Properties

Label 2760.3.g.a.2161.6
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 2161.6
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.43

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} -6.71584i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} -6.71584i q^{7} +3.00000 q^{9} -10.6175i q^{11} +11.0715 q^{13} +3.87298i q^{15} -30.7855i q^{17} +32.5373i q^{19} +11.6322i q^{21} +(22.7195 + 3.58142i) q^{23} -5.00000 q^{25} -5.19615 q^{27} +26.7694 q^{29} +10.6841 q^{31} +18.3901i q^{33} -15.0171 q^{35} -65.6685i q^{37} -19.1765 q^{39} +58.5907 q^{41} -50.0380i q^{43} -6.70820i q^{45} +86.0576 q^{47} +3.89754 q^{49} +53.3221i q^{51} +80.6213i q^{53} -23.7415 q^{55} -56.3563i q^{57} +46.8627 q^{59} +65.5994i q^{61} -20.1475i q^{63} -24.7567i q^{65} -10.4393i q^{67} +(-39.3512 - 6.20320i) q^{69} +10.8534 q^{71} -130.680 q^{73} +8.66025 q^{75} -71.3055 q^{77} +121.872i q^{79} +9.00000 q^{81} +104.864i q^{83} -68.8385 q^{85} -46.3659 q^{87} -106.114i q^{89} -74.3546i q^{91} -18.5055 q^{93} +72.7556 q^{95} -104.436i q^{97} -31.8526i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 6.71584i 0.959405i −0.877431 0.479703i \(-0.840745\pi\)
0.877431 0.479703i \(-0.159255\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 10.6175i 0.965229i −0.875833 0.482615i \(-0.839687\pi\)
0.875833 0.482615i \(-0.160313\pi\)
\(12\) 0 0
\(13\) 11.0715 0.851657 0.425828 0.904804i \(-0.359983\pi\)
0.425828 + 0.904804i \(0.359983\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 30.7855i 1.81091i −0.424440 0.905456i \(-0.639529\pi\)
0.424440 0.905456i \(-0.360471\pi\)
\(18\) 0 0
\(19\) 32.5373i 1.71249i 0.516570 + 0.856245i \(0.327209\pi\)
−0.516570 + 0.856245i \(0.672791\pi\)
\(20\) 0 0
\(21\) 11.6322i 0.553913i
\(22\) 0 0
\(23\) 22.7195 + 3.58142i 0.987802 + 0.155714i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 26.7694 0.923081 0.461541 0.887119i \(-0.347297\pi\)
0.461541 + 0.887119i \(0.347297\pi\)
\(30\) 0 0
\(31\) 10.6841 0.344650 0.172325 0.985040i \(-0.444872\pi\)
0.172325 + 0.985040i \(0.444872\pi\)
\(32\) 0 0
\(33\) 18.3901i 0.557275i
\(34\) 0 0
\(35\) −15.0171 −0.429059
\(36\) 0 0
\(37\) 65.6685i 1.77483i −0.460976 0.887413i \(-0.652501\pi\)
0.460976 0.887413i \(-0.347499\pi\)
\(38\) 0 0
\(39\) −19.1765 −0.491704
\(40\) 0 0
\(41\) 58.5907 1.42904 0.714521 0.699614i \(-0.246645\pi\)
0.714521 + 0.699614i \(0.246645\pi\)
\(42\) 0 0
\(43\) 50.0380i 1.16367i −0.813306 0.581837i \(-0.802334\pi\)
0.813306 0.581837i \(-0.197666\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) 86.0576 1.83101 0.915507 0.402303i \(-0.131790\pi\)
0.915507 + 0.402303i \(0.131790\pi\)
\(48\) 0 0
\(49\) 3.89754 0.0795416
\(50\) 0 0
\(51\) 53.3221i 1.04553i
\(52\) 0 0
\(53\) 80.6213i 1.52116i 0.649246 + 0.760578i \(0.275084\pi\)
−0.649246 + 0.760578i \(0.724916\pi\)
\(54\) 0 0
\(55\) −23.7415 −0.431664
\(56\) 0 0
\(57\) 56.3563i 0.988707i
\(58\) 0 0
\(59\) 46.8627 0.794283 0.397141 0.917757i \(-0.370002\pi\)
0.397141 + 0.917757i \(0.370002\pi\)
\(60\) 0 0
\(61\) 65.5994i 1.07540i 0.843136 + 0.537700i \(0.180707\pi\)
−0.843136 + 0.537700i \(0.819293\pi\)
\(62\) 0 0
\(63\) 20.1475i 0.319802i
\(64\) 0 0
\(65\) 24.7567i 0.380872i
\(66\) 0 0
\(67\) 10.4393i 0.155811i −0.996961 0.0779055i \(-0.975177\pi\)
0.996961 0.0779055i \(-0.0248233\pi\)
\(68\) 0 0
\(69\) −39.3512 6.20320i −0.570308 0.0899015i
\(70\) 0 0
\(71\) 10.8534 0.152865 0.0764324 0.997075i \(-0.475647\pi\)
0.0764324 + 0.997075i \(0.475647\pi\)
\(72\) 0 0
\(73\) −130.680 −1.79014 −0.895072 0.445922i \(-0.852876\pi\)
−0.895072 + 0.445922i \(0.852876\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) −71.3055 −0.926046
\(78\) 0 0
\(79\) 121.872i 1.54268i 0.636421 + 0.771342i \(0.280414\pi\)
−0.636421 + 0.771342i \(0.719586\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 104.864i 1.26342i 0.775206 + 0.631709i \(0.217646\pi\)
−0.775206 + 0.631709i \(0.782354\pi\)
\(84\) 0 0
\(85\) −68.8385 −0.809864
\(86\) 0 0
\(87\) −46.3659 −0.532941
\(88\) 0 0
\(89\) 106.114i 1.19229i −0.802875 0.596147i \(-0.796697\pi\)
0.802875 0.596147i \(-0.203303\pi\)
\(90\) 0 0
\(91\) 74.3546i 0.817084i
\(92\) 0 0
\(93\) −18.5055 −0.198983
\(94\) 0 0
\(95\) 72.7556 0.765849
\(96\) 0 0
\(97\) 104.436i 1.07666i −0.842733 0.538331i \(-0.819055\pi\)
0.842733 0.538331i \(-0.180945\pi\)
\(98\) 0 0
\(99\) 31.8526i 0.321743i
\(100\) 0 0
\(101\) −100.062 −0.990709 −0.495355 0.868691i \(-0.664962\pi\)
−0.495355 + 0.868691i \(0.664962\pi\)
\(102\) 0 0
\(103\) 11.1895i 0.108636i 0.998524 + 0.0543180i \(0.0172985\pi\)
−0.998524 + 0.0543180i \(0.982702\pi\)
\(104\) 0 0
\(105\) 26.0103 0.247717
\(106\) 0 0
\(107\) 38.5091i 0.359898i −0.983676 0.179949i \(-0.942407\pi\)
0.983676 0.179949i \(-0.0575933\pi\)
\(108\) 0 0
\(109\) 130.220i 1.19468i −0.801989 0.597339i \(-0.796225\pi\)
0.801989 0.597339i \(-0.203775\pi\)
\(110\) 0 0
\(111\) 113.741i 1.02470i
\(112\) 0 0
\(113\) 99.8590i 0.883708i 0.897087 + 0.441854i \(0.145679\pi\)
−0.897087 + 0.441854i \(0.854321\pi\)
\(114\) 0 0
\(115\) 8.00830 50.8022i 0.0696374 0.441759i
\(116\) 0 0
\(117\) 33.2146 0.283886
\(118\) 0 0
\(119\) −206.750 −1.73740
\(120\) 0 0
\(121\) 8.26821 0.0683323
\(122\) 0 0
\(123\) −101.482 −0.825058
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) 197.059 1.55165 0.775823 0.630950i \(-0.217335\pi\)
0.775823 + 0.630950i \(0.217335\pi\)
\(128\) 0 0
\(129\) 86.6683i 0.671847i
\(130\) 0 0
\(131\) −57.4131 −0.438268 −0.219134 0.975695i \(-0.570323\pi\)
−0.219134 + 0.975695i \(0.570323\pi\)
\(132\) 0 0
\(133\) 218.515 1.64297
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 42.9712i 0.313658i −0.987626 0.156829i \(-0.949873\pi\)
0.987626 0.156829i \(-0.0501272\pi\)
\(138\) 0 0
\(139\) 191.272 1.37606 0.688029 0.725683i \(-0.258476\pi\)
0.688029 + 0.725683i \(0.258476\pi\)
\(140\) 0 0
\(141\) −149.056 −1.05714
\(142\) 0 0
\(143\) 117.552i 0.822044i
\(144\) 0 0
\(145\) 59.8581i 0.412814i
\(146\) 0 0
\(147\) −6.75073 −0.0459234
\(148\) 0 0
\(149\) 59.2649i 0.397751i 0.980025 + 0.198875i \(0.0637289\pi\)
−0.980025 + 0.198875i \(0.936271\pi\)
\(150\) 0 0
\(151\) 135.330 0.896228 0.448114 0.893977i \(-0.352096\pi\)
0.448114 + 0.893977i \(0.352096\pi\)
\(152\) 0 0
\(153\) 92.3565i 0.603637i
\(154\) 0 0
\(155\) 23.8905i 0.154132i
\(156\) 0 0
\(157\) 101.775i 0.648246i −0.946015 0.324123i \(-0.894931\pi\)
0.946015 0.324123i \(-0.105069\pi\)
\(158\) 0 0
\(159\) 139.640i 0.878240i
\(160\) 0 0
\(161\) 24.0522 152.580i 0.149393 0.947703i
\(162\) 0 0
\(163\) −1.73710 −0.0106570 −0.00532851 0.999986i \(-0.501696\pi\)
−0.00532851 + 0.999986i \(0.501696\pi\)
\(164\) 0 0
\(165\) 41.1215 0.249221
\(166\) 0 0
\(167\) −222.286 −1.33105 −0.665526 0.746374i \(-0.731793\pi\)
−0.665526 + 0.746374i \(0.731793\pi\)
\(168\) 0 0
\(169\) −46.4211 −0.274681
\(170\) 0 0
\(171\) 97.6119i 0.570830i
\(172\) 0 0
\(173\) 81.1902 0.469308 0.234654 0.972079i \(-0.424604\pi\)
0.234654 + 0.972079i \(0.424604\pi\)
\(174\) 0 0
\(175\) 33.5792i 0.191881i
\(176\) 0 0
\(177\) −81.1685 −0.458579
\(178\) 0 0
\(179\) −250.438 −1.39909 −0.699546 0.714587i \(-0.746615\pi\)
−0.699546 + 0.714587i \(0.746615\pi\)
\(180\) 0 0
\(181\) 211.909i 1.17077i −0.810757 0.585383i \(-0.800944\pi\)
0.810757 0.585383i \(-0.199056\pi\)
\(182\) 0 0
\(183\) 113.622i 0.620883i
\(184\) 0 0
\(185\) −146.839 −0.793726
\(186\) 0 0
\(187\) −326.866 −1.74795
\(188\) 0 0
\(189\) 34.8965i 0.184638i
\(190\) 0 0
\(191\) 1.51437i 0.00792862i 0.999992 + 0.00396431i \(0.00126188\pi\)
−0.999992 + 0.00396431i \(0.998738\pi\)
\(192\) 0 0
\(193\) 104.644 0.542196 0.271098 0.962552i \(-0.412613\pi\)
0.271098 + 0.962552i \(0.412613\pi\)
\(194\) 0 0
\(195\) 42.8799i 0.219897i
\(196\) 0 0
\(197\) −0.785066 −0.00398511 −0.00199255 0.999998i \(-0.500634\pi\)
−0.00199255 + 0.999998i \(0.500634\pi\)
\(198\) 0 0
\(199\) 157.211i 0.790004i −0.918680 0.395002i \(-0.870744\pi\)
0.918680 0.395002i \(-0.129256\pi\)
\(200\) 0 0
\(201\) 18.0815i 0.0899576i
\(202\) 0 0
\(203\) 179.779i 0.885609i
\(204\) 0 0
\(205\) 131.013i 0.639087i
\(206\) 0 0
\(207\) 68.1584 + 10.7443i 0.329267 + 0.0519046i
\(208\) 0 0
\(209\) 345.466 1.65295
\(210\) 0 0
\(211\) −329.572 −1.56195 −0.780976 0.624561i \(-0.785278\pi\)
−0.780976 + 0.624561i \(0.785278\pi\)
\(212\) 0 0
\(213\) −18.7986 −0.0882565
\(214\) 0 0
\(215\) −111.888 −0.520411
\(216\) 0 0
\(217\) 71.7529i 0.330659i
\(218\) 0 0
\(219\) 226.345 1.03354
\(220\) 0 0
\(221\) 340.843i 1.54228i
\(222\) 0 0
\(223\) 57.8098 0.259237 0.129618 0.991564i \(-0.458625\pi\)
0.129618 + 0.991564i \(0.458625\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 71.7582i 0.316115i 0.987430 + 0.158058i \(0.0505232\pi\)
−0.987430 + 0.158058i \(0.949477\pi\)
\(228\) 0 0
\(229\) 195.294i 0.852811i 0.904532 + 0.426405i \(0.140220\pi\)
−0.904532 + 0.426405i \(0.859780\pi\)
\(230\) 0 0
\(231\) 123.505 0.534653
\(232\) 0 0
\(233\) 415.948 1.78518 0.892591 0.450867i \(-0.148885\pi\)
0.892591 + 0.450867i \(0.148885\pi\)
\(234\) 0 0
\(235\) 192.431i 0.818854i
\(236\) 0 0
\(237\) 211.088i 0.890669i
\(238\) 0 0
\(239\) −136.892 −0.572769 −0.286385 0.958115i \(-0.592453\pi\)
−0.286385 + 0.958115i \(0.592453\pi\)
\(240\) 0 0
\(241\) 368.014i 1.52703i −0.645790 0.763515i \(-0.723472\pi\)
0.645790 0.763515i \(-0.276528\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 8.71516i 0.0355721i
\(246\) 0 0
\(247\) 360.238i 1.45845i
\(248\) 0 0
\(249\) 181.629i 0.729435i
\(250\) 0 0
\(251\) 65.6254i 0.261456i 0.991418 + 0.130728i \(0.0417314\pi\)
−0.991418 + 0.130728i \(0.958269\pi\)
\(252\) 0 0
\(253\) 38.0258 241.224i 0.150300 0.953456i
\(254\) 0 0
\(255\) 119.232 0.467575
\(256\) 0 0
\(257\) −8.59659 −0.0334498 −0.0167249 0.999860i \(-0.505324\pi\)
−0.0167249 + 0.999860i \(0.505324\pi\)
\(258\) 0 0
\(259\) −441.019 −1.70278
\(260\) 0 0
\(261\) 80.3081 0.307694
\(262\) 0 0
\(263\) 201.224i 0.765110i 0.923933 + 0.382555i \(0.124956\pi\)
−0.923933 + 0.382555i \(0.875044\pi\)
\(264\) 0 0
\(265\) 180.275 0.680282
\(266\) 0 0
\(267\) 183.795i 0.688372i
\(268\) 0 0
\(269\) −348.814 −1.29671 −0.648353 0.761340i \(-0.724542\pi\)
−0.648353 + 0.761340i \(0.724542\pi\)
\(270\) 0 0
\(271\) 77.2223 0.284953 0.142477 0.989798i \(-0.454493\pi\)
0.142477 + 0.989798i \(0.454493\pi\)
\(272\) 0 0
\(273\) 128.786i 0.471744i
\(274\) 0 0
\(275\) 53.0876i 0.193046i
\(276\) 0 0
\(277\) 182.061 0.657260 0.328630 0.944459i \(-0.393413\pi\)
0.328630 + 0.944459i \(0.393413\pi\)
\(278\) 0 0
\(279\) 32.0524 0.114883
\(280\) 0 0
\(281\) 179.247i 0.637889i −0.947773 0.318944i \(-0.896672\pi\)
0.947773 0.318944i \(-0.103328\pi\)
\(282\) 0 0
\(283\) 210.837i 0.745008i 0.928030 + 0.372504i \(0.121501\pi\)
−0.928030 + 0.372504i \(0.878499\pi\)
\(284\) 0 0
\(285\) −126.016 −0.442163
\(286\) 0 0
\(287\) 393.486i 1.37103i
\(288\) 0 0
\(289\) −658.747 −2.27940
\(290\) 0 0
\(291\) 180.889i 0.621611i
\(292\) 0 0
\(293\) 310.850i 1.06092i −0.847709 0.530461i \(-0.822019\pi\)
0.847709 0.530461i \(-0.177981\pi\)
\(294\) 0 0
\(295\) 104.788i 0.355214i
\(296\) 0 0
\(297\) 55.1703i 0.185758i
\(298\) 0 0
\(299\) 251.539 + 39.6518i 0.841268 + 0.132615i
\(300\) 0 0
\(301\) −336.047 −1.11643
\(302\) 0 0
\(303\) 173.312 0.571986
\(304\) 0 0
\(305\) 146.685 0.480934
\(306\) 0 0
\(307\) −90.1435 −0.293627 −0.146813 0.989164i \(-0.546902\pi\)
−0.146813 + 0.989164i \(0.546902\pi\)
\(308\) 0 0
\(309\) 19.3808i 0.0627211i
\(310\) 0 0
\(311\) −533.391 −1.71508 −0.857541 0.514415i \(-0.828009\pi\)
−0.857541 + 0.514415i \(0.828009\pi\)
\(312\) 0 0
\(313\) 191.852i 0.612945i −0.951879 0.306473i \(-0.900851\pi\)
0.951879 0.306473i \(-0.0991488\pi\)
\(314\) 0 0
\(315\) −45.0512 −0.143020
\(316\) 0 0
\(317\) 281.798 0.888953 0.444476 0.895791i \(-0.353390\pi\)
0.444476 + 0.895791i \(0.353390\pi\)
\(318\) 0 0
\(319\) 284.224i 0.890985i
\(320\) 0 0
\(321\) 66.6997i 0.207787i
\(322\) 0 0
\(323\) 1001.68 3.10117
\(324\) 0 0
\(325\) −55.3577 −0.170331
\(326\) 0 0
\(327\) 225.547i 0.689747i
\(328\) 0 0
\(329\) 577.949i 1.75668i
\(330\) 0 0
\(331\) 583.695 1.76343 0.881714 0.471785i \(-0.156390\pi\)
0.881714 + 0.471785i \(0.156390\pi\)
\(332\) 0 0
\(333\) 197.006i 0.591608i
\(334\) 0 0
\(335\) −23.3431 −0.0696808
\(336\) 0 0
\(337\) 455.763i 1.35241i −0.736712 0.676206i \(-0.763623\pi\)
0.736712 0.676206i \(-0.236377\pi\)
\(338\) 0 0
\(339\) 172.961i 0.510209i
\(340\) 0 0
\(341\) 113.439i 0.332666i
\(342\) 0 0
\(343\) 355.251i 1.03572i
\(344\) 0 0
\(345\) −13.8708 + 87.9921i −0.0402052 + 0.255049i
\(346\) 0 0
\(347\) 196.143 0.565253 0.282627 0.959230i \(-0.408794\pi\)
0.282627 + 0.959230i \(0.408794\pi\)
\(348\) 0 0
\(349\) 346.354 0.992419 0.496210 0.868203i \(-0.334725\pi\)
0.496210 + 0.868203i \(0.334725\pi\)
\(350\) 0 0
\(351\) −57.5294 −0.163901
\(352\) 0 0
\(353\) −608.361 −1.72340 −0.861702 0.507415i \(-0.830601\pi\)
−0.861702 + 0.507415i \(0.830601\pi\)
\(354\) 0 0
\(355\) 24.2689i 0.0683632i
\(356\) 0 0
\(357\) 358.102 1.00309
\(358\) 0 0
\(359\) 368.847i 1.02743i 0.857961 + 0.513714i \(0.171731\pi\)
−0.857961 + 0.513714i \(0.828269\pi\)
\(360\) 0 0
\(361\) −697.676 −1.93262
\(362\) 0 0
\(363\) −14.3210 −0.0394517
\(364\) 0 0
\(365\) 292.210i 0.800577i
\(366\) 0 0
\(367\) 250.075i 0.681402i −0.940172 0.340701i \(-0.889336\pi\)
0.940172 0.340701i \(-0.110664\pi\)
\(368\) 0 0
\(369\) 175.772 0.476347
\(370\) 0 0
\(371\) 541.439 1.45941
\(372\) 0 0
\(373\) 629.629i 1.68801i −0.536332 0.844007i \(-0.680190\pi\)
0.536332 0.844007i \(-0.319810\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) 296.378 0.786148
\(378\) 0 0
\(379\) 159.573i 0.421036i 0.977590 + 0.210518i \(0.0675150\pi\)
−0.977590 + 0.210518i \(0.932485\pi\)
\(380\) 0 0
\(381\) −341.316 −0.895843
\(382\) 0 0
\(383\) 216.548i 0.565400i 0.959208 + 0.282700i \(0.0912302\pi\)
−0.959208 + 0.282700i \(0.908770\pi\)
\(384\) 0 0
\(385\) 159.444i 0.414140i
\(386\) 0 0
\(387\) 150.114i 0.387891i
\(388\) 0 0
\(389\) 47.3874i 0.121819i 0.998143 + 0.0609093i \(0.0194000\pi\)
−0.998143 + 0.0609093i \(0.980600\pi\)
\(390\) 0 0
\(391\) 110.256 699.430i 0.281984 1.78882i
\(392\) 0 0
\(393\) 99.4423 0.253034
\(394\) 0 0
\(395\) 272.514 0.689909
\(396\) 0 0
\(397\) −619.329 −1.56002 −0.780012 0.625765i \(-0.784787\pi\)
−0.780012 + 0.625765i \(0.784787\pi\)
\(398\) 0 0
\(399\) −378.480 −0.948570
\(400\) 0 0
\(401\) 120.360i 0.300149i −0.988675 0.150074i \(-0.952049\pi\)
0.988675 0.150074i \(-0.0479513\pi\)
\(402\) 0 0
\(403\) 118.290 0.293523
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) −697.237 −1.71311
\(408\) 0 0
\(409\) −10.0890 −0.0246675 −0.0123337 0.999924i \(-0.503926\pi\)
−0.0123337 + 0.999924i \(0.503926\pi\)
\(410\) 0 0
\(411\) 74.4283i 0.181091i
\(412\) 0 0
\(413\) 314.722i 0.762039i
\(414\) 0 0
\(415\) 234.482 0.565018
\(416\) 0 0
\(417\) −331.293 −0.794468
\(418\) 0 0
\(419\) 417.580i 0.996610i 0.867002 + 0.498305i \(0.166044\pi\)
−0.867002 + 0.498305i \(0.833956\pi\)
\(420\) 0 0
\(421\) 710.109i 1.68672i 0.537350 + 0.843360i \(0.319426\pi\)
−0.537350 + 0.843360i \(0.680574\pi\)
\(422\) 0 0
\(423\) 258.173 0.610338
\(424\) 0 0
\(425\) 153.928i 0.362182i
\(426\) 0 0
\(427\) 440.555 1.03174
\(428\) 0 0
\(429\) 203.607i 0.474607i
\(430\) 0 0
\(431\) 147.520i 0.342275i −0.985247 0.171137i \(-0.945256\pi\)
0.985247 0.171137i \(-0.0547442\pi\)
\(432\) 0 0
\(433\) 141.831i 0.327555i 0.986497 + 0.163778i \(0.0523680\pi\)
−0.986497 + 0.163778i \(0.947632\pi\)
\(434\) 0 0
\(435\) 103.677i 0.238339i
\(436\) 0 0
\(437\) −116.530 + 739.230i −0.266658 + 1.69160i
\(438\) 0 0
\(439\) −611.160 −1.39216 −0.696082 0.717962i \(-0.745075\pi\)
−0.696082 + 0.717962i \(0.745075\pi\)
\(440\) 0 0
\(441\) 11.6926 0.0265139
\(442\) 0 0
\(443\) 585.986 1.32277 0.661383 0.750048i \(-0.269970\pi\)
0.661383 + 0.750048i \(0.269970\pi\)
\(444\) 0 0
\(445\) −237.279 −0.533210
\(446\) 0 0
\(447\) 102.650i 0.229642i
\(448\) 0 0
\(449\) −43.6068 −0.0971197 −0.0485599 0.998820i \(-0.515463\pi\)
−0.0485599 + 0.998820i \(0.515463\pi\)
\(450\) 0 0
\(451\) 622.088i 1.37935i
\(452\) 0 0
\(453\) −234.399 −0.517437
\(454\) 0 0
\(455\) −166.262 −0.365411
\(456\) 0 0
\(457\) 768.985i 1.68268i −0.540505 0.841341i \(-0.681767\pi\)
0.540505 0.841341i \(-0.318233\pi\)
\(458\) 0 0
\(459\) 159.966i 0.348510i
\(460\) 0 0
\(461\) 405.147 0.878844 0.439422 0.898281i \(-0.355183\pi\)
0.439422 + 0.898281i \(0.355183\pi\)
\(462\) 0 0
\(463\) −409.951 −0.885423 −0.442712 0.896664i \(-0.645983\pi\)
−0.442712 + 0.896664i \(0.645983\pi\)
\(464\) 0 0
\(465\) 41.3795i 0.0889881i
\(466\) 0 0
\(467\) 258.310i 0.553127i 0.960996 + 0.276563i \(0.0891956\pi\)
−0.960996 + 0.276563i \(0.910804\pi\)
\(468\) 0 0
\(469\) −70.1089 −0.149486
\(470\) 0 0
\(471\) 176.279i 0.374265i
\(472\) 0 0
\(473\) −531.279 −1.12321
\(474\) 0 0
\(475\) 162.687i 0.342498i
\(476\) 0 0
\(477\) 241.864i 0.507052i
\(478\) 0 0
\(479\) 82.1055i 0.171410i 0.996321 + 0.0857051i \(0.0273143\pi\)
−0.996321 + 0.0857051i \(0.972686\pi\)
\(480\) 0 0
\(481\) 727.051i 1.51154i
\(482\) 0 0
\(483\) −41.6597 + 264.277i −0.0862519 + 0.547156i
\(484\) 0 0
\(485\) −233.527 −0.481498
\(486\) 0 0
\(487\) −14.0287 −0.0288063 −0.0144032 0.999896i \(-0.504585\pi\)
−0.0144032 + 0.999896i \(0.504585\pi\)
\(488\) 0 0
\(489\) 3.00874 0.00615284
\(490\) 0 0
\(491\) −778.780 −1.58611 −0.793055 0.609150i \(-0.791511\pi\)
−0.793055 + 0.609150i \(0.791511\pi\)
\(492\) 0 0
\(493\) 824.108i 1.67162i
\(494\) 0 0
\(495\) −71.2245 −0.143888
\(496\) 0 0
\(497\) 72.8897i 0.146659i
\(498\) 0 0
\(499\) 319.510 0.640301 0.320150 0.947367i \(-0.396267\pi\)
0.320150 + 0.947367i \(0.396267\pi\)
\(500\) 0 0
\(501\) 385.010 0.768484
\(502\) 0 0
\(503\) 153.969i 0.306101i −0.988218 0.153051i \(-0.951090\pi\)
0.988218 0.153051i \(-0.0489098\pi\)
\(504\) 0 0
\(505\) 223.745i 0.443059i
\(506\) 0 0
\(507\) 80.4037 0.158587
\(508\) 0 0
\(509\) −135.390 −0.265991 −0.132996 0.991117i \(-0.542460\pi\)
−0.132996 + 0.991117i \(0.542460\pi\)
\(510\) 0 0
\(511\) 877.629i 1.71747i
\(512\) 0 0
\(513\) 169.069i 0.329569i
\(514\) 0 0
\(515\) 25.0205 0.0485835
\(516\) 0 0
\(517\) 913.719i 1.76735i
\(518\) 0 0
\(519\) −140.626 −0.270955
\(520\) 0 0
\(521\) 619.112i 1.18831i 0.804349 + 0.594157i \(0.202514\pi\)
−0.804349 + 0.594157i \(0.797486\pi\)
\(522\) 0 0
\(523\) 769.129i 1.47061i 0.677736 + 0.735305i \(0.262961\pi\)
−0.677736 + 0.735305i \(0.737039\pi\)
\(524\) 0 0
\(525\) 58.1609i 0.110783i
\(526\) 0 0
\(527\) 328.916i 0.624130i
\(528\) 0 0
\(529\) 503.347 + 162.736i 0.951506 + 0.307629i
\(530\) 0 0
\(531\) 140.588 0.264761
\(532\) 0 0
\(533\) 648.689 1.21705
\(534\) 0 0
\(535\) −86.1090 −0.160951
\(536\) 0 0
\(537\) 433.771 0.807767
\(538\) 0 0
\(539\) 41.3822i 0.0767759i
\(540\) 0 0
\(541\) −366.675 −0.677773 −0.338886 0.940827i \(-0.610050\pi\)
−0.338886 + 0.940827i \(0.610050\pi\)
\(542\) 0 0
\(543\) 367.037i 0.675942i
\(544\) 0 0
\(545\) −291.180 −0.534276
\(546\) 0 0
\(547\) −663.846 −1.21361 −0.606806 0.794850i \(-0.707549\pi\)
−0.606806 + 0.794850i \(0.707549\pi\)
\(548\) 0 0
\(549\) 196.798i 0.358467i
\(550\) 0 0
\(551\) 871.003i 1.58077i
\(552\) 0 0
\(553\) 818.472 1.48006
\(554\) 0 0
\(555\) 254.333 0.458258
\(556\) 0 0
\(557\) 988.104i 1.77397i 0.461794 + 0.886987i \(0.347206\pi\)
−0.461794 + 0.886987i \(0.652794\pi\)
\(558\) 0 0
\(559\) 553.997i 0.991050i
\(560\) 0 0
\(561\) 566.148 1.00918
\(562\) 0 0
\(563\) 422.251i 0.750002i −0.927024 0.375001i \(-0.877642\pi\)
0.927024 0.375001i \(-0.122358\pi\)
\(564\) 0 0
\(565\) 223.292 0.395206
\(566\) 0 0
\(567\) 60.4425i 0.106601i
\(568\) 0 0
\(569\) 13.0661i 0.0229633i 0.999934 + 0.0114817i \(0.00365481\pi\)
−0.999934 + 0.0114817i \(0.996345\pi\)
\(570\) 0 0
\(571\) 435.663i 0.762982i 0.924373 + 0.381491i \(0.124589\pi\)
−0.924373 + 0.381491i \(0.875411\pi\)
\(572\) 0 0
\(573\) 2.62296i 0.00457759i
\(574\) 0 0
\(575\) −113.597 17.9071i −0.197560 0.0311428i
\(576\) 0 0
\(577\) −585.239 −1.01428 −0.507140 0.861864i \(-0.669297\pi\)
−0.507140 + 0.861864i \(0.669297\pi\)
\(578\) 0 0
\(579\) −181.248 −0.313037
\(580\) 0 0
\(581\) 704.247 1.21213
\(582\) 0 0
\(583\) 855.998 1.46826
\(584\) 0 0
\(585\) 74.2701i 0.126957i
\(586\) 0 0
\(587\) −401.932 −0.684722 −0.342361 0.939569i \(-0.611227\pi\)
−0.342361 + 0.939569i \(0.611227\pi\)
\(588\) 0 0
\(589\) 347.633i 0.590209i
\(590\) 0 0
\(591\) 1.35977 0.00230080
\(592\) 0 0
\(593\) 573.328 0.966826 0.483413 0.875393i \(-0.339397\pi\)
0.483413 + 0.875393i \(0.339397\pi\)
\(594\) 0 0
\(595\) 462.308i 0.776988i
\(596\) 0 0
\(597\) 272.297i 0.456109i
\(598\) 0 0
\(599\) −996.359 −1.66337 −0.831686 0.555247i \(-0.812624\pi\)
−0.831686 + 0.555247i \(0.812624\pi\)
\(600\) 0 0
\(601\) 635.428 1.05728 0.528642 0.848845i \(-0.322701\pi\)
0.528642 + 0.848845i \(0.322701\pi\)
\(602\) 0 0
\(603\) 31.3180i 0.0519370i
\(604\) 0 0
\(605\) 18.4883i 0.0305591i
\(606\) 0 0
\(607\) 572.546 0.943240 0.471620 0.881802i \(-0.343669\pi\)
0.471620 + 0.881802i \(0.343669\pi\)
\(608\) 0 0
\(609\) 311.386i 0.511306i
\(610\) 0 0
\(611\) 952.790 1.55939
\(612\) 0 0
\(613\) 249.053i 0.406285i 0.979149 + 0.203143i \(0.0651155\pi\)
−0.979149 + 0.203143i \(0.934885\pi\)
\(614\) 0 0
\(615\) 226.921i 0.368977i
\(616\) 0 0
\(617\) 144.424i 0.234075i 0.993127 + 0.117038i \(0.0373398\pi\)
−0.993127 + 0.117038i \(0.962660\pi\)
\(618\) 0 0
\(619\) 357.665i 0.577811i 0.957358 + 0.288905i \(0.0932913\pi\)
−0.957358 + 0.288905i \(0.906709\pi\)
\(620\) 0 0
\(621\) −118.054 18.6096i −0.190103 0.0299672i
\(622\) 0 0
\(623\) −712.646 −1.14389
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −598.364 −0.954329
\(628\) 0 0
\(629\) −2021.64 −3.21405
\(630\) 0 0
\(631\) 311.184i 0.493160i 0.969122 + 0.246580i \(0.0793068\pi\)
−0.969122 + 0.246580i \(0.920693\pi\)
\(632\) 0 0
\(633\) 570.835 0.901794
\(634\) 0 0
\(635\) 440.637i 0.693917i
\(636\) 0 0
\(637\) 43.1517 0.0677421
\(638\) 0 0
\(639\) 32.5602 0.0509549
\(640\) 0 0
\(641\) 188.628i 0.294272i −0.989116 0.147136i \(-0.952994\pi\)
0.989116 0.147136i \(-0.0470055\pi\)
\(642\) 0 0
\(643\) 253.530i 0.394292i −0.980374 0.197146i \(-0.936833\pi\)
0.980374 0.197146i \(-0.0631674\pi\)
\(644\) 0 0
\(645\) 193.796 0.300459
\(646\) 0 0
\(647\) −175.568 −0.271357 −0.135679 0.990753i \(-0.543321\pi\)
−0.135679 + 0.990753i \(0.543321\pi\)
\(648\) 0 0
\(649\) 497.565i 0.766665i
\(650\) 0 0
\(651\) 124.280i 0.190906i
\(652\) 0 0
\(653\) 288.665 0.442060 0.221030 0.975267i \(-0.429058\pi\)
0.221030 + 0.975267i \(0.429058\pi\)
\(654\) 0 0
\(655\) 128.380i 0.195999i
\(656\) 0 0
\(657\) −392.041 −0.596715
\(658\) 0 0
\(659\) 54.7309i 0.0830514i −0.999137 0.0415257i \(-0.986778\pi\)
0.999137 0.0415257i \(-0.0132218\pi\)
\(660\) 0 0
\(661\) 632.757i 0.957272i 0.878013 + 0.478636i \(0.158869\pi\)
−0.878013 + 0.478636i \(0.841131\pi\)
\(662\) 0 0
\(663\) 590.357i 0.890433i
\(664\) 0 0
\(665\) 488.615i 0.734759i
\(666\) 0 0
\(667\) 608.185 + 95.8723i 0.911821 + 0.143737i
\(668\) 0 0
\(669\) −100.130 −0.149670
\(670\) 0 0
\(671\) 696.503 1.03801
\(672\) 0 0
\(673\) −1102.60 −1.63834 −0.819168 0.573554i \(-0.805564\pi\)
−0.819168 + 0.573554i \(0.805564\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 1235.01i 1.82424i 0.409928 + 0.912118i \(0.365554\pi\)
−0.409928 + 0.912118i \(0.634446\pi\)
\(678\) 0 0
\(679\) −701.377 −1.03296
\(680\) 0 0
\(681\) 124.289i 0.182509i
\(682\) 0 0
\(683\) 461.977 0.676393 0.338197 0.941075i \(-0.390183\pi\)
0.338197 + 0.941075i \(0.390183\pi\)
\(684\) 0 0
\(685\) −96.0865 −0.140272
\(686\) 0 0
\(687\) 338.259i 0.492370i
\(688\) 0 0
\(689\) 892.601i 1.29550i
\(690\) 0 0
\(691\) −874.609 −1.26571 −0.632857 0.774269i \(-0.718118\pi\)
−0.632857 + 0.774269i \(0.718118\pi\)
\(692\) 0 0
\(693\) −213.917 −0.308682
\(694\) 0 0
\(695\) 427.697i 0.615392i
\(696\) 0 0
\(697\) 1803.74i 2.58787i
\(698\) 0 0
\(699\) −720.442 −1.03068
\(700\) 0 0
\(701\) 488.670i 0.697104i −0.937290 0.348552i \(-0.886674\pi\)
0.937290 0.348552i \(-0.113326\pi\)
\(702\) 0 0
\(703\) 2136.68 3.03937
\(704\) 0 0
\(705\) 333.300i 0.472766i
\(706\) 0 0
\(707\) 671.998i 0.950492i
\(708\) 0 0
\(709\) 673.368i 0.949743i −0.880055 0.474871i \(-0.842495\pi\)
0.880055 0.474871i \(-0.157505\pi\)
\(710\) 0 0
\(711\) 365.616i 0.514228i
\(712\) 0 0
\(713\) 242.738 + 38.2644i 0.340446 + 0.0536667i
\(714\) 0 0
\(715\) −262.855 −0.367629
\(716\) 0 0
\(717\) 237.104 0.330688
\(718\) 0 0
\(719\) 1396.62 1.94244 0.971221 0.238181i \(-0.0765513\pi\)
0.971221 + 0.238181i \(0.0765513\pi\)
\(720\) 0 0
\(721\) 75.1470 0.104226
\(722\) 0 0
\(723\) 637.420i 0.881631i
\(724\) 0 0
\(725\) −133.847 −0.184616
\(726\) 0 0
\(727\) 251.423i 0.345836i 0.984936 + 0.172918i \(0.0553196\pi\)
−0.984936 + 0.172918i \(0.944680\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) −1540.44 −2.10731
\(732\) 0 0
\(733\) 51.2884i 0.0699705i −0.999388 0.0349853i \(-0.988862\pi\)
0.999388 0.0349853i \(-0.0111384\pi\)
\(734\) 0 0
\(735\) 15.0951i 0.0205375i
\(736\) 0 0
\(737\) −110.840 −0.150393
\(738\) 0 0
\(739\) −373.243 −0.505064 −0.252532 0.967589i \(-0.581263\pi\)
−0.252532 + 0.967589i \(0.581263\pi\)
\(740\) 0 0
\(741\) 623.950i 0.842038i
\(742\) 0 0
\(743\) 1.55367i 0.00209108i 0.999999 + 0.00104554i \(0.000332805\pi\)
−0.999999 + 0.00104554i \(0.999667\pi\)
\(744\) 0 0
\(745\) 132.520 0.177880
\(746\) 0 0
\(747\) 314.591i 0.421139i
\(748\) 0 0
\(749\) −258.621 −0.345288
\(750\) 0 0
\(751\) 171.616i 0.228516i 0.993451 + 0.114258i \(0.0364491\pi\)
−0.993451 + 0.114258i \(0.963551\pi\)
\(752\) 0 0
\(753\) 113.667i 0.150952i
\(754\) 0 0
\(755\) 302.608i 0.400805i
\(756\) 0 0
\(757\) 611.407i 0.807672i 0.914831 + 0.403836i \(0.132323\pi\)
−0.914831 + 0.403836i \(0.867677\pi\)
\(758\) 0 0
\(759\) −65.8626 + 417.813i −0.0867755 + 0.550478i
\(760\) 0 0
\(761\) −595.991 −0.783168 −0.391584 0.920142i \(-0.628073\pi\)
−0.391584 + 0.920142i \(0.628073\pi\)
\(762\) 0 0
\(763\) −874.535 −1.14618
\(764\) 0 0
\(765\) −206.515 −0.269955
\(766\) 0 0
\(767\) 518.842 0.676456
\(768\) 0 0
\(769\) 1455.94i 1.89329i 0.322282 + 0.946644i \(0.395550\pi\)
−0.322282 + 0.946644i \(0.604450\pi\)
\(770\) 0 0
\(771\) 14.8897 0.0193122
\(772\) 0 0
\(773\) 213.523i 0.276226i −0.990416 0.138113i \(-0.955896\pi\)
0.990416 0.138113i \(-0.0441038\pi\)
\(774\) 0 0
\(775\) −53.4207 −0.0689299
\(776\) 0 0
\(777\) 763.868 0.983099
\(778\) 0 0
\(779\) 1906.38i 2.44722i
\(780\) 0 0
\(781\) 115.236i 0.147550i
\(782\) 0 0
\(783\) −139.098 −0.177647
\(784\) 0 0
\(785\) −227.575 −0.289904
\(786\) 0 0
\(787\) 691.213i 0.878289i 0.898416 + 0.439145i \(0.144718\pi\)
−0.898416 + 0.439145i \(0.855282\pi\)
\(788\) 0 0
\(789\) 348.530i 0.441736i
\(790\) 0 0
\(791\) 670.637 0.847834
\(792\) 0 0
\(793\) 726.286i 0.915872i
\(794\) 0 0
\(795\) −312.245 −0.392761
\(796\) 0 0
\(797\) 1280.04i 1.60607i 0.595929 + 0.803037i \(0.296784\pi\)
−0.595929 + 0.803037i \(0.703216\pi\)
\(798\) 0 0
\(799\)