Properties

Label 2760.3.g.a.2161.15
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 2161.15
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.34

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} +4.06988i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} +4.06988i q^{7} +3.00000 q^{9} +17.3647i q^{11} -4.19646 q^{13} +3.87298i q^{15} -30.0302i q^{17} +21.4527i q^{19} -7.04924i q^{21} +(3.97679 + 22.6536i) q^{23} -5.00000 q^{25} -5.19615 q^{27} +4.66787 q^{29} -9.39975 q^{31} -30.0765i q^{33} +9.10052 q^{35} +28.9451i q^{37} +7.26849 q^{39} -7.61913 q^{41} +25.5568i q^{43} -6.70820i q^{45} -71.5501 q^{47} +32.4361 q^{49} +52.0137i q^{51} -34.8695i q^{53} +38.8287 q^{55} -37.1571i q^{57} +21.1760 q^{59} -29.4644i q^{61} +12.2096i q^{63} +9.38358i q^{65} -25.5514i q^{67} +(-6.88800 - 39.2372i) q^{69} +57.5699 q^{71} -40.9959 q^{73} +8.66025 q^{75} -70.6722 q^{77} -69.0568i q^{79} +9.00000 q^{81} -4.53246i q^{83} -67.1495 q^{85} -8.08499 q^{87} -40.4744i q^{89} -17.0791i q^{91} +16.2808 q^{93} +47.9696 q^{95} +62.9396i q^{97} +52.0941i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 4.06988i 0.581411i 0.956813 + 0.290706i \(0.0938900\pi\)
−0.956813 + 0.290706i \(0.906110\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 17.3647i 1.57861i 0.614002 + 0.789305i \(0.289559\pi\)
−0.614002 + 0.789305i \(0.710441\pi\)
\(12\) 0 0
\(13\) −4.19646 −0.322805 −0.161402 0.986889i \(-0.551602\pi\)
−0.161402 + 0.986889i \(0.551602\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 30.0302i 1.76648i −0.468922 0.883240i \(-0.655357\pi\)
0.468922 0.883240i \(-0.344643\pi\)
\(18\) 0 0
\(19\) 21.4527i 1.12909i 0.825403 + 0.564543i \(0.190948\pi\)
−0.825403 + 0.564543i \(0.809052\pi\)
\(20\) 0 0
\(21\) 7.04924i 0.335678i
\(22\) 0 0
\(23\) 3.97679 + 22.6536i 0.172904 + 0.984939i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 4.66787 0.160961 0.0804806 0.996756i \(-0.474355\pi\)
0.0804806 + 0.996756i \(0.474355\pi\)
\(30\) 0 0
\(31\) −9.39975 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(32\) 0 0
\(33\) 30.0765i 0.911411i
\(34\) 0 0
\(35\) 9.10052 0.260015
\(36\) 0 0
\(37\) 28.9451i 0.782300i 0.920327 + 0.391150i \(0.127923\pi\)
−0.920327 + 0.391150i \(0.872077\pi\)
\(38\) 0 0
\(39\) 7.26849 0.186372
\(40\) 0 0
\(41\) −7.61913 −0.185832 −0.0929162 0.995674i \(-0.529619\pi\)
−0.0929162 + 0.995674i \(0.529619\pi\)
\(42\) 0 0
\(43\) 25.5568i 0.594343i 0.954824 + 0.297172i \(0.0960434\pi\)
−0.954824 + 0.297172i \(0.903957\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) −71.5501 −1.52234 −0.761172 0.648551i \(-0.775376\pi\)
−0.761172 + 0.648551i \(0.775376\pi\)
\(48\) 0 0
\(49\) 32.4361 0.661961
\(50\) 0 0
\(51\) 52.0137i 1.01988i
\(52\) 0 0
\(53\) 34.8695i 0.657915i −0.944345 0.328958i \(-0.893303\pi\)
0.944345 0.328958i \(-0.106697\pi\)
\(54\) 0 0
\(55\) 38.8287 0.705976
\(56\) 0 0
\(57\) 37.1571i 0.651879i
\(58\) 0 0
\(59\) 21.1760 0.358915 0.179458 0.983766i \(-0.442566\pi\)
0.179458 + 0.983766i \(0.442566\pi\)
\(60\) 0 0
\(61\) 29.4644i 0.483023i −0.970398 0.241511i \(-0.922357\pi\)
0.970398 0.241511i \(-0.0776431\pi\)
\(62\) 0 0
\(63\) 12.2096i 0.193804i
\(64\) 0 0
\(65\) 9.38358i 0.144363i
\(66\) 0 0
\(67\) 25.5514i 0.381365i −0.981652 0.190682i \(-0.938930\pi\)
0.981652 0.190682i \(-0.0610701\pi\)
\(68\) 0 0
\(69\) −6.88800 39.2372i −0.0998260 0.568655i
\(70\) 0 0
\(71\) 57.5699 0.810843 0.405422 0.914130i \(-0.367125\pi\)
0.405422 + 0.914130i \(0.367125\pi\)
\(72\) 0 0
\(73\) −40.9959 −0.561588 −0.280794 0.959768i \(-0.590598\pi\)
−0.280794 + 0.959768i \(0.590598\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) −70.6722 −0.917821
\(78\) 0 0
\(79\) 69.0568i 0.874137i −0.899428 0.437068i \(-0.856017\pi\)
0.899428 0.437068i \(-0.143983\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 4.53246i 0.0546079i −0.999627 0.0273040i \(-0.991308\pi\)
0.999627 0.0273040i \(-0.00869220\pi\)
\(84\) 0 0
\(85\) −67.1495 −0.789994
\(86\) 0 0
\(87\) −8.08499 −0.0929310
\(88\) 0 0
\(89\) 40.4744i 0.454769i −0.973805 0.227384i \(-0.926983\pi\)
0.973805 0.227384i \(-0.0730174\pi\)
\(90\) 0 0
\(91\) 17.0791i 0.187682i
\(92\) 0 0
\(93\) 16.2808 0.175063
\(94\) 0 0
\(95\) 47.9696 0.504943
\(96\) 0 0
\(97\) 62.9396i 0.648862i 0.945909 + 0.324431i \(0.105173\pi\)
−0.945909 + 0.324431i \(0.894827\pi\)
\(98\) 0 0
\(99\) 52.0941i 0.526203i
\(100\) 0 0
\(101\) 18.6235 0.184391 0.0921955 0.995741i \(-0.470612\pi\)
0.0921955 + 0.995741i \(0.470612\pi\)
\(102\) 0 0
\(103\) 186.358i 1.80930i −0.426159 0.904648i \(-0.640134\pi\)
0.426159 0.904648i \(-0.359866\pi\)
\(104\) 0 0
\(105\) −15.7626 −0.150120
\(106\) 0 0
\(107\) 10.4962i 0.0980955i −0.998796 0.0490478i \(-0.984381\pi\)
0.998796 0.0490478i \(-0.0156187\pi\)
\(108\) 0 0
\(109\) 142.457i 1.30695i 0.756949 + 0.653474i \(0.226689\pi\)
−0.756949 + 0.653474i \(0.773311\pi\)
\(110\) 0 0
\(111\) 50.1344i 0.451661i
\(112\) 0 0
\(113\) 18.1285i 0.160429i −0.996778 0.0802146i \(-0.974439\pi\)
0.996778 0.0802146i \(-0.0255606\pi\)
\(114\) 0 0
\(115\) 50.6550 8.89237i 0.440478 0.0773249i
\(116\) 0 0
\(117\) −12.5894 −0.107602
\(118\) 0 0
\(119\) 122.219 1.02705
\(120\) 0 0
\(121\) −180.533 −1.49201
\(122\) 0 0
\(123\) 13.1967 0.107290
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −3.31482 −0.0261009 −0.0130505 0.999915i \(-0.504154\pi\)
−0.0130505 + 0.999915i \(0.504154\pi\)
\(128\) 0 0
\(129\) 44.2656i 0.343144i
\(130\) 0 0
\(131\) −71.8354 −0.548362 −0.274181 0.961678i \(-0.588407\pi\)
−0.274181 + 0.961678i \(0.588407\pi\)
\(132\) 0 0
\(133\) −87.3097 −0.656464
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 39.2827i 0.286735i −0.989670 0.143367i \(-0.954207\pi\)
0.989670 0.143367i \(-0.0457931\pi\)
\(138\) 0 0
\(139\) 67.1523 0.483110 0.241555 0.970387i \(-0.422343\pi\)
0.241555 + 0.970387i \(0.422343\pi\)
\(140\) 0 0
\(141\) 123.928 0.878925
\(142\) 0 0
\(143\) 72.8704i 0.509583i
\(144\) 0 0
\(145\) 10.4377i 0.0719840i
\(146\) 0 0
\(147\) −56.1810 −0.382183
\(148\) 0 0
\(149\) 68.1840i 0.457611i 0.973472 + 0.228805i \(0.0734820\pi\)
−0.973472 + 0.228805i \(0.926518\pi\)
\(150\) 0 0
\(151\) −180.073 −1.19254 −0.596268 0.802785i \(-0.703350\pi\)
−0.596268 + 0.802785i \(0.703350\pi\)
\(152\) 0 0
\(153\) 90.0905i 0.588826i
\(154\) 0 0
\(155\) 21.0185i 0.135603i
\(156\) 0 0
\(157\) 9.34337i 0.0595119i 0.999557 + 0.0297559i \(0.00947301\pi\)
−0.999557 + 0.0297559i \(0.990527\pi\)
\(158\) 0 0
\(159\) 60.3958i 0.379848i
\(160\) 0 0
\(161\) −92.1974 + 16.1850i −0.572654 + 0.100528i
\(162\) 0 0
\(163\) −147.116 −0.902551 −0.451276 0.892385i \(-0.649031\pi\)
−0.451276 + 0.892385i \(0.649031\pi\)
\(164\) 0 0
\(165\) −67.2532 −0.407595
\(166\) 0 0
\(167\) −148.803 −0.891037 −0.445518 0.895273i \(-0.646981\pi\)
−0.445518 + 0.895273i \(0.646981\pi\)
\(168\) 0 0
\(169\) −151.390 −0.895797
\(170\) 0 0
\(171\) 64.3580i 0.376362i
\(172\) 0 0
\(173\) −62.7591 −0.362770 −0.181385 0.983412i \(-0.558058\pi\)
−0.181385 + 0.983412i \(0.558058\pi\)
\(174\) 0 0
\(175\) 20.3494i 0.116282i
\(176\) 0 0
\(177\) −36.6779 −0.207220
\(178\) 0 0
\(179\) 84.1049 0.469860 0.234930 0.972012i \(-0.424514\pi\)
0.234930 + 0.972012i \(0.424514\pi\)
\(180\) 0 0
\(181\) 65.8162i 0.363626i 0.983333 + 0.181813i \(0.0581965\pi\)
−0.983333 + 0.181813i \(0.941804\pi\)
\(182\) 0 0
\(183\) 51.0338i 0.278873i
\(184\) 0 0
\(185\) 64.7233 0.349855
\(186\) 0 0
\(187\) 521.465 2.78858
\(188\) 0 0
\(189\) 21.1477i 0.111893i
\(190\) 0 0
\(191\) 223.741i 1.17142i 0.810520 + 0.585711i \(0.199185\pi\)
−0.810520 + 0.585711i \(0.800815\pi\)
\(192\) 0 0
\(193\) −156.930 −0.813111 −0.406556 0.913626i \(-0.633270\pi\)
−0.406556 + 0.913626i \(0.633270\pi\)
\(194\) 0 0
\(195\) 16.2528i 0.0833479i
\(196\) 0 0
\(197\) −185.239 −0.940298 −0.470149 0.882587i \(-0.655800\pi\)
−0.470149 + 0.882587i \(0.655800\pi\)
\(198\) 0 0
\(199\) 304.120i 1.52824i 0.645075 + 0.764119i \(0.276826\pi\)
−0.645075 + 0.764119i \(0.723174\pi\)
\(200\) 0 0
\(201\) 44.2564i 0.220181i
\(202\) 0 0
\(203\) 18.9977i 0.0935846i
\(204\) 0 0
\(205\) 17.0369i 0.0831068i
\(206\) 0 0
\(207\) 11.9304 + 67.9608i 0.0576346 + 0.328313i
\(208\) 0 0
\(209\) −372.519 −1.78239
\(210\) 0 0
\(211\) −374.217 −1.77354 −0.886771 0.462210i \(-0.847057\pi\)
−0.886771 + 0.462210i \(0.847057\pi\)
\(212\) 0 0
\(213\) −99.7139 −0.468141
\(214\) 0 0
\(215\) 57.1467 0.265798
\(216\) 0 0
\(217\) 38.2558i 0.176294i
\(218\) 0 0
\(219\) 71.0070 0.324233
\(220\) 0 0
\(221\) 126.020i 0.570228i
\(222\) 0 0
\(223\) −298.901 −1.34036 −0.670181 0.742197i \(-0.733784\pi\)
−0.670181 + 0.742197i \(0.733784\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 56.4171i 0.248533i 0.992249 + 0.124267i \(0.0396578\pi\)
−0.992249 + 0.124267i \(0.960342\pi\)
\(228\) 0 0
\(229\) 298.462i 1.30333i −0.758508 0.651664i \(-0.774071\pi\)
0.758508 0.651664i \(-0.225929\pi\)
\(230\) 0 0
\(231\) 122.408 0.529904
\(232\) 0 0
\(233\) −41.4523 −0.177907 −0.0889533 0.996036i \(-0.528352\pi\)
−0.0889533 + 0.996036i \(0.528352\pi\)
\(234\) 0 0
\(235\) 159.991i 0.680813i
\(236\) 0 0
\(237\) 119.610i 0.504683i
\(238\) 0 0
\(239\) −341.193 −1.42758 −0.713792 0.700357i \(-0.753024\pi\)
−0.713792 + 0.700357i \(0.753024\pi\)
\(240\) 0 0
\(241\) 125.010i 0.518713i 0.965782 + 0.259357i \(0.0835105\pi\)
−0.965782 + 0.259357i \(0.916489\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 72.5293i 0.296038i
\(246\) 0 0
\(247\) 90.0253i 0.364475i
\(248\) 0 0
\(249\) 7.85045i 0.0315279i
\(250\) 0 0
\(251\) 342.666i 1.36520i −0.730790 0.682602i \(-0.760848\pi\)
0.730790 0.682602i \(-0.239152\pi\)
\(252\) 0 0
\(253\) −393.373 + 69.0557i −1.55483 + 0.272947i
\(254\) 0 0
\(255\) 116.306 0.456103
\(256\) 0 0
\(257\) 224.294 0.872738 0.436369 0.899768i \(-0.356264\pi\)
0.436369 + 0.899768i \(0.356264\pi\)
\(258\) 0 0
\(259\) −117.803 −0.454838
\(260\) 0 0
\(261\) 14.0036 0.0536537
\(262\) 0 0
\(263\) 213.175i 0.810551i −0.914195 0.405275i \(-0.867176\pi\)
0.914195 0.405275i \(-0.132824\pi\)
\(264\) 0 0
\(265\) −77.9706 −0.294229
\(266\) 0 0
\(267\) 70.1037i 0.262561i
\(268\) 0 0
\(269\) −159.673 −0.593579 −0.296790 0.954943i \(-0.595916\pi\)
−0.296790 + 0.954943i \(0.595916\pi\)
\(270\) 0 0
\(271\) −252.216 −0.930686 −0.465343 0.885131i \(-0.654069\pi\)
−0.465343 + 0.885131i \(0.654069\pi\)
\(272\) 0 0
\(273\) 29.5819i 0.108358i
\(274\) 0 0
\(275\) 86.8235i 0.315722i
\(276\) 0 0
\(277\) 74.0765 0.267424 0.133712 0.991020i \(-0.457310\pi\)
0.133712 + 0.991020i \(0.457310\pi\)
\(278\) 0 0
\(279\) −28.1992 −0.101073
\(280\) 0 0
\(281\) 287.169i 1.02195i −0.859594 0.510977i \(-0.829284\pi\)
0.859594 0.510977i \(-0.170716\pi\)
\(282\) 0 0
\(283\) 459.469i 1.62357i −0.583960 0.811783i \(-0.698497\pi\)
0.583960 0.811783i \(-0.301503\pi\)
\(284\) 0 0
\(285\) −83.0858 −0.291529
\(286\) 0 0
\(287\) 31.0089i 0.108045i
\(288\) 0 0
\(289\) −612.810 −2.12045
\(290\) 0 0
\(291\) 109.015i 0.374621i
\(292\) 0 0
\(293\) 282.896i 0.965516i −0.875754 0.482758i \(-0.839635\pi\)
0.875754 0.482758i \(-0.160365\pi\)
\(294\) 0 0
\(295\) 47.3510i 0.160512i
\(296\) 0 0
\(297\) 90.2296i 0.303804i
\(298\) 0 0
\(299\) −16.6884 95.0650i −0.0558142 0.317943i
\(300\) 0 0
\(301\) −104.013 −0.345558
\(302\) 0 0
\(303\) −32.2568 −0.106458
\(304\) 0 0
\(305\) −65.8844 −0.216014
\(306\) 0 0
\(307\) 418.749 1.36400 0.682001 0.731351i \(-0.261110\pi\)
0.682001 + 0.731351i \(0.261110\pi\)
\(308\) 0 0
\(309\) 322.781i 1.04460i
\(310\) 0 0
\(311\) 144.043 0.463161 0.231581 0.972816i \(-0.425610\pi\)
0.231581 + 0.972816i \(0.425610\pi\)
\(312\) 0 0
\(313\) 217.434i 0.694676i −0.937740 0.347338i \(-0.887086\pi\)
0.937740 0.347338i \(-0.112914\pi\)
\(314\) 0 0
\(315\) 27.3016 0.0866717
\(316\) 0 0
\(317\) 223.248 0.704253 0.352126 0.935952i \(-0.385459\pi\)
0.352126 + 0.935952i \(0.385459\pi\)
\(318\) 0 0
\(319\) 81.0562i 0.254095i
\(320\) 0 0
\(321\) 18.1800i 0.0566355i
\(322\) 0 0
\(323\) 644.226 1.99451
\(324\) 0 0
\(325\) 20.9823 0.0645610
\(326\) 0 0
\(327\) 246.743i 0.754567i
\(328\) 0 0
\(329\) 291.200i 0.885107i
\(330\) 0 0
\(331\) −645.945 −1.95149 −0.975747 0.218901i \(-0.929753\pi\)
−0.975747 + 0.218901i \(0.929753\pi\)
\(332\) 0 0
\(333\) 86.8354i 0.260767i
\(334\) 0 0
\(335\) −57.1348 −0.170552
\(336\) 0 0
\(337\) 94.1513i 0.279381i 0.990195 + 0.139690i \(0.0446107\pi\)
−0.990195 + 0.139690i \(0.955389\pi\)
\(338\) 0 0
\(339\) 31.3995i 0.0926238i
\(340\) 0 0
\(341\) 163.224i 0.478662i
\(342\) 0 0
\(343\) 331.435i 0.966283i
\(344\) 0 0
\(345\) −87.7370 + 15.4020i −0.254310 + 0.0446436i
\(346\) 0 0
\(347\) −332.408 −0.957947 −0.478974 0.877829i \(-0.658991\pi\)
−0.478974 + 0.877829i \(0.658991\pi\)
\(348\) 0 0
\(349\) −321.360 −0.920801 −0.460401 0.887711i \(-0.652294\pi\)
−0.460401 + 0.887711i \(0.652294\pi\)
\(350\) 0 0
\(351\) 21.8055 0.0621238
\(352\) 0 0
\(353\) 17.6190 0.0499121 0.0249561 0.999689i \(-0.492055\pi\)
0.0249561 + 0.999689i \(0.492055\pi\)
\(354\) 0 0
\(355\) 128.730i 0.362620i
\(356\) 0 0
\(357\) −211.690 −0.592968
\(358\) 0 0
\(359\) 255.615i 0.712020i −0.934482 0.356010i \(-0.884137\pi\)
0.934482 0.356010i \(-0.115863\pi\)
\(360\) 0 0
\(361\) −99.2162 −0.274837
\(362\) 0 0
\(363\) 312.692 0.861411
\(364\) 0 0
\(365\) 91.6696i 0.251150i
\(366\) 0 0
\(367\) 417.221i 1.13684i 0.822738 + 0.568421i \(0.192445\pi\)
−0.822738 + 0.568421i \(0.807555\pi\)
\(368\) 0 0
\(369\) −22.8574 −0.0619441
\(370\) 0 0
\(371\) 141.915 0.382519
\(372\) 0 0
\(373\) 508.704i 1.36382i −0.731437 0.681909i \(-0.761150\pi\)
0.731437 0.681909i \(-0.238850\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) −19.5886 −0.0519591
\(378\) 0 0
\(379\) 152.462i 0.402274i 0.979563 + 0.201137i \(0.0644636\pi\)
−0.979563 + 0.201137i \(0.935536\pi\)
\(380\) 0 0
\(381\) 5.74143 0.0150694
\(382\) 0 0
\(383\) 162.088i 0.423205i −0.977356 0.211603i \(-0.932132\pi\)
0.977356 0.211603i \(-0.0678683\pi\)
\(384\) 0 0
\(385\) 158.028i 0.410462i
\(386\) 0 0
\(387\) 76.6703i 0.198114i
\(388\) 0 0
\(389\) 388.032i 0.997512i 0.866742 + 0.498756i \(0.166210\pi\)
−0.866742 + 0.498756i \(0.833790\pi\)
\(390\) 0 0
\(391\) 680.291 119.423i 1.73987 0.305431i
\(392\) 0 0
\(393\) 124.423 0.316597
\(394\) 0 0
\(395\) −154.416 −0.390926
\(396\) 0 0
\(397\) −367.782 −0.926403 −0.463201 0.886253i \(-0.653299\pi\)
−0.463201 + 0.886253i \(0.653299\pi\)
\(398\) 0 0
\(399\) 151.225 0.379010
\(400\) 0 0
\(401\) 519.817i 1.29630i 0.761512 + 0.648151i \(0.224458\pi\)
−0.761512 + 0.648151i \(0.775542\pi\)
\(402\) 0 0
\(403\) 39.4457 0.0978801
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) −502.623 −1.23495
\(408\) 0 0
\(409\) −606.573 −1.48306 −0.741532 0.670917i \(-0.765901\pi\)
−0.741532 + 0.670917i \(0.765901\pi\)
\(410\) 0 0
\(411\) 68.0396i 0.165547i
\(412\) 0 0
\(413\) 86.1838i 0.208677i
\(414\) 0 0
\(415\) −10.1349 −0.0244214
\(416\) 0 0
\(417\) −116.311 −0.278924
\(418\) 0 0
\(419\) 16.0282i 0.0382534i −0.999817 0.0191267i \(-0.993911\pi\)
0.999817 0.0191267i \(-0.00608858\pi\)
\(420\) 0 0
\(421\) 280.347i 0.665908i −0.942943 0.332954i \(-0.891955\pi\)
0.942943 0.332954i \(-0.108045\pi\)
\(422\) 0 0
\(423\) −214.650 −0.507448
\(424\) 0 0
\(425\) 150.151i 0.353296i
\(426\) 0 0
\(427\) 119.916 0.280835
\(428\) 0 0
\(429\) 126.215i 0.294208i
\(430\) 0 0
\(431\) 322.272i 0.747730i 0.927483 + 0.373865i \(0.121968\pi\)
−0.927483 + 0.373865i \(0.878032\pi\)
\(432\) 0 0
\(433\) 117.301i 0.270903i 0.990784 + 0.135452i \(0.0432486\pi\)
−0.990784 + 0.135452i \(0.956751\pi\)
\(434\) 0 0
\(435\) 18.0786i 0.0415600i
\(436\) 0 0
\(437\) −485.980 + 85.3126i −1.11208 + 0.195223i
\(438\) 0 0
\(439\) −349.607 −0.796372 −0.398186 0.917305i \(-0.630360\pi\)
−0.398186 + 0.917305i \(0.630360\pi\)
\(440\) 0 0
\(441\) 97.3083 0.220654
\(442\) 0 0
\(443\) 284.068 0.641238 0.320619 0.947208i \(-0.396109\pi\)
0.320619 + 0.947208i \(0.396109\pi\)
\(444\) 0 0
\(445\) −90.5035 −0.203379
\(446\) 0 0
\(447\) 118.098i 0.264202i
\(448\) 0 0
\(449\) 87.9963 0.195983 0.0979914 0.995187i \(-0.468758\pi\)
0.0979914 + 0.995187i \(0.468758\pi\)
\(450\) 0 0
\(451\) 132.304i 0.293357i
\(452\) 0 0
\(453\) 311.895 0.688511
\(454\) 0 0
\(455\) −38.1900 −0.0839341
\(456\) 0 0
\(457\) 133.171i 0.291402i −0.989329 0.145701i \(-0.953456\pi\)
0.989329 0.145701i \(-0.0465438\pi\)
\(458\) 0 0
\(459\) 156.041i 0.339959i
\(460\) 0 0
\(461\) −90.0146 −0.195260 −0.0976298 0.995223i \(-0.531126\pi\)
−0.0976298 + 0.995223i \(0.531126\pi\)
\(462\) 0 0
\(463\) 792.467 1.71159 0.855795 0.517314i \(-0.173068\pi\)
0.855795 + 0.517314i \(0.173068\pi\)
\(464\) 0 0
\(465\) 36.4051i 0.0782904i
\(466\) 0 0
\(467\) 50.3184i 0.107748i 0.998548 + 0.0538741i \(0.0171570\pi\)
−0.998548 + 0.0538741i \(0.982843\pi\)
\(468\) 0 0
\(469\) 103.991 0.221730
\(470\) 0 0
\(471\) 16.1832i 0.0343592i
\(472\) 0 0
\(473\) −443.786 −0.938236
\(474\) 0 0
\(475\) 107.263i 0.225817i
\(476\) 0 0
\(477\) 104.609i 0.219305i
\(478\) 0 0
\(479\) 722.886i 1.50916i 0.656210 + 0.754578i \(0.272158\pi\)
−0.656210 + 0.754578i \(0.727842\pi\)
\(480\) 0 0
\(481\) 121.467i 0.252530i
\(482\) 0 0
\(483\) 159.690 28.0333i 0.330622 0.0580400i
\(484\) 0 0
\(485\) 140.737 0.290180
\(486\) 0 0
\(487\) −874.984 −1.79668 −0.898341 0.439298i \(-0.855227\pi\)
−0.898341 + 0.439298i \(0.855227\pi\)
\(488\) 0 0
\(489\) 254.812 0.521088
\(490\) 0 0
\(491\) 667.621 1.35972 0.679858 0.733343i \(-0.262041\pi\)
0.679858 + 0.733343i \(0.262041\pi\)
\(492\) 0 0
\(493\) 140.177i 0.284335i
\(494\) 0 0
\(495\) 116.486 0.235325
\(496\) 0 0
\(497\) 234.302i 0.471433i
\(498\) 0 0
\(499\) −162.867 −0.326386 −0.163193 0.986594i \(-0.552179\pi\)
−0.163193 + 0.986594i \(0.552179\pi\)
\(500\) 0 0
\(501\) 257.735 0.514440
\(502\) 0 0
\(503\) 162.385i 0.322833i −0.986886 0.161417i \(-0.948394\pi\)
0.986886 0.161417i \(-0.0516063\pi\)
\(504\) 0 0
\(505\) 41.6434i 0.0824621i
\(506\) 0 0
\(507\) 262.215 0.517189
\(508\) 0 0
\(509\) 719.868 1.41428 0.707140 0.707074i \(-0.249985\pi\)
0.707140 + 0.707074i \(0.249985\pi\)
\(510\) 0 0
\(511\) 166.848i 0.326513i
\(512\) 0 0
\(513\) 111.471i 0.217293i
\(514\) 0 0
\(515\) −416.708 −0.809142
\(516\) 0 0
\(517\) 1242.45i 2.40319i
\(518\) 0 0
\(519\) 108.702 0.209445
\(520\) 0 0
\(521\) 34.5837i 0.0663795i −0.999449 0.0331897i \(-0.989433\pi\)
0.999449 0.0331897i \(-0.0105666\pi\)
\(522\) 0 0
\(523\) 569.000i 1.08795i −0.839100 0.543977i \(-0.816918\pi\)
0.839100 0.543977i \(-0.183082\pi\)
\(524\) 0 0
\(525\) 35.2462i 0.0671356i
\(526\) 0 0
\(527\) 282.276i 0.535628i
\(528\) 0 0
\(529\) −497.370 + 180.177i −0.940209 + 0.340599i
\(530\) 0 0
\(531\) 63.5280 0.119638
\(532\) 0 0
\(533\) 31.9734 0.0599876
\(534\) 0 0
\(535\) −23.4703 −0.0438697
\(536\) 0 0
\(537\) −145.674 −0.271274
\(538\) 0 0
\(539\) 563.243i 1.04498i
\(540\) 0 0
\(541\) 384.769 0.711219 0.355609 0.934635i \(-0.384273\pi\)
0.355609 + 0.934635i \(0.384273\pi\)
\(542\) 0 0
\(543\) 113.997i 0.209939i
\(544\) 0 0
\(545\) 318.544 0.584485
\(546\) 0 0
\(547\) 27.1406 0.0496171 0.0248086 0.999692i \(-0.492102\pi\)
0.0248086 + 0.999692i \(0.492102\pi\)
\(548\) 0 0
\(549\) 88.3932i 0.161008i
\(550\) 0 0
\(551\) 100.138i 0.181739i
\(552\) 0 0
\(553\) 281.053 0.508233
\(554\) 0 0
\(555\) −112.104 −0.201989
\(556\) 0 0
\(557\) 80.6970i 0.144878i 0.997373 + 0.0724389i \(0.0230782\pi\)
−0.997373 + 0.0724389i \(0.976922\pi\)
\(558\) 0 0
\(559\) 107.248i 0.191857i
\(560\) 0 0
\(561\) −903.203 −1.60999
\(562\) 0 0
\(563\) 301.705i 0.535889i 0.963434 + 0.267944i \(0.0863444\pi\)
−0.963434 + 0.267944i \(0.913656\pi\)
\(564\) 0 0
\(565\) −40.5366 −0.0717461
\(566\) 0 0
\(567\) 36.6289i 0.0646012i
\(568\) 0 0
\(569\) 341.572i 0.600302i 0.953892 + 0.300151i \(0.0970370\pi\)
−0.953892 + 0.300151i \(0.902963\pi\)
\(570\) 0 0
\(571\) 477.489i 0.836232i −0.908394 0.418116i \(-0.862691\pi\)
0.908394 0.418116i \(-0.137309\pi\)
\(572\) 0 0
\(573\) 387.532i 0.676320i
\(574\) 0 0
\(575\) −19.8839 113.268i −0.0345808 0.196988i
\(576\) 0 0
\(577\) −191.370 −0.331664 −0.165832 0.986154i \(-0.553031\pi\)
−0.165832 + 0.986154i \(0.553031\pi\)
\(578\) 0 0
\(579\) 271.811 0.469450
\(580\) 0 0
\(581\) 18.4465 0.0317497
\(582\) 0 0
\(583\) 605.499 1.03859
\(584\) 0 0
\(585\) 28.1507i 0.0481209i
\(586\) 0 0
\(587\) −336.539 −0.573320 −0.286660 0.958032i \(-0.592545\pi\)
−0.286660 + 0.958032i \(0.592545\pi\)
\(588\) 0 0
\(589\) 201.649i 0.342359i
\(590\) 0 0
\(591\) 320.843 0.542881
\(592\) 0 0
\(593\) 307.076 0.517835 0.258917 0.965899i \(-0.416634\pi\)
0.258917 + 0.965899i \(0.416634\pi\)
\(594\) 0 0
\(595\) 273.290i 0.459311i
\(596\) 0 0
\(597\) 526.750i 0.882329i
\(598\) 0 0
\(599\) 416.609 0.695508 0.347754 0.937586i \(-0.386944\pi\)
0.347754 + 0.937586i \(0.386944\pi\)
\(600\) 0 0
\(601\) −815.774 −1.35736 −0.678681 0.734434i \(-0.737448\pi\)
−0.678681 + 0.734434i \(0.737448\pi\)
\(602\) 0 0
\(603\) 76.6543i 0.127122i
\(604\) 0 0
\(605\) 403.684i 0.667246i
\(606\) 0 0
\(607\) −260.733 −0.429544 −0.214772 0.976664i \(-0.568901\pi\)
−0.214772 + 0.976664i \(0.568901\pi\)
\(608\) 0 0
\(609\) 32.9049i 0.0540311i
\(610\) 0 0
\(611\) 300.258 0.491420
\(612\) 0 0
\(613\) 257.869i 0.420667i −0.977630 0.210334i \(-0.932545\pi\)
0.977630 0.210334i \(-0.0674550\pi\)
\(614\) 0 0
\(615\) 29.5088i 0.0479817i
\(616\) 0 0
\(617\) 794.369i 1.28747i 0.765249 + 0.643735i \(0.222616\pi\)
−0.765249 + 0.643735i \(0.777384\pi\)
\(618\) 0 0
\(619\) 362.177i 0.585099i 0.956250 + 0.292550i \(0.0945037\pi\)
−0.956250 + 0.292550i \(0.905496\pi\)
\(620\) 0 0
\(621\) −20.6640 117.712i −0.0332753 0.189552i
\(622\) 0 0
\(623\) 164.726 0.264408
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 645.222 1.02906
\(628\) 0 0
\(629\) 869.226 1.38192
\(630\) 0 0
\(631\) 1011.86i 1.60358i 0.597608 + 0.801788i \(0.296118\pi\)
−0.597608 + 0.801788i \(0.703882\pi\)
\(632\) 0 0
\(633\) 648.163 1.02395
\(634\) 0 0
\(635\) 7.41216i 0.0116727i
\(636\) 0 0
\(637\) −136.117 −0.213684
\(638\) 0 0
\(639\) 172.710 0.270281
\(640\) 0 0
\(641\) 254.146i 0.396483i −0.980153 0.198242i \(-0.936477\pi\)
0.980153 0.198242i \(-0.0635231\pi\)
\(642\) 0 0
\(643\) 1131.31i 1.75943i −0.475499 0.879716i \(-0.657733\pi\)
0.475499 0.879716i \(-0.342267\pi\)
\(644\) 0 0
\(645\) −98.9809 −0.153459
\(646\) 0 0
\(647\) 471.801 0.729214 0.364607 0.931162i \(-0.381203\pi\)
0.364607 + 0.931162i \(0.381203\pi\)
\(648\) 0 0
\(649\) 367.715i 0.566587i
\(650\) 0 0
\(651\) 66.2610i 0.101783i
\(652\) 0 0
\(653\) 190.527 0.291771 0.145886 0.989301i \(-0.453397\pi\)
0.145886 + 0.989301i \(0.453397\pi\)
\(654\) 0 0
\(655\) 160.629i 0.245235i
\(656\) 0 0
\(657\) −122.988 −0.187196
\(658\) 0 0
\(659\) 880.248i 1.33573i 0.744281 + 0.667866i \(0.232792\pi\)
−0.744281 + 0.667866i \(0.767208\pi\)
\(660\) 0 0
\(661\) 423.407i 0.640556i −0.947324 0.320278i \(-0.896224\pi\)
0.947324 0.320278i \(-0.103776\pi\)
\(662\) 0 0
\(663\) 218.274i 0.329222i
\(664\) 0 0
\(665\) 195.230i 0.293580i
\(666\) 0 0
\(667\) 18.5631 + 105.744i 0.0278308 + 0.158537i
\(668\) 0 0
\(669\) 517.711 0.773859
\(670\) 0 0
\(671\) 511.640 0.762504
\(672\) 0 0
\(673\) −448.792 −0.666854 −0.333427 0.942776i \(-0.608205\pi\)
−0.333427 + 0.942776i \(0.608205\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 411.720i 0.608154i −0.952648 0.304077i \(-0.901652\pi\)
0.952648 0.304077i \(-0.0983480\pi\)
\(678\) 0 0
\(679\) −256.157 −0.377256
\(680\) 0 0
\(681\) 97.7173i 0.143491i
\(682\) 0 0
\(683\) 67.7191 0.0991495 0.0495748 0.998770i \(-0.484213\pi\)
0.0495748 + 0.998770i \(0.484213\pi\)
\(684\) 0 0
\(685\) −87.8388 −0.128232
\(686\) 0 0
\(687\) 516.952i 0.752477i
\(688\) 0 0
\(689\) 146.329i 0.212378i
\(690\) 0 0
\(691\) 1090.32 1.57789 0.788943 0.614466i \(-0.210629\pi\)
0.788943 + 0.614466i \(0.210629\pi\)
\(692\) 0 0
\(693\) −212.017 −0.305940
\(694\) 0 0
\(695\) 150.157i 0.216054i
\(696\) 0 0
\(697\) 228.804i 0.328269i
\(698\) 0 0
\(699\) 71.7974 0.102714
\(700\) 0 0
\(701\) 619.938i 0.884362i 0.896926 + 0.442181i \(0.145795\pi\)
−0.896926 + 0.442181i \(0.854205\pi\)
\(702\) 0 0
\(703\) −620.950 −0.883285
\(704\) 0 0
\(705\) 277.112i 0.393067i
\(706\) 0 0
\(707\) 75.7953i 0.107207i
\(708\) 0 0
\(709\) 751.400i 1.05980i 0.848059 + 0.529901i \(0.177771\pi\)
−0.848059 + 0.529901i \(0.822229\pi\)
\(710\) 0 0
\(711\) 207.170i 0.291379i
\(712\) 0 0
\(713\) −37.3808 212.938i −0.0524275 0.298651i
\(714\) 0 0
\(715\) −162.943 −0.227892
\(716\) 0 0
\(717\) 590.963 0.824216
\(718\) 0 0
\(719\) −449.466 −0.625126 −0.312563 0.949897i \(-0.601188\pi\)
−0.312563 + 0.949897i \(0.601188\pi\)
\(720\) 0 0
\(721\) 758.452 1.05195
\(722\) 0 0
\(723\) 216.524i 0.299479i
\(724\) 0 0
\(725\) −23.3394 −0.0321922
\(726\) 0 0
\(727\) 650.350i 0.894567i 0.894392 + 0.447284i \(0.147609\pi\)
−0.894392 + 0.447284i \(0.852391\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 767.473 1.04990
\(732\) 0 0
\(733\) 1051.10i 1.43397i 0.697087 + 0.716987i \(0.254479\pi\)
−0.697087 + 0.716987i \(0.745521\pi\)
\(734\) 0 0
\(735\) 125.624i 0.170918i
\(736\) 0 0
\(737\) 443.693 0.602026
\(738\) 0 0
\(739\) 790.804 1.07010 0.535050 0.844820i \(-0.320293\pi\)
0.535050 + 0.844820i \(0.320293\pi\)
\(740\) 0 0
\(741\) 155.928i 0.210430i
\(742\) 0 0
\(743\) 1255.78i 1.69014i −0.534653 0.845072i \(-0.679558\pi\)
0.534653 0.845072i \(-0.320442\pi\)
\(744\) 0 0
\(745\) 152.464 0.204650
\(746\) 0 0
\(747\) 13.5974i 0.0182026i
\(748\) 0 0
\(749\) 42.7183 0.0570338
\(750\) 0 0
\(751\) 612.688i 0.815830i −0.913020 0.407915i \(-0.866256\pi\)
0.913020 0.407915i \(-0.133744\pi\)
\(752\) 0 0
\(753\) 593.515i 0.788201i
\(754\) 0 0
\(755\) 402.655i 0.533318i
\(756\) 0 0
\(757\) 633.623i 0.837019i 0.908212 + 0.418510i \(0.137447\pi\)
−0.908212 + 0.418510i \(0.862553\pi\)
\(758\) 0 0
\(759\) 681.342 119.608i 0.897684 0.157586i
\(760\) 0 0
\(761\) −57.9001 −0.0760843 −0.0380421 0.999276i \(-0.512112\pi\)
−0.0380421 + 0.999276i \(0.512112\pi\)
\(762\) 0 0
\(763\) −579.784 −0.759874
\(764\) 0 0
\(765\) −201.448 −0.263331
\(766\) 0 0
\(767\) −88.8644 −0.115860
\(768\) 0 0
\(769\) 259.436i 0.337367i −0.985670 0.168684i \(-0.946048\pi\)
0.985670 0.168684i \(-0.0539517\pi\)
\(770\) 0 0
\(771\) −388.488 −0.503876
\(772\) 0 0
\(773\) 368.152i 0.476264i −0.971233 0.238132i \(-0.923465\pi\)
0.971233 0.238132i \(-0.0765350\pi\)
\(774\) 0 0
\(775\) 46.9987 0.0606435
\(776\) 0 0
\(777\) 204.041 0.262601
\(778\) 0 0
\(779\) 163.451i 0.209821i
\(780\) 0 0
\(781\) 999.684i 1.28000i
\(782\) 0 0
\(783\) −24.2550 −0.0309770
\(784\) 0 0
\(785\) 20.8924 0.0266145
\(786\) 0 0
\(787\) 412.841i 0.524576i 0.964990 + 0.262288i \(0.0844771\pi\)
−0.964990 + 0.262288i \(0.915523\pi\)
\(788\) 0 0
\(789\) 369.230i 0.467972i
\(790\) 0 0
\(791\) 73.7808 0.0932753
\(792\) 0 0
\(793\) 123.646i 0.155922i
\(794\) 0 0
\(795\) 135.049 0.169873
\(796\) 0 0
\(797\) 425.929i 0.534416i −0.963639 0.267208i \(-0.913899\pi\)
0.963639 0.267208i \(-0.0861010\pi\)
\(798\) 0 0