Properties

Label 2760.3.g.a.2161.19
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 2161.19
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.30

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} +6.39488i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} +6.39488i q^{7} +3.00000 q^{9} +2.14475i q^{11} +23.8081 q^{13} +3.87298i q^{15} +7.36156i q^{17} -7.64208i q^{19} -11.0763i q^{21} +(-2.73940 - 22.8363i) q^{23} -5.00000 q^{25} -5.19615 q^{27} +41.2159 q^{29} +16.4553 q^{31} -3.71482i q^{33} +14.2994 q^{35} +12.9080i q^{37} -41.2369 q^{39} -29.3570 q^{41} -70.1044i q^{43} -6.70820i q^{45} -80.3689 q^{47} +8.10546 q^{49} -12.7506i q^{51} +84.5512i q^{53} +4.79581 q^{55} +13.2365i q^{57} +41.0693 q^{59} +19.8626i q^{61} +19.1847i q^{63} -53.2365i q^{65} +7.96687i q^{67} +(4.74478 + 39.5536i) q^{69} -33.7474 q^{71} +79.6030 q^{73} +8.66025 q^{75} -13.7154 q^{77} -78.0646i q^{79} +9.00000 q^{81} +56.2341i q^{83} +16.4609 q^{85} -71.3880 q^{87} +45.9891i q^{89} +152.250i q^{91} -28.5015 q^{93} -17.0882 q^{95} +10.4324i q^{97} +6.43425i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 6.39488i 0.913555i 0.889581 + 0.456777i \(0.150996\pi\)
−0.889581 + 0.456777i \(0.849004\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 2.14475i 0.194977i 0.995237 + 0.0974887i \(0.0310810\pi\)
−0.995237 + 0.0974887i \(0.968919\pi\)
\(12\) 0 0
\(13\) 23.8081 1.83139 0.915696 0.401871i \(-0.131640\pi\)
0.915696 + 0.401871i \(0.131640\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 7.36156i 0.433033i 0.976279 + 0.216516i \(0.0694695\pi\)
−0.976279 + 0.216516i \(0.930530\pi\)
\(18\) 0 0
\(19\) 7.64208i 0.402215i −0.979569 0.201107i \(-0.935546\pi\)
0.979569 0.201107i \(-0.0644541\pi\)
\(20\) 0 0
\(21\) 11.0763i 0.527441i
\(22\) 0 0
\(23\) −2.73940 22.8363i −0.119104 0.992882i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 41.2159 1.42124 0.710619 0.703577i \(-0.248415\pi\)
0.710619 + 0.703577i \(0.248415\pi\)
\(30\) 0 0
\(31\) 16.4553 0.530817 0.265408 0.964136i \(-0.414493\pi\)
0.265408 + 0.964136i \(0.414493\pi\)
\(32\) 0 0
\(33\) 3.71482i 0.112570i
\(34\) 0 0
\(35\) 14.2994 0.408554
\(36\) 0 0
\(37\) 12.9080i 0.348865i 0.984669 + 0.174432i \(0.0558090\pi\)
−0.984669 + 0.174432i \(0.944191\pi\)
\(38\) 0 0
\(39\) −41.2369 −1.05736
\(40\) 0 0
\(41\) −29.3570 −0.716023 −0.358012 0.933717i \(-0.616545\pi\)
−0.358012 + 0.933717i \(0.616545\pi\)
\(42\) 0 0
\(43\) 70.1044i 1.63034i −0.579225 0.815168i \(-0.696645\pi\)
0.579225 0.815168i \(-0.303355\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) −80.3689 −1.70998 −0.854989 0.518647i \(-0.826436\pi\)
−0.854989 + 0.518647i \(0.826436\pi\)
\(48\) 0 0
\(49\) 8.10546 0.165418
\(50\) 0 0
\(51\) 12.7506i 0.250012i
\(52\) 0 0
\(53\) 84.5512i 1.59531i 0.603116 + 0.797653i \(0.293925\pi\)
−0.603116 + 0.797653i \(0.706075\pi\)
\(54\) 0 0
\(55\) 4.79581 0.0871965
\(56\) 0 0
\(57\) 13.2365i 0.232219i
\(58\) 0 0
\(59\) 41.0693 0.696090 0.348045 0.937478i \(-0.386846\pi\)
0.348045 + 0.937478i \(0.386846\pi\)
\(60\) 0 0
\(61\) 19.8626i 0.325617i 0.986658 + 0.162808i \(0.0520552\pi\)
−0.986658 + 0.162808i \(0.947945\pi\)
\(62\) 0 0
\(63\) 19.1847i 0.304518i
\(64\) 0 0
\(65\) 53.2365i 0.819024i
\(66\) 0 0
\(67\) 7.96687i 0.118908i 0.998231 + 0.0594542i \(0.0189360\pi\)
−0.998231 + 0.0594542i \(0.981064\pi\)
\(68\) 0 0
\(69\) 4.74478 + 39.5536i 0.0687649 + 0.573241i
\(70\) 0 0
\(71\) −33.7474 −0.475315 −0.237657 0.971349i \(-0.576380\pi\)
−0.237657 + 0.971349i \(0.576380\pi\)
\(72\) 0 0
\(73\) 79.6030 1.09045 0.545226 0.838289i \(-0.316444\pi\)
0.545226 + 0.838289i \(0.316444\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) −13.7154 −0.178122
\(78\) 0 0
\(79\) 78.0646i 0.988160i −0.869416 0.494080i \(-0.835505\pi\)
0.869416 0.494080i \(-0.164495\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 56.2341i 0.677519i 0.940873 + 0.338759i \(0.110007\pi\)
−0.940873 + 0.338759i \(0.889993\pi\)
\(84\) 0 0
\(85\) 16.4609 0.193658
\(86\) 0 0
\(87\) −71.3880 −0.820552
\(88\) 0 0
\(89\) 45.9891i 0.516732i 0.966047 + 0.258366i \(0.0831840\pi\)
−0.966047 + 0.258366i \(0.916816\pi\)
\(90\) 0 0
\(91\) 152.250i 1.67308i
\(92\) 0 0
\(93\) −28.5015 −0.306467
\(94\) 0 0
\(95\) −17.0882 −0.179876
\(96\) 0 0
\(97\) 10.4324i 0.107551i 0.998553 + 0.0537754i \(0.0171255\pi\)
−0.998553 + 0.0537754i \(0.982875\pi\)
\(98\) 0 0
\(99\) 6.43425i 0.0649924i
\(100\) 0 0
\(101\) −131.321 −1.30020 −0.650102 0.759847i \(-0.725274\pi\)
−0.650102 + 0.759847i \(0.725274\pi\)
\(102\) 0 0
\(103\) 177.356i 1.72191i −0.508684 0.860954i \(-0.669868\pi\)
0.508684 0.860954i \(-0.330132\pi\)
\(104\) 0 0
\(105\) −24.7673 −0.235879
\(106\) 0 0
\(107\) 7.69385i 0.0719052i −0.999353 0.0359526i \(-0.988553\pi\)
0.999353 0.0359526i \(-0.0114465\pi\)
\(108\) 0 0
\(109\) 114.657i 1.05190i 0.850517 + 0.525948i \(0.176289\pi\)
−0.850517 + 0.525948i \(0.823711\pi\)
\(110\) 0 0
\(111\) 22.3573i 0.201417i
\(112\) 0 0
\(113\) 27.7793i 0.245835i −0.992417 0.122917i \(-0.960775\pi\)
0.992417 0.122917i \(-0.0392250\pi\)
\(114\) 0 0
\(115\) −51.0635 + 6.12548i −0.444030 + 0.0532650i
\(116\) 0 0
\(117\) 71.4243 0.610464
\(118\) 0 0
\(119\) −47.0763 −0.395599
\(120\) 0 0
\(121\) 116.400 0.961984
\(122\) 0 0
\(123\) 50.8477 0.413396
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −27.8575 −0.219351 −0.109675 0.993967i \(-0.534981\pi\)
−0.109675 + 0.993967i \(0.534981\pi\)
\(128\) 0 0
\(129\) 121.424i 0.941275i
\(130\) 0 0
\(131\) 171.391 1.30833 0.654165 0.756352i \(-0.273020\pi\)
0.654165 + 0.756352i \(0.273020\pi\)
\(132\) 0 0
\(133\) 48.8702 0.367445
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 117.514i 0.857766i 0.903360 + 0.428883i \(0.141093\pi\)
−0.903360 + 0.428883i \(0.858907\pi\)
\(138\) 0 0
\(139\) 108.286 0.779036 0.389518 0.921019i \(-0.372642\pi\)
0.389518 + 0.921019i \(0.372642\pi\)
\(140\) 0 0
\(141\) 139.203 0.987256
\(142\) 0 0
\(143\) 51.0625i 0.357080i
\(144\) 0 0
\(145\) 92.1615i 0.635597i
\(146\) 0 0
\(147\) −14.0391 −0.0955039
\(148\) 0 0
\(149\) 171.330i 1.14987i −0.818201 0.574933i \(-0.805028\pi\)
0.818201 0.574933i \(-0.194972\pi\)
\(150\) 0 0
\(151\) 125.055 0.828180 0.414090 0.910236i \(-0.364100\pi\)
0.414090 + 0.910236i \(0.364100\pi\)
\(152\) 0 0
\(153\) 22.0847i 0.144344i
\(154\) 0 0
\(155\) 36.7952i 0.237389i
\(156\) 0 0
\(157\) 78.4202i 0.499491i −0.968311 0.249746i \(-0.919653\pi\)
0.968311 0.249746i \(-0.0803470\pi\)
\(158\) 0 0
\(159\) 146.447i 0.921051i
\(160\) 0 0
\(161\) 146.035 17.5181i 0.907052 0.108808i
\(162\) 0 0
\(163\) −69.0368 −0.423539 −0.211769 0.977320i \(-0.567923\pi\)
−0.211769 + 0.977320i \(0.567923\pi\)
\(164\) 0 0
\(165\) −8.30658 −0.0503429
\(166\) 0 0
\(167\) 317.592 1.90175 0.950875 0.309575i \(-0.100187\pi\)
0.950875 + 0.309575i \(0.100187\pi\)
\(168\) 0 0
\(169\) 397.826 2.35400
\(170\) 0 0
\(171\) 22.9263i 0.134072i
\(172\) 0 0
\(173\) 70.9148 0.409912 0.204956 0.978771i \(-0.434295\pi\)
0.204956 + 0.978771i \(0.434295\pi\)
\(174\) 0 0
\(175\) 31.9744i 0.182711i
\(176\) 0 0
\(177\) −71.1341 −0.401888
\(178\) 0 0
\(179\) 152.432 0.851575 0.425788 0.904823i \(-0.359997\pi\)
0.425788 + 0.904823i \(0.359997\pi\)
\(180\) 0 0
\(181\) 179.513i 0.991787i −0.868383 0.495894i \(-0.834841\pi\)
0.868383 0.495894i \(-0.165159\pi\)
\(182\) 0 0
\(183\) 34.4030i 0.187995i
\(184\) 0 0
\(185\) 28.8632 0.156017
\(186\) 0 0
\(187\) −15.7887 −0.0844316
\(188\) 0 0
\(189\) 33.2288i 0.175814i
\(190\) 0 0
\(191\) 373.577i 1.95590i 0.208832 + 0.977952i \(0.433034\pi\)
−0.208832 + 0.977952i \(0.566966\pi\)
\(192\) 0 0
\(193\) −18.8712 −0.0977783 −0.0488892 0.998804i \(-0.515568\pi\)
−0.0488892 + 0.998804i \(0.515568\pi\)
\(194\) 0 0
\(195\) 92.2084i 0.472864i
\(196\) 0 0
\(197\) −8.64382 −0.0438772 −0.0219386 0.999759i \(-0.506984\pi\)
−0.0219386 + 0.999759i \(0.506984\pi\)
\(198\) 0 0
\(199\) 3.93956i 0.0197968i −0.999951 0.00989839i \(-0.996849\pi\)
0.999951 0.00989839i \(-0.00315081\pi\)
\(200\) 0 0
\(201\) 13.7990i 0.0686519i
\(202\) 0 0
\(203\) 263.571i 1.29838i
\(204\) 0 0
\(205\) 65.6442i 0.320215i
\(206\) 0 0
\(207\) −8.21819 68.5088i −0.0397014 0.330961i
\(208\) 0 0
\(209\) 16.3904 0.0784228
\(210\) 0 0
\(211\) −341.304 −1.61756 −0.808778 0.588114i \(-0.799871\pi\)
−0.808778 + 0.588114i \(0.799871\pi\)
\(212\) 0 0
\(213\) 58.4521 0.274423
\(214\) 0 0
\(215\) −156.758 −0.729108
\(216\) 0 0
\(217\) 105.230i 0.484930i
\(218\) 0 0
\(219\) −137.876 −0.629573
\(220\) 0 0
\(221\) 175.265i 0.793053i
\(222\) 0 0
\(223\) 79.8407 0.358030 0.179015 0.983846i \(-0.442709\pi\)
0.179015 + 0.983846i \(0.442709\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 126.103i 0.555522i −0.960650 0.277761i \(-0.910408\pi\)
0.960650 0.277761i \(-0.0895923\pi\)
\(228\) 0 0
\(229\) 6.28614i 0.0274504i 0.999906 + 0.0137252i \(0.00436901\pi\)
−0.999906 + 0.0137252i \(0.995631\pi\)
\(230\) 0 0
\(231\) 23.7558 0.102839
\(232\) 0 0
\(233\) 386.172 1.65739 0.828695 0.559701i \(-0.189084\pi\)
0.828695 + 0.559701i \(0.189084\pi\)
\(234\) 0 0
\(235\) 179.710i 0.764725i
\(236\) 0 0
\(237\) 135.212i 0.570514i
\(238\) 0 0
\(239\) −92.4413 −0.386784 −0.193392 0.981122i \(-0.561949\pi\)
−0.193392 + 0.981122i \(0.561949\pi\)
\(240\) 0 0
\(241\) 365.924i 1.51836i −0.650882 0.759179i \(-0.725601\pi\)
0.650882 0.759179i \(-0.274399\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 18.1244i 0.0739770i
\(246\) 0 0
\(247\) 181.944i 0.736614i
\(248\) 0 0
\(249\) 97.4003i 0.391166i
\(250\) 0 0
\(251\) 316.118i 1.25944i −0.776824 0.629718i \(-0.783171\pi\)
0.776824 0.629718i \(-0.216829\pi\)
\(252\) 0 0
\(253\) 48.9781 5.87533i 0.193589 0.0232226i
\(254\) 0 0
\(255\) −28.5112 −0.111809
\(256\) 0 0
\(257\) 5.51488 0.0214587 0.0107293 0.999942i \(-0.496585\pi\)
0.0107293 + 0.999942i \(0.496585\pi\)
\(258\) 0 0
\(259\) −82.5451 −0.318707
\(260\) 0 0
\(261\) 123.648 0.473746
\(262\) 0 0
\(263\) 394.828i 1.50125i 0.660731 + 0.750623i \(0.270247\pi\)
−0.660731 + 0.750623i \(0.729753\pi\)
\(264\) 0 0
\(265\) 189.062 0.713443
\(266\) 0 0
\(267\) 79.6555i 0.298335i
\(268\) 0 0
\(269\) 241.967 0.899506 0.449753 0.893153i \(-0.351512\pi\)
0.449753 + 0.893153i \(0.351512\pi\)
\(270\) 0 0
\(271\) 38.6016 0.142441 0.0712206 0.997461i \(-0.477311\pi\)
0.0712206 + 0.997461i \(0.477311\pi\)
\(272\) 0 0
\(273\) 263.705i 0.965952i
\(274\) 0 0
\(275\) 10.7238i 0.0389955i
\(276\) 0 0
\(277\) 169.974 0.613624 0.306812 0.951770i \(-0.400738\pi\)
0.306812 + 0.951770i \(0.400738\pi\)
\(278\) 0 0
\(279\) 49.3660 0.176939
\(280\) 0 0
\(281\) 362.263i 1.28919i −0.764523 0.644596i \(-0.777025\pi\)
0.764523 0.644596i \(-0.222975\pi\)
\(282\) 0 0
\(283\) 304.879i 1.07731i 0.842527 + 0.538655i \(0.181067\pi\)
−0.842527 + 0.538655i \(0.818933\pi\)
\(284\) 0 0
\(285\) 29.5977 0.103851
\(286\) 0 0
\(287\) 187.734i 0.654127i
\(288\) 0 0
\(289\) 234.807 0.812482
\(290\) 0 0
\(291\) 18.0695i 0.0620944i
\(292\) 0 0
\(293\) 387.148i 1.32132i 0.750683 + 0.660662i \(0.229724\pi\)
−0.750683 + 0.660662i \(0.770276\pi\)
\(294\) 0 0
\(295\) 91.8338i 0.311301i
\(296\) 0 0
\(297\) 11.1445i 0.0375234i
\(298\) 0 0
\(299\) −65.2199 543.689i −0.218127 1.81836i
\(300\) 0 0
\(301\) 448.310 1.48940
\(302\) 0 0
\(303\) 227.454 0.750673
\(304\) 0 0
\(305\) 44.4141 0.145620
\(306\) 0 0
\(307\) 270.200 0.880129 0.440064 0.897966i \(-0.354956\pi\)
0.440064 + 0.897966i \(0.354956\pi\)
\(308\) 0 0
\(309\) 307.190i 0.994144i
\(310\) 0 0
\(311\) −10.1209 −0.0325430 −0.0162715 0.999868i \(-0.505180\pi\)
−0.0162715 + 0.999868i \(0.505180\pi\)
\(312\) 0 0
\(313\) 97.2834i 0.310810i 0.987851 + 0.155405i \(0.0496682\pi\)
−0.987851 + 0.155405i \(0.950332\pi\)
\(314\) 0 0
\(315\) 42.8982 0.136185
\(316\) 0 0
\(317\) −35.8613 −0.113127 −0.0565636 0.998399i \(-0.518014\pi\)
−0.0565636 + 0.998399i \(0.518014\pi\)
\(318\) 0 0
\(319\) 88.3978i 0.277109i
\(320\) 0 0
\(321\) 13.3261i 0.0415145i
\(322\) 0 0
\(323\) 56.2577 0.174172
\(324\) 0 0
\(325\) −119.041 −0.366279
\(326\) 0 0
\(327\) 198.591i 0.607312i
\(328\) 0 0
\(329\) 513.950i 1.56216i
\(330\) 0 0
\(331\) 375.264 1.13373 0.566864 0.823811i \(-0.308156\pi\)
0.566864 + 0.823811i \(0.308156\pi\)
\(332\) 0 0
\(333\) 38.7240i 0.116288i
\(334\) 0 0
\(335\) 17.8145 0.0531775
\(336\) 0 0
\(337\) 309.610i 0.918724i 0.888249 + 0.459362i \(0.151922\pi\)
−0.888249 + 0.459362i \(0.848078\pi\)
\(338\) 0 0
\(339\) 48.1152i 0.141933i
\(340\) 0 0
\(341\) 35.2926i 0.103497i
\(342\) 0 0
\(343\) 365.183i 1.06467i
\(344\) 0 0
\(345\) 88.4445 10.6096i 0.256361 0.0307526i
\(346\) 0 0
\(347\) −492.980 −1.42069 −0.710345 0.703853i \(-0.751461\pi\)
−0.710345 + 0.703853i \(0.751461\pi\)
\(348\) 0 0
\(349\) 621.786 1.78162 0.890811 0.454374i \(-0.150137\pi\)
0.890811 + 0.454374i \(0.150137\pi\)
\(350\) 0 0
\(351\) −123.711 −0.352452
\(352\) 0 0
\(353\) −66.7112 −0.188984 −0.0944918 0.995526i \(-0.530123\pi\)
−0.0944918 + 0.995526i \(0.530123\pi\)
\(354\) 0 0
\(355\) 75.4614i 0.212567i
\(356\) 0 0
\(357\) 81.5386 0.228399
\(358\) 0 0
\(359\) 221.326i 0.616507i −0.951304 0.308253i \(-0.900256\pi\)
0.951304 0.308253i \(-0.0997444\pi\)
\(360\) 0 0
\(361\) 302.599 0.838223
\(362\) 0 0
\(363\) −201.611 −0.555402
\(364\) 0 0
\(365\) 177.998i 0.487665i
\(366\) 0 0
\(367\) 148.108i 0.403564i 0.979430 + 0.201782i \(0.0646733\pi\)
−0.979430 + 0.201782i \(0.935327\pi\)
\(368\) 0 0
\(369\) −88.0709 −0.238674
\(370\) 0 0
\(371\) −540.695 −1.45740
\(372\) 0 0
\(373\) 452.543i 1.21325i 0.794988 + 0.606626i \(0.207477\pi\)
−0.794988 + 0.606626i \(0.792523\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) 981.272 2.60284
\(378\) 0 0
\(379\) 354.076i 0.934236i 0.884195 + 0.467118i \(0.154708\pi\)
−0.884195 + 0.467118i \(0.845292\pi\)
\(380\) 0 0
\(381\) 48.2507 0.126642
\(382\) 0 0
\(383\) 522.893i 1.36526i 0.730766 + 0.682628i \(0.239163\pi\)
−0.730766 + 0.682628i \(0.760837\pi\)
\(384\) 0 0
\(385\) 30.6686i 0.0796588i
\(386\) 0 0
\(387\) 210.313i 0.543445i
\(388\) 0 0
\(389\) 161.981i 0.416403i −0.978086 0.208202i \(-0.933239\pi\)
0.978086 0.208202i \(-0.0667610\pi\)
\(390\) 0 0
\(391\) 168.111 20.1662i 0.429951 0.0515761i
\(392\) 0 0
\(393\) −296.858 −0.755365
\(394\) 0 0
\(395\) −174.558 −0.441919
\(396\) 0 0
\(397\) 295.931 0.745418 0.372709 0.927948i \(-0.378429\pi\)
0.372709 + 0.927948i \(0.378429\pi\)
\(398\) 0 0
\(399\) −84.6457 −0.212145
\(400\) 0 0
\(401\) 213.814i 0.533202i 0.963807 + 0.266601i \(0.0859006\pi\)
−0.963807 + 0.266601i \(0.914099\pi\)
\(402\) 0 0
\(403\) 391.770 0.972134
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) −27.6844 −0.0680207
\(408\) 0 0
\(409\) 183.539 0.448750 0.224375 0.974503i \(-0.427966\pi\)
0.224375 + 0.974503i \(0.427966\pi\)
\(410\) 0 0
\(411\) 203.540i 0.495231i
\(412\) 0 0
\(413\) 262.633i 0.635916i
\(414\) 0 0
\(415\) 125.743 0.302996
\(416\) 0 0
\(417\) −187.557 −0.449776
\(418\) 0 0
\(419\) 560.084i 1.33672i 0.743840 + 0.668358i \(0.233003\pi\)
−0.743840 + 0.668358i \(0.766997\pi\)
\(420\) 0 0
\(421\) 536.211i 1.27366i −0.771005 0.636830i \(-0.780245\pi\)
0.771005 0.636830i \(-0.219755\pi\)
\(422\) 0 0
\(423\) −241.107 −0.569992
\(424\) 0 0
\(425\) 36.8078i 0.0866066i
\(426\) 0 0
\(427\) −127.019 −0.297469
\(428\) 0 0
\(429\) 88.4428i 0.206160i
\(430\) 0 0
\(431\) 404.974i 0.939616i −0.882769 0.469808i \(-0.844323\pi\)
0.882769 0.469808i \(-0.155677\pi\)
\(432\) 0 0
\(433\) 632.410i 1.46053i 0.683164 + 0.730265i \(0.260604\pi\)
−0.683164 + 0.730265i \(0.739396\pi\)
\(434\) 0 0
\(435\) 159.628i 0.366962i
\(436\) 0 0
\(437\) −174.517 + 20.9347i −0.399352 + 0.0479055i
\(438\) 0 0
\(439\) 169.127 0.385255 0.192628 0.981272i \(-0.438299\pi\)
0.192628 + 0.981272i \(0.438299\pi\)
\(440\) 0 0
\(441\) 24.3164 0.0551392
\(442\) 0 0
\(443\) −209.960 −0.473950 −0.236975 0.971516i \(-0.576156\pi\)
−0.236975 + 0.971516i \(0.576156\pi\)
\(444\) 0 0
\(445\) 102.835 0.231089
\(446\) 0 0
\(447\) 296.752i 0.663875i
\(448\) 0 0
\(449\) 131.345 0.292529 0.146264 0.989246i \(-0.453275\pi\)
0.146264 + 0.989246i \(0.453275\pi\)
\(450\) 0 0
\(451\) 62.9634i 0.139608i
\(452\) 0 0
\(453\) −216.602 −0.478150
\(454\) 0 0
\(455\) 340.442 0.748223
\(456\) 0 0
\(457\) 354.892i 0.776569i −0.921540 0.388284i \(-0.873068\pi\)
0.921540 0.388284i \(-0.126932\pi\)
\(458\) 0 0
\(459\) 38.2518i 0.0833372i
\(460\) 0 0
\(461\) −586.971 −1.27326 −0.636628 0.771171i \(-0.719671\pi\)
−0.636628 + 0.771171i \(0.719671\pi\)
\(462\) 0 0
\(463\) 356.547 0.770080 0.385040 0.922900i \(-0.374188\pi\)
0.385040 + 0.922900i \(0.374188\pi\)
\(464\) 0 0
\(465\) 63.7312i 0.137056i
\(466\) 0 0
\(467\) 350.300i 0.750106i 0.927003 + 0.375053i \(0.122376\pi\)
−0.927003 + 0.375053i \(0.877624\pi\)
\(468\) 0 0
\(469\) −50.9472 −0.108629
\(470\) 0 0
\(471\) 135.828i 0.288382i
\(472\) 0 0
\(473\) 150.357 0.317878
\(474\) 0 0
\(475\) 38.2104i 0.0804430i
\(476\) 0 0
\(477\) 253.654i 0.531769i
\(478\) 0 0
\(479\) 833.219i 1.73950i −0.493496 0.869748i \(-0.664281\pi\)
0.493496 0.869748i \(-0.335719\pi\)
\(480\) 0 0
\(481\) 307.315i 0.638908i
\(482\) 0 0
\(483\) −252.941 + 30.3423i −0.523687 + 0.0628205i
\(484\) 0 0
\(485\) 23.3276 0.0480981
\(486\) 0 0
\(487\) 377.705 0.775574 0.387787 0.921749i \(-0.373240\pi\)
0.387787 + 0.921749i \(0.373240\pi\)
\(488\) 0 0
\(489\) 119.575 0.244530
\(490\) 0 0
\(491\) 782.837 1.59437 0.797187 0.603733i \(-0.206321\pi\)
0.797187 + 0.603733i \(0.206321\pi\)
\(492\) 0 0
\(493\) 303.413i 0.615443i
\(494\) 0 0
\(495\) 14.3874 0.0290655
\(496\) 0 0
\(497\) 215.810i 0.434226i
\(498\) 0 0
\(499\) −556.310 −1.11485 −0.557424 0.830228i \(-0.688210\pi\)
−0.557424 + 0.830228i \(0.688210\pi\)
\(500\) 0 0
\(501\) −550.086 −1.09798
\(502\) 0 0
\(503\) 241.516i 0.480152i 0.970754 + 0.240076i \(0.0771724\pi\)
−0.970754 + 0.240076i \(0.922828\pi\)
\(504\) 0 0
\(505\) 293.642i 0.581469i
\(506\) 0 0
\(507\) −689.055 −1.35908
\(508\) 0 0
\(509\) −99.6188 −0.195715 −0.0978573 0.995200i \(-0.531199\pi\)
−0.0978573 + 0.995200i \(0.531199\pi\)
\(510\) 0 0
\(511\) 509.052i 0.996188i
\(512\) 0 0
\(513\) 39.7094i 0.0774063i
\(514\) 0 0
\(515\) −396.581 −0.770060
\(516\) 0 0
\(517\) 172.371i 0.333407i
\(518\) 0 0
\(519\) −122.828 −0.236663
\(520\) 0 0
\(521\) 686.711i 1.31806i 0.752116 + 0.659031i \(0.229034\pi\)
−0.752116 + 0.659031i \(0.770966\pi\)
\(522\) 0 0
\(523\) 77.0470i 0.147317i −0.997284 0.0736587i \(-0.976532\pi\)
0.997284 0.0736587i \(-0.0234676\pi\)
\(524\) 0 0
\(525\) 55.3813i 0.105488i
\(526\) 0 0
\(527\) 121.137i 0.229861i
\(528\) 0 0
\(529\) −513.991 + 125.115i −0.971628 + 0.236513i
\(530\) 0 0
\(531\) 123.208 0.232030
\(532\) 0 0
\(533\) −698.934 −1.31132
\(534\) 0 0
\(535\) −17.2040 −0.0321570
\(536\) 0 0
\(537\) −264.020 −0.491657
\(538\) 0 0
\(539\) 17.3842i 0.0322527i
\(540\) 0 0
\(541\) 414.356 0.765908 0.382954 0.923767i \(-0.374907\pi\)
0.382954 + 0.923767i \(0.374907\pi\)
\(542\) 0 0
\(543\) 310.926i 0.572608i
\(544\) 0 0
\(545\) 256.380 0.470422
\(546\) 0 0
\(547\) 603.491 1.10328 0.551638 0.834084i \(-0.314003\pi\)
0.551638 + 0.834084i \(0.314003\pi\)
\(548\) 0 0
\(549\) 59.5878i 0.108539i
\(550\) 0 0
\(551\) 314.975i 0.571643i
\(552\) 0 0
\(553\) 499.214 0.902738
\(554\) 0 0
\(555\) −49.9925 −0.0900765
\(556\) 0 0
\(557\) 577.330i 1.03650i −0.855230 0.518249i \(-0.826584\pi\)
0.855230 0.518249i \(-0.173416\pi\)
\(558\) 0 0
\(559\) 1669.05i 2.98578i
\(560\) 0 0
\(561\) 27.3469 0.0487466
\(562\) 0 0
\(563\) 722.038i 1.28248i −0.767339 0.641241i \(-0.778420\pi\)
0.767339 0.641241i \(-0.221580\pi\)
\(564\) 0 0
\(565\) −62.1165 −0.109941
\(566\) 0 0
\(567\) 57.5540i 0.101506i
\(568\) 0 0
\(569\) 596.301i 1.04798i −0.851724 0.523990i \(-0.824443\pi\)
0.851724 0.523990i \(-0.175557\pi\)
\(570\) 0 0
\(571\) 44.7220i 0.0783223i −0.999233 0.0391612i \(-0.987531\pi\)
0.999233 0.0391612i \(-0.0124686\pi\)
\(572\) 0 0
\(573\) 647.055i 1.12924i
\(574\) 0 0
\(575\) 13.6970 + 114.181i 0.0238209 + 0.198576i
\(576\) 0 0
\(577\) −582.802 −1.01006 −0.505028 0.863103i \(-0.668518\pi\)
−0.505028 + 0.863103i \(0.668518\pi\)
\(578\) 0 0
\(579\) 32.6859 0.0564524
\(580\) 0 0
\(581\) −359.610 −0.618951
\(582\) 0 0
\(583\) −181.341 −0.311049
\(584\) 0 0
\(585\) 159.710i 0.273008i
\(586\) 0 0
\(587\) −58.7601 −0.100102 −0.0500512 0.998747i \(-0.515938\pi\)
−0.0500512 + 0.998747i \(0.515938\pi\)
\(588\) 0 0
\(589\) 125.753i 0.213503i
\(590\) 0 0
\(591\) 14.9715 0.0253325
\(592\) 0 0
\(593\) −622.657 −1.05001 −0.525006 0.851099i \(-0.675937\pi\)
−0.525006 + 0.851099i \(0.675937\pi\)
\(594\) 0 0
\(595\) 105.266i 0.176917i
\(596\) 0 0
\(597\) 6.82352i 0.0114297i
\(598\) 0 0
\(599\) −125.459 −0.209448 −0.104724 0.994501i \(-0.533396\pi\)
−0.104724 + 0.994501i \(0.533396\pi\)
\(600\) 0 0
\(601\) 119.413 0.198691 0.0993456 0.995053i \(-0.468325\pi\)
0.0993456 + 0.995053i \(0.468325\pi\)
\(602\) 0 0
\(603\) 23.9006i 0.0396362i
\(604\) 0 0
\(605\) 260.278i 0.430212i
\(606\) 0 0
\(607\) −509.436 −0.839269 −0.419635 0.907693i \(-0.637842\pi\)
−0.419635 + 0.907693i \(0.637842\pi\)
\(608\) 0 0
\(609\) 456.518i 0.749619i
\(610\) 0 0
\(611\) −1913.43 −3.13164
\(612\) 0 0
\(613\) 11.4881i 0.0187408i −0.999956 0.00937039i \(-0.997017\pi\)
0.999956 0.00937039i \(-0.00298273\pi\)
\(614\) 0 0
\(615\) 113.699i 0.184876i
\(616\) 0 0
\(617\) 289.273i 0.468839i −0.972136 0.234419i \(-0.924681\pi\)
0.972136 0.234419i \(-0.0753189\pi\)
\(618\) 0 0
\(619\) 211.722i 0.342040i −0.985268 0.171020i \(-0.945294\pi\)
0.985268 0.171020i \(-0.0547062\pi\)
\(620\) 0 0
\(621\) 14.2343 + 118.661i 0.0229216 + 0.191080i
\(622\) 0 0
\(623\) −294.095 −0.472063
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −28.3889 −0.0452774
\(628\) 0 0
\(629\) −95.0230 −0.151070
\(630\) 0 0
\(631\) 970.495i 1.53803i 0.639232 + 0.769014i \(0.279252\pi\)
−0.639232 + 0.769014i \(0.720748\pi\)
\(632\) 0 0
\(633\) 591.157 0.933897
\(634\) 0 0
\(635\) 62.2914i 0.0980966i
\(636\) 0 0
\(637\) 192.976 0.302945
\(638\) 0 0
\(639\) −101.242 −0.158438
\(640\) 0 0
\(641\) 523.389i 0.816520i 0.912866 + 0.408260i \(0.133864\pi\)
−0.912866 + 0.408260i \(0.866136\pi\)
\(642\) 0 0
\(643\) 100.620i 0.156485i 0.996934 + 0.0782423i \(0.0249308\pi\)
−0.996934 + 0.0782423i \(0.975069\pi\)
\(644\) 0 0
\(645\) 271.513 0.420951
\(646\) 0 0
\(647\) 364.738 0.563738 0.281869 0.959453i \(-0.409046\pi\)
0.281869 + 0.959453i \(0.409046\pi\)
\(648\) 0 0
\(649\) 88.0834i 0.135722i
\(650\) 0 0
\(651\) 182.264i 0.279975i
\(652\) 0 0
\(653\) −1016.24 −1.55626 −0.778131 0.628102i \(-0.783832\pi\)
−0.778131 + 0.628102i \(0.783832\pi\)
\(654\) 0 0
\(655\) 383.242i 0.585103i
\(656\) 0 0
\(657\) 238.809 0.363484
\(658\) 0 0
\(659\) 281.233i 0.426757i −0.976970 0.213379i \(-0.931553\pi\)
0.976970 0.213379i \(-0.0684468\pi\)
\(660\) 0 0
\(661\) 974.582i 1.47441i 0.675671 + 0.737203i \(0.263854\pi\)
−0.675671 + 0.737203i \(0.736146\pi\)
\(662\) 0 0
\(663\) 303.568i 0.457870i
\(664\) 0 0
\(665\) 109.277i 0.164327i
\(666\) 0 0
\(667\) −112.907 941.217i −0.169275 1.41112i
\(668\) 0 0
\(669\) −138.288 −0.206709
\(670\) 0 0
\(671\) −42.6003 −0.0634878
\(672\) 0 0
\(673\) 946.163 1.40589 0.702944 0.711245i \(-0.251868\pi\)
0.702944 + 0.711245i \(0.251868\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 628.111i 0.927785i 0.885891 + 0.463893i \(0.153548\pi\)
−0.885891 + 0.463893i \(0.846452\pi\)
\(678\) 0 0
\(679\) −66.7141 −0.0982535
\(680\) 0 0
\(681\) 218.417i 0.320731i
\(682\) 0 0
\(683\) −50.3076 −0.0736569 −0.0368284 0.999322i \(-0.511726\pi\)
−0.0368284 + 0.999322i \(0.511726\pi\)
\(684\) 0 0
\(685\) 262.769 0.383605
\(686\) 0 0
\(687\) 10.8879i 0.0158485i
\(688\) 0 0
\(689\) 2013.01i 2.92163i
\(690\) 0 0
\(691\) −947.991 −1.37191 −0.685956 0.727643i \(-0.740616\pi\)
−0.685956 + 0.727643i \(0.740616\pi\)
\(692\) 0 0
\(693\) −41.1463 −0.0593742
\(694\) 0 0
\(695\) 242.135i 0.348395i
\(696\) 0 0
\(697\) 216.113i 0.310062i
\(698\) 0 0
\(699\) −668.869 −0.956894
\(700\) 0 0
\(701\) 74.5851i 0.106398i −0.998584 0.0531991i \(-0.983058\pi\)
0.998584 0.0531991i \(-0.0169418\pi\)
\(702\) 0 0
\(703\) 98.6440 0.140319
\(704\) 0 0
\(705\) 311.268i 0.441514i
\(706\) 0 0
\(707\) 839.780i 1.18781i
\(708\) 0 0
\(709\) 742.618i 1.04742i −0.851898 0.523708i \(-0.824548\pi\)
0.851898 0.523708i \(-0.175452\pi\)
\(710\) 0 0
\(711\) 234.194i 0.329387i
\(712\) 0 0
\(713\) −45.0777 375.778i −0.0632226 0.527038i
\(714\) 0 0
\(715\) 114.179 0.159691
\(716\) 0 0
\(717\) 160.113 0.223310
\(718\) 0 0
\(719\) 736.387 1.02418 0.512091 0.858931i \(-0.328871\pi\)
0.512091 + 0.858931i \(0.328871\pi\)
\(720\) 0 0
\(721\) 1134.17 1.57306
\(722\) 0 0
\(723\) 633.799i 0.876624i
\(724\) 0 0
\(725\) −206.079 −0.284247
\(726\) 0 0
\(727\) 329.605i 0.453376i 0.973967 + 0.226688i \(0.0727898\pi\)
−0.973967 + 0.226688i \(0.927210\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 516.078 0.705989
\(732\) 0 0
\(733\) 787.046i 1.07373i −0.843667 0.536866i \(-0.819608\pi\)
0.843667 0.536866i \(-0.180392\pi\)
\(734\) 0 0
\(735\) 31.3923i 0.0427106i
\(736\) 0 0
\(737\) −17.0869 −0.0231845
\(738\) 0 0
\(739\) −1087.38 −1.47143 −0.735713 0.677293i \(-0.763153\pi\)
−0.735713 + 0.677293i \(0.763153\pi\)
\(740\) 0 0
\(741\) 315.136i 0.425284i
\(742\) 0 0
\(743\) 1019.63i 1.37231i 0.727456 + 0.686154i \(0.240703\pi\)
−0.727456 + 0.686154i \(0.759297\pi\)
\(744\) 0 0
\(745\) −383.106 −0.514236
\(746\) 0 0
\(747\) 168.702i 0.225840i
\(748\) 0 0
\(749\) 49.2013 0.0656893
\(750\) 0 0
\(751\) 877.346i 1.16824i −0.811668 0.584119i \(-0.801440\pi\)
0.811668 0.584119i \(-0.198560\pi\)
\(752\) 0 0
\(753\) 547.533i 0.727136i
\(754\) 0 0
\(755\) 279.632i 0.370373i
\(756\) 0 0
\(757\) 158.385i 0.209227i 0.994513 + 0.104614i \(0.0333606\pi\)
−0.994513 + 0.104614i \(0.966639\pi\)
\(758\) 0 0
\(759\) −84.8326 + 10.1764i −0.111769 + 0.0134076i
\(760\) 0 0
\(761\) −412.068 −0.541482 −0.270741 0.962652i \(-0.587269\pi\)
−0.270741 + 0.962652i \(0.587269\pi\)
\(762\) 0 0
\(763\) −733.216 −0.960964
\(764\) 0 0
\(765\) 49.3828 0.0645527
\(766\) 0 0
\(767\) 977.783 1.27481
\(768\) 0 0
\(769\) 1488.58i 1.93573i −0.251466 0.967866i \(-0.580913\pi\)
0.251466 0.967866i \(-0.419087\pi\)
\(770\) 0 0
\(771\) −9.55206 −0.0123892
\(772\) 0 0
\(773\) 987.615i 1.27764i −0.769357 0.638820i \(-0.779423\pi\)
0.769357 0.638820i \(-0.220577\pi\)
\(774\) 0 0
\(775\) −82.2766 −0.106163
\(776\) 0 0
\(777\) 142.972 0.184006
\(778\) 0 0
\(779\) 224.348i 0.287995i
\(780\) 0 0
\(781\) 72.3797i 0.0926757i
\(782\) 0 0
\(783\) −214.164 −0.273517
\(784\) 0 0
\(785\) −175.353 −0.223379
\(786\) 0 0
\(787\) 1384.16i 1.75878i 0.476106 + 0.879388i \(0.342048\pi\)
−0.476106 + 0.879388i \(0.657952\pi\)
\(788\) 0 0
\(789\) 683.862i 0.866745i
\(790\) 0 0
\(791\) 177.646 0.224584
\(792\) 0 0
\(793\) 472.891i 0.596332i
\(794\) 0 0
\(795\) −327.466 −0.411906
\(796\) 0 0
\(797\) 667.685i 0.837748i 0.908044 + 0.418874i \(0.137575\pi\)
−0.908044 + 0.418874i \(0.862425\pi\)
\(798\) 0 0
\(799\)