Properties

Label 2760.3.g.a.2161.14
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 2161.14
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.35

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} +3.56454i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} +3.56454i q^{7} +3.00000 q^{9} -4.23478i q^{11} -15.5305 q^{13} +3.87298i q^{15} +9.96060i q^{17} +9.40924i q^{19} -6.17397i q^{21} +(18.1725 - 14.0982i) q^{23} -5.00000 q^{25} -5.19615 q^{27} +29.5135 q^{29} -57.3969 q^{31} +7.33486i q^{33} +7.97056 q^{35} +17.4527i q^{37} +26.8997 q^{39} -19.0465 q^{41} +2.74875i q^{43} -6.70820i q^{45} +25.1833 q^{47} +36.2940 q^{49} -17.2523i q^{51} -36.1205i q^{53} -9.46927 q^{55} -16.2973i q^{57} +13.6254 q^{59} +28.2845i q^{61} +10.6936i q^{63} +34.7274i q^{65} +19.6093i q^{67} +(-31.4757 + 24.4188i) q^{69} -42.1382 q^{71} +49.2899 q^{73} +8.66025 q^{75} +15.0951 q^{77} -124.063i q^{79} +9.00000 q^{81} +55.4945i q^{83} +22.2726 q^{85} -51.1189 q^{87} -47.5283i q^{89} -55.3593i q^{91} +99.4144 q^{93} +21.0397 q^{95} -91.3507i q^{97} -12.7044i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 3.56454i 0.509220i 0.967044 + 0.254610i \(0.0819472\pi\)
−0.967044 + 0.254610i \(0.918053\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 4.23478i 0.384980i −0.981299 0.192490i \(-0.938344\pi\)
0.981299 0.192490i \(-0.0616564\pi\)
\(12\) 0 0
\(13\) −15.5305 −1.19466 −0.597329 0.801996i \(-0.703771\pi\)
−0.597329 + 0.801996i \(0.703771\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 9.96060i 0.585917i 0.956125 + 0.292959i \(0.0946399\pi\)
−0.956125 + 0.292959i \(0.905360\pi\)
\(18\) 0 0
\(19\) 9.40924i 0.495223i 0.968859 + 0.247612i \(0.0796457\pi\)
−0.968859 + 0.247612i \(0.920354\pi\)
\(20\) 0 0
\(21\) 6.17397i 0.293999i
\(22\) 0 0
\(23\) 18.1725 14.0982i 0.790110 0.612965i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 29.5135 1.01771 0.508853 0.860853i \(-0.330070\pi\)
0.508853 + 0.860853i \(0.330070\pi\)
\(30\) 0 0
\(31\) −57.3969 −1.85151 −0.925757 0.378119i \(-0.876571\pi\)
−0.925757 + 0.378119i \(0.876571\pi\)
\(32\) 0 0
\(33\) 7.33486i 0.222269i
\(34\) 0 0
\(35\) 7.97056 0.227730
\(36\) 0 0
\(37\) 17.4527i 0.471695i 0.971790 + 0.235847i \(0.0757866\pi\)
−0.971790 + 0.235847i \(0.924213\pi\)
\(38\) 0 0
\(39\) 26.8997 0.689736
\(40\) 0 0
\(41\) −19.0465 −0.464548 −0.232274 0.972650i \(-0.574617\pi\)
−0.232274 + 0.972650i \(0.574617\pi\)
\(42\) 0 0
\(43\) 2.74875i 0.0639244i 0.999489 + 0.0319622i \(0.0101756\pi\)
−0.999489 + 0.0319622i \(0.989824\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) 25.1833 0.535816 0.267908 0.963445i \(-0.413668\pi\)
0.267908 + 0.963445i \(0.413668\pi\)
\(48\) 0 0
\(49\) 36.2940 0.740695
\(50\) 0 0
\(51\) 17.2523i 0.338280i
\(52\) 0 0
\(53\) 36.1205i 0.681520i −0.940150 0.340760i \(-0.889316\pi\)
0.940150 0.340760i \(-0.110684\pi\)
\(54\) 0 0
\(55\) −9.46927 −0.172168
\(56\) 0 0
\(57\) 16.2973i 0.285917i
\(58\) 0 0
\(59\) 13.6254 0.230939 0.115469 0.993311i \(-0.463163\pi\)
0.115469 + 0.993311i \(0.463163\pi\)
\(60\) 0 0
\(61\) 28.2845i 0.463680i 0.972754 + 0.231840i \(0.0744746\pi\)
−0.972754 + 0.231840i \(0.925525\pi\)
\(62\) 0 0
\(63\) 10.6936i 0.169740i
\(64\) 0 0
\(65\) 34.7274i 0.534267i
\(66\) 0 0
\(67\) 19.6093i 0.292676i 0.989235 + 0.146338i \(0.0467488\pi\)
−0.989235 + 0.146338i \(0.953251\pi\)
\(68\) 0 0
\(69\) −31.4757 + 24.4188i −0.456170 + 0.353896i
\(70\) 0 0
\(71\) −42.1382 −0.593496 −0.296748 0.954956i \(-0.595902\pi\)
−0.296748 + 0.954956i \(0.595902\pi\)
\(72\) 0 0
\(73\) 49.2899 0.675204 0.337602 0.941289i \(-0.390384\pi\)
0.337602 + 0.941289i \(0.390384\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) 15.0951 0.196040
\(78\) 0 0
\(79\) 124.063i 1.57041i −0.619234 0.785207i \(-0.712557\pi\)
0.619234 0.785207i \(-0.287443\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 55.4945i 0.668608i 0.942465 + 0.334304i \(0.108501\pi\)
−0.942465 + 0.334304i \(0.891499\pi\)
\(84\) 0 0
\(85\) 22.2726 0.262030
\(86\) 0 0
\(87\) −51.1189 −0.587573
\(88\) 0 0
\(89\) 47.5283i 0.534026i −0.963693 0.267013i \(-0.913963\pi\)
0.963693 0.267013i \(-0.0860366\pi\)
\(90\) 0 0
\(91\) 55.3593i 0.608344i
\(92\) 0 0
\(93\) 99.4144 1.06897
\(94\) 0 0
\(95\) 21.0397 0.221471
\(96\) 0 0
\(97\) 91.3507i 0.941759i −0.882197 0.470880i \(-0.843937\pi\)
0.882197 0.470880i \(-0.156063\pi\)
\(98\) 0 0
\(99\) 12.7044i 0.128327i
\(100\) 0 0
\(101\) 98.5205 0.975450 0.487725 0.872997i \(-0.337827\pi\)
0.487725 + 0.872997i \(0.337827\pi\)
\(102\) 0 0
\(103\) 65.7326i 0.638181i 0.947724 + 0.319090i \(0.103377\pi\)
−0.947724 + 0.319090i \(0.896623\pi\)
\(104\) 0 0
\(105\) −13.8054 −0.131480
\(106\) 0 0
\(107\) 202.746i 1.89482i 0.320015 + 0.947412i \(0.396312\pi\)
−0.320015 + 0.947412i \(0.603688\pi\)
\(108\) 0 0
\(109\) 88.4157i 0.811153i −0.914061 0.405576i \(-0.867071\pi\)
0.914061 0.405576i \(-0.132929\pi\)
\(110\) 0 0
\(111\) 30.2290i 0.272333i
\(112\) 0 0
\(113\) 2.51013i 0.0222136i 0.999938 + 0.0111068i \(0.00353547\pi\)
−0.999938 + 0.0111068i \(0.996465\pi\)
\(114\) 0 0
\(115\) −31.5245 40.6350i −0.274126 0.353348i
\(116\) 0 0
\(117\) −46.5916 −0.398219
\(118\) 0 0
\(119\) −35.5050 −0.298361
\(120\) 0 0
\(121\) 103.067 0.851790
\(122\) 0 0
\(123\) 32.9894 0.268207
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −26.8952 −0.211773 −0.105886 0.994378i \(-0.533768\pi\)
−0.105886 + 0.994378i \(0.533768\pi\)
\(128\) 0 0
\(129\) 4.76097i 0.0369068i
\(130\) 0 0
\(131\) −138.064 −1.05393 −0.526964 0.849888i \(-0.676670\pi\)
−0.526964 + 0.849888i \(0.676670\pi\)
\(132\) 0 0
\(133\) −33.5397 −0.252178
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 220.431i 1.60899i −0.593962 0.804493i \(-0.702437\pi\)
0.593962 0.804493i \(-0.297563\pi\)
\(138\) 0 0
\(139\) −142.183 −1.02290 −0.511450 0.859313i \(-0.670892\pi\)
−0.511450 + 0.859313i \(0.670892\pi\)
\(140\) 0 0
\(141\) −43.6188 −0.309353
\(142\) 0 0
\(143\) 65.7685i 0.459920i
\(144\) 0 0
\(145\) 65.9942i 0.455132i
\(146\) 0 0
\(147\) −62.8631 −0.427640
\(148\) 0 0
\(149\) 80.4962i 0.540243i −0.962826 0.270121i \(-0.912936\pi\)
0.962826 0.270121i \(-0.0870638\pi\)
\(150\) 0 0
\(151\) −30.4233 −0.201479 −0.100739 0.994913i \(-0.532121\pi\)
−0.100739 + 0.994913i \(0.532121\pi\)
\(152\) 0 0
\(153\) 29.8818i 0.195306i
\(154\) 0 0
\(155\) 128.343i 0.828022i
\(156\) 0 0
\(157\) 6.66993i 0.0424836i −0.999774 0.0212418i \(-0.993238\pi\)
0.999774 0.0212418i \(-0.00676199\pi\)
\(158\) 0 0
\(159\) 62.5626i 0.393476i
\(160\) 0 0
\(161\) 50.2536 + 64.7767i 0.312134 + 0.402340i
\(162\) 0 0
\(163\) 201.010 1.23319 0.616594 0.787281i \(-0.288512\pi\)
0.616594 + 0.787281i \(0.288512\pi\)
\(164\) 0 0
\(165\) 16.4013 0.0994015
\(166\) 0 0
\(167\) 13.2877 0.0795671 0.0397836 0.999208i \(-0.487333\pi\)
0.0397836 + 0.999208i \(0.487333\pi\)
\(168\) 0 0
\(169\) 72.1979 0.427206
\(170\) 0 0
\(171\) 28.2277i 0.165074i
\(172\) 0 0
\(173\) 187.705 1.08500 0.542499 0.840057i \(-0.317478\pi\)
0.542499 + 0.840057i \(0.317478\pi\)
\(174\) 0 0
\(175\) 17.8227i 0.101844i
\(176\) 0 0
\(177\) −23.5999 −0.133333
\(178\) 0 0
\(179\) 54.1869 0.302720 0.151360 0.988479i \(-0.451635\pi\)
0.151360 + 0.988479i \(0.451635\pi\)
\(180\) 0 0
\(181\) 21.9178i 0.121093i 0.998165 + 0.0605463i \(0.0192843\pi\)
−0.998165 + 0.0605463i \(0.980716\pi\)
\(182\) 0 0
\(183\) 48.9902i 0.267706i
\(184\) 0 0
\(185\) 39.0255 0.210948
\(186\) 0 0
\(187\) 42.1810 0.225567
\(188\) 0 0
\(189\) 18.5219i 0.0979995i
\(190\) 0 0
\(191\) 344.941i 1.80597i −0.429668 0.902987i \(-0.641370\pi\)
0.429668 0.902987i \(-0.358630\pi\)
\(192\) 0 0
\(193\) 206.578 1.07035 0.535175 0.844741i \(-0.320246\pi\)
0.535175 + 0.844741i \(0.320246\pi\)
\(194\) 0 0
\(195\) 60.1496i 0.308459i
\(196\) 0 0
\(197\) 154.646 0.785003 0.392502 0.919751i \(-0.371610\pi\)
0.392502 + 0.919751i \(0.371610\pi\)
\(198\) 0 0
\(199\) 128.107i 0.643754i −0.946781 0.321877i \(-0.895686\pi\)
0.946781 0.321877i \(-0.104314\pi\)
\(200\) 0 0
\(201\) 33.9643i 0.168977i
\(202\) 0 0
\(203\) 105.202i 0.518237i
\(204\) 0 0
\(205\) 42.5892i 0.207752i
\(206\) 0 0
\(207\) 54.5176 42.2946i 0.263370 0.204322i
\(208\) 0 0
\(209\) 39.8461 0.190651
\(210\) 0 0
\(211\) 40.8218 0.193468 0.0967341 0.995310i \(-0.469160\pi\)
0.0967341 + 0.995310i \(0.469160\pi\)
\(212\) 0 0
\(213\) 72.9855 0.342655
\(214\) 0 0
\(215\) 6.14639 0.0285879
\(216\) 0 0
\(217\) 204.594i 0.942829i
\(218\) 0 0
\(219\) −85.3726 −0.389829
\(220\) 0 0
\(221\) 154.694i 0.699971i
\(222\) 0 0
\(223\) 193.085 0.865851 0.432926 0.901430i \(-0.357481\pi\)
0.432926 + 0.901430i \(0.357481\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 87.9176i 0.387302i −0.981070 0.193651i \(-0.937967\pi\)
0.981070 0.193651i \(-0.0620330\pi\)
\(228\) 0 0
\(229\) 387.053i 1.69019i −0.534618 0.845094i \(-0.679545\pi\)
0.534618 0.845094i \(-0.320455\pi\)
\(230\) 0 0
\(231\) −26.1454 −0.113184
\(232\) 0 0
\(233\) −135.592 −0.581940 −0.290970 0.956732i \(-0.593978\pi\)
−0.290970 + 0.956732i \(0.593978\pi\)
\(234\) 0 0
\(235\) 56.3116i 0.239624i
\(236\) 0 0
\(237\) 214.883i 0.906679i
\(238\) 0 0
\(239\) 461.517 1.93103 0.965516 0.260344i \(-0.0838359\pi\)
0.965516 + 0.260344i \(0.0838359\pi\)
\(240\) 0 0
\(241\) 368.653i 1.52968i −0.644221 0.764840i \(-0.722818\pi\)
0.644221 0.764840i \(-0.277182\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 81.1559i 0.331249i
\(246\) 0 0
\(247\) 146.131i 0.591622i
\(248\) 0 0
\(249\) 96.1192i 0.386021i
\(250\) 0 0
\(251\) 77.7032i 0.309575i 0.987948 + 0.154787i \(0.0494692\pi\)
−0.987948 + 0.154787i \(0.950531\pi\)
\(252\) 0 0
\(253\) −59.7029 76.9567i −0.235980 0.304177i
\(254\) 0 0
\(255\) −38.5772 −0.151283
\(256\) 0 0
\(257\) −157.196 −0.611659 −0.305829 0.952086i \(-0.598934\pi\)
−0.305829 + 0.952086i \(0.598934\pi\)
\(258\) 0 0
\(259\) −62.2109 −0.240197
\(260\) 0 0
\(261\) 88.5405 0.339235
\(262\) 0 0
\(263\) 452.069i 1.71889i −0.511225 0.859447i \(-0.670808\pi\)
0.511225 0.859447i \(-0.329192\pi\)
\(264\) 0 0
\(265\) −80.7680 −0.304785
\(266\) 0 0
\(267\) 82.3214i 0.308320i
\(268\) 0 0
\(269\) −18.6332 −0.0692685 −0.0346343 0.999400i \(-0.511027\pi\)
−0.0346343 + 0.999400i \(0.511027\pi\)
\(270\) 0 0
\(271\) 238.072 0.878493 0.439247 0.898367i \(-0.355245\pi\)
0.439247 + 0.898367i \(0.355245\pi\)
\(272\) 0 0
\(273\) 95.8851i 0.351228i
\(274\) 0 0
\(275\) 21.1739i 0.0769961i
\(276\) 0 0
\(277\) 223.227 0.805872 0.402936 0.915228i \(-0.367990\pi\)
0.402936 + 0.915228i \(0.367990\pi\)
\(278\) 0 0
\(279\) −172.191 −0.617171
\(280\) 0 0
\(281\) 436.729i 1.55420i −0.629380 0.777098i \(-0.716691\pi\)
0.629380 0.777098i \(-0.283309\pi\)
\(282\) 0 0
\(283\) 23.3890i 0.0826466i 0.999146 + 0.0413233i \(0.0131574\pi\)
−0.999146 + 0.0413233i \(0.986843\pi\)
\(284\) 0 0
\(285\) −36.4418 −0.127866
\(286\) 0 0
\(287\) 67.8919i 0.236557i
\(288\) 0 0
\(289\) 189.787 0.656701
\(290\) 0 0
\(291\) 158.224i 0.543725i
\(292\) 0 0
\(293\) 156.918i 0.535558i 0.963480 + 0.267779i \(0.0862897\pi\)
−0.963480 + 0.267779i \(0.913710\pi\)
\(294\) 0 0
\(295\) 30.4673i 0.103279i
\(296\) 0 0
\(297\) 22.0046i 0.0740895i
\(298\) 0 0
\(299\) −282.229 + 218.953i −0.943911 + 0.732284i
\(300\) 0 0
\(301\) −9.79803 −0.0325516
\(302\) 0 0
\(303\) −170.642 −0.563176
\(304\) 0 0
\(305\) 63.2460 0.207364
\(306\) 0 0
\(307\) 271.068 0.882957 0.441479 0.897272i \(-0.354454\pi\)
0.441479 + 0.897272i \(0.354454\pi\)
\(308\) 0 0
\(309\) 113.852i 0.368454i
\(310\) 0 0
\(311\) −111.747 −0.359315 −0.179657 0.983729i \(-0.557499\pi\)
−0.179657 + 0.983729i \(0.557499\pi\)
\(312\) 0 0
\(313\) 599.676i 1.91590i −0.286939 0.957949i \(-0.592638\pi\)
0.286939 0.957949i \(-0.407362\pi\)
\(314\) 0 0
\(315\) 23.9117 0.0759101
\(316\) 0 0
\(317\) 54.9678 0.173400 0.0867000 0.996234i \(-0.472368\pi\)
0.0867000 + 0.996234i \(0.472368\pi\)
\(318\) 0 0
\(319\) 124.983i 0.391797i
\(320\) 0 0
\(321\) 351.167i 1.09398i
\(322\) 0 0
\(323\) −93.7217 −0.290160
\(324\) 0 0
\(325\) 77.6527 0.238931
\(326\) 0 0
\(327\) 153.140i 0.468319i
\(328\) 0 0
\(329\) 89.7671i 0.272848i
\(330\) 0 0
\(331\) −30.2790 −0.0914774 −0.0457387 0.998953i \(-0.514564\pi\)
−0.0457387 + 0.998953i \(0.514564\pi\)
\(332\) 0 0
\(333\) 52.3581i 0.157232i
\(334\) 0 0
\(335\) 43.8478 0.130889
\(336\) 0 0
\(337\) 527.421i 1.56505i −0.622621 0.782523i \(-0.713932\pi\)
0.622621 0.782523i \(-0.286068\pi\)
\(338\) 0 0
\(339\) 4.34768i 0.0128250i
\(340\) 0 0
\(341\) 243.064i 0.712797i
\(342\) 0 0
\(343\) 304.034i 0.886397i
\(344\) 0 0
\(345\) 54.6021 + 70.3819i 0.158267 + 0.204005i
\(346\) 0 0
\(347\) −94.2652 −0.271658 −0.135829 0.990732i \(-0.543370\pi\)
−0.135829 + 0.990732i \(0.543370\pi\)
\(348\) 0 0
\(349\) 70.5283 0.202087 0.101043 0.994882i \(-0.467782\pi\)
0.101043 + 0.994882i \(0.467782\pi\)
\(350\) 0 0
\(351\) 80.6991 0.229912
\(352\) 0 0
\(353\) 53.2357 0.150809 0.0754047 0.997153i \(-0.475975\pi\)
0.0754047 + 0.997153i \(0.475975\pi\)
\(354\) 0 0
\(355\) 94.2238i 0.265419i
\(356\) 0 0
\(357\) 61.4964 0.172259
\(358\) 0 0
\(359\) 406.022i 1.13098i 0.824755 + 0.565491i \(0.191313\pi\)
−0.824755 + 0.565491i \(0.808687\pi\)
\(360\) 0 0
\(361\) 272.466 0.754754
\(362\) 0 0
\(363\) −178.517 −0.491781
\(364\) 0 0
\(365\) 110.216i 0.301961i
\(366\) 0 0
\(367\) 59.1029i 0.161043i 0.996753 + 0.0805216i \(0.0256586\pi\)
−0.996753 + 0.0805216i \(0.974341\pi\)
\(368\) 0 0
\(369\) −57.1394 −0.154849
\(370\) 0 0
\(371\) 128.753 0.347044
\(372\) 0 0
\(373\) 165.580i 0.443915i −0.975056 0.221957i \(-0.928755\pi\)
0.975056 0.221957i \(-0.0712446\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) −458.361 −1.21581
\(378\) 0 0
\(379\) 104.043i 0.274521i 0.990535 + 0.137260i \(0.0438297\pi\)
−0.990535 + 0.137260i \(0.956170\pi\)
\(380\) 0 0
\(381\) 46.5838 0.122267
\(382\) 0 0
\(383\) 197.855i 0.516593i −0.966066 0.258296i \(-0.916839\pi\)
0.966066 0.258296i \(-0.0831611\pi\)
\(384\) 0 0
\(385\) 33.7536i 0.0876717i
\(386\) 0 0
\(387\) 8.24625i 0.0213081i
\(388\) 0 0
\(389\) 290.651i 0.747174i −0.927595 0.373587i \(-0.878128\pi\)
0.927595 0.373587i \(-0.121872\pi\)
\(390\) 0 0
\(391\) 140.427 + 181.009i 0.359147 + 0.462939i
\(392\) 0 0
\(393\) 239.135 0.608485
\(394\) 0 0
\(395\) −277.413 −0.702310
\(396\) 0 0
\(397\) 590.952 1.48854 0.744272 0.667877i \(-0.232797\pi\)
0.744272 + 0.667877i \(0.232797\pi\)
\(398\) 0 0
\(399\) 58.0924 0.145595
\(400\) 0 0
\(401\) 541.538i 1.35047i 0.737603 + 0.675234i \(0.235957\pi\)
−0.737603 + 0.675234i \(0.764043\pi\)
\(402\) 0 0
\(403\) 891.406 2.21193
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) 73.9085 0.181593
\(408\) 0 0
\(409\) 17.9115 0.0437934 0.0218967 0.999760i \(-0.493030\pi\)
0.0218967 + 0.999760i \(0.493030\pi\)
\(410\) 0 0
\(411\) 381.798i 0.928948i
\(412\) 0 0
\(413\) 48.5683i 0.117599i
\(414\) 0 0
\(415\) 124.089 0.299011
\(416\) 0 0
\(417\) 246.268 0.590572
\(418\) 0 0
\(419\) 744.342i 1.77647i −0.459387 0.888236i \(-0.651931\pi\)
0.459387 0.888236i \(-0.348069\pi\)
\(420\) 0 0
\(421\) 172.505i 0.409751i −0.978788 0.204876i \(-0.934321\pi\)
0.978788 0.204876i \(-0.0656790\pi\)
\(422\) 0 0
\(423\) 75.5500 0.178605
\(424\) 0 0
\(425\) 49.8030i 0.117183i
\(426\) 0 0
\(427\) −100.821 −0.236115
\(428\) 0 0
\(429\) 113.914i 0.265535i
\(430\) 0 0
\(431\) 371.378i 0.861666i 0.902432 + 0.430833i \(0.141780\pi\)
−0.902432 + 0.430833i \(0.858220\pi\)
\(432\) 0 0
\(433\) 329.060i 0.759955i 0.924996 + 0.379977i \(0.124068\pi\)
−0.924996 + 0.379977i \(0.875932\pi\)
\(434\) 0 0
\(435\) 114.305i 0.262771i
\(436\) 0 0
\(437\) 132.653 + 170.990i 0.303555 + 0.391281i
\(438\) 0 0
\(439\) −666.702 −1.51868 −0.759341 0.650693i \(-0.774479\pi\)
−0.759341 + 0.650693i \(0.774479\pi\)
\(440\) 0 0
\(441\) 108.882 0.246898
\(442\) 0 0
\(443\) 532.068 1.20106 0.600528 0.799604i \(-0.294957\pi\)
0.600528 + 0.799604i \(0.294957\pi\)
\(444\) 0 0
\(445\) −106.277 −0.238824
\(446\) 0 0
\(447\) 139.423i 0.311909i
\(448\) 0 0
\(449\) 298.428 0.664650 0.332325 0.943165i \(-0.392167\pi\)
0.332325 + 0.943165i \(0.392167\pi\)
\(450\) 0 0
\(451\) 80.6576i 0.178842i
\(452\) 0 0
\(453\) 52.6947 0.116324
\(454\) 0 0
\(455\) −123.787 −0.272060
\(456\) 0 0
\(457\) 641.898i 1.40459i 0.711886 + 0.702295i \(0.247841\pi\)
−0.711886 + 0.702295i \(0.752159\pi\)
\(458\) 0 0
\(459\) 51.7568i 0.112760i
\(460\) 0 0
\(461\) −199.215 −0.432137 −0.216069 0.976378i \(-0.569323\pi\)
−0.216069 + 0.976378i \(0.569323\pi\)
\(462\) 0 0
\(463\) 89.7770 0.193903 0.0969514 0.995289i \(-0.469091\pi\)
0.0969514 + 0.995289i \(0.469091\pi\)
\(464\) 0 0
\(465\) 222.297i 0.478059i
\(466\) 0 0
\(467\) 393.352i 0.842295i −0.906992 0.421148i \(-0.861627\pi\)
0.906992 0.421148i \(-0.138373\pi\)
\(468\) 0 0
\(469\) −69.8983 −0.149037
\(470\) 0 0
\(471\) 11.5527i 0.0245279i
\(472\) 0 0
\(473\) 11.6404 0.0246096
\(474\) 0 0
\(475\) 47.0462i 0.0990447i
\(476\) 0 0
\(477\) 108.362i 0.227173i
\(478\) 0 0
\(479\) 166.947i 0.348533i 0.984699 + 0.174266i \(0.0557554\pi\)
−0.984699 + 0.174266i \(0.944245\pi\)
\(480\) 0 0
\(481\) 271.050i 0.563514i
\(482\) 0 0
\(483\) −87.0419 112.197i −0.180211 0.232291i
\(484\) 0 0
\(485\) −204.266 −0.421168
\(486\) 0 0
\(487\) −16.4883 −0.0338568 −0.0169284 0.999857i \(-0.505389\pi\)
−0.0169284 + 0.999857i \(0.505389\pi\)
\(488\) 0 0
\(489\) −348.159 −0.711981
\(490\) 0 0
\(491\) 278.396 0.566997 0.283499 0.958973i \(-0.408505\pi\)
0.283499 + 0.958973i \(0.408505\pi\)
\(492\) 0 0
\(493\) 293.972i 0.596292i
\(494\) 0 0
\(495\) −28.4078 −0.0573895
\(496\) 0 0
\(497\) 150.203i 0.302220i
\(498\) 0 0
\(499\) −627.776 −1.25807 −0.629034 0.777378i \(-0.716549\pi\)
−0.629034 + 0.777378i \(0.716549\pi\)
\(500\) 0 0
\(501\) −23.0150 −0.0459381
\(502\) 0 0
\(503\) 328.082i 0.652250i 0.945327 + 0.326125i \(0.105743\pi\)
−0.945327 + 0.326125i \(0.894257\pi\)
\(504\) 0 0
\(505\) 220.298i 0.436235i
\(506\) 0 0
\(507\) −125.050 −0.246648
\(508\) 0 0
\(509\) −77.9340 −0.153112 −0.0765560 0.997065i \(-0.524392\pi\)
−0.0765560 + 0.997065i \(0.524392\pi\)
\(510\) 0 0
\(511\) 175.696i 0.343828i
\(512\) 0 0
\(513\) 48.8919i 0.0953058i
\(514\) 0 0
\(515\) 146.983 0.285403
\(516\) 0 0
\(517\) 106.646i 0.206279i
\(518\) 0 0
\(519\) −325.114 −0.626423
\(520\) 0 0
\(521\) 202.724i 0.389105i 0.980892 + 0.194552i \(0.0623254\pi\)
−0.980892 + 0.194552i \(0.937675\pi\)
\(522\) 0 0
\(523\) 253.141i 0.484018i 0.970274 + 0.242009i \(0.0778063\pi\)
−0.970274 + 0.242009i \(0.922194\pi\)
\(524\) 0 0
\(525\) 30.8698i 0.0587997i
\(526\) 0 0
\(527\) 571.708i 1.08483i
\(528\) 0 0
\(529\) 131.481 512.400i 0.248547 0.968620i
\(530\) 0 0
\(531\) 40.8762 0.0769796
\(532\) 0 0
\(533\) 295.802 0.554975
\(534\) 0 0
\(535\) 453.354 0.847391
\(536\) 0 0
\(537\) −93.8544 −0.174775
\(538\) 0 0
\(539\) 153.697i 0.285153i
\(540\) 0 0
\(541\) 254.574 0.470561 0.235281 0.971928i \(-0.424399\pi\)
0.235281 + 0.971928i \(0.424399\pi\)
\(542\) 0 0
\(543\) 37.9627i 0.0699128i
\(544\) 0 0
\(545\) −197.703 −0.362759
\(546\) 0 0
\(547\) −499.840 −0.913784 −0.456892 0.889522i \(-0.651037\pi\)
−0.456892 + 0.889522i \(0.651037\pi\)
\(548\) 0 0
\(549\) 84.8534i 0.154560i
\(550\) 0 0
\(551\) 277.700i 0.503992i
\(552\) 0 0
\(553\) 442.227 0.799687
\(554\) 0 0
\(555\) −67.5941 −0.121791
\(556\) 0 0
\(557\) 530.592i 0.952589i −0.879286 0.476295i \(-0.841980\pi\)
0.879286 0.476295i \(-0.158020\pi\)
\(558\) 0 0
\(559\) 42.6896i 0.0763678i
\(560\) 0 0
\(561\) −73.0596 −0.130231
\(562\) 0 0
\(563\) 907.847i 1.61252i 0.591564 + 0.806258i \(0.298511\pi\)
−0.591564 + 0.806258i \(0.701489\pi\)
\(564\) 0 0
\(565\) 5.61283 0.00993421
\(566\) 0 0
\(567\) 32.0809i 0.0565800i
\(568\) 0 0
\(569\) 132.208i 0.232352i 0.993229 + 0.116176i \(0.0370637\pi\)
−0.993229 + 0.116176i \(0.962936\pi\)
\(570\) 0 0
\(571\) 314.412i 0.550635i −0.961353 0.275317i \(-0.911217\pi\)
0.961353 0.275317i \(-0.0887829\pi\)
\(572\) 0 0
\(573\) 597.455i 1.04268i
\(574\) 0 0
\(575\) −90.8626 + 70.4910i −0.158022 + 0.122593i
\(576\) 0 0
\(577\) 928.302 1.60884 0.804421 0.594060i \(-0.202476\pi\)
0.804421 + 0.594060i \(0.202476\pi\)
\(578\) 0 0
\(579\) −357.803 −0.617967
\(580\) 0 0
\(581\) −197.812 −0.340469
\(582\) 0 0
\(583\) −152.963 −0.262372
\(584\) 0 0
\(585\) 104.182i 0.178089i
\(586\) 0 0
\(587\) −696.989 −1.18737 −0.593687 0.804696i \(-0.702328\pi\)
−0.593687 + 0.804696i \(0.702328\pi\)
\(588\) 0 0
\(589\) 540.062i 0.916913i
\(590\) 0 0
\(591\) −267.854 −0.453222
\(592\) 0 0
\(593\) 26.7574 0.0451221 0.0225610 0.999745i \(-0.492818\pi\)
0.0225610 + 0.999745i \(0.492818\pi\)
\(594\) 0 0
\(595\) 79.3915i 0.133431i
\(596\) 0 0
\(597\) 221.888i 0.371672i
\(598\) 0 0
\(599\) −589.134 −0.983529 −0.491765 0.870728i \(-0.663648\pi\)
−0.491765 + 0.870728i \(0.663648\pi\)
\(600\) 0 0
\(601\) 161.884 0.269358 0.134679 0.990889i \(-0.457000\pi\)
0.134679 + 0.990889i \(0.457000\pi\)
\(602\) 0 0
\(603\) 58.8280i 0.0975588i
\(604\) 0 0
\(605\) 230.464i 0.380932i
\(606\) 0 0
\(607\) −440.997 −0.726519 −0.363259 0.931688i \(-0.618336\pi\)
−0.363259 + 0.931688i \(0.618336\pi\)
\(608\) 0 0
\(609\) 182.215i 0.299204i
\(610\) 0 0
\(611\) −391.111 −0.640116
\(612\) 0 0
\(613\) 900.764i 1.46944i −0.678373 0.734718i \(-0.737315\pi\)
0.678373 0.734718i \(-0.262685\pi\)
\(614\) 0 0
\(615\) 73.7666i 0.119946i
\(616\) 0 0
\(617\) 252.328i 0.408959i 0.978871 + 0.204480i \(0.0655502\pi\)
−0.978871 + 0.204480i \(0.934450\pi\)
\(618\) 0 0
\(619\) 65.5081i 0.105829i 0.998599 + 0.0529144i \(0.0168511\pi\)
−0.998599 + 0.0529144i \(0.983149\pi\)
\(620\) 0 0
\(621\) −94.4272 + 73.2564i −0.152057 + 0.117965i
\(622\) 0 0
\(623\) 169.417 0.271937
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −69.0155 −0.110073
\(628\) 0 0
\(629\) −173.839 −0.276374
\(630\) 0 0
\(631\) 619.416i 0.981642i −0.871261 0.490821i \(-0.836697\pi\)
0.871261 0.490821i \(-0.163303\pi\)
\(632\) 0 0
\(633\) −70.7054 −0.111699
\(634\) 0 0
\(635\) 60.1394i 0.0947078i
\(636\) 0 0
\(637\) −563.666 −0.884876
\(638\) 0 0
\(639\) −126.415 −0.197832
\(640\) 0 0
\(641\) 204.217i 0.318592i −0.987231 0.159296i \(-0.949078\pi\)
0.987231 0.159296i \(-0.0509224\pi\)
\(642\) 0 0
\(643\) 955.967i 1.48673i 0.668886 + 0.743365i \(0.266771\pi\)
−0.668886 + 0.743365i \(0.733229\pi\)
\(644\) 0 0
\(645\) −10.6459 −0.0165052
\(646\) 0 0
\(647\) −845.973 −1.30753 −0.653766 0.756697i \(-0.726812\pi\)
−0.653766 + 0.756697i \(0.726812\pi\)
\(648\) 0 0
\(649\) 57.7006i 0.0889069i
\(650\) 0 0
\(651\) 354.367i 0.544342i
\(652\) 0 0
\(653\) 337.119 0.516261 0.258131 0.966110i \(-0.416894\pi\)
0.258131 + 0.966110i \(0.416894\pi\)
\(654\) 0 0
\(655\) 308.722i 0.471331i
\(656\) 0 0
\(657\) 147.870 0.225068
\(658\) 0 0
\(659\) 56.3995i 0.0855835i −0.999084 0.0427917i \(-0.986375\pi\)
0.999084 0.0427917i \(-0.0136252\pi\)
\(660\) 0 0
\(661\) 563.767i 0.852900i −0.904511 0.426450i \(-0.859764\pi\)
0.904511 0.426450i \(-0.140236\pi\)
\(662\) 0 0
\(663\) 267.937i 0.404128i
\(664\) 0 0
\(665\) 74.9969i 0.112777i
\(666\) 0 0
\(667\) 536.335 416.087i 0.804100 0.623819i
\(668\) 0 0
\(669\) −334.433 −0.499899
\(670\) 0 0
\(671\) 119.779 0.178508
\(672\) 0 0
\(673\) 576.120 0.856048 0.428024 0.903767i \(-0.359210\pi\)
0.428024 + 0.903767i \(0.359210\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 1003.34i 1.48204i 0.671482 + 0.741020i \(0.265658\pi\)
−0.671482 + 0.741020i \(0.734342\pi\)
\(678\) 0 0
\(679\) 325.623 0.479563
\(680\) 0 0
\(681\) 152.278i 0.223609i
\(682\) 0 0
\(683\) 603.291 0.883296 0.441648 0.897188i \(-0.354394\pi\)
0.441648 + 0.897188i \(0.354394\pi\)
\(684\) 0 0
\(685\) −492.899 −0.719560
\(686\) 0 0
\(687\) 670.395i 0.975830i
\(688\) 0 0
\(689\) 560.972i 0.814183i
\(690\) 0 0
\(691\) −710.505 −1.02823 −0.514113 0.857722i \(-0.671879\pi\)
−0.514113 + 0.857722i \(0.671879\pi\)
\(692\) 0 0
\(693\) 45.2852 0.0653466
\(694\) 0 0
\(695\) 317.931i 0.457455i
\(696\) 0 0
\(697\) 189.714i 0.272187i
\(698\) 0 0
\(699\) 234.852 0.335983
\(700\) 0 0
\(701\) 1303.64i 1.85968i 0.367961 + 0.929841i \(0.380056\pi\)
−0.367961 + 0.929841i \(0.619944\pi\)
\(702\) 0 0
\(703\) −164.217 −0.233594
\(704\) 0 0
\(705\) 97.5346i 0.138347i
\(706\) 0 0
\(707\) 351.180i 0.496719i
\(708\) 0 0
\(709\) 322.823i 0.455322i −0.973740 0.227661i \(-0.926892\pi\)
0.973740 0.227661i \(-0.0731077\pi\)
\(710\) 0 0
\(711\) 372.188i 0.523471i
\(712\) 0 0
\(713\) −1043.05 + 809.194i −1.46290 + 1.13491i
\(714\) 0 0
\(715\) 147.063 0.205682
\(716\) 0 0
\(717\) −799.370 −1.11488
\(718\) 0 0
\(719\) −972.662 −1.35280 −0.676399 0.736535i \(-0.736461\pi\)
−0.676399 + 0.736535i \(0.736461\pi\)
\(720\) 0 0
\(721\) −234.307 −0.324975
\(722\) 0 0
\(723\) 638.525i 0.883161i
\(724\) 0 0
\(725\) −147.567 −0.203541
\(726\) 0 0
\(727\) 1114.68i 1.53326i −0.642092 0.766628i \(-0.721933\pi\)
0.642092 0.766628i \(-0.278067\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) −27.3792 −0.0374544
\(732\) 0 0
\(733\) 689.439i 0.940572i 0.882514 + 0.470286i \(0.155849\pi\)
−0.882514 + 0.470286i \(0.844151\pi\)
\(734\) 0 0
\(735\) 140.566i 0.191247i
\(736\) 0 0
\(737\) 83.0413 0.112675
\(738\) 0 0
\(739\) −1116.55 −1.51090 −0.755448 0.655209i \(-0.772581\pi\)
−0.755448 + 0.655209i \(0.772581\pi\)
\(740\) 0 0
\(741\) 253.106i 0.341573i
\(742\) 0 0
\(743\) 551.192i 0.741847i 0.928663 + 0.370924i \(0.120959\pi\)
−0.928663 + 0.370924i \(0.879041\pi\)
\(744\) 0 0
\(745\) −179.995 −0.241604
\(746\) 0 0
\(747\) 166.483i 0.222869i
\(748\) 0 0
\(749\) −722.698 −0.964883
\(750\) 0 0
\(751\) 405.850i 0.540413i −0.962802 0.270207i \(-0.912908\pi\)
0.962802 0.270207i \(-0.0870920\pi\)
\(752\) 0 0
\(753\) 134.586i 0.178733i
\(754\) 0 0
\(755\) 68.0285i 0.0901040i
\(756\) 0 0
\(757\) 793.111i 1.04770i 0.851810 + 0.523852i \(0.175505\pi\)
−0.851810 + 0.523852i \(0.824495\pi\)
\(758\) 0 0
\(759\) 103.408 + 133.293i 0.136243 + 0.175617i
\(760\) 0 0
\(761\) −563.287 −0.740193 −0.370097 0.928993i \(-0.620675\pi\)
−0.370097 + 0.928993i \(0.620675\pi\)
\(762\) 0 0
\(763\) 315.161 0.413055
\(764\) 0 0
\(765\) 66.8177 0.0873434
\(766\) 0 0
\(767\) −211.610 −0.275893
\(768\) 0 0
\(769\) 342.516i 0.445405i 0.974886 + 0.222702i \(0.0714878\pi\)
−0.974886 + 0.222702i \(0.928512\pi\)
\(770\) 0 0
\(771\) 272.272 0.353141
\(772\) 0 0
\(773\) 927.041i 1.19928i 0.800271 + 0.599638i \(0.204689\pi\)
−0.800271 + 0.599638i \(0.795311\pi\)
\(774\) 0 0
\(775\) 286.985 0.370303
\(776\) 0 0
\(777\) 107.752 0.138678
\(778\) 0 0
\(779\) 179.213i 0.230055i
\(780\) 0 0
\(781\) 178.446i 0.228484i
\(782\) 0 0
\(783\) −153.357 −0.195858
\(784\) 0 0
\(785\) −14.9144 −0.0189992
\(786\) 0 0
\(787\) 866.621i 1.10117i −0.834779 0.550585i \(-0.814405\pi\)
0.834779 0.550585i \(-0.185595\pi\)
\(788\) 0 0
\(789\) 783.006i 0.992404i
\(790\) 0 0
\(791\) −8.94748 −0.0113116
\(792\) 0 0
\(793\) 439.273i 0.553939i
\(794\) 0 0
\(795\) 139.894 0.175968
\(796\) 0 0
\(797\) 425.944i 0.534434i −0.963636 0.267217i \(-0.913896\pi\)
0.963636 0.267217i \(-0.0861041\pi\)
\(798\) 0 0
\(799\)