(259\) −396.955 −1.53265
\(260\) 0 0
\(261\) 58.2818 0.223302
\(262\) 0 0
\(263\) 415.194i 1.57869i −0.613953 0.789343i \(-0.710422\pi\)
0.613953 0.789343i \(-0.289578\pi\)
\(264\) 0 0
\(265\) −127.002 −0.479254
\(266\) 0 0
\(267\) 105.286i 0.394329i
\(268\) 0 0
\(269\) −237.575 −0.883179 −0.441589 0.897217i \(-0.645585\pi\)
−0.441589 + 0.897217i \(0.645585\pi\)
\(270\) 0 0
\(271\) −218.766 −0.807253 −0.403627 0.914924i \(-0.632250\pi\)
−0.403627 + 0.914924i \(0.632250\pi\)
\(272\) 0 0
\(273\) 51.8569i 0.189952i
\(274\) 0 0
\(275\) 39.5570i 0.143844i
\(276\) 0 0
\(277\) −435.549 −1.57238 −0.786189 0.617987i \(-0.787949\pi\)
−0.786189 + 0.617987i \(0.787949\pi\)
\(278\) 0 0
\(279\) 78.9786 0.283077
\(280\) 0 0
\(281\) 126.751i 0.451070i −0.974235 0.225535i \(-0.927587\pi\)
0.974235 0.225535i \(-0.0724131\pi\)
\(282\) 0 0
\(283\) 87.2964i 0.308468i −0.988034 0.154234i \(-0.950709\pi\)
0.988034 0.154234i \(-0.0492910\pi\)
\(284\) 0 0
\(285\) −55.0002 −0.192983
\(286\) 0 0
\(287\) 199.032i 0.693490i
\(288\) 0 0
\(289\) 105.198 0.364006
\(290\) 0 0
\(291\) 240.929i 0.827936i
\(292\) 0 0
\(293\) 26.6052i 0.0908027i −0.998969 0.0454013i \(-0.985543\pi\)
0.998969 0.0454013i \(-0.0144567\pi\)
\(294\) 0 0
\(295\) 8.67029i 0.0293908i
\(296\) 0 0
\(297\) 41.1089i 0.138414i
\(298\) 0 0
\(299\) 39.8021 + 87.6554i 0.133117 + 0.293162i
\(300\) 0 0
\(301\) −201.850 −0.670597
\(302\) 0 0
\(303\) 174.397 0.575569
\(304\) 0 0
\(305\) 30.9141 0.101358
\(306\) 0 0
\(307\) −128.834 −0.419656 −0.209828 0.977738i \(-0.567290\pi\)
−0.209828 + 0.977738i \(0.567290\pi\)
\(308\) 0 0
\(309\) 111.058i 0.359412i
\(310\) 0 0
\(311\) 231.829 0.745430 0.372715 0.927946i \(-0.378427\pi\)
0.372715 + 0.927946i \(0.378427\pi\)
\(312\) 0 0
\(313\) 29.9680i 0.0957443i 0.998853 + 0.0478722i \(0.0152440\pi\)
−0.998853 + 0.0478722i \(0.984756\pi\)
\(314\) 0 0
\(315\) −47.9838 −0.152329
\(316\) 0 0
\(317\) 42.3589 0.133624 0.0668121 0.997766i \(-0.478717\pi\)
0.0668121 + 0.997766i \(0.478717\pi\)
\(318\) 0 0
\(319\) 153.697i 0.481809i
\(320\) 0 0
\(321\) 3.20186i 0.00997464i
\(322\) 0 0
\(323\) −192.528 −0.596063
\(324\) 0 0
\(325\) 20.9280 0.0643938
\(326\) 0 0
\(327\) 190.571i 0.582784i
\(328\) 0 0
\(329\) 121.888i 0.370480i
\(330\) 0 0
\(331\) −295.202 −0.891849 −0.445924 0.895071i \(-0.647125\pi\)
−0.445924 + 0.895071i \(0.647125\pi\)
\(332\) 0 0
\(333\) 166.485i 0.499954i
\(334\) 0 0
\(335\) −82.8596 −0.247342
\(336\) 0 0
\(337\) 36.6799i 0.108842i 0.998518 + 0.0544212i \(0.0173314\pi\)
−0.998518 + 0.0544212i \(0.982669\pi\)
\(338\) 0 0
\(339\) 160.007i 0.471998i
\(340\) 0 0
\(341\) 208.277i 0.610784i
\(342\) 0 0
\(343\) 335.008i 0.976700i
\(344\) 0 0
\(345\) 81.1086 36.8293i 0.235097 0.106752i
\(346\) 0 0
\(347\) −234.840 −0.676773 −0.338386 0.941007i \(-0.609881\pi\)
−0.338386 + 0.941007i \(0.609881\pi\)
\(348\) 0 0
\(349\) −281.131 −0.805533 −0.402767 0.915303i \(-0.631951\pi\)
−0.402767 + 0.915303i \(0.631951\pi\)
\(350\) 0 0
\(351\) 21.7490 0.0619630
\(352\) 0 0
\(353\) 196.822 0.557570 0.278785 0.960354i \(-0.410068\pi\)
0.278785 + 0.960354i \(0.410068\pi\)
\(354\) 0 0
\(355\) 129.781i 0.365580i
\(356\) 0 0
\(357\) −167.967 −0.470496
\(358\) 0 0
\(359\) 429.346i 1.19595i −0.801515 0.597975i \(-0.795972\pi\)
0.801515 0.597975i \(-0.204028\pi\)
\(360\) 0 0
\(361\) 159.332 0.441362
\(362\) 0 0
\(363\) −101.168 −0.278701
\(364\) 0 0
\(365\) 28.9706i 0.0793715i
\(366\) 0 0
\(367\) 347.990i 0.948200i −0.880471 0.474100i \(-0.842773\pi\)
0.880471 0.474100i \(-0.157227\pi\)
\(368\) 0 0
\(369\) 83.4748 0.226219
\(370\) 0 0
\(371\) −406.270 −1.09507
\(372\) 0 0
\(373\) 322.048i 0.863400i −0.902017 0.431700i \(-0.857914\pi\)
0.902017 0.431700i \(-0.142086\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) −81.3147 −0.215689
\(378\) 0 0
\(379\) 75.9107i 0.200292i −0.994973 0.100146i \(-0.968069\pi\)
0.994973 0.100146i \(-0.0319310\pi\)
\(380\) 0 0
\(381\) 364.874 0.957673
\(382\) 0 0
\(383\) 331.984i 0.866799i 0.901202 + 0.433399i \(0.142686\pi\)
−0.901202 + 0.433399i \(0.857314\pi\)
\(384\) 0 0
\(385\) 126.540i 0.328675i
\(386\) 0 0
\(387\) 84.6567i 0.218751i
\(388\) 0 0
\(389\) 23.0806i 0.0593332i 0.999560 + 0.0296666i \(0.00944456\pi\)
−0.999560 + 0.0296666i \(0.990555\pi\)
\(390\) 0 0
\(391\) 283.920 128.921i 0.726139 0.329721i
\(392\) 0 0
\(393\) 371.881 0.946262
\(394\) 0 0
\(395\) −125.048 −0.316576
\(396\) 0 0
\(397\) 278.493 0.701494 0.350747 0.936470i \(-0.385928\pi\)
0.350747 + 0.936470i \(0.385928\pi\)
\(398\) 0 0
\(399\) −175.941 −0.440956
\(400\) 0 0
\(401\) 178.343i 0.444745i 0.974962 + 0.222373i \(0.0713801\pi\)
−0.974962 + 0.222373i \(0.928620\pi\)
\(402\) 0 0
\(403\) −110.191 −0.273427
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) 439.043 1.07873
\(408\) 0 0
\(409\) −94.0543 −0.229962 −0.114981 0.993368i \(-0.536681\pi\)
−0.114981 + 0.993368i \(0.536681\pi\)
\(410\) 0 0
\(411\) 120.658i 0.293571i
\(412\) 0 0
\(413\) 27.7355i 0.0671562i
\(414\) 0 0
\(415\) 49.6110 0.119545
\(416\) 0 0
\(417\) 205.414 0.492601
\(418\) 0 0
\(419\) 525.568i 1.25434i 0.778883 + 0.627169i \(0.215787\pi\)
−0.778883 + 0.627169i \(0.784213\pi\)
\(420\) 0 0
\(421\) 175.791i 0.417556i −0.977963 0.208778i \(-0.933051\pi\)
0.977963 0.208778i \(-0.0669486\pi\)
\(422\) 0 0
\(423\) −51.1203 −0.120852
\(424\) 0 0
\(425\) 67.7869i 0.159498i
\(426\) 0 0
\(427\) 98.8917 0.231596
\(428\) 0 0
\(429\) 57.3551i 0.133695i
\(430\) 0 0
\(431\) 216.554i 0.502444i 0.967929 + 0.251222i \(0.0808325\pi\)
−0.967929 + 0.251222i \(0.919168\pi\)
\(432\) 0 0
\(433\) 674.973i 1.55883i −0.626508 0.779415i \(-0.715517\pi\)
0.626508 0.779415i \(-0.284483\pi\)
\(434\) 0 0
\(435\) 75.2414i 0.172969i
\(436\) 0 0
\(437\) 297.399 135.041i 0.680548 0.309019i
\(438\) 0 0
\(439\) −744.144 −1.69509 −0.847545 0.530724i \(-0.821920\pi\)
−0.847545 + 0.530724i \(0.821920\pi\)
\(440\) 0 0
\(441\) −6.49614 −0.0147305
\(442\) 0 0
\(443\) −437.171 −0.986841 −0.493420 0.869791i \(-0.664254\pi\)
−0.493420 + 0.869791i \(0.664254\pi\)
\(444\) 0 0
\(445\) 135.923 0.305446
\(446\) 0 0
\(447\) 8.20892i 0.0183645i
\(448\) 0 0
\(449\) 247.503 0.551232 0.275616 0.961268i \(-0.411118\pi\)
0.275616 + 0.961268i \(0.411118\pi\)
\(450\) 0 0
\(451\) 220.134i 0.488103i
\(452\) 0 0
\(453\) −239.575 −0.528863
\(454\) 0 0
\(455\) 66.9469 0.147136
\(456\) 0 0
\(457\) 799.537i 1.74953i 0.484544 + 0.874767i \(0.338986\pi\)
−0.484544 + 0.874767i \(0.661014\pi\)
\(458\) 0 0
\(459\) 70.4462i 0.153477i
\(460\) 0 0
\(461\) −112.380 −0.243775 −0.121888 0.992544i \(-0.538895\pi\)
−0.121888 + 0.992544i \(0.538895\pi\)
\(462\) 0 0
\(463\) −465.657 −1.00574 −0.502869 0.864363i \(-0.667722\pi\)
−0.502869 + 0.864363i \(0.667722\pi\)
\(464\) 0 0
\(465\) 101.961i 0.219271i
\(466\) 0 0
\(467\) 715.028i 1.53111i 0.643371 + 0.765555i \(0.277535\pi\)
−0.643371 + 0.765555i \(0.722465\pi\)
\(468\) 0 0
\(469\) −265.061 −0.565162
\(470\) 0 0
\(471\) 256.784i 0.545189i
\(472\) 0 0
\(473\) 223.251 0.471990
\(474\) 0 0
\(475\) 71.0050i 0.149484i
\(476\) 0 0
\(477\) 170.391i 0.357215i
\(478\) 0 0
\(479\) 375.058i 0.783002i 0.920178 + 0.391501i \(0.128044\pi\)
−0.920178 + 0.391501i \(0.871956\pi\)
\(480\) 0 0
\(481\) 232.280i 0.482910i
\(482\) 0 0
\(483\) 259.460 117.814i 0.537184 0.243921i
\(484\) 0 0
\(485\) 311.038 0.641316
\(486\) 0 0
\(487\) −566.914 −1.16409 −0.582047 0.813155i \(-0.697748\pi\)
−0.582047 + 0.813155i \(0.697748\pi\)
\(488\) 0 0
\(489\) 424.959 0.869037
\(490\) 0 0
\(491\) 35.6700 0.0726477 0.0363238 0.999340i \(-0.488435\pi\)
0.0363238 + 0.999340i \(0.488435\pi\)
\(492\) 0 0
\(493\) 263.383i 0.534245i
\(494\) 0 0
\(495\) 53.0713 0.107215
\(496\) 0 0
\(497\) 415.159i 0.835329i
\(498\) 0 0
\(499\) −590.157 −1.18268 −0.591340 0.806422i \(-0.701401\pi\)
−0.591340 + 0.806422i \(0.701401\pi\)
\(500\) 0 0
\(501\) 404.997 0.808378
\(502\) 0 0
\(503\) 590.702i 1.17436i −0.809457 0.587179i \(-0.800238\pi\)
0.809457 0.587179i \(-0.199762\pi\)
\(504\) 0 0
\(505\) 225.146i 0.445834i
\(506\) 0 0
\(507\) 262.372 0.517500
\(508\) 0 0
\(509\) −822.658 −1.61622 −0.808112 0.589029i \(-0.799510\pi\)
−0.808112 + 0.589029i \(0.799510\pi\)
\(510\) 0 0
\(511\) 92.6746i 0.181359i
\(512\) 0 0
\(513\) 73.7906i 0.143841i
\(514\) 0 0
\(515\) −143.376 −0.278399
\(516\) 0 0
\(517\) 134.811i 0.260757i
\(518\) 0 0
\(519\) −318.579 −0.613833
\(520\) 0 0
\(521\) 525.022i 1.00772i −0.863786 0.503860i \(-0.831913\pi\)
0.863786 0.503860i \(-0.168087\pi\)
\(522\) 0 0
\(523\) 161.456i 0.308712i −0.988015 0.154356i \(-0.950670\pi\)
0.988015 0.154356i \(-0.0493302\pi\)
\(524\) 0 0
\(525\) 61.9468i 0.117994i
\(526\) 0 0
\(527\) 356.914i 0.677256i
\(528\) 0 0
\(529\) −348.147 + 398.290i −0.658122 + 0.752911i
\(530\) 0 0
\(531\) 11.6324 0.0219066
\(532\) 0 0
\(533\) −116.464 −0.218506
\(534\) 0 0
\(535\) 4.13358 0.00772632
\(536\) 0 0
\(537\) −413.517 −0.770051
\(538\) 0 0
\(539\) 17.1312i 0.0317833i
\(540\) 0 0
\(541\) −303.462 −0.560928 −0.280464 0.959865i \(-0.590488\pi\)
−0.280464 + 0.959865i \(0.590488\pi\)
\(542\) 0 0
\(543\) 300.154i 0.552769i
\(544\) 0 0
\(545\) −246.025 −0.451423
\(546\) 0 0
\(547\) 135.210 0.247185 0.123592 0.992333i \(-0.460559\pi\)
0.123592 + 0.992333i \(0.460559\pi\)
\(548\) 0 0
\(549\) 41.4756i 0.0755476i
\(550\) 0 0
\(551\) 275.887i 0.500701i
\(552\) 0 0
\(553\) −400.017 −0.723359
\(554\) 0 0
\(555\) 214.931 0.387263
\(556\) 0 0
\(557\) 298.873i 0.536576i 0.963339 + 0.268288i \(0.0864579\pi\)
−0.963339 + 0.268288i \(0.913542\pi\)
\(558\) 0 0
\(559\) 118.113i 0.211293i
\(560\) 0 0
\(561\) 185.776 0.331152
\(562\) 0 0
\(563\) 113.012i 0.200731i −0.994951 0.100366i \(-0.967999\pi\)
0.994951 0.100366i \(-0.0320013\pi\)
\(564\) 0 0
\(565\) 206.568 0.365608
\(566\) 0 0
\(567\) 64.3770i 0.113540i
\(568\) 0 0
\(569\) 50.6364i 0.0889918i 0.999010 + 0.0444959i \(0.0141682\pi\)
−0.999010 + 0.0444959i \(0.985832\pi\)
\(570\) 0 0
\(571\) 463.896i 0.812428i −0.913778 0.406214i \(-0.866849\pi\)
0.913778 0.406214i \(-0.133151\pi\)
\(572\) 0 0
\(573\) 280.356i 0.489278i
\(574\) 0 0
\(575\) 47.5465 + 104.711i 0.0826895 + 0.182106i
\(576\) 0 0
\(577\) 824.880 1.42960 0.714801 0.699328i \(-0.246517\pi\)
0.714801 + 0.699328i \(0.246517\pi\)
\(578\) 0 0
\(579\) −14.7998 −0.0255610
\(580\) 0 0
\(581\) 158.702 0.273152
\(582\) 0 0
\(583\) 449.345 0.770747
\(584\) 0 0
\(585\) 28.0779i 0.0479963i
\(586\) 0 0
\(587\) 1080.60 1.84089 0.920444 0.390874i \(-0.127827\pi\)
0.920444 + 0.390874i \(0.127827\pi\)
\(588\) 0 0
\(589\) 373.858i 0.634734i
\(590\) 0 0
\(591\) 53.4653 0.0904658
\(592\) 0 0
\(593\) 742.032 1.25132 0.625659 0.780097i \(-0.284830\pi\)
0.625659 + 0.780097i \(0.284830\pi\)
\(594\) 0 0
\(595\) 216.845i 0.364445i
\(596\) 0 0
\(597\) 113.568i 0.190232i
\(598\) 0 0
\(599\) −311.435 −0.519924 −0.259962 0.965619i \(-0.583710\pi\)
−0.259962 + 0.965619i \(0.583710\pi\)
\(600\) 0 0
\(601\) 621.128 1.03349 0.516745 0.856139i \(-0.327143\pi\)
0.516745 + 0.856139i \(0.327143\pi\)
\(602\) 0 0
\(603\) 111.168i 0.184358i
\(604\) 0 0
\(605\) 130.608i 0.215881i
\(606\) 0 0
\(607\) −546.389 −0.900147 −0.450073 0.892992i \(-0.648602\pi\)
−0.450073 + 0.892992i \(0.648602\pi\)
\(608\) 0 0
\(609\) 240.691i 0.395224i
\(610\) 0 0
\(611\) 71.3230 0.116732
\(612\) 0 0
\(613\) 990.815i 1.61634i −0.588951 0.808169i \(-0.700459\pi\)
0.588951 0.808169i \(-0.299541\pi\)
\(614\) 0 0
\(615\) 107.765i 0.175228i
\(616\) 0 0
\(617\) 46.0578i 0.0746480i −0.999303 0.0373240i \(-0.988117\pi\)
0.999303 0.0373240i \(-0.0118834\pi\)
\(618\) 0 0
\(619\) 499.837i 0.807492i −0.914871 0.403746i \(-0.867708\pi\)
0.914871 0.403746i \(-0.132292\pi\)
\(620\) 0 0
\(621\) 49.4117 + 108.819i 0.0795680 + 0.175231i
\(622\) 0 0
\(623\) 434.808 0.697926
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 194.596 0.310360
\(628\) 0 0
\(629\) 752.366 1.19613
\(630\) 0 0
\(631\) 113.732i 0.180241i −0.995931 0.0901205i \(-0.971275\pi\)
0.995931 0.0901205i \(-0.0287252\pi\)
\(632\) 0 0
\(633\) 407.988 0.644531
\(634\) 0 0
\(635\) 471.050i 0.741811i
\(636\) 0 0
\(637\) 9.06342 0.0142283
\(638\) 0 0
\(639\) −174.119 −0.272487
\(640\) 0 0
\(641\) 864.720i 1.34902i −0.738267 0.674508i \(-0.764356\pi\)
0.738267 0.674508i \(-0.235644\pi\)
\(642\) 0 0
\(643\) 178.936i 0.278282i −0.990273 0.139141i \(-0.955566\pi\)
0.990273 0.139141i \(-0.0444342\pi\)
\(644\) 0 0
\(645\) 109.291 0.169444
\(646\) 0 0
\(647\) 581.856 0.899313 0.449657 0.893202i \(-0.351546\pi\)
0.449657 + 0.893202i \(0.351546\pi\)
\(648\) 0 0
\(649\) 30.6762i 0.0472669i
\(650\) 0 0
\(651\) 326.165i 0.501021i
\(652\) 0 0
\(653\) 367.973 0.563511 0.281755 0.959486i \(-0.409083\pi\)
0.281755 + 0.959486i \(0.409083\pi\)
\(654\) 0 0
\(655\) 480.096i 0.732971i
\(656\) 0 0
\(657\) −38.8682 −0.0591601
\(658\) 0 0
\(659\) 598.038i 0.907493i 0.891131 + 0.453747i \(0.149913\pi\)
−0.891131 + 0.453747i \(0.850087\pi\)
\(660\) 0 0
\(661\) 320.615i 0.485045i −0.970146 0.242522i \(-0.922025\pi\)
0.970146 0.242522i \(-0.0779748\pi\)
\(662\) 0 0
\(663\) 98.2865i 0.148245i
\(664\) 0 0
\(665\) 227.139i 0.341563i
\(666\) 0 0
\(667\) −184.740 406.848i −0.276971 0.609968i
\(668\) 0 0
\(669\) 502.985 0.751845
\(670\) 0 0
\(671\) −109.377 −0.163006
\(672\) 0 0
\(673\) 257.176 0.382134 0.191067 0.981577i \(-0.438805\pi\)
0.191067 + 0.981577i \(0.438805\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 523.172i 0.772780i −0.922336 0.386390i \(-0.873722\pi\)
0.922336 0.386390i \(-0.126278\pi\)
\(678\) 0 0
\(679\) 994.986 1.46537
\(680\) 0 0
\(681\) 231.640i 0.340146i
\(682\) 0 0
\(683\) 609.415 0.892262 0.446131 0.894968i \(-0.352801\pi\)
0.446131 + 0.894968i \(0.352801\pi\)
\(684\) 0 0
\(685\) 155.769 0.227399
\(686\) 0 0
\(687\) 182.307i 0.265366i
\(688\) 0 0
\(689\) 237.730i 0.345036i
\(690\) 0 0
\(691\) −697.307 −1.00913 −0.504564 0.863374i \(-0.668347\pi\)
−0.504564 + 0.863374i \(0.668347\pi\)
\(692\) 0 0
\(693\) 169.771 0.244980
\(694\) 0 0
\(695\) 265.189i 0.381567i
\(696\) 0 0
\(697\) 377.233i 0.541223i
\(698\) 0 0
\(699\) −372.194 −0.532467
\(700\) 0 0
\(701\) 803.156i 1.14573i 0.819650 + 0.572865i \(0.194168\pi\)
−0.819650 + 0.572865i \(0.805832\pi\)
\(702\) 0 0
\(703\) 788.084 1.12103
\(704\) 0 0
\(705\) 65.9960i 0.0936114i
\(706\) 0 0
\(707\) 720.224i 1.01870i
\(708\) 0 0
\(709\) 600.093i 0.846394i 0.906038 + 0.423197i \(0.139092\pi\)
−0.906038 + 0.423197i \(0.860908\pi\)
\(710\) 0 0
\(711\) 167.769i 0.235962i
\(712\) 0 0
\(713\) −250.344 551.327i −0.351113 0.773250i
\(714\) 0 0
\(715\) −74.0451 −0.103560
\(716\) 0 0
\(717\) −232.889 −0.324811
\(718\) 0 0
\(719\) 242.742 0.337611 0.168806 0.985649i \(-0.446009\pi\)
0.168806 + 0.985649i \(0.446009\pi\)
\(720\) 0 0
\(721\) −458.647 −0.636126
\(722\) 0 0
\(723\) 491.169i 0.679349i
\(724\) 0 0
\(725\) −97.1363 −0.133981
\(726\) 0 0
\(727\) 394.554i 0.542715i −0.962479 0.271357i \(-0.912527\pi\)
0.962479 0.271357i \(-0.0874725\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 382.574 0.523357
\(732\) 0 0
\(733\) 806.874i 1.10078i −0.834907 0.550392i \(-0.814478\pi\)
0.834907 0.550392i \(-0.185522\pi\)
\(734\) 0 0
\(735\) 8.38648i 0.0114102i
\(736\) 0 0
\(737\) 293.165 0.397781
\(738\) 0 0
\(739\) 111.860 0.151366 0.0756832 0.997132i \(-0.475886\pi\)
0.0756832 + 0.997132i \(0.475886\pi\)
\(740\) 0 0
\(741\) 102.953i 0.138937i
\(742\) 0 0
\(743\) 275.780i 0.371171i 0.982628 + 0.185586i \(0.0594182\pi\)
−0.982628 + 0.185586i \(0.940582\pi\)
\(744\) 0 0
\(745\) −10.5977 −0.0142251
\(746\) 0 0
\(747\) 66.5602i 0.0891033i
\(748\) 0 0
\(749\) 13.2230 0.0176542
\(750\) 0 0
\(751\) 889.185i 1.18400i −0.805937 0.592001i \(-0.798338\pi\)
0.805937 0.592001i \(-0.201662\pi\)
\(752\) 0 0
\(753\) 372.450i 0.494621i
\(754\) 0 0
\(755\) 309.290i 0.409655i
\(756\) 0 0
\(757\) 909.789i 1.20184i 0.799311 + 0.600918i \(0.205198\pi\)
−0.799311 + 0.600918i \(0.794802\pi\)
\(758\) 0 0
\(759\) −286.969 + 130.306i −0.378089 + 0.171681i
\(760\) 0 0
\(761\) 945.275 1.24215 0.621074 0.783752i \(-0.286697\pi\)
0.621074 + 0.783752i \(0.286697\pi\)
\(762\) 0 0
\(763\) −787.015 −1.03147
\(764\) 0 0
\(765\) 90.9456 0.118883
\(766\) 0 0
\(767\) −16.2295 −0.0211598
\(768\) 0 0
\(769\) 229.849i 0.298894i 0.988770 + 0.149447i \(0.0477493\pi\)
−0.988770 + 0.149447i \(0.952251\pi\)
\(770\) 0 0
\(771\) 240.352 0.311741
\(772\) 0 0
\(773\) 316.488i 0.409428i −0.978822 0.204714i \(-0.934374\pi\)
0.978822 0.204714i \(-0.0656265\pi\)
\(774\) 0 0
\(775\) −131.631 −0.169846
\(776\) 0 0
\(777\) 687.547 0.884873
\(778\) 0 0
\(779\) 395.142i 0.507242i
\(780\) 0 0
\(781\) 459.177i 0.587934i
\(782\) 0 0
\(783\) −100.947 −0.128923
\(784\) 0 0
\(785\) −331.507 −0.422302
\(786\) 0 0
\(787\) 539.232i 0.685174i −0.939486 0.342587i \(-0.888697\pi\)
0.939486 0.342587i \(-0.111303\pi\)
\(788\) 0 0
\(789\) 719.138i 0.911454i
\(790\) 0 0
\(791\) 660.795 0.835392
\(792\) 0 0
\(793\) 57.8668i 0.0729720i
\(794\) 0 0
\(795\) 219.974 0.276697
\(796\) 0 0
\(797\) 1323.72i 1.66088i 0.557106 + 0.830442i \(0.311912\pi\)
−0.557106 + 0.830442i \(0.688088\pi\)
\(798\) 0 0