Properties

Label 2760.3.g.a.2161.18
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 2161.18
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.31

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} +5.37611i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} +5.37611i q^{7} +3.00000 q^{9} -2.93010i q^{11} -1.21554 q^{13} +3.87298i q^{15} -0.710115i q^{17} -16.4246i q^{19} -9.31170i q^{21} +(0.135784 + 22.9996i) q^{23} -5.00000 q^{25} -5.19615 q^{27} -2.03426 q^{29} +27.2298 q^{31} +5.07509i q^{33} +12.0214 q^{35} +11.9372i q^{37} +2.10538 q^{39} -25.8264 q^{41} +6.15738i q^{43} -6.70820i q^{45} -50.0760 q^{47} +20.0974 q^{49} +1.22996i q^{51} -93.8852i q^{53} -6.55191 q^{55} +28.4483i q^{57} +41.7448 q^{59} +9.92596i q^{61} +16.1283i q^{63} +2.71803i q^{65} -34.0783i q^{67} +(-0.235185 - 39.8365i) q^{69} +135.416 q^{71} -86.6398 q^{73} +8.66025 q^{75} +15.7526 q^{77} +30.9645i q^{79} +9.00000 q^{81} +90.3663i q^{83} -1.58787 q^{85} +3.52344 q^{87} +134.517i q^{89} -6.53487i q^{91} -47.1634 q^{93} -36.7266 q^{95} -95.7494i q^{97} -8.79031i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 5.37611i 0.768016i 0.923330 + 0.384008i \(0.125457\pi\)
−0.923330 + 0.384008i \(0.874543\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 2.93010i 0.266373i −0.991091 0.133186i \(-0.957479\pi\)
0.991091 0.133186i \(-0.0425209\pi\)
\(12\) 0 0
\(13\) −1.21554 −0.0935030 −0.0467515 0.998907i \(-0.514887\pi\)
−0.0467515 + 0.998907i \(0.514887\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 0.710115i 0.0417715i −0.999782 0.0208857i \(-0.993351\pi\)
0.999782 0.0208857i \(-0.00664862\pi\)
\(18\) 0 0
\(19\) 16.4246i 0.864455i −0.901765 0.432227i \(-0.857728\pi\)
0.901765 0.432227i \(-0.142272\pi\)
\(20\) 0 0
\(21\) 9.31170i 0.443414i
\(22\) 0 0
\(23\) 0.135784 + 22.9996i 0.00590366 + 0.999983i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) −2.03426 −0.0701468 −0.0350734 0.999385i \(-0.511166\pi\)
−0.0350734 + 0.999385i \(0.511166\pi\)
\(30\) 0 0
\(31\) 27.2298 0.878380 0.439190 0.898394i \(-0.355265\pi\)
0.439190 + 0.898394i \(0.355265\pi\)
\(32\) 0 0
\(33\) 5.07509i 0.153790i
\(34\) 0 0
\(35\) 12.0214 0.343467
\(36\) 0 0
\(37\) 11.9372i 0.322628i 0.986903 + 0.161314i \(0.0515732\pi\)
−0.986903 + 0.161314i \(0.948427\pi\)
\(38\) 0 0
\(39\) 2.10538 0.0539840
\(40\) 0 0
\(41\) −25.8264 −0.629911 −0.314956 0.949106i \(-0.601990\pi\)
−0.314956 + 0.949106i \(0.601990\pi\)
\(42\) 0 0
\(43\) 6.15738i 0.143195i 0.997434 + 0.0715975i \(0.0228097\pi\)
−0.997434 + 0.0715975i \(0.977190\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) −50.0760 −1.06545 −0.532724 0.846289i \(-0.678831\pi\)
−0.532724 + 0.846289i \(0.678831\pi\)
\(48\) 0 0
\(49\) 20.0974 0.410151
\(50\) 0 0
\(51\) 1.22996i 0.0241168i
\(52\) 0 0
\(53\) 93.8852i 1.77142i −0.464240 0.885709i \(-0.653673\pi\)
0.464240 0.885709i \(-0.346327\pi\)
\(54\) 0 0
\(55\) −6.55191 −0.119126
\(56\) 0 0
\(57\) 28.4483i 0.499093i
\(58\) 0 0
\(59\) 41.7448 0.707539 0.353770 0.935333i \(-0.384900\pi\)
0.353770 + 0.935333i \(0.384900\pi\)
\(60\) 0 0
\(61\) 9.92596i 0.162721i 0.996685 + 0.0813604i \(0.0259265\pi\)
−0.996685 + 0.0813604i \(0.974074\pi\)
\(62\) 0 0
\(63\) 16.1283i 0.256005i
\(64\) 0 0
\(65\) 2.71803i 0.0418158i
\(66\) 0 0
\(67\) 34.0783i 0.508631i −0.967121 0.254316i \(-0.918150\pi\)
0.967121 0.254316i \(-0.0818502\pi\)
\(68\) 0 0
\(69\) −0.235185 39.8365i −0.00340848 0.577340i
\(70\) 0 0
\(71\) 135.416 1.90727 0.953636 0.300962i \(-0.0973078\pi\)
0.953636 + 0.300962i \(0.0973078\pi\)
\(72\) 0 0
\(73\) −86.6398 −1.18685 −0.593423 0.804891i \(-0.702224\pi\)
−0.593423 + 0.804891i \(0.702224\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) 15.7526 0.204579
\(78\) 0 0
\(79\) 30.9645i 0.391956i 0.980608 + 0.195978i \(0.0627881\pi\)
−0.980608 + 0.195978i \(0.937212\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 90.3663i 1.08875i 0.838842 + 0.544375i \(0.183233\pi\)
−0.838842 + 0.544375i \(0.816767\pi\)
\(84\) 0 0
\(85\) −1.58787 −0.0186808
\(86\) 0 0
\(87\) 3.52344 0.0404993
\(88\) 0 0
\(89\) 134.517i 1.51143i 0.654901 + 0.755715i \(0.272710\pi\)
−0.654901 + 0.755715i \(0.727290\pi\)
\(90\) 0 0
\(91\) 6.53487i 0.0718118i
\(92\) 0 0
\(93\) −47.1634 −0.507133
\(94\) 0 0
\(95\) −36.7266 −0.386596
\(96\) 0 0
\(97\) 95.7494i 0.987108i −0.869715 0.493554i \(-0.835698\pi\)
0.869715 0.493554i \(-0.164302\pi\)
\(98\) 0 0
\(99\) 8.79031i 0.0887910i
\(100\) 0 0
\(101\) −73.2934 −0.725677 −0.362839 0.931852i \(-0.618192\pi\)
−0.362839 + 0.931852i \(0.618192\pi\)
\(102\) 0 0
\(103\) 15.6450i 0.151893i −0.997112 0.0759465i \(-0.975802\pi\)
0.997112 0.0759465i \(-0.0241978\pi\)
\(104\) 0 0
\(105\) −20.8216 −0.198301
\(106\) 0 0
\(107\) 95.4436i 0.891996i 0.895034 + 0.445998i \(0.147151\pi\)
−0.895034 + 0.445998i \(0.852849\pi\)
\(108\) 0 0
\(109\) 143.437i 1.31594i −0.753044 0.657970i \(-0.771415\pi\)
0.753044 0.657970i \(-0.228585\pi\)
\(110\) 0 0
\(111\) 20.6759i 0.186269i
\(112\) 0 0
\(113\) 198.490i 1.75655i 0.478159 + 0.878273i \(0.341304\pi\)
−0.478159 + 0.878273i \(0.658696\pi\)
\(114\) 0 0
\(115\) 51.4287 0.303622i 0.447206 0.00264020i
\(116\) 0 0
\(117\) −3.64662 −0.0311677
\(118\) 0 0
\(119\) 3.81766 0.0320812
\(120\) 0 0
\(121\) 112.415 0.929045
\(122\) 0 0
\(123\) 44.7326 0.363679
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −17.2829 −0.136086 −0.0680431 0.997682i \(-0.521676\pi\)
−0.0680431 + 0.997682i \(0.521676\pi\)
\(128\) 0 0
\(129\) 10.6649i 0.0826737i
\(130\) 0 0
\(131\) −123.355 −0.941642 −0.470821 0.882229i \(-0.656042\pi\)
−0.470821 + 0.882229i \(0.656042\pi\)
\(132\) 0 0
\(133\) 88.3007 0.663915
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 98.3089i 0.717583i −0.933418 0.358792i \(-0.883189\pi\)
0.933418 0.358792i \(-0.116811\pi\)
\(138\) 0 0
\(139\) 254.990 1.83446 0.917229 0.398359i \(-0.130420\pi\)
0.917229 + 0.398359i \(0.130420\pi\)
\(140\) 0 0
\(141\) 86.7342 0.615136
\(142\) 0 0
\(143\) 3.56165i 0.0249067i
\(144\) 0 0
\(145\) 4.54874i 0.0313706i
\(146\) 0 0
\(147\) −34.8097 −0.236801
\(148\) 0 0
\(149\) 53.5300i 0.359261i −0.983734 0.179631i \(-0.942510\pi\)
0.983734 0.179631i \(-0.0574903\pi\)
\(150\) 0 0
\(151\) 5.13530 0.0340086 0.0170043 0.999855i \(-0.494587\pi\)
0.0170043 + 0.999855i \(0.494587\pi\)
\(152\) 0 0
\(153\) 2.13035i 0.0139238i
\(154\) 0 0
\(155\) 60.8877i 0.392824i
\(156\) 0 0
\(157\) 144.585i 0.920922i 0.887680 + 0.460461i \(0.152316\pi\)
−0.887680 + 0.460461i \(0.847684\pi\)
\(158\) 0 0
\(159\) 162.614i 1.02273i
\(160\) 0 0
\(161\) −123.648 + 0.729991i −0.768003 + 0.00453410i
\(162\) 0 0
\(163\) 182.630 1.12043 0.560215 0.828347i \(-0.310718\pi\)
0.560215 + 0.828347i \(0.310718\pi\)
\(164\) 0 0
\(165\) 11.3482 0.0687772
\(166\) 0 0
\(167\) 164.842 0.987079 0.493540 0.869723i \(-0.335703\pi\)
0.493540 + 0.869723i \(0.335703\pi\)
\(168\) 0 0
\(169\) −167.522 −0.991257
\(170\) 0 0
\(171\) 49.2739i 0.288152i
\(172\) 0 0
\(173\) 296.615 1.71454 0.857270 0.514867i \(-0.172159\pi\)
0.857270 + 0.514867i \(0.172159\pi\)
\(174\) 0 0
\(175\) 26.8806i 0.153603i
\(176\) 0 0
\(177\) −72.3042 −0.408498
\(178\) 0 0
\(179\) 219.030 1.22363 0.611816 0.791000i \(-0.290439\pi\)
0.611816 + 0.791000i \(0.290439\pi\)
\(180\) 0 0
\(181\) 348.203i 1.92377i −0.273448 0.961887i \(-0.588164\pi\)
0.273448 0.961887i \(-0.411836\pi\)
\(182\) 0 0
\(183\) 17.1923i 0.0939468i
\(184\) 0 0
\(185\) 26.6925 0.144284
\(186\) 0 0
\(187\) −2.08071 −0.0111268
\(188\) 0 0
\(189\) 27.9351i 0.147805i
\(190\) 0 0
\(191\) 67.7395i 0.354657i 0.984152 + 0.177329i \(0.0567455\pi\)
−0.984152 + 0.177329i \(0.943254\pi\)
\(192\) 0 0
\(193\) 139.053 0.720484 0.360242 0.932859i \(-0.382694\pi\)
0.360242 + 0.932859i \(0.382694\pi\)
\(194\) 0 0
\(195\) 4.70776i 0.0241424i
\(196\) 0 0
\(197\) −170.784 −0.866922 −0.433461 0.901172i \(-0.642708\pi\)
−0.433461 + 0.901172i \(0.642708\pi\)
\(198\) 0 0
\(199\) 167.126i 0.839832i −0.907563 0.419916i \(-0.862060\pi\)
0.907563 0.419916i \(-0.137940\pi\)
\(200\) 0 0
\(201\) 59.0253i 0.293658i
\(202\) 0 0
\(203\) 10.9364i 0.0538739i
\(204\) 0 0
\(205\) 57.7495i 0.281705i
\(206\) 0 0
\(207\) 0.407352 + 68.9988i 0.00196789 + 0.333328i
\(208\) 0 0
\(209\) −48.1259 −0.230267
\(210\) 0 0
\(211\) 227.020 1.07592 0.537961 0.842970i \(-0.319195\pi\)
0.537961 + 0.842970i \(0.319195\pi\)
\(212\) 0 0
\(213\) −234.548 −1.10116
\(214\) 0 0
\(215\) 13.7683 0.0640387
\(216\) 0 0
\(217\) 146.390i 0.674610i
\(218\) 0 0
\(219\) 150.065 0.685226
\(220\) 0 0
\(221\) 0.863172i 0.00390576i
\(222\) 0 0
\(223\) 283.038 1.26923 0.634615 0.772828i \(-0.281159\pi\)
0.634615 + 0.772828i \(0.281159\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 134.311i 0.591679i −0.955238 0.295840i \(-0.904401\pi\)
0.955238 0.295840i \(-0.0955994\pi\)
\(228\) 0 0
\(229\) 51.1600i 0.223406i 0.993742 + 0.111703i \(0.0356305\pi\)
−0.993742 + 0.111703i \(0.964369\pi\)
\(230\) 0 0
\(231\) −27.2842 −0.118114
\(232\) 0 0
\(233\) 151.864 0.651776 0.325888 0.945408i \(-0.394337\pi\)
0.325888 + 0.945408i \(0.394337\pi\)
\(234\) 0 0
\(235\) 111.973i 0.476482i
\(236\) 0 0
\(237\) 53.6321i 0.226296i
\(238\) 0 0
\(239\) 296.004 1.23851 0.619256 0.785189i \(-0.287434\pi\)
0.619256 + 0.785189i \(0.287434\pi\)
\(240\) 0 0
\(241\) 123.298i 0.511610i 0.966728 + 0.255805i \(0.0823405\pi\)
−0.966728 + 0.255805i \(0.917660\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 44.9392i 0.183425i
\(246\) 0 0
\(247\) 19.9648i 0.0808291i
\(248\) 0 0
\(249\) 156.519i 0.628590i
\(250\) 0 0
\(251\) 359.681i 1.43299i −0.697591 0.716496i \(-0.745745\pi\)
0.697591 0.716496i \(-0.254255\pi\)
\(252\) 0 0
\(253\) 67.3912 0.397861i 0.266368 0.00157257i
\(254\) 0 0
\(255\) 2.75026 0.0107853
\(256\) 0 0
\(257\) 266.271 1.03607 0.518037 0.855358i \(-0.326663\pi\)
0.518037 + 0.855358i \(0.326663\pi\)
\(258\) 0 0
\(259\) −64.1759 −0.247784
\(260\) 0 0
\(261\) −6.10277 −0.0233823
\(262\) 0 0
\(263\) 169.219i 0.643420i −0.946838 0.321710i \(-0.895742\pi\)
0.946838 0.321710i \(-0.104258\pi\)
\(264\) 0 0
\(265\) −209.934 −0.792202
\(266\) 0 0
\(267\) 232.991i 0.872624i
\(268\) 0 0
\(269\) 166.876 0.620357 0.310179 0.950678i \(-0.399611\pi\)
0.310179 + 0.950678i \(0.399611\pi\)
\(270\) 0 0
\(271\) 243.321 0.897863 0.448932 0.893566i \(-0.351805\pi\)
0.448932 + 0.893566i \(0.351805\pi\)
\(272\) 0 0
\(273\) 11.3187i 0.0414606i
\(274\) 0 0
\(275\) 14.6505i 0.0532746i
\(276\) 0 0
\(277\) 513.948 1.85541 0.927704 0.373316i \(-0.121779\pi\)
0.927704 + 0.373316i \(0.121779\pi\)
\(278\) 0 0
\(279\) 81.6894 0.292793
\(280\) 0 0
\(281\) 314.663i 1.11980i 0.828561 + 0.559898i \(0.189160\pi\)
−0.828561 + 0.559898i \(0.810840\pi\)
\(282\) 0 0
\(283\) 138.123i 0.488065i 0.969767 + 0.244033i \(0.0784704\pi\)
−0.969767 + 0.244033i \(0.921530\pi\)
\(284\) 0 0
\(285\) 63.6123 0.223201
\(286\) 0 0
\(287\) 138.845i 0.483782i
\(288\) 0 0
\(289\) 288.496 0.998255
\(290\) 0 0
\(291\) 165.843i 0.569907i
\(292\) 0 0
\(293\) 368.443i 1.25748i 0.777614 + 0.628742i \(0.216430\pi\)
−0.777614 + 0.628742i \(0.783570\pi\)
\(294\) 0 0
\(295\) 93.3443i 0.316421i
\(296\) 0 0
\(297\) 15.2253i 0.0512635i
\(298\) 0 0
\(299\) −0.165051 27.9569i −0.000552010 0.0935014i
\(300\) 0 0
\(301\) −33.1028 −0.109976
\(302\) 0 0
\(303\) 126.948 0.418970
\(304\) 0 0
\(305\) 22.1951 0.0727709
\(306\) 0 0
\(307\) 454.749 1.48127 0.740633 0.671909i \(-0.234526\pi\)
0.740633 + 0.671909i \(0.234526\pi\)
\(308\) 0 0
\(309\) 27.0979i 0.0876955i
\(310\) 0 0
\(311\) −330.174 −1.06165 −0.530826 0.847481i \(-0.678118\pi\)
−0.530826 + 0.847481i \(0.678118\pi\)
\(312\) 0 0
\(313\) 405.397i 1.29520i 0.761982 + 0.647599i \(0.224227\pi\)
−0.761982 + 0.647599i \(0.775773\pi\)
\(314\) 0 0
\(315\) 36.0641 0.114489
\(316\) 0 0
\(317\) −545.175 −1.71979 −0.859897 0.510467i \(-0.829472\pi\)
−0.859897 + 0.510467i \(0.829472\pi\)
\(318\) 0 0
\(319\) 5.96058i 0.0186852i
\(320\) 0 0
\(321\) 165.313i 0.514994i
\(322\) 0 0
\(323\) −11.6634 −0.0361095
\(324\) 0 0
\(325\) 6.07769 0.0187006
\(326\) 0 0
\(327\) 248.441i 0.759758i
\(328\) 0 0
\(329\) 269.214i 0.818281i
\(330\) 0 0
\(331\) 552.652 1.66964 0.834822 0.550520i \(-0.185571\pi\)
0.834822 + 0.550520i \(0.185571\pi\)
\(332\) 0 0
\(333\) 35.8117i 0.107543i
\(334\) 0 0
\(335\) −76.2014 −0.227467
\(336\) 0 0
\(337\) 392.041i 1.16333i −0.813430 0.581663i \(-0.802402\pi\)
0.813430 0.581663i \(-0.197598\pi\)
\(338\) 0 0
\(339\) 343.794i 1.01414i
\(340\) 0 0
\(341\) 79.7861i 0.233977i
\(342\) 0 0
\(343\) 371.475i 1.08302i
\(344\) 0 0
\(345\) −89.0771 + 0.525890i −0.258194 + 0.00152432i
\(346\) 0 0
\(347\) 436.686 1.25846 0.629230 0.777219i \(-0.283370\pi\)
0.629230 + 0.777219i \(0.283370\pi\)
\(348\) 0 0
\(349\) −238.327 −0.682885 −0.341443 0.939903i \(-0.610916\pi\)
−0.341443 + 0.939903i \(0.610916\pi\)
\(350\) 0 0
\(351\) 6.31613 0.0179947
\(352\) 0 0
\(353\) −151.374 −0.428822 −0.214411 0.976744i \(-0.568783\pi\)
−0.214411 + 0.976744i \(0.568783\pi\)
\(354\) 0 0
\(355\) 302.800i 0.852958i
\(356\) 0 0
\(357\) −6.61238 −0.0185221
\(358\) 0 0
\(359\) 550.110i 1.53234i −0.642639 0.766169i \(-0.722160\pi\)
0.642639 0.766169i \(-0.277840\pi\)
\(360\) 0 0
\(361\) 91.2313 0.252718
\(362\) 0 0
\(363\) −194.708 −0.536385
\(364\) 0 0
\(365\) 193.732i 0.530774i
\(366\) 0 0
\(367\) 119.225i 0.324865i 0.986720 + 0.162432i \(0.0519340\pi\)
−0.986720 + 0.162432i \(0.948066\pi\)
\(368\) 0 0
\(369\) −77.4791 −0.209970
\(370\) 0 0
\(371\) 504.737 1.36048
\(372\) 0 0
\(373\) 39.6098i 0.106192i 0.998589 + 0.0530962i \(0.0169090\pi\)
−0.998589 + 0.0530962i \(0.983091\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) 2.47272 0.00655893
\(378\) 0 0
\(379\) 10.3346i 0.0272682i −0.999907 0.0136341i \(-0.995660\pi\)
0.999907 0.0136341i \(-0.00434000\pi\)
\(380\) 0 0
\(381\) 29.9349 0.0785694
\(382\) 0 0
\(383\) 302.985i 0.791083i 0.918448 + 0.395541i \(0.129443\pi\)
−0.918448 + 0.395541i \(0.870557\pi\)
\(384\) 0 0
\(385\) 35.2238i 0.0914904i
\(386\) 0 0
\(387\) 18.4722i 0.0477317i
\(388\) 0 0
\(389\) 304.811i 0.783577i −0.920055 0.391789i \(-0.871856\pi\)
0.920055 0.391789i \(-0.128144\pi\)
\(390\) 0 0
\(391\) 16.3324 0.0964223i 0.0417707 0.000246604i
\(392\) 0 0
\(393\) 213.657 0.543657
\(394\) 0 0
\(395\) 69.2388 0.175288
\(396\) 0 0
\(397\) −366.059 −0.922063 −0.461031 0.887384i \(-0.652520\pi\)
−0.461031 + 0.887384i \(0.652520\pi\)
\(398\) 0 0
\(399\) −152.941 −0.383312
\(400\) 0 0
\(401\) 492.905i 1.22919i −0.788843 0.614595i \(-0.789319\pi\)
0.788843 0.614595i \(-0.210681\pi\)
\(402\) 0 0
\(403\) −33.0989 −0.0821312
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) 34.9773 0.0859394
\(408\) 0 0
\(409\) 83.4770 0.204100 0.102050 0.994779i \(-0.467460\pi\)
0.102050 + 0.994779i \(0.467460\pi\)
\(410\) 0 0
\(411\) 170.276i 0.414297i
\(412\) 0 0
\(413\) 224.425i 0.543402i
\(414\) 0 0
\(415\) 202.065 0.486904
\(416\) 0 0
\(417\) −441.655 −1.05913
\(418\) 0 0
\(419\) 555.121i 1.32487i 0.749119 + 0.662436i \(0.230477\pi\)
−0.749119 + 0.662436i \(0.769523\pi\)
\(420\) 0 0
\(421\) 733.651i 1.74264i 0.490717 + 0.871319i \(0.336735\pi\)
−0.490717 + 0.871319i \(0.663265\pi\)
\(422\) 0 0
\(423\) −150.228 −0.355149
\(424\) 0 0
\(425\) 3.55058i 0.00835429i
\(426\) 0 0
\(427\) −53.3631 −0.124972
\(428\) 0 0
\(429\) 6.16896i 0.0143799i
\(430\) 0 0
\(431\) 373.120i 0.865708i 0.901464 + 0.432854i \(0.142493\pi\)
−0.901464 + 0.432854i \(0.857507\pi\)
\(432\) 0 0
\(433\) 219.625i 0.507217i 0.967307 + 0.253609i \(0.0816176\pi\)
−0.967307 + 0.253609i \(0.918382\pi\)
\(434\) 0 0
\(435\) 7.87864i 0.0181118i
\(436\) 0 0
\(437\) 377.760 2.23020i 0.864440 0.00510344i
\(438\) 0 0
\(439\) 840.036 1.91352 0.956761 0.290876i \(-0.0939468\pi\)
0.956761 + 0.290876i \(0.0939468\pi\)
\(440\) 0 0
\(441\) 60.2922 0.136717
\(442\) 0 0
\(443\) −52.0995 −0.117606 −0.0588030 0.998270i \(-0.518728\pi\)
−0.0588030 + 0.998270i \(0.518728\pi\)
\(444\) 0 0
\(445\) 300.790 0.675932
\(446\) 0 0
\(447\) 92.7166i 0.207420i
\(448\) 0 0
\(449\) −740.113 −1.64836 −0.824179 0.566329i \(-0.808363\pi\)
−0.824179 + 0.566329i \(0.808363\pi\)
\(450\) 0 0
\(451\) 75.6738i 0.167791i
\(452\) 0 0
\(453\) −8.89460 −0.0196349
\(454\) 0 0
\(455\) −14.6124 −0.0321152
\(456\) 0 0
\(457\) 87.5933i 0.191670i −0.995397 0.0958351i \(-0.969448\pi\)
0.995397 0.0958351i \(-0.0305522\pi\)
\(458\) 0 0
\(459\) 3.68987i 0.00803892i
\(460\) 0 0
\(461\) 479.987 1.04119 0.520593 0.853805i \(-0.325711\pi\)
0.520593 + 0.853805i \(0.325711\pi\)
\(462\) 0 0
\(463\) 164.223 0.354692 0.177346 0.984149i \(-0.443249\pi\)
0.177346 + 0.984149i \(0.443249\pi\)
\(464\) 0 0
\(465\) 105.461i 0.226797i
\(466\) 0 0
\(467\) 235.058i 0.503336i 0.967814 + 0.251668i \(0.0809792\pi\)
−0.967814 + 0.251668i \(0.919021\pi\)
\(468\) 0 0
\(469\) 183.209 0.390637
\(470\) 0 0
\(471\) 250.428i 0.531695i
\(472\) 0 0
\(473\) 18.0418 0.0381433
\(474\) 0 0
\(475\) 82.1232i 0.172891i
\(476\) 0 0
\(477\) 281.656i 0.590473i
\(478\) 0 0
\(479\) 222.596i 0.464710i 0.972631 + 0.232355i \(0.0746431\pi\)
−0.972631 + 0.232355i \(0.925357\pi\)
\(480\) 0 0
\(481\) 14.5102i 0.0301667i
\(482\) 0 0
\(483\) 214.165 1.26438i 0.443407 0.00261777i
\(484\) 0 0
\(485\) −214.102 −0.441448
\(486\) 0 0
\(487\) 117.381 0.241028 0.120514 0.992712i \(-0.461546\pi\)
0.120514 + 0.992712i \(0.461546\pi\)
\(488\) 0 0
\(489\) −316.325 −0.646881
\(490\) 0 0
\(491\) −953.112 −1.94117 −0.970583 0.240768i \(-0.922601\pi\)
−0.970583 + 0.240768i \(0.922601\pi\)
\(492\) 0 0
\(493\) 1.44456i 0.00293013i
\(494\) 0 0
\(495\) −19.6557 −0.0397085
\(496\) 0 0
\(497\) 728.013i 1.46482i
\(498\) 0 0
\(499\) 305.953 0.613131 0.306566 0.951850i \(-0.400820\pi\)
0.306566 + 0.951850i \(0.400820\pi\)
\(500\) 0 0
\(501\) −285.515 −0.569890
\(502\) 0 0
\(503\) 539.182i 1.07193i −0.844239 0.535966i \(-0.819947\pi\)
0.844239 0.535966i \(-0.180053\pi\)
\(504\) 0 0
\(505\) 163.889i 0.324533i
\(506\) 0 0
\(507\) 290.157 0.572303
\(508\) 0 0
\(509\) 207.429 0.407523 0.203762 0.979021i \(-0.434683\pi\)
0.203762 + 0.979021i \(0.434683\pi\)
\(510\) 0 0
\(511\) 465.785i 0.911517i
\(512\) 0 0
\(513\) 85.3449i 0.166364i
\(514\) 0 0
\(515\) −34.9832 −0.0679286
\(516\) 0 0
\(517\) 146.728i 0.283806i
\(518\) 0 0
\(519\) −513.753 −0.989890
\(520\) 0 0
\(521\) 218.933i 0.420217i 0.977678 + 0.210109i \(0.0673817\pi\)
−0.977678 + 0.210109i \(0.932618\pi\)
\(522\) 0 0
\(523\) 203.163i 0.388457i 0.980956 + 0.194229i \(0.0622204\pi\)
−0.980956 + 0.194229i \(0.937780\pi\)
\(524\) 0 0
\(525\) 46.5585i 0.0886829i
\(526\) 0 0
\(527\) 19.3363i 0.0366912i
\(528\) 0 0
\(529\) −528.963 + 6.24596i −0.999930 + 0.0118071i
\(530\) 0 0
\(531\) 125.234 0.235846
\(532\) 0 0
\(533\) 31.3929 0.0588986
\(534\) 0 0
\(535\) 213.418 0.398913
\(536\) 0 0
\(537\) −379.371 −0.706464
\(538\) 0 0
\(539\) 58.8874i 0.109253i
\(540\) 0 0
\(541\) 92.9862 0.171878 0.0859392 0.996300i \(-0.472611\pi\)
0.0859392 + 0.996300i \(0.472611\pi\)
\(542\) 0 0
\(543\) 603.105i 1.11069i
\(544\) 0 0
\(545\) −320.736 −0.588506
\(546\) 0 0
\(547\) 367.033 0.670992 0.335496 0.942042i \(-0.391096\pi\)
0.335496 + 0.942042i \(0.391096\pi\)
\(548\) 0 0
\(549\) 29.7779i 0.0542402i
\(550\) 0 0
\(551\) 33.4119i 0.0606387i
\(552\) 0 0
\(553\) −166.469 −0.301028
\(554\) 0 0
\(555\) −46.2327 −0.0833022
\(556\) 0 0
\(557\) 613.017i 1.10057i −0.834977 0.550284i \(-0.814519\pi\)
0.834977 0.550284i \(-0.185481\pi\)
\(558\) 0 0
\(559\) 7.48454i 0.0133892i
\(560\) 0 0
\(561\) 3.60389 0.00642405
\(562\) 0 0
\(563\) 924.483i 1.64206i 0.570881 + 0.821032i \(0.306602\pi\)
−0.570881 + 0.821032i \(0.693398\pi\)
\(564\) 0 0
\(565\) 443.837 0.785552
\(566\) 0 0
\(567\) 48.3850i 0.0853351i
\(568\) 0 0
\(569\) 729.146i 1.28145i −0.767770 0.640726i \(-0.778633\pi\)
0.767770 0.640726i \(-0.221367\pi\)
\(570\) 0 0
\(571\) 326.391i 0.571613i −0.958287 0.285807i \(-0.907739\pi\)
0.958287 0.285807i \(-0.0922615\pi\)
\(572\) 0 0
\(573\) 117.328i 0.204761i
\(574\) 0 0
\(575\) −0.678920 114.998i −0.00118073 0.199997i
\(576\) 0 0
\(577\) 370.825 0.642678 0.321339 0.946964i \(-0.395867\pi\)
0.321339 + 0.946964i \(0.395867\pi\)
\(578\) 0 0
\(579\) −240.847 −0.415971
\(580\) 0 0
\(581\) −485.819 −0.836178
\(582\) 0 0
\(583\) −275.093 −0.471858
\(584\) 0 0
\(585\) 8.15408i 0.0139386i
\(586\) 0 0
\(587\) 8.31967 0.0141732 0.00708660 0.999975i \(-0.497744\pi\)
0.00708660 + 0.999975i \(0.497744\pi\)
\(588\) 0 0
\(589\) 447.239i 0.759320i
\(590\) 0 0
\(591\) 295.806 0.500518
\(592\) 0 0
\(593\) −334.917 −0.564783 −0.282392 0.959299i \(-0.591128\pi\)
−0.282392 + 0.959299i \(0.591128\pi\)
\(594\) 0 0
\(595\) 8.53655i 0.0143471i
\(596\) 0 0
\(597\) 289.472i 0.484877i
\(598\) 0 0
\(599\) −665.958 −1.11178 −0.555891 0.831255i \(-0.687623\pi\)
−0.555891 + 0.831255i \(0.687623\pi\)
\(600\) 0 0
\(601\) −207.240 −0.344825 −0.172412 0.985025i \(-0.555156\pi\)
−0.172412 + 0.985025i \(0.555156\pi\)
\(602\) 0 0
\(603\) 102.235i 0.169544i
\(604\) 0 0
\(605\) 251.366i 0.415482i
\(606\) 0 0
\(607\) −440.061 −0.724978 −0.362489 0.931988i \(-0.618073\pi\)
−0.362489 + 0.931988i \(0.618073\pi\)
\(608\) 0 0
\(609\) 18.9424i 0.0311041i
\(610\) 0 0
\(611\) 60.8693 0.0996225
\(612\) 0 0
\(613\) 191.955i 0.313140i 0.987667 + 0.156570i \(0.0500437\pi\)
−0.987667 + 0.156570i \(0.949956\pi\)
\(614\) 0 0
\(615\) 100.025i 0.162642i
\(616\) 0 0
\(617\) 852.003i 1.38088i −0.723390 0.690440i \(-0.757417\pi\)
0.723390 0.690440i \(-0.242583\pi\)
\(618\) 0 0
\(619\) 912.672i 1.47443i −0.675658 0.737215i \(-0.736140\pi\)
0.675658 0.737215i \(-0.263860\pi\)
\(620\) 0 0
\(621\) −0.705555 119.509i −0.00113616 0.192447i
\(622\) 0 0
\(623\) −723.180 −1.16080
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 83.3564 0.132945
\(628\) 0 0
\(629\) 8.47681 0.0134766
\(630\) 0 0
\(631\) 691.828i 1.09640i 0.836347 + 0.548200i \(0.184687\pi\)
−0.836347 + 0.548200i \(0.815313\pi\)
\(632\) 0 0
\(633\) −393.210 −0.621184
\(634\) 0 0
\(635\) 38.6458i 0.0608596i
\(636\) 0 0
\(637\) −24.4292 −0.0383504
\(638\) 0 0
\(639\) 406.249 0.635757
\(640\) 0 0
\(641\) 135.634i 0.211598i −0.994388 0.105799i \(-0.966260\pi\)
0.994388 0.105799i \(-0.0337401\pi\)
\(642\) 0 0
\(643\) 605.034i 0.940956i −0.882412 0.470478i \(-0.844082\pi\)
0.882412 0.470478i \(-0.155918\pi\)
\(644\) 0 0
\(645\) −23.8474 −0.0369728
\(646\) 0 0
\(647\) −254.682 −0.393635 −0.196817 0.980440i \(-0.563061\pi\)
−0.196817 + 0.980440i \(0.563061\pi\)
\(648\) 0 0
\(649\) 122.317i 0.188469i
\(650\) 0 0
\(651\) 253.556i 0.389486i
\(652\) 0 0
\(653\) 92.8857 0.142245 0.0711223 0.997468i \(-0.477342\pi\)
0.0711223 + 0.997468i \(0.477342\pi\)
\(654\) 0 0
\(655\) 275.830i 0.421115i
\(656\) 0 0
\(657\) −259.919 −0.395615
\(658\) 0 0
\(659\) 795.363i 1.20692i −0.797392 0.603462i \(-0.793787\pi\)
0.797392 0.603462i \(-0.206213\pi\)
\(660\) 0 0
\(661\) 32.0723i 0.0485209i −0.999706 0.0242605i \(-0.992277\pi\)
0.999706 0.0242605i \(-0.00772310\pi\)
\(662\) 0 0
\(663\) 1.49506i 0.00225499i
\(664\) 0 0
\(665\) 197.446i 0.296912i
\(666\) 0 0
\(667\) −0.276220 46.7871i −0.000414122 0.0701456i
\(668\) 0 0
\(669\) −490.237 −0.732791
\(670\) 0 0
\(671\) 29.0841 0.0433444
\(672\) 0 0
\(673\) −532.710 −0.791545 −0.395773 0.918348i \(-0.629523\pi\)
−0.395773 + 0.918348i \(0.629523\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 820.502i 1.21197i −0.795477 0.605983i \(-0.792780\pi\)
0.795477 0.605983i \(-0.207220\pi\)
\(678\) 0 0
\(679\) 514.760 0.758115
\(680\) 0 0
\(681\) 232.634i 0.341606i
\(682\) 0 0
\(683\) −45.1174 −0.0660577 −0.0330288 0.999454i \(-0.510515\pi\)
−0.0330288 + 0.999454i \(0.510515\pi\)
\(684\) 0 0
\(685\) −219.825 −0.320913
\(686\) 0 0
\(687\) 88.6116i 0.128983i
\(688\) 0 0
\(689\) 114.121i 0.165633i
\(690\) 0 0
\(691\) 14.3095 0.0207084 0.0103542 0.999946i \(-0.496704\pi\)
0.0103542 + 0.999946i \(0.496704\pi\)
\(692\) 0 0
\(693\) 47.2577 0.0681929
\(694\) 0 0
\(695\) 570.174i 0.820395i
\(696\) 0 0
\(697\) 18.3397i 0.0263123i
\(698\) 0 0
\(699\) −263.036 −0.376303
\(700\) 0 0
\(701\) 112.502i 0.160488i −0.996775 0.0802441i \(-0.974430\pi\)
0.996775 0.0802441i \(-0.0255700\pi\)
\(702\) 0 0
\(703\) 196.065 0.278897
\(704\) 0 0
\(705\) 193.944i 0.275097i
\(706\) 0 0
\(707\) 394.034i 0.557332i
\(708\) 0 0
\(709\) 1089.05i 1.53604i −0.640427 0.768019i \(-0.721243\pi\)
0.640427 0.768019i \(-0.278757\pi\)
\(710\) 0 0
\(711\) 92.8935i 0.130652i
\(712\) 0 0
\(713\) 3.69737 + 626.274i 0.00518566 + 0.878365i
\(714\) 0 0
\(715\) 7.96410 0.0111386
\(716\) 0 0
\(717\) −512.694 −0.715055
\(718\) 0 0
\(719\) −1169.16 −1.62609 −0.813046 0.582199i \(-0.802193\pi\)
−0.813046 + 0.582199i \(0.802193\pi\)
\(720\) 0 0
\(721\) 84.1092 0.116656
\(722\) 0 0
\(723\) 213.558i 0.295378i
\(724\) 0 0
\(725\) 10.1713 0.0140294
\(726\) 0 0
\(727\) 132.969i 0.182901i −0.995810 0.0914506i \(-0.970850\pi\)
0.995810 0.0914506i \(-0.0291504\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 4.37245 0.00598147
\(732\) 0 0
\(733\) 1279.17i 1.74511i 0.488514 + 0.872556i \(0.337539\pi\)
−0.488514 + 0.872556i \(0.662461\pi\)
\(734\) 0 0
\(735\) 77.8369i 0.105901i
\(736\) 0 0
\(737\) −99.8529 −0.135486
\(738\) 0 0
\(739\) −458.877 −0.620944 −0.310472 0.950583i \(-0.600487\pi\)
−0.310472 + 0.950583i \(0.600487\pi\)
\(740\) 0 0
\(741\) 34.5800i 0.0466667i
\(742\) 0 0
\(743\) 1226.68i 1.65098i 0.564419 + 0.825489i \(0.309100\pi\)
−0.564419 + 0.825489i \(0.690900\pi\)
\(744\) 0 0
\(745\) −119.697 −0.160667
\(746\) 0 0
\(747\) 271.099i 0.362917i
\(748\) 0 0
\(749\) −513.116 −0.685067
\(750\) 0 0
\(751\) 381.375i 0.507823i 0.967227 + 0.253912i \(0.0817172\pi\)
−0.967227 + 0.253912i \(0.918283\pi\)
\(752\) 0 0
\(753\) 622.986i 0.827338i
\(754\) 0 0
\(755\) 11.4829i 0.0152091i
\(756\) 0 0
\(757\) 464.497i 0.613603i −0.951774 0.306801i \(-0.900741\pi\)
0.951774 0.306801i \(-0.0992588\pi\)
\(758\) 0 0
\(759\) −116.725 + 0.689116i −0.153788 + 0.000907926i
\(760\) 0 0
\(761\) −431.433 −0.566929 −0.283465 0.958983i \(-0.591484\pi\)
−0.283465 + 0.958983i \(0.591484\pi\)
\(762\) 0 0
\(763\) 771.136 1.01066
\(764\) 0 0
\(765\) −4.76360 −0.00622692
\(766\) 0 0
\(767\) −50.7425 −0.0661571
\(768\) 0 0
\(769\) 837.825i 1.08950i −0.838599 0.544750i \(-0.816625\pi\)
0.838599 0.544750i \(-0.183375\pi\)
\(770\) 0 0
\(771\) −461.195 −0.598178
\(772\) 0 0
\(773\) 409.257i 0.529440i −0.964325 0.264720i \(-0.914720\pi\)
0.964325 0.264720i \(-0.0852796\pi\)
\(774\) 0 0
\(775\) −136.149 −0.175676
\(776\) 0 0
\(777\) 111.156 0.143058
\(778\) 0 0
\(779\) 424.188i 0.544529i
\(780\) 0 0
\(781\) 396.784i 0.508046i
\(782\) 0 0
\(783\) 10.5703 0.0134998
\(784\) 0 0
\(785\) 323.301 0.411849
\(786\) 0 0
\(787\) 1375.46i 1.74773i 0.486170 + 0.873864i \(0.338394\pi\)
−0.486170 + 0.873864i \(0.661606\pi\)
\(788\) 0 0
\(789\) 293.097i 0.371479i
\(790\) 0 0
\(791\) −1067.10 −1.34906
\(792\) 0 0
\(793\) 12.0654i 0.0152149i
\(794\) 0 0
\(795\) 363.616 0.457378
\(796\) 0 0
\(797\) 483.005i 0.606029i 0.952986 + 0.303014i \(0.0979930\pi\)
−0.952986 + 0.303014i \(0.902007\pi\)
\(798\) 0 0