Properties

Label 2760.3.g.a.2161.3
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 2161.3
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.46

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} -10.5413i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} -10.5413i q^{7} +3.00000 q^{9} +6.70196i q^{11} -22.5434 q^{13} +3.87298i q^{15} -2.57844i q^{17} -22.5074i q^{19} +18.2581i q^{21} +(22.6148 + 4.19184i) q^{23} -5.00000 q^{25} -5.19615 q^{27} -28.9269 q^{29} -43.4297 q^{31} -11.6081i q^{33} -23.5711 q^{35} +15.5489i q^{37} +39.0464 q^{39} +48.2836 q^{41} +10.3366i q^{43} -6.70820i q^{45} -52.9469 q^{47} -62.1194 q^{49} +4.46599i q^{51} +31.7564i q^{53} +14.9860 q^{55} +38.9839i q^{57} +20.6145 q^{59} -46.6394i q^{61} -31.6239i q^{63} +50.4086i q^{65} +32.4906i q^{67} +(-39.1700 - 7.26047i) q^{69} +8.24993 q^{71} -142.331 q^{73} +8.66025 q^{75} +70.6475 q^{77} -104.916i q^{79} +9.00000 q^{81} +112.969i q^{83} -5.76556 q^{85} +50.1028 q^{87} -147.454i q^{89} +237.637i q^{91} +75.2225 q^{93} -50.3280 q^{95} -25.1105i q^{97} +20.1059i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 10.5413i 1.50590i −0.658076 0.752951i \(-0.728630\pi\)
0.658076 0.752951i \(-0.271370\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 6.70196i 0.609269i 0.952469 + 0.304635i \(0.0985344\pi\)
−0.952469 + 0.304635i \(0.901466\pi\)
\(12\) 0 0
\(13\) −22.5434 −1.73411 −0.867055 0.498213i \(-0.833990\pi\)
−0.867055 + 0.498213i \(0.833990\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 2.57844i 0.151673i −0.997120 0.0758364i \(-0.975837\pi\)
0.997120 0.0758364i \(-0.0241627\pi\)
\(18\) 0 0
\(19\) 22.5074i 1.18460i −0.805718 0.592299i \(-0.798220\pi\)
0.805718 0.592299i \(-0.201780\pi\)
\(20\) 0 0
\(21\) 18.2581i 0.869433i
\(22\) 0 0
\(23\) 22.6148 + 4.19184i 0.983252 + 0.182254i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) −28.9269 −0.997479 −0.498739 0.866752i \(-0.666204\pi\)
−0.498739 + 0.866752i \(0.666204\pi\)
\(30\) 0 0
\(31\) −43.4297 −1.40096 −0.700479 0.713673i \(-0.747030\pi\)
−0.700479 + 0.713673i \(0.747030\pi\)
\(32\) 0 0
\(33\) 11.6081i 0.351762i
\(34\) 0 0
\(35\) −23.5711 −0.673460
\(36\) 0 0
\(37\) 15.5489i 0.420240i 0.977676 + 0.210120i \(0.0673855\pi\)
−0.977676 + 0.210120i \(0.932614\pi\)
\(38\) 0 0
\(39\) 39.0464 1.00119
\(40\) 0 0
\(41\) 48.2836 1.17765 0.588824 0.808261i \(-0.299591\pi\)
0.588824 + 0.808261i \(0.299591\pi\)
\(42\) 0 0
\(43\) 10.3366i 0.240385i 0.992751 + 0.120193i \(0.0383512\pi\)
−0.992751 + 0.120193i \(0.961649\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) −52.9469 −1.12653 −0.563264 0.826277i \(-0.690455\pi\)
−0.563264 + 0.826277i \(0.690455\pi\)
\(48\) 0 0
\(49\) −62.1194 −1.26774
\(50\) 0 0
\(51\) 4.46599i 0.0875684i
\(52\) 0 0
\(53\) 31.7564i 0.599178i 0.954068 + 0.299589i \(0.0968496\pi\)
−0.954068 + 0.299589i \(0.903150\pi\)
\(54\) 0 0
\(55\) 14.9860 0.272474
\(56\) 0 0
\(57\) 38.9839i 0.683929i
\(58\) 0 0
\(59\) 20.6145 0.349398 0.174699 0.984622i \(-0.444105\pi\)
0.174699 + 0.984622i \(0.444105\pi\)
\(60\) 0 0
\(61\) 46.6394i 0.764580i −0.924042 0.382290i \(-0.875136\pi\)
0.924042 0.382290i \(-0.124864\pi\)
\(62\) 0 0
\(63\) 31.6239i 0.501967i
\(64\) 0 0
\(65\) 50.4086i 0.775517i
\(66\) 0 0
\(67\) 32.4906i 0.484934i 0.970160 + 0.242467i \(0.0779567\pi\)
−0.970160 + 0.242467i \(0.922043\pi\)
\(68\) 0 0
\(69\) −39.1700 7.26047i −0.567681 0.105224i
\(70\) 0 0
\(71\) 8.24993 0.116196 0.0580981 0.998311i \(-0.481496\pi\)
0.0580981 + 0.998311i \(0.481496\pi\)
\(72\) 0 0
\(73\) −142.331 −1.94974 −0.974870 0.222775i \(-0.928488\pi\)
−0.974870 + 0.222775i \(0.928488\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) 70.6475 0.917500
\(78\) 0 0
\(79\) 104.916i 1.32805i −0.747711 0.664024i \(-0.768847\pi\)
0.747711 0.664024i \(-0.231153\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 112.969i 1.36107i 0.732717 + 0.680534i \(0.238252\pi\)
−0.732717 + 0.680534i \(0.761748\pi\)
\(84\) 0 0
\(85\) −5.76556 −0.0678302
\(86\) 0 0
\(87\) 50.1028 0.575895
\(88\) 0 0
\(89\) 147.454i 1.65679i −0.560144 0.828395i \(-0.689254\pi\)
0.560144 0.828395i \(-0.310746\pi\)
\(90\) 0 0
\(91\) 237.637i 2.61140i
\(92\) 0 0
\(93\) 75.2225 0.808844
\(94\) 0 0
\(95\) −50.3280 −0.529769
\(96\) 0 0
\(97\) 25.1105i 0.258871i −0.991588 0.129436i \(-0.958683\pi\)
0.991588 0.129436i \(-0.0413166\pi\)
\(98\) 0 0
\(99\) 20.1059i 0.203090i
\(100\) 0 0
\(101\) 136.267 1.34918 0.674589 0.738194i \(-0.264321\pi\)
0.674589 + 0.738194i \(0.264321\pi\)
\(102\) 0 0
\(103\) 107.313i 1.04188i 0.853594 + 0.520939i \(0.174418\pi\)
−0.853594 + 0.520939i \(0.825582\pi\)
\(104\) 0 0
\(105\) 40.8263 0.388822
\(106\) 0 0
\(107\) 139.793i 1.30647i 0.757153 + 0.653237i \(0.226589\pi\)
−0.757153 + 0.653237i \(0.773411\pi\)
\(108\) 0 0
\(109\) 14.3881i 0.132001i −0.997820 0.0660005i \(-0.978976\pi\)
0.997820 0.0660005i \(-0.0210239\pi\)
\(110\) 0 0
\(111\) 26.9315i 0.242626i
\(112\) 0 0
\(113\) 117.464i 1.03950i −0.854317 0.519752i \(-0.826024\pi\)
0.854317 0.519752i \(-0.173976\pi\)
\(114\) 0 0
\(115\) 9.37323 50.5682i 0.0815064 0.439723i
\(116\) 0 0
\(117\) −67.6303 −0.578036
\(118\) 0 0
\(119\) −27.1801 −0.228405
\(120\) 0 0
\(121\) 76.0837 0.628791
\(122\) 0 0
\(123\) −83.6296 −0.679916
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) 206.779 1.62818 0.814090 0.580739i \(-0.197236\pi\)
0.814090 + 0.580739i \(0.197236\pi\)
\(128\) 0 0
\(129\) 17.9034i 0.138786i
\(130\) 0 0
\(131\) 35.5136 0.271096 0.135548 0.990771i \(-0.456720\pi\)
0.135548 + 0.990771i \(0.456720\pi\)
\(132\) 0 0
\(133\) −237.257 −1.78389
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 36.7299i 0.268101i 0.990974 + 0.134051i \(0.0427985\pi\)
−0.990974 + 0.134051i \(0.957202\pi\)
\(138\) 0 0
\(139\) 77.5405 0.557845 0.278923 0.960314i \(-0.410023\pi\)
0.278923 + 0.960314i \(0.410023\pi\)
\(140\) 0 0
\(141\) 91.7067 0.650402
\(142\) 0 0
\(143\) 151.085i 1.05654i
\(144\) 0 0
\(145\) 64.6825i 0.446086i
\(146\) 0 0
\(147\) 107.594 0.731931
\(148\) 0 0
\(149\) 77.7870i 0.522061i −0.965331 0.261030i \(-0.915938\pi\)
0.965331 0.261030i \(-0.0840622\pi\)
\(150\) 0 0
\(151\) 137.905 0.913280 0.456640 0.889652i \(-0.349053\pi\)
0.456640 + 0.889652i \(0.349053\pi\)
\(152\) 0 0
\(153\) 7.73532i 0.0505576i
\(154\) 0 0
\(155\) 97.1118i 0.626528i
\(156\) 0 0
\(157\) 139.467i 0.888324i 0.895947 + 0.444162i \(0.146498\pi\)
−0.895947 + 0.444162i \(0.853502\pi\)
\(158\) 0 0
\(159\) 55.0038i 0.345936i
\(160\) 0 0
\(161\) 44.1875 238.390i 0.274456 1.48068i
\(162\) 0 0
\(163\) 39.5024 0.242346 0.121173 0.992631i \(-0.461334\pi\)
0.121173 + 0.992631i \(0.461334\pi\)
\(164\) 0 0
\(165\) −25.9566 −0.157313
\(166\) 0 0
\(167\) 2.79650 0.0167455 0.00837276 0.999965i \(-0.497335\pi\)
0.00837276 + 0.999965i \(0.497335\pi\)
\(168\) 0 0
\(169\) 339.206 2.00714
\(170\) 0 0
\(171\) 67.5221i 0.394866i
\(172\) 0 0
\(173\) −141.902 −0.820241 −0.410120 0.912031i \(-0.634513\pi\)
−0.410120 + 0.912031i \(0.634513\pi\)
\(174\) 0 0
\(175\) 52.7066i 0.301180i
\(176\) 0 0
\(177\) −35.7053 −0.201725
\(178\) 0 0
\(179\) −23.9238 −0.133652 −0.0668262 0.997765i \(-0.521287\pi\)
−0.0668262 + 0.997765i \(0.521287\pi\)
\(180\) 0 0
\(181\) 147.209i 0.813309i −0.913582 0.406655i \(-0.866695\pi\)
0.913582 0.406655i \(-0.133305\pi\)
\(182\) 0 0
\(183\) 80.7818i 0.441430i
\(184\) 0 0
\(185\) 34.7684 0.187937
\(186\) 0 0
\(187\) 17.2806 0.0924096
\(188\) 0 0
\(189\) 54.7743i 0.289811i
\(190\) 0 0
\(191\) 142.182i 0.744407i 0.928151 + 0.372204i \(0.121398\pi\)
−0.928151 + 0.372204i \(0.878602\pi\)
\(192\) 0 0
\(193\) −50.1235 −0.259707 −0.129854 0.991533i \(-0.541451\pi\)
−0.129854 + 0.991533i \(0.541451\pi\)
\(194\) 0 0
\(195\) 87.3103i 0.447745i
\(196\) 0 0
\(197\) −59.1582 −0.300295 −0.150148 0.988664i \(-0.547975\pi\)
−0.150148 + 0.988664i \(0.547975\pi\)
\(198\) 0 0
\(199\) 289.195i 1.45324i 0.687040 + 0.726620i \(0.258910\pi\)
−0.687040 + 0.726620i \(0.741090\pi\)
\(200\) 0 0
\(201\) 56.2754i 0.279977i
\(202\) 0 0
\(203\) 304.927i 1.50211i
\(204\) 0 0
\(205\) 107.965i 0.526660i
\(206\) 0 0
\(207\) 67.8444 + 12.5755i 0.327751 + 0.0607512i
\(208\) 0 0
\(209\) 150.844 0.721740
\(210\) 0 0
\(211\) 54.4739 0.258170 0.129085 0.991634i \(-0.458796\pi\)
0.129085 + 0.991634i \(0.458796\pi\)
\(212\) 0 0
\(213\) −14.2893 −0.0670859
\(214\) 0 0
\(215\) 23.1132 0.107503
\(216\) 0 0
\(217\) 457.806i 2.10971i
\(218\) 0 0
\(219\) 246.525 1.12568
\(220\) 0 0
\(221\) 58.1268i 0.263017i
\(222\) 0 0
\(223\) −303.076 −1.35909 −0.679543 0.733636i \(-0.737822\pi\)
−0.679543 + 0.733636i \(0.737822\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 236.140i 1.04027i 0.854085 + 0.520133i \(0.174118\pi\)
−0.854085 + 0.520133i \(0.825882\pi\)
\(228\) 0 0
\(229\) 417.555i 1.82338i 0.410874 + 0.911692i \(0.365223\pi\)
−0.410874 + 0.911692i \(0.634777\pi\)
\(230\) 0 0
\(231\) −122.365 −0.529719
\(232\) 0 0
\(233\) 387.074 1.66126 0.830631 0.556823i \(-0.187980\pi\)
0.830631 + 0.556823i \(0.187980\pi\)
\(234\) 0 0
\(235\) 118.393i 0.503799i
\(236\) 0 0
\(237\) 181.719i 0.766749i
\(238\) 0 0
\(239\) −181.340 −0.758744 −0.379372 0.925244i \(-0.623860\pi\)
−0.379372 + 0.925244i \(0.623860\pi\)
\(240\) 0 0
\(241\) 245.520i 1.01876i 0.860543 + 0.509378i \(0.170125\pi\)
−0.860543 + 0.509378i \(0.829875\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 138.903i 0.566951i
\(246\) 0 0
\(247\) 507.393i 2.05422i
\(248\) 0 0
\(249\) 195.667i 0.785812i
\(250\) 0 0
\(251\) 157.305i 0.626712i 0.949636 + 0.313356i \(0.101453\pi\)
−0.949636 + 0.313356i \(0.898547\pi\)
\(252\) 0 0
\(253\) −28.0935 + 151.563i −0.111042 + 0.599065i
\(254\) 0 0
\(255\) 9.98625 0.0391618
\(256\) 0 0
\(257\) −225.229 −0.876378 −0.438189 0.898883i \(-0.644380\pi\)
−0.438189 + 0.898883i \(0.644380\pi\)
\(258\) 0 0
\(259\) 163.906 0.632841
\(260\) 0 0
\(261\) −86.7807 −0.332493
\(262\) 0 0
\(263\) 426.803i 1.62283i 0.584473 + 0.811413i \(0.301301\pi\)
−0.584473 + 0.811413i \(0.698699\pi\)
\(264\) 0 0
\(265\) 71.0096 0.267961
\(266\) 0 0
\(267\) 255.398i 0.956548i
\(268\) 0 0
\(269\) −314.327 −1.16850 −0.584252 0.811573i \(-0.698612\pi\)
−0.584252 + 0.811573i \(0.698612\pi\)
\(270\) 0 0
\(271\) −194.568 −0.717962 −0.358981 0.933345i \(-0.616876\pi\)
−0.358981 + 0.933345i \(0.616876\pi\)
\(272\) 0 0
\(273\) 411.600i 1.50769i
\(274\) 0 0
\(275\) 33.5098i 0.121854i
\(276\) 0 0
\(277\) −341.136 −1.23154 −0.615769 0.787927i \(-0.711154\pi\)
−0.615769 + 0.787927i \(0.711154\pi\)
\(278\) 0 0
\(279\) −130.289 −0.466986
\(280\) 0 0
\(281\) 428.279i 1.52413i 0.647503 + 0.762063i \(0.275813\pi\)
−0.647503 + 0.762063i \(0.724187\pi\)
\(282\) 0 0
\(283\) 373.272i 1.31898i 0.751712 + 0.659491i \(0.229228\pi\)
−0.751712 + 0.659491i \(0.770772\pi\)
\(284\) 0 0
\(285\) 87.1707 0.305862
\(286\) 0 0
\(287\) 508.973i 1.77342i
\(288\) 0 0
\(289\) 282.352 0.976995
\(290\) 0 0
\(291\) 43.4927i 0.149459i
\(292\) 0 0
\(293\) 399.057i 1.36197i −0.732297 0.680985i \(-0.761552\pi\)
0.732297 0.680985i \(-0.238448\pi\)
\(294\) 0 0
\(295\) 46.0954i 0.156256i
\(296\) 0 0
\(297\) 34.8244i 0.117254i
\(298\) 0 0
\(299\) −509.815 94.4983i −1.70507 0.316048i
\(300\) 0 0
\(301\) 108.961 0.361996
\(302\) 0 0
\(303\) −236.021 −0.778948
\(304\) 0 0
\(305\) −104.289 −0.341930
\(306\) 0 0
\(307\) 129.417 0.421554 0.210777 0.977534i \(-0.432401\pi\)
0.210777 + 0.977534i \(0.432401\pi\)
\(308\) 0 0
\(309\) 185.872i 0.601529i
\(310\) 0 0
\(311\) −478.099 −1.53730 −0.768648 0.639672i \(-0.779070\pi\)
−0.768648 + 0.639672i \(0.779070\pi\)
\(312\) 0 0
\(313\) 155.225i 0.495926i −0.968770 0.247963i \(-0.920239\pi\)
0.968770 0.247963i \(-0.0797611\pi\)
\(314\) 0 0
\(315\) −70.7133 −0.224487
\(316\) 0 0
\(317\) 212.964 0.671812 0.335906 0.941896i \(-0.390958\pi\)
0.335906 + 0.941896i \(0.390958\pi\)
\(318\) 0 0
\(319\) 193.867i 0.607733i
\(320\) 0 0
\(321\) 242.128i 0.754293i
\(322\) 0 0
\(323\) −58.0339 −0.179672
\(324\) 0 0
\(325\) 112.717 0.346822
\(326\) 0 0
\(327\) 24.9209i 0.0762108i
\(328\) 0 0
\(329\) 558.130i 1.69644i
\(330\) 0 0
\(331\) 18.7734 0.0567172 0.0283586 0.999598i \(-0.490972\pi\)
0.0283586 + 0.999598i \(0.490972\pi\)
\(332\) 0 0
\(333\) 46.6467i 0.140080i
\(334\) 0 0
\(335\) 72.6512 0.216869
\(336\) 0 0
\(337\) 561.997i 1.66765i −0.552031 0.833823i \(-0.686147\pi\)
0.552031 0.833823i \(-0.313853\pi\)
\(338\) 0 0
\(339\) 203.453i 0.600158i
\(340\) 0 0
\(341\) 291.064i 0.853561i
\(342\) 0 0
\(343\) 138.295i 0.403193i
\(344\) 0 0
\(345\) −16.2349 + 87.5867i −0.0470577 + 0.253874i
\(346\) 0 0
\(347\) 51.4163 0.148174 0.0740869 0.997252i \(-0.476396\pi\)
0.0740869 + 0.997252i \(0.476396\pi\)
\(348\) 0 0
\(349\) −248.996 −0.713456 −0.356728 0.934208i \(-0.616108\pi\)
−0.356728 + 0.934208i \(0.616108\pi\)
\(350\) 0 0
\(351\) 117.139 0.333729
\(352\) 0 0
\(353\) −31.6635 −0.0896983 −0.0448491 0.998994i \(-0.514281\pi\)
−0.0448491 + 0.998994i \(0.514281\pi\)
\(354\) 0 0
\(355\) 18.4474i 0.0519645i
\(356\) 0 0
\(357\) 47.0774 0.131869
\(358\) 0 0
\(359\) 151.946i 0.423248i −0.977351 0.211624i \(-0.932125\pi\)
0.977351 0.211624i \(-0.0678752\pi\)
\(360\) 0 0
\(361\) −145.582 −0.403275
\(362\) 0 0
\(363\) −131.781 −0.363033
\(364\) 0 0
\(365\) 318.262i 0.871950i
\(366\) 0 0
\(367\) 717.946i 1.95625i 0.208007 + 0.978127i \(0.433302\pi\)
−0.208007 + 0.978127i \(0.566698\pi\)
\(368\) 0 0
\(369\) 144.851 0.392549
\(370\) 0 0
\(371\) 334.755 0.902304
\(372\) 0 0
\(373\) 581.182i 1.55813i 0.626945 + 0.779064i \(0.284305\pi\)
−0.626945 + 0.779064i \(0.715695\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) 652.111 1.72974
\(378\) 0 0
\(379\) 369.888i 0.975959i 0.872855 + 0.487979i \(0.162266\pi\)
−0.872855 + 0.487979i \(0.837734\pi\)
\(380\) 0 0
\(381\) −358.152 −0.940030
\(382\) 0 0
\(383\) 12.5052i 0.0326508i 0.999867 + 0.0163254i \(0.00519676\pi\)
−0.999867 + 0.0163254i \(0.994803\pi\)
\(384\) 0 0
\(385\) 157.973i 0.410319i
\(386\) 0 0
\(387\) 31.0097i 0.0801284i
\(388\) 0 0
\(389\) 184.277i 0.473720i 0.971544 + 0.236860i \(0.0761182\pi\)
−0.971544 + 0.236860i \(0.923882\pi\)
\(390\) 0 0
\(391\) 10.8084 58.3108i 0.0276429 0.149133i
\(392\) 0 0
\(393\) −61.5113 −0.156517
\(394\) 0 0
\(395\) −234.599 −0.593921
\(396\) 0 0
\(397\) 153.024 0.385451 0.192725 0.981253i \(-0.438267\pi\)
0.192725 + 0.981253i \(0.438267\pi\)
\(398\) 0 0
\(399\) 410.942 1.02993
\(400\) 0 0
\(401\) 499.402i 1.24539i 0.782464 + 0.622696i \(0.213963\pi\)
−0.782464 + 0.622696i \(0.786037\pi\)
\(402\) 0 0
\(403\) 979.054 2.42942
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) −104.208 −0.256039
\(408\) 0 0
\(409\) 575.323 1.40666 0.703329 0.710865i \(-0.251696\pi\)
0.703329 + 0.710865i \(0.251696\pi\)
\(410\) 0 0
\(411\) 63.6180i 0.154788i
\(412\) 0 0
\(413\) 217.304i 0.526159i
\(414\) 0 0
\(415\) 252.605 0.608688
\(416\) 0 0
\(417\) −134.304 −0.322072
\(418\) 0 0
\(419\) 39.3004i 0.0937956i −0.998900 0.0468978i \(-0.985066\pi\)
0.998900 0.0468978i \(-0.0149335\pi\)
\(420\) 0 0
\(421\) 140.762i 0.334352i −0.985927 0.167176i \(-0.946535\pi\)
0.985927 0.167176i \(-0.0534648\pi\)
\(422\) 0 0
\(423\) −158.841 −0.375510
\(424\) 0 0
\(425\) 12.8922i 0.0303346i
\(426\) 0 0
\(427\) −491.640 −1.15138
\(428\) 0 0
\(429\) 261.687i 0.609993i
\(430\) 0 0
\(431\) 45.0746i 0.104582i 0.998632 + 0.0522908i \(0.0166523\pi\)
−0.998632 + 0.0522908i \(0.983348\pi\)
\(432\) 0 0
\(433\) 724.380i 1.67293i −0.548017 0.836467i \(-0.684617\pi\)
0.548017 0.836467i \(-0.315383\pi\)
\(434\) 0 0
\(435\) 112.033i 0.257548i
\(436\) 0 0
\(437\) 94.3473 509.000i 0.215898 1.16476i
\(438\) 0 0
\(439\) 569.280 1.29677 0.648383 0.761314i \(-0.275446\pi\)
0.648383 + 0.761314i \(0.275446\pi\)
\(440\) 0 0
\(441\) −186.358 −0.422581
\(442\) 0 0
\(443\) 314.822 0.710658 0.355329 0.934741i \(-0.384369\pi\)
0.355329 + 0.934741i \(0.384369\pi\)
\(444\) 0 0
\(445\) −329.718 −0.740939
\(446\) 0 0
\(447\) 134.731i 0.301412i
\(448\) 0 0
\(449\) 115.216 0.256606 0.128303 0.991735i \(-0.459047\pi\)
0.128303 + 0.991735i \(0.459047\pi\)
\(450\) 0 0
\(451\) 323.595i 0.717505i
\(452\) 0 0
\(453\) −238.859 −0.527282
\(454\) 0 0
\(455\) 531.373 1.16785
\(456\) 0 0
\(457\) 403.276i 0.882442i −0.897399 0.441221i \(-0.854546\pi\)
0.897399 0.441221i \(-0.145454\pi\)
\(458\) 0 0
\(459\) 13.3980i 0.0291895i
\(460\) 0 0
\(461\) −858.437 −1.86212 −0.931060 0.364867i \(-0.881115\pi\)
−0.931060 + 0.364867i \(0.881115\pi\)
\(462\) 0 0
\(463\) 342.521 0.739786 0.369893 0.929074i \(-0.379394\pi\)
0.369893 + 0.929074i \(0.379394\pi\)
\(464\) 0 0
\(465\) 168.203i 0.361726i
\(466\) 0 0
\(467\) 498.514i 1.06748i −0.845648 0.533741i \(-0.820786\pi\)
0.845648 0.533741i \(-0.179214\pi\)
\(468\) 0 0
\(469\) 342.494 0.730264
\(470\) 0 0
\(471\) 241.564i 0.512874i
\(472\) 0 0
\(473\) −69.2752 −0.146459
\(474\) 0 0
\(475\) 112.537i 0.236920i
\(476\) 0 0
\(477\) 95.2693i 0.199726i
\(478\) 0 0
\(479\) 641.001i 1.33821i −0.743170 0.669103i \(-0.766678\pi\)
0.743170 0.669103i \(-0.233322\pi\)
\(480\) 0 0
\(481\) 350.525i 0.728742i
\(482\) 0 0
\(483\) −76.5349 + 412.903i −0.158457 + 0.854871i
\(484\) 0 0
\(485\) −56.1488 −0.115771
\(486\) 0 0
\(487\) −337.691 −0.693410 −0.346705 0.937974i \(-0.612699\pi\)
−0.346705 + 0.937974i \(0.612699\pi\)
\(488\) 0 0
\(489\) −68.4202 −0.139919
\(490\) 0 0
\(491\) 371.537 0.756694 0.378347 0.925664i \(-0.376493\pi\)
0.378347 + 0.925664i \(0.376493\pi\)
\(492\) 0 0
\(493\) 74.5862i 0.151290i
\(494\) 0 0
\(495\) 44.9581 0.0908245
\(496\) 0 0
\(497\) 86.9651i 0.174980i
\(498\) 0 0
\(499\) 74.5902 0.149479 0.0747397 0.997203i \(-0.476187\pi\)
0.0747397 + 0.997203i \(0.476187\pi\)
\(500\) 0 0
\(501\) −4.84368 −0.00966803
\(502\) 0 0
\(503\) 265.676i 0.528182i −0.964498 0.264091i \(-0.914928\pi\)
0.964498 0.264091i \(-0.0850719\pi\)
\(504\) 0 0
\(505\) 304.702i 0.603370i
\(506\) 0 0
\(507\) −587.522 −1.15882
\(508\) 0 0
\(509\) 112.469 0.220960 0.110480 0.993878i \(-0.464761\pi\)
0.110480 + 0.993878i \(0.464761\pi\)
\(510\) 0 0
\(511\) 1500.36i 2.93612i
\(512\) 0 0
\(513\) 116.952i 0.227976i
\(514\) 0 0
\(515\) 239.960 0.465942
\(516\) 0 0
\(517\) 354.848i 0.686360i
\(518\) 0 0
\(519\) 245.781 0.473566
\(520\) 0 0
\(521\) 321.945i 0.617937i 0.951072 + 0.308968i \(0.0999837\pi\)
−0.951072 + 0.308968i \(0.900016\pi\)
\(522\) 0 0
\(523\) 618.055i 1.18175i 0.806763 + 0.590875i \(0.201217\pi\)
−0.806763 + 0.590875i \(0.798783\pi\)
\(524\) 0 0
\(525\) 91.2905i 0.173887i
\(526\) 0 0
\(527\) 111.981i 0.212487i
\(528\) 0 0
\(529\) 493.857 + 189.595i 0.933567 + 0.358403i
\(530\) 0 0
\(531\) 61.8435 0.116466
\(532\) 0 0
\(533\) −1088.48 −2.04217
\(534\) 0 0
\(535\) 312.586 0.584273
\(536\) 0 0
\(537\) 41.4372 0.0771642
\(538\) 0 0
\(539\) 416.322i 0.772396i
\(540\) 0 0
\(541\) −257.136 −0.475297 −0.237649 0.971351i \(-0.576377\pi\)
−0.237649 + 0.971351i \(0.576377\pi\)
\(542\) 0 0
\(543\) 254.973i 0.469564i
\(544\) 0 0
\(545\) −32.1728 −0.0590327
\(546\) 0 0
\(547\) 848.386 1.55098 0.775490 0.631360i \(-0.217503\pi\)
0.775490 + 0.631360i \(0.217503\pi\)
\(548\) 0 0
\(549\) 139.918i 0.254860i
\(550\) 0 0
\(551\) 651.068i 1.18161i
\(552\) 0 0
\(553\) −1105.95 −1.99991
\(554\) 0 0
\(555\) −60.2206 −0.108506
\(556\) 0 0
\(557\) 480.111i 0.861958i 0.902362 + 0.430979i \(0.141832\pi\)
−0.902362 + 0.430979i \(0.858168\pi\)
\(558\) 0 0
\(559\) 233.021i 0.416854i
\(560\) 0 0
\(561\) −29.9309 −0.0533527
\(562\) 0 0
\(563\) 201.278i 0.357510i −0.983894 0.178755i \(-0.942793\pi\)
0.983894 0.178755i \(-0.0572069\pi\)
\(564\) 0 0
\(565\) −262.657 −0.464880
\(566\) 0 0
\(567\) 94.8718i 0.167322i
\(568\) 0 0
\(569\) 724.981i 1.27413i −0.770810 0.637066i \(-0.780148\pi\)
0.770810 0.637066i \(-0.219852\pi\)
\(570\) 0 0
\(571\) 208.655i 0.365421i −0.983167 0.182710i \(-0.941513\pi\)
0.983167 0.182710i \(-0.0584871\pi\)
\(572\) 0 0
\(573\) 246.266i 0.429784i
\(574\) 0 0
\(575\) −113.074 20.9592i −0.196650 0.0364507i
\(576\) 0 0
\(577\) −128.435 −0.222590 −0.111295 0.993787i \(-0.535500\pi\)
−0.111295 + 0.993787i \(0.535500\pi\)
\(578\) 0 0
\(579\) 86.8164 0.149942
\(580\) 0 0
\(581\) 1190.84 2.04963
\(582\) 0 0
\(583\) −212.831 −0.365061
\(584\) 0 0
\(585\) 151.226i 0.258506i
\(586\) 0 0
\(587\) 781.503 1.33135 0.665676 0.746241i \(-0.268143\pi\)
0.665676 + 0.746241i \(0.268143\pi\)
\(588\) 0 0
\(589\) 977.489i 1.65957i
\(590\) 0 0
\(591\) 102.465 0.173376
\(592\) 0 0
\(593\) −453.876 −0.765389 −0.382695 0.923875i \(-0.625004\pi\)
−0.382695 + 0.923875i \(0.625004\pi\)
\(594\) 0 0
\(595\) 60.7766i 0.102146i
\(596\) 0 0
\(597\) 500.900i 0.839029i
\(598\) 0 0
\(599\) 901.941 1.50574 0.752872 0.658167i \(-0.228668\pi\)
0.752872 + 0.658167i \(0.228668\pi\)
\(600\) 0 0
\(601\) −366.374 −0.609607 −0.304803 0.952415i \(-0.598591\pi\)
−0.304803 + 0.952415i \(0.598591\pi\)
\(602\) 0 0
\(603\) 97.4718i 0.161645i
\(604\) 0 0
\(605\) 170.128i 0.281204i
\(606\) 0 0
\(607\) −684.073 −1.12697 −0.563487 0.826125i \(-0.690540\pi\)
−0.563487 + 0.826125i \(0.690540\pi\)
\(608\) 0 0
\(609\) 528.150i 0.867241i
\(610\) 0 0
\(611\) 1193.60 1.95352
\(612\) 0 0
\(613\) 259.950i 0.424062i 0.977263 + 0.212031i \(0.0680078\pi\)
−0.977263 + 0.212031i \(0.931992\pi\)
\(614\) 0 0
\(615\) 187.002i 0.304067i
\(616\) 0 0
\(617\) 1010.22i 1.63731i −0.574285 0.818656i \(-0.694720\pi\)
0.574285 0.818656i \(-0.305280\pi\)
\(618\) 0 0
\(619\) 226.888i 0.366540i 0.983063 + 0.183270i \(0.0586682\pi\)
−0.983063 + 0.183270i \(0.941332\pi\)
\(620\) 0 0
\(621\) −117.510 21.7814i −0.189227 0.0350748i
\(622\) 0 0
\(623\) −1554.36 −2.49496
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −261.269 −0.416697
\(628\) 0 0
\(629\) 40.0919 0.0637390
\(630\) 0 0
\(631\) 885.121i 1.40273i −0.712803 0.701364i \(-0.752575\pi\)
0.712803 0.701364i \(-0.247425\pi\)
\(632\) 0 0
\(633\) −94.3516 −0.149055
\(634\) 0 0
\(635\) 462.372i 0.728144i
\(636\) 0 0
\(637\) 1400.38 2.19840
\(638\) 0 0
\(639\) 24.7498 0.0387321
\(640\) 0 0
\(641\) 533.910i 0.832934i 0.909151 + 0.416467i \(0.136732\pi\)
−0.909151 + 0.416467i \(0.863268\pi\)
\(642\) 0 0
\(643\) 67.0255i 0.104239i 0.998641 + 0.0521194i \(0.0165976\pi\)
−0.998641 + 0.0521194i \(0.983402\pi\)
\(644\) 0 0
\(645\) −40.0333 −0.0620672
\(646\) 0 0
\(647\) −679.146 −1.04968 −0.524842 0.851199i \(-0.675876\pi\)
−0.524842 + 0.851199i \(0.675876\pi\)
\(648\) 0 0
\(649\) 138.158i 0.212878i
\(650\) 0 0
\(651\) 792.944i 1.21804i
\(652\) 0 0
\(653\) 132.816 0.203394 0.101697 0.994815i \(-0.467573\pi\)
0.101697 + 0.994815i \(0.467573\pi\)
\(654\) 0 0
\(655\) 79.4108i 0.121238i
\(656\) 0 0
\(657\) −426.993 −0.649913
\(658\) 0 0
\(659\) 1152.93i 1.74952i 0.484555 + 0.874761i \(0.338982\pi\)
−0.484555 + 0.874761i \(0.661018\pi\)
\(660\) 0 0
\(661\) 903.470i 1.36682i −0.730033 0.683412i \(-0.760495\pi\)
0.730033 0.683412i \(-0.239505\pi\)
\(662\) 0 0
\(663\) 100.679i 0.151853i
\(664\) 0 0
\(665\) 530.524i 0.797780i
\(666\) 0 0
\(667\) −654.175 121.257i −0.980773 0.181794i
\(668\) 0 0
\(669\) 524.943 0.784668
\(670\) 0 0
\(671\) 312.575 0.465835
\(672\) 0 0
\(673\) −1209.10 −1.79658 −0.898289 0.439406i \(-0.855189\pi\)
−0.898289 + 0.439406i \(0.855189\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 930.759i 1.37483i −0.726266 0.687414i \(-0.758746\pi\)
0.726266 0.687414i \(-0.241254\pi\)
\(678\) 0 0
\(679\) −264.698 −0.389835
\(680\) 0 0
\(681\) 409.007i 0.600598i
\(682\) 0 0
\(683\) −1028.45 −1.50578 −0.752892 0.658144i \(-0.771342\pi\)
−0.752892 + 0.658144i \(0.771342\pi\)
\(684\) 0 0
\(685\) 82.1305 0.119899
\(686\) 0 0
\(687\) 723.226i 1.05273i
\(688\) 0 0
\(689\) 715.899i 1.03904i
\(690\) 0 0
\(691\) −825.577 −1.19476 −0.597378 0.801960i \(-0.703791\pi\)
−0.597378 + 0.801960i \(0.703791\pi\)
\(692\) 0 0
\(693\) 211.943 0.305833
\(694\) 0 0
\(695\) 173.386i 0.249476i
\(696\) 0 0
\(697\) 124.496i 0.178617i
\(698\) 0 0
\(699\) −670.432 −0.959131
\(700\) 0 0
\(701\) 974.830i 1.39063i −0.718706 0.695314i \(-0.755265\pi\)
0.718706 0.695314i \(-0.244735\pi\)
\(702\) 0 0
\(703\) 349.965 0.497816
\(704\) 0 0
\(705\) 205.062i 0.290869i
\(706\) 0 0
\(707\) 1436.43i 2.03173i
\(708\) 0 0
\(709\) 1214.02i 1.71230i 0.516730 + 0.856148i \(0.327149\pi\)
−0.516730 + 0.856148i \(0.672851\pi\)
\(710\) 0 0
\(711\) 314.747i 0.442683i
\(712\) 0 0
\(713\) −982.154 182.050i −1.37749 0.255330i
\(714\) 0 0
\(715\) −337.837 −0.472499
\(716\) 0 0
\(717\) 314.090 0.438061
\(718\) 0 0
\(719\) 592.289 0.823767 0.411884 0.911236i \(-0.364871\pi\)
0.411884 + 0.911236i \(0.364871\pi\)
\(720\) 0 0
\(721\) 1131.23 1.56897
\(722\) 0 0
\(723\) 425.254i 0.588180i
\(724\) 0 0
\(725\) 144.634 0.199496
\(726\) 0 0
\(727\) 81.9727i 0.112755i 0.998410 + 0.0563774i \(0.0179550\pi\)
−0.998410 + 0.0563774i \(0.982045\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 26.6522 0.0364599
\(732\) 0 0
\(733\) 1015.21i 1.38500i 0.721416 + 0.692502i \(0.243491\pi\)
−0.721416 + 0.692502i \(0.756509\pi\)
\(734\) 0 0
\(735\) 240.587i 0.327330i
\(736\) 0 0
\(737\) −217.751 −0.295456
\(738\) 0 0
\(739\) −318.591 −0.431111 −0.215555 0.976492i \(-0.569156\pi\)
−0.215555 + 0.976492i \(0.569156\pi\)
\(740\) 0 0
\(741\) 878.831i 1.18601i
\(742\) 0 0
\(743\) 89.8810i 0.120970i −0.998169 0.0604852i \(-0.980735\pi\)
0.998169 0.0604852i \(-0.0192648\pi\)
\(744\) 0 0
\(745\) −173.937 −0.233473
\(746\) 0 0
\(747\) 338.906i 0.453689i
\(748\) 0 0
\(749\) 1473.60 1.96742
\(750\) 0 0
\(751\) 178.081i 0.237125i −0.992947 0.118562i \(-0.962171\pi\)
0.992947 0.118562i \(-0.0378285\pi\)
\(752\) 0 0
\(753\) 272.460i 0.361832i
\(754\) 0 0
\(755\) 308.365i 0.408431i
\(756\) 0 0
\(757\) 184.101i 0.243198i 0.992579 + 0.121599i \(0.0388022\pi\)
−0.992579 + 0.121599i \(0.961198\pi\)
\(758\) 0 0
\(759\) 48.6594 262.516i 0.0641099 0.345870i
\(760\) 0 0
\(761\) −423.841 −0.556952 −0.278476 0.960443i \(-0.589829\pi\)
−0.278476 + 0.960443i \(0.589829\pi\)
\(762\) 0 0
\(763\) −151.670 −0.198781
\(764\) 0 0
\(765\) −17.2967 −0.0226101
\(766\) 0 0
\(767\) −464.721 −0.605894
\(768\) 0 0
\(769\) 167.137i 0.217343i 0.994078 + 0.108672i \(0.0346597\pi\)
−0.994078 + 0.108672i \(0.965340\pi\)
\(770\) 0 0
\(771\) 390.108 0.505977
\(772\) 0 0
\(773\) 90.4285i 0.116984i −0.998288 0.0584919i \(-0.981371\pi\)
0.998288 0.0584919i \(-0.0186292\pi\)
\(774\) 0 0
\(775\) 217.149 0.280192
\(776\) 0 0
\(777\) −283.893 −0.365371
\(778\) 0 0
\(779\) 1086.74i 1.39504i
\(780\) 0 0
\(781\) 55.2907i 0.0707948i
\(782\) 0 0
\(783\) 150.309 0.191965
\(784\) 0 0
\(785\) 311.857 0.397270
\(786\) 0 0
\(787\) 751.440i 0.954816i 0.878682 + 0.477408i \(0.158424\pi\)
−0.878682 + 0.477408i \(0.841576\pi\)
\(788\) 0 0
\(789\) 739.245i 0.936939i
\(790\) 0 0
\(791\) −1238.22 −1.56539
\(792\) 0 0
\(793\) 1051.41i 1.32586i
\(794\) 0 0
\(795\) −122.992 −0.154707
\(796\) 0 0
\(797\) 242.077i 0.303735i −0.988401 0.151868i \(-0.951471\pi\)
0.988401 0.151868i \(-0.0485287\pi\)
\(798\) 0 0
\(799\)