Properties

Label 2760.3.g.a.2161.12
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 2161.12
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.37

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} +0.00406525i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} +0.00406525i q^{7} +3.00000 q^{9} -15.2374i q^{11} -17.0059 q^{13} +3.87298i q^{15} -29.9683i q^{17} +2.23468i q^{19} -0.00704122i q^{21} +(8.01839 - 21.5570i) q^{23} -5.00000 q^{25} -5.19615 q^{27} +3.32327 q^{29} +50.1303 q^{31} +26.3920i q^{33} +0.00909018 q^{35} +9.14013i q^{37} +29.4550 q^{39} -25.2881 q^{41} -41.4922i q^{43} -6.70820i q^{45} -71.5342 q^{47} +49.0000 q^{49} +51.9066i q^{51} -35.1120i q^{53} -34.0719 q^{55} -3.87057i q^{57} -113.791 q^{59} +51.7271i q^{61} +0.0121958i q^{63} +38.0262i q^{65} +78.1826i q^{67} +(-13.8883 + 37.3379i) q^{69} -6.44793 q^{71} +116.232 q^{73} +8.66025 q^{75} +0.0619440 q^{77} +153.554i q^{79} +9.00000 q^{81} -94.1572i q^{83} -67.0112 q^{85} -5.75608 q^{87} -151.007i q^{89} -0.0691331i q^{91} -86.8281 q^{93} +4.99689 q^{95} -80.0084i q^{97} -45.7123i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 0.00406525i 0.000580750i 1.00000 0.000290375i \(9.24293e-5\pi\)
−1.00000 0.000290375i \(0.999908\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 15.2374i 1.38522i −0.721312 0.692610i \(-0.756461\pi\)
0.721312 0.692610i \(-0.243539\pi\)
\(12\) 0 0
\(13\) −17.0059 −1.30814 −0.654071 0.756433i \(-0.726940\pi\)
−0.654071 + 0.756433i \(0.726940\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 29.9683i 1.76284i −0.472331 0.881421i \(-0.656587\pi\)
0.472331 0.881421i \(-0.343413\pi\)
\(18\) 0 0
\(19\) 2.23468i 0.117614i 0.998269 + 0.0588072i \(0.0187297\pi\)
−0.998269 + 0.0588072i \(0.981270\pi\)
\(20\) 0 0
\(21\) 0.00704122i 0.000335296i
\(22\) 0 0
\(23\) 8.01839 21.5570i 0.348626 0.937262i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 3.32327 0.114596 0.0572978 0.998357i \(-0.481752\pi\)
0.0572978 + 0.998357i \(0.481752\pi\)
\(30\) 0 0
\(31\) 50.1303 1.61710 0.808552 0.588424i \(-0.200251\pi\)
0.808552 + 0.588424i \(0.200251\pi\)
\(32\) 0 0
\(33\) 26.3920i 0.799758i
\(34\) 0 0
\(35\) 0.00909018 0.000259719
\(36\) 0 0
\(37\) 9.14013i 0.247031i 0.992343 + 0.123515i \(0.0394168\pi\)
−0.992343 + 0.123515i \(0.960583\pi\)
\(38\) 0 0
\(39\) 29.4550 0.755256
\(40\) 0 0
\(41\) −25.2881 −0.616782 −0.308391 0.951260i \(-0.599790\pi\)
−0.308391 + 0.951260i \(0.599790\pi\)
\(42\) 0 0
\(43\) 41.4922i 0.964935i −0.875914 0.482467i \(-0.839741\pi\)
0.875914 0.482467i \(-0.160259\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) −71.5342 −1.52200 −0.761002 0.648750i \(-0.775292\pi\)
−0.761002 + 0.648750i \(0.775292\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 51.9066i 1.01778i
\(52\) 0 0
\(53\) 35.1120i 0.662491i −0.943545 0.331245i \(-0.892531\pi\)
0.943545 0.331245i \(-0.107469\pi\)
\(54\) 0 0
\(55\) −34.0719 −0.619490
\(56\) 0 0
\(57\) 3.87057i 0.0679048i
\(58\) 0 0
\(59\) −113.791 −1.92867 −0.964334 0.264689i \(-0.914731\pi\)
−0.964334 + 0.264689i \(0.914731\pi\)
\(60\) 0 0
\(61\) 51.7271i 0.847985i 0.905666 + 0.423993i \(0.139372\pi\)
−0.905666 + 0.423993i \(0.860628\pi\)
\(62\) 0 0
\(63\) 0.0121958i 0.000193583i
\(64\) 0 0
\(65\) 38.0262i 0.585019i
\(66\) 0 0
\(67\) 78.1826i 1.16690i 0.812148 + 0.583452i \(0.198298\pi\)
−0.812148 + 0.583452i \(0.801702\pi\)
\(68\) 0 0
\(69\) −13.8883 + 37.3379i −0.201279 + 0.541129i
\(70\) 0 0
\(71\) −6.44793 −0.0908159 −0.0454080 0.998969i \(-0.514459\pi\)
−0.0454080 + 0.998969i \(0.514459\pi\)
\(72\) 0 0
\(73\) 116.232 1.59221 0.796107 0.605157i \(-0.206889\pi\)
0.796107 + 0.605157i \(0.206889\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) 0.0619440 0.000804467
\(78\) 0 0
\(79\) 153.554i 1.94373i 0.235546 + 0.971863i \(0.424312\pi\)
−0.235546 + 0.971863i \(0.575688\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 94.1572i 1.13442i −0.823572 0.567212i \(-0.808022\pi\)
0.823572 0.567212i \(-0.191978\pi\)
\(84\) 0 0
\(85\) −67.0112 −0.788367
\(86\) 0 0
\(87\) −5.75608 −0.0661618
\(88\) 0 0
\(89\) 151.007i 1.69671i −0.529426 0.848356i \(-0.677593\pi\)
0.529426 0.848356i \(-0.322407\pi\)
\(90\) 0 0
\(91\) 0.0691331i 0.000759704i
\(92\) 0 0
\(93\) −86.8281 −0.933636
\(94\) 0 0
\(95\) 4.99689 0.0525988
\(96\) 0 0
\(97\) 80.0084i 0.824828i −0.910996 0.412414i \(-0.864686\pi\)
0.910996 0.412414i \(-0.135314\pi\)
\(98\) 0 0
\(99\) 45.7123i 0.461740i
\(100\) 0 0
\(101\) 172.731 1.71021 0.855106 0.518453i \(-0.173492\pi\)
0.855106 + 0.518453i \(0.173492\pi\)
\(102\) 0 0
\(103\) 160.984i 1.56295i −0.623935 0.781476i \(-0.714467\pi\)
0.623935 0.781476i \(-0.285533\pi\)
\(104\) 0 0
\(105\) −0.0157447 −0.000149949
\(106\) 0 0
\(107\) 147.142i 1.37516i 0.726108 + 0.687580i \(0.241327\pi\)
−0.726108 + 0.687580i \(0.758673\pi\)
\(108\) 0 0
\(109\) 15.9321i 0.146166i 0.997326 + 0.0730832i \(0.0232839\pi\)
−0.997326 + 0.0730832i \(0.976716\pi\)
\(110\) 0 0
\(111\) 15.8312i 0.142623i
\(112\) 0 0
\(113\) 26.5616i 0.235059i −0.993069 0.117529i \(-0.962503\pi\)
0.993069 0.117529i \(-0.0374974\pi\)
\(114\) 0 0
\(115\) −48.2030 17.9297i −0.419156 0.155910i
\(116\) 0 0
\(117\) −51.0176 −0.436048
\(118\) 0 0
\(119\) 0.121829 0.00102377
\(120\) 0 0
\(121\) −111.179 −0.918837
\(122\) 0 0
\(123\) 43.8002 0.356099
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) 27.5246 0.216729 0.108365 0.994111i \(-0.465439\pi\)
0.108365 + 0.994111i \(0.465439\pi\)
\(128\) 0 0
\(129\) 71.8666i 0.557105i
\(130\) 0 0
\(131\) 38.1375 0.291126 0.145563 0.989349i \(-0.453501\pi\)
0.145563 + 0.989349i \(0.453501\pi\)
\(132\) 0 0
\(133\) −0.00908452 −6.83047e−5
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 146.494i 1.06930i 0.845075 + 0.534649i \(0.179556\pi\)
−0.845075 + 0.534649i \(0.820444\pi\)
\(138\) 0 0
\(139\) −154.014 −1.10801 −0.554005 0.832513i \(-0.686901\pi\)
−0.554005 + 0.832513i \(0.686901\pi\)
\(140\) 0 0
\(141\) 123.901 0.878729
\(142\) 0 0
\(143\) 259.125i 1.81207i
\(144\) 0 0
\(145\) 7.43106i 0.0512487i
\(146\) 0 0
\(147\) −84.8705 −0.577350
\(148\) 0 0
\(149\) 52.1230i 0.349819i −0.984585 0.174910i \(-0.944037\pi\)
0.984585 0.174910i \(-0.0559633\pi\)
\(150\) 0 0
\(151\) −183.173 −1.21307 −0.606534 0.795057i \(-0.707441\pi\)
−0.606534 + 0.795057i \(0.707441\pi\)
\(152\) 0 0
\(153\) 89.9050i 0.587614i
\(154\) 0 0
\(155\) 112.095i 0.723191i
\(156\) 0 0
\(157\) 38.4422i 0.244855i −0.992477 0.122427i \(-0.960932\pi\)
0.992477 0.122427i \(-0.0390678\pi\)
\(158\) 0 0
\(159\) 60.8158i 0.382489i
\(160\) 0 0
\(161\) 0.0876348 + 0.0325968i 0.000544315 + 0.000202464i
\(162\) 0 0
\(163\) 27.3583 0.167842 0.0839211 0.996472i \(-0.473256\pi\)
0.0839211 + 0.996472i \(0.473256\pi\)
\(164\) 0 0
\(165\) 59.0143 0.357662
\(166\) 0 0
\(167\) 103.548 0.620050 0.310025 0.950728i \(-0.399663\pi\)
0.310025 + 0.950728i \(0.399663\pi\)
\(168\) 0 0
\(169\) 120.199 0.711237
\(170\) 0 0
\(171\) 6.70403i 0.0392048i
\(172\) 0 0
\(173\) −37.0300 −0.214046 −0.107023 0.994257i \(-0.534132\pi\)
−0.107023 + 0.994257i \(0.534132\pi\)
\(174\) 0 0
\(175\) 0.0203263i 0.000116150i
\(176\) 0 0
\(177\) 197.092 1.11352
\(178\) 0 0
\(179\) −331.734 −1.85326 −0.926631 0.375973i \(-0.877309\pi\)
−0.926631 + 0.375973i \(0.877309\pi\)
\(180\) 0 0
\(181\) 286.944i 1.58533i 0.609659 + 0.792664i \(0.291306\pi\)
−0.609659 + 0.792664i \(0.708694\pi\)
\(182\) 0 0
\(183\) 89.5940i 0.489584i
\(184\) 0 0
\(185\) 20.4380 0.110475
\(186\) 0 0
\(187\) −456.640 −2.44193
\(188\) 0 0
\(189\) 0.0211237i 0.000111765i
\(190\) 0 0
\(191\) 10.2870i 0.0538584i −0.999637 0.0269292i \(-0.991427\pi\)
0.999637 0.0269292i \(-0.00857287\pi\)
\(192\) 0 0
\(193\) −212.862 −1.10291 −0.551455 0.834205i \(-0.685927\pi\)
−0.551455 + 0.834205i \(0.685927\pi\)
\(194\) 0 0
\(195\) 65.8634i 0.337761i
\(196\) 0 0
\(197\) 119.362 0.605901 0.302950 0.953006i \(-0.402028\pi\)
0.302950 + 0.953006i \(0.402028\pi\)
\(198\) 0 0
\(199\) 75.6515i 0.380158i 0.981769 + 0.190079i \(0.0608744\pi\)
−0.981769 + 0.190079i \(0.939126\pi\)
\(200\) 0 0
\(201\) 135.416i 0.673712i
\(202\) 0 0
\(203\) 0.0135099i 6.65514e-5i
\(204\) 0 0
\(205\) 56.5458i 0.275833i
\(206\) 0 0
\(207\) 24.0552 64.6711i 0.116209 0.312421i
\(208\) 0 0
\(209\) 34.0507 0.162922
\(210\) 0 0
\(211\) −302.528 −1.43378 −0.716890 0.697186i \(-0.754435\pi\)
−0.716890 + 0.697186i \(0.754435\pi\)
\(212\) 0 0
\(213\) 11.1681 0.0524326
\(214\) 0 0
\(215\) −92.7794 −0.431532
\(216\) 0 0
\(217\) 0.203792i 0.000939134i
\(218\) 0 0
\(219\) −201.319 −0.919265
\(220\) 0 0
\(221\) 509.637i 2.30605i
\(222\) 0 0
\(223\) 41.3082 0.185238 0.0926192 0.995702i \(-0.470476\pi\)
0.0926192 + 0.995702i \(0.470476\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 124.502i 0.548466i −0.961663 0.274233i \(-0.911576\pi\)
0.961663 0.274233i \(-0.0884240\pi\)
\(228\) 0 0
\(229\) 253.060i 1.10506i 0.833492 + 0.552532i \(0.186338\pi\)
−0.833492 + 0.552532i \(0.813662\pi\)
\(230\) 0 0
\(231\) −0.107290 −0.000464459
\(232\) 0 0
\(233\) −332.346 −1.42638 −0.713188 0.700973i \(-0.752750\pi\)
−0.713188 + 0.700973i \(0.752750\pi\)
\(234\) 0 0
\(235\) 159.955i 0.680661i
\(236\) 0 0
\(237\) 265.964i 1.12221i
\(238\) 0 0
\(239\) −73.2581 −0.306519 −0.153260 0.988186i \(-0.548977\pi\)
−0.153260 + 0.988186i \(0.548977\pi\)
\(240\) 0 0
\(241\) 132.842i 0.551212i −0.961271 0.275606i \(-0.911121\pi\)
0.961271 0.275606i \(-0.0888786\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 109.567i 0.447213i
\(246\) 0 0
\(247\) 38.0026i 0.153857i
\(248\) 0 0
\(249\) 163.085i 0.654960i
\(250\) 0 0
\(251\) 148.237i 0.590587i 0.955407 + 0.295294i \(0.0954174\pi\)
−0.955407 + 0.295294i \(0.904583\pi\)
\(252\) 0 0
\(253\) −328.474 122.180i −1.29831 0.482923i
\(254\) 0 0
\(255\) 116.067 0.455164
\(256\) 0 0
\(257\) −51.5948 −0.200758 −0.100379 0.994949i \(-0.532006\pi\)
−0.100379 + 0.994949i \(0.532006\pi\)
\(258\) 0 0
\(259\) −0.0371569 −0.000143463
\(260\) 0 0
\(261\) 9.96982 0.0381985
\(262\) 0 0
\(263\) 344.329i 1.30924i −0.755959 0.654619i \(-0.772829\pi\)
0.755959 0.654619i \(-0.227171\pi\)
\(264\) 0 0
\(265\) −78.5128 −0.296275
\(266\) 0 0
\(267\) 261.553i 0.979598i
\(268\) 0 0
\(269\) 120.491 0.447924 0.223962 0.974598i \(-0.428101\pi\)
0.223962 + 0.974598i \(0.428101\pi\)
\(270\) 0 0
\(271\) −278.237 −1.02670 −0.513352 0.858178i \(-0.671596\pi\)
−0.513352 + 0.858178i \(0.671596\pi\)
\(272\) 0 0
\(273\) 0.119742i 0.000438615i
\(274\) 0 0
\(275\) 76.1871i 0.277044i
\(276\) 0 0
\(277\) 85.0056 0.306879 0.153440 0.988158i \(-0.450965\pi\)
0.153440 + 0.988158i \(0.450965\pi\)
\(278\) 0 0
\(279\) 150.391 0.539035
\(280\) 0 0
\(281\) 176.685i 0.628773i 0.949295 + 0.314387i \(0.101799\pi\)
−0.949295 + 0.314387i \(0.898201\pi\)
\(282\) 0 0
\(283\) 335.765i 1.18645i 0.805037 + 0.593224i \(0.202145\pi\)
−0.805037 + 0.593224i \(0.797855\pi\)
\(284\) 0 0
\(285\) −8.65486 −0.0303679
\(286\) 0 0
\(287\) 0.102802i 0.000358196i
\(288\) 0 0
\(289\) −609.100 −2.10761
\(290\) 0 0
\(291\) 138.579i 0.476215i
\(292\) 0 0
\(293\) 70.7243i 0.241380i 0.992690 + 0.120690i \(0.0385107\pi\)
−0.992690 + 0.120690i \(0.961489\pi\)
\(294\) 0 0
\(295\) 254.445i 0.862526i
\(296\) 0 0
\(297\) 79.1760i 0.266586i
\(298\) 0 0
\(299\) −136.360 + 366.596i −0.456052 + 1.22607i
\(300\) 0 0
\(301\) 0.168676 0.000560386
\(302\) 0 0
\(303\) −299.180 −0.987391
\(304\) 0 0
\(305\) 115.665 0.379230
\(306\) 0 0
\(307\) −100.740 −0.328145 −0.164072 0.986448i \(-0.552463\pi\)
−0.164072 + 0.986448i \(0.552463\pi\)
\(308\) 0 0
\(309\) 278.833i 0.902371i
\(310\) 0 0
\(311\) −564.238 −1.81427 −0.907136 0.420838i \(-0.861736\pi\)
−0.907136 + 0.420838i \(0.861736\pi\)
\(312\) 0 0
\(313\) 220.516i 0.704524i −0.935901 0.352262i \(-0.885413\pi\)
0.935901 0.352262i \(-0.114587\pi\)
\(314\) 0 0
\(315\) 0.0272705 8.65731e−5
\(316\) 0 0
\(317\) −467.676 −1.47532 −0.737659 0.675173i \(-0.764069\pi\)
−0.737659 + 0.675173i \(0.764069\pi\)
\(318\) 0 0
\(319\) 50.6381i 0.158740i
\(320\) 0 0
\(321\) 254.858i 0.793949i
\(322\) 0 0
\(323\) 66.9695 0.207336
\(324\) 0 0
\(325\) 85.0293 0.261629
\(326\) 0 0
\(327\) 27.5953i 0.0843892i
\(328\) 0 0
\(329\) 0.290804i 0.000883904i
\(330\) 0 0
\(331\) −35.3046 −0.106660 −0.0533302 0.998577i \(-0.516984\pi\)
−0.0533302 + 0.998577i \(0.516984\pi\)
\(332\) 0 0
\(333\) 27.4204i 0.0823435i
\(334\) 0 0
\(335\) 174.822 0.521855
\(336\) 0 0
\(337\) 55.7782i 0.165514i 0.996570 + 0.0827570i \(0.0263725\pi\)
−0.996570 + 0.0827570i \(0.973627\pi\)
\(338\) 0 0
\(339\) 46.0061i 0.135711i
\(340\) 0 0
\(341\) 763.856i 2.24005i
\(342\) 0 0
\(343\) 0.398395i 0.00116150i
\(344\) 0 0
\(345\) 83.4900 + 31.0551i 0.242000 + 0.0900147i
\(346\) 0 0
\(347\) 400.411 1.15392 0.576961 0.816772i \(-0.304238\pi\)
0.576961 + 0.816772i \(0.304238\pi\)
\(348\) 0 0
\(349\) 349.176 1.00051 0.500253 0.865880i \(-0.333240\pi\)
0.500253 + 0.865880i \(0.333240\pi\)
\(350\) 0 0
\(351\) 88.3650 0.251752
\(352\) 0 0
\(353\) 170.286 0.482396 0.241198 0.970476i \(-0.422460\pi\)
0.241198 + 0.970476i \(0.422460\pi\)
\(354\) 0 0
\(355\) 14.4180i 0.0406141i
\(356\) 0 0
\(357\) −0.211014 −0.000591075
\(358\) 0 0
\(359\) 339.758i 0.946402i −0.880954 0.473201i \(-0.843098\pi\)
0.880954 0.473201i \(-0.156902\pi\)
\(360\) 0 0
\(361\) 356.006 0.986167
\(362\) 0 0
\(363\) 192.568 0.530491
\(364\) 0 0
\(365\) 259.902i 0.712059i
\(366\) 0 0
\(367\) 359.169i 0.978662i 0.872098 + 0.489331i \(0.162759\pi\)
−0.872098 + 0.489331i \(0.837241\pi\)
\(368\) 0 0
\(369\) −75.8642 −0.205594
\(370\) 0 0
\(371\) 0.142739 0.000384742
\(372\) 0 0
\(373\) 123.592i 0.331345i −0.986181 0.165672i \(-0.947021\pi\)
0.986181 0.165672i \(-0.0529794\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) −56.5151 −0.149907
\(378\) 0 0
\(379\) 195.626i 0.516163i 0.966123 + 0.258082i \(0.0830903\pi\)
−0.966123 + 0.258082i \(0.916910\pi\)
\(380\) 0 0
\(381\) −47.6740 −0.125129
\(382\) 0 0
\(383\) 438.104i 1.14387i 0.820297 + 0.571937i \(0.193808\pi\)
−0.820297 + 0.571937i \(0.806192\pi\)
\(384\) 0 0
\(385\) 0.138511i 0.000359769i
\(386\) 0 0
\(387\) 124.477i 0.321645i
\(388\) 0 0
\(389\) 189.600i 0.487403i 0.969850 + 0.243701i \(0.0783617\pi\)
−0.969850 + 0.243701i \(0.921638\pi\)
\(390\) 0 0
\(391\) −646.028 240.298i −1.65225 0.614572i
\(392\) 0 0
\(393\) −66.0560 −0.168082
\(394\) 0 0
\(395\) 343.358 0.869261
\(396\) 0 0
\(397\) −202.827 −0.510899 −0.255450 0.966822i \(-0.582224\pi\)
−0.255450 + 0.966822i \(0.582224\pi\)
\(398\) 0 0
\(399\) 0.0157348 3.94357e−5
\(400\) 0 0
\(401\) 634.552i 1.58242i −0.611541 0.791212i \(-0.709450\pi\)
0.611541 0.791212i \(-0.290550\pi\)
\(402\) 0 0
\(403\) −852.508 −2.11540
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) 139.272 0.342192
\(408\) 0 0
\(409\) 409.129 1.00031 0.500157 0.865934i \(-0.333275\pi\)
0.500157 + 0.865934i \(0.333275\pi\)
\(410\) 0 0
\(411\) 253.735i 0.617359i
\(412\) 0 0
\(413\) 0.462591i 0.00112007i
\(414\) 0 0
\(415\) −210.542 −0.507330
\(416\) 0 0
\(417\) 266.759 0.639710
\(418\) 0 0
\(419\) 533.372i 1.27296i 0.771292 + 0.636482i \(0.219611\pi\)
−0.771292 + 0.636482i \(0.780389\pi\)
\(420\) 0 0
\(421\) 227.712i 0.540883i 0.962736 + 0.270442i \(0.0871697\pi\)
−0.962736 + 0.270442i \(0.912830\pi\)
\(422\) 0 0
\(423\) −214.603 −0.507335
\(424\) 0 0
\(425\) 149.842i 0.352568i
\(426\) 0 0
\(427\) −0.210284 −0.000492468
\(428\) 0 0
\(429\) 448.819i 1.04620i
\(430\) 0 0
\(431\) 131.051i 0.304062i 0.988376 + 0.152031i \(0.0485814\pi\)
−0.988376 + 0.152031i \(0.951419\pi\)
\(432\) 0 0
\(433\) 517.145i 1.19433i −0.802118 0.597166i \(-0.796294\pi\)
0.802118 0.597166i \(-0.203706\pi\)
\(434\) 0 0
\(435\) 12.8710i 0.0295885i
\(436\) 0 0
\(437\) 48.1730 + 17.9185i 0.110236 + 0.0410034i
\(438\) 0 0
\(439\) 527.882 1.20247 0.601233 0.799074i \(-0.294676\pi\)
0.601233 + 0.799074i \(0.294676\pi\)
\(440\) 0 0
\(441\) 147.000 0.333333
\(442\) 0 0
\(443\) −800.259 −1.80645 −0.903227 0.429164i \(-0.858808\pi\)
−0.903227 + 0.429164i \(0.858808\pi\)
\(444\) 0 0
\(445\) −337.663 −0.758793
\(446\) 0 0
\(447\) 90.2798i 0.201968i
\(448\) 0 0
\(449\) 476.816 1.06195 0.530975 0.847387i \(-0.321826\pi\)
0.530975 + 0.847387i \(0.321826\pi\)
\(450\) 0 0
\(451\) 385.325i 0.854379i
\(452\) 0 0
\(453\) 317.266 0.700365
\(454\) 0 0
\(455\) −0.154586 −0.000339750
\(456\) 0 0
\(457\) 334.499i 0.731946i −0.930625 0.365973i \(-0.880736\pi\)
0.930625 0.365973i \(-0.119264\pi\)
\(458\) 0 0
\(459\) 155.720i 0.339259i
\(460\) 0 0
\(461\) −352.571 −0.764796 −0.382398 0.923998i \(-0.624902\pi\)
−0.382398 + 0.923998i \(0.624902\pi\)
\(462\) 0 0
\(463\) −47.7318 −0.103092 −0.0515462 0.998671i \(-0.516415\pi\)
−0.0515462 + 0.998671i \(0.516415\pi\)
\(464\) 0 0
\(465\) 194.154i 0.417535i
\(466\) 0 0
\(467\) 447.079i 0.957343i 0.877994 + 0.478672i \(0.158882\pi\)
−0.877994 + 0.478672i \(0.841118\pi\)
\(468\) 0 0
\(469\) −0.317832 −0.000677680
\(470\) 0 0
\(471\) 66.5838i 0.141367i
\(472\) 0 0
\(473\) −632.234 −1.33665
\(474\) 0 0
\(475\) 11.1734i 0.0235229i
\(476\) 0 0
\(477\) 105.336i 0.220830i
\(478\) 0 0
\(479\) 131.715i 0.274979i −0.990503 0.137490i \(-0.956097\pi\)
0.990503 0.137490i \(-0.0439034\pi\)
\(480\) 0 0
\(481\) 155.436i 0.323151i
\(482\) 0 0
\(483\) −0.151788 0.0564592i −0.000314261 0.000116893i
\(484\) 0 0
\(485\) −178.904 −0.368874
\(486\) 0 0
\(487\) −910.350 −1.86930 −0.934651 0.355566i \(-0.884288\pi\)
−0.934651 + 0.355566i \(0.884288\pi\)
\(488\) 0 0
\(489\) −47.3859 −0.0969037
\(490\) 0 0
\(491\) 851.950 1.73513 0.867567 0.497321i \(-0.165683\pi\)
0.867567 + 0.497321i \(0.165683\pi\)
\(492\) 0 0
\(493\) 99.5929i 0.202014i
\(494\) 0 0
\(495\) −102.216 −0.206497
\(496\) 0 0
\(497\) 0.0262125i 5.27414e-5i
\(498\) 0 0
\(499\) 816.691 1.63665 0.818327 0.574752i \(-0.194902\pi\)
0.818327 + 0.574752i \(0.194902\pi\)
\(500\) 0 0
\(501\) −179.351 −0.357986
\(502\) 0 0
\(503\) 395.328i 0.785940i −0.919551 0.392970i \(-0.871448\pi\)
0.919551 0.392970i \(-0.128552\pi\)
\(504\) 0 0
\(505\) 386.239i 0.764830i
\(506\) 0 0
\(507\) −208.191 −0.410633
\(508\) 0 0
\(509\) 543.733 1.06824 0.534119 0.845409i \(-0.320643\pi\)
0.534119 + 0.845409i \(0.320643\pi\)
\(510\) 0 0
\(511\) 0.472511i 0.000924678i
\(512\) 0 0
\(513\) 11.6117i 0.0226349i
\(514\) 0 0
\(515\) −359.971 −0.698973
\(516\) 0 0
\(517\) 1090.00i 2.10831i
\(518\) 0 0
\(519\) 64.1379 0.123580
\(520\) 0 0
\(521\) 532.334i 1.02175i 0.859654 + 0.510877i \(0.170679\pi\)
−0.859654 + 0.510877i \(0.829321\pi\)
\(522\) 0 0
\(523\) 76.9293i 0.147092i 0.997292 + 0.0735462i \(0.0234316\pi\)
−0.997292 + 0.0735462i \(0.976568\pi\)
\(524\) 0 0
\(525\) 0.0352061i 6.70593e-5i
\(526\) 0 0
\(527\) 1502.32i 2.85070i
\(528\) 0 0
\(529\) −400.411 345.705i −0.756921 0.653507i
\(530\) 0 0
\(531\) −341.374 −0.642889
\(532\) 0 0
\(533\) 430.045 0.806839
\(534\) 0 0
\(535\) 329.020 0.614990
\(536\) 0 0
\(537\) 574.580 1.06998
\(538\) 0 0
\(539\) 746.634i 1.38522i
\(540\) 0 0
\(541\) −571.727 −1.05680 −0.528398 0.848997i \(-0.677207\pi\)
−0.528398 + 0.848997i \(0.677207\pi\)
\(542\) 0 0
\(543\) 497.002i 0.915290i
\(544\) 0 0
\(545\) 35.6254 0.0653676
\(546\) 0 0
\(547\) −1044.99 −1.91039 −0.955197 0.295971i \(-0.904357\pi\)
−0.955197 + 0.295971i \(0.904357\pi\)
\(548\) 0 0
\(549\) 155.181i 0.282662i
\(550\) 0 0
\(551\) 7.42644i 0.0134781i
\(552\) 0 0
\(553\) −0.624237 −0.00112882
\(554\) 0 0
\(555\) −35.3996 −0.0637830
\(556\) 0 0
\(557\) 108.890i 0.195494i −0.995211 0.0977468i \(-0.968836\pi\)
0.995211 0.0977468i \(-0.0311635\pi\)
\(558\) 0 0
\(559\) 705.610i 1.26227i
\(560\) 0 0
\(561\) 790.924 1.40985
\(562\) 0 0
\(563\) 81.5720i 0.144888i −0.997372 0.0724441i \(-0.976920\pi\)
0.997372 0.0724441i \(-0.0230799\pi\)
\(564\) 0 0
\(565\) −59.3936 −0.105121
\(566\) 0 0
\(567\) 0.0365873i 6.45278e-5i
\(568\) 0 0
\(569\) 450.698i 0.792089i 0.918231 + 0.396044i \(0.129617\pi\)
−0.918231 + 0.396044i \(0.870383\pi\)
\(570\) 0 0
\(571\) 24.2937i 0.0425459i 0.999774 + 0.0212730i \(0.00677191\pi\)
−0.999774 + 0.0212730i \(0.993228\pi\)
\(572\) 0 0
\(573\) 17.8175i 0.0310952i
\(574\) 0 0
\(575\) −40.0919 + 107.785i −0.0697251 + 0.187452i
\(576\) 0 0
\(577\) 928.710 1.60955 0.804775 0.593581i \(-0.202286\pi\)
0.804775 + 0.593581i \(0.202286\pi\)
\(578\) 0 0
\(579\) 368.687 0.636765
\(580\) 0 0
\(581\) 0.382773 0.000658817
\(582\) 0 0
\(583\) −535.017 −0.917696
\(584\) 0 0
\(585\) 114.079i 0.195006i
\(586\) 0 0
\(587\) −573.154 −0.976413 −0.488207 0.872728i \(-0.662349\pi\)
−0.488207 + 0.872728i \(0.662349\pi\)
\(588\) 0 0
\(589\) 112.025i 0.190195i
\(590\) 0 0
\(591\) −206.742 −0.349817
\(592\) 0 0
\(593\) 853.534 1.43935 0.719674 0.694312i \(-0.244291\pi\)
0.719674 + 0.694312i \(0.244291\pi\)
\(594\) 0 0
\(595\) 0.272417i 0.000457844i
\(596\) 0 0
\(597\) 131.032i 0.219484i
\(598\) 0 0
\(599\) −354.069 −0.591100 −0.295550 0.955327i \(-0.595503\pi\)
−0.295550 + 0.955327i \(0.595503\pi\)
\(600\) 0 0
\(601\) 801.780 1.33408 0.667038 0.745024i \(-0.267562\pi\)
0.667038 + 0.745024i \(0.267562\pi\)
\(602\) 0 0
\(603\) 234.548i 0.388968i
\(604\) 0 0
\(605\) 248.604i 0.410916i
\(606\) 0 0
\(607\) −249.091 −0.410365 −0.205182 0.978724i \(-0.565779\pi\)
−0.205182 + 0.978724i \(0.565779\pi\)
\(608\) 0 0
\(609\) 0.0233999i 3.84235e-5i
\(610\) 0 0
\(611\) 1216.50 1.99100
\(612\) 0 0
\(613\) 588.293i 0.959694i −0.877352 0.479847i \(-0.840692\pi\)
0.877352 0.479847i \(-0.159308\pi\)
\(614\) 0 0
\(615\) 97.9402i 0.159252i
\(616\) 0 0
\(617\) 797.182i 1.29203i 0.763325 + 0.646014i \(0.223565\pi\)
−0.763325 + 0.646014i \(0.776435\pi\)
\(618\) 0 0
\(619\) 1018.87i 1.64599i −0.568047 0.822996i \(-0.692301\pi\)
0.568047 0.822996i \(-0.307699\pi\)
\(620\) 0 0
\(621\) −41.6648 + 112.014i −0.0670930 + 0.180376i
\(622\) 0 0
\(623\) 0.613883 0.000985366
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −58.9776 −0.0940631
\(628\) 0 0
\(629\) 273.914 0.435476
\(630\) 0 0
\(631\) 627.621i 0.994645i 0.867566 + 0.497322i \(0.165683\pi\)
−0.867566 + 0.497322i \(0.834317\pi\)
\(632\) 0 0
\(633\) 523.993 0.827793
\(634\) 0 0
\(635\) 61.5468i 0.0969242i
\(636\) 0 0
\(637\) −833.287 −1.30814
\(638\) 0 0
\(639\) −19.3438 −0.0302720
\(640\) 0 0
\(641\) 892.382i 1.39217i 0.717958 + 0.696086i \(0.245077\pi\)
−0.717958 + 0.696086i \(0.754923\pi\)
\(642\) 0 0
\(643\) 233.345i 0.362900i 0.983400 + 0.181450i \(0.0580791\pi\)
−0.983400 + 0.181450i \(0.941921\pi\)
\(644\) 0 0
\(645\) 160.699 0.249145
\(646\) 0 0
\(647\) 1070.00 1.65379 0.826894 0.562358i \(-0.190106\pi\)
0.826894 + 0.562358i \(0.190106\pi\)
\(648\) 0 0
\(649\) 1733.89i 2.67163i
\(650\) 0 0
\(651\) 0.352978i 0.000542209i
\(652\) 0 0
\(653\) −99.1847 −0.151891 −0.0759454 0.997112i \(-0.524197\pi\)
−0.0759454 + 0.997112i \(0.524197\pi\)
\(654\) 0 0
\(655\) 85.2780i 0.130195i
\(656\) 0 0
\(657\) 348.695 0.530738
\(658\) 0 0
\(659\) 338.990i 0.514400i 0.966358 + 0.257200i \(0.0828000\pi\)
−0.966358 + 0.257200i \(0.917200\pi\)
\(660\) 0 0
\(661\) 1195.78i 1.80905i −0.426421 0.904525i \(-0.640226\pi\)
0.426421 0.904525i \(-0.359774\pi\)
\(662\) 0 0
\(663\) 882.717i 1.33140i
\(664\) 0 0
\(665\) 0.0203136i 3.05468e-5i
\(666\) 0 0
\(667\) 26.6473 71.6399i 0.0399510 0.107406i
\(668\) 0 0
\(669\) −71.5478 −0.106947
\(670\) 0 0
\(671\) 788.188 1.17465
\(672\) 0 0
\(673\) 381.479 0.566834 0.283417 0.958997i \(-0.408532\pi\)
0.283417 + 0.958997i \(0.408532\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 844.496i 1.24741i 0.781660 + 0.623704i \(0.214373\pi\)
−0.781660 + 0.623704i \(0.785627\pi\)
\(678\) 0 0
\(679\) 0.325254 0.000479019
\(680\) 0 0
\(681\) 215.643i 0.316657i
\(682\) 0 0
\(683\) 1080.14 1.58147 0.790733 0.612162i \(-0.209700\pi\)
0.790733 + 0.612162i \(0.209700\pi\)
\(684\) 0 0
\(685\) 327.570 0.478204
\(686\) 0 0
\(687\) 438.312i 0.638009i
\(688\) 0 0
\(689\) 597.110i 0.866632i
\(690\) 0 0
\(691\) 241.213 0.349078 0.174539 0.984650i \(-0.444156\pi\)
0.174539 + 0.984650i \(0.444156\pi\)
\(692\) 0 0
\(693\) 0.185832 0.000268156
\(694\) 0 0
\(695\) 344.385i 0.495518i
\(696\) 0 0
\(697\) 757.840i 1.08729i
\(698\) 0 0
\(699\) 575.640 0.823519
\(700\) 0 0
\(701\) 1286.89i 1.83580i −0.396814 0.917899i \(-0.629884\pi\)
0.396814 0.917899i \(-0.370116\pi\)
\(702\) 0 0
\(703\) −20.4252 −0.0290544
\(704\) 0 0
\(705\) 277.051i 0.392980i
\(706\) 0 0
\(707\) 0.702197i 0.000993206i
\(708\) 0 0
\(709\) 661.784i 0.933404i −0.884415 0.466702i \(-0.845442\pi\)
0.884415 0.466702i \(-0.154558\pi\)
\(710\) 0 0
\(711\) 460.663i 0.647909i
\(712\) 0 0
\(713\) 401.964 1080.66i 0.563764 1.51565i
\(714\) 0 0
\(715\) 579.422 0.810381
\(716\) 0 0
\(717\) 126.887 0.176969
\(718\) 0 0
\(719\) 512.499 0.712794 0.356397 0.934335i \(-0.384005\pi\)
0.356397 + 0.934335i \(0.384005\pi\)
\(720\) 0 0
\(721\) 0.654441 0.000907685
\(722\) 0 0
\(723\) 230.089i 0.318243i
\(724\) 0 0
\(725\) −16.6164 −0.0229191
\(726\) 0 0
\(727\) 401.070i 0.551678i 0.961204 + 0.275839i \(0.0889556\pi\)
−0.961204 + 0.275839i \(0.911044\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) −1243.45 −1.70103
\(732\) 0 0
\(733\) 990.923i 1.35187i −0.736960 0.675936i \(-0.763739\pi\)
0.736960 0.675936i \(-0.236261\pi\)
\(734\) 0 0
\(735\) 189.776i 0.258199i
\(736\) 0 0
\(737\) 1191.30 1.61642
\(738\) 0 0
\(739\) −798.831 −1.08096 −0.540481 0.841356i \(-0.681758\pi\)
−0.540481 + 0.841356i \(0.681758\pi\)
\(740\) 0 0
\(741\) 65.8224i 0.0888291i
\(742\) 0 0
\(743\) 1118.17i 1.50494i −0.658624 0.752472i \(-0.728861\pi\)
0.658624 0.752472i \(-0.271139\pi\)
\(744\) 0 0
\(745\) −116.551 −0.156444
\(746\) 0 0
\(747\) 282.472i 0.378141i
\(748\) 0 0
\(749\) −0.598170 −0.000798625
\(750\) 0 0
\(751\) 356.585i 0.474813i 0.971410 + 0.237407i \(0.0762973\pi\)
−0.971410 + 0.237407i \(0.923703\pi\)
\(752\) 0 0
\(753\) 256.755i 0.340976i
\(754\) 0 0
\(755\) 409.588i 0.542501i
\(756\) 0 0
\(757\) 1.81410i 0.00239643i −0.999999 0.00119822i \(-0.999619\pi\)
0.999999 0.00119822i \(-0.000381404\pi\)
\(758\) 0 0
\(759\) 568.933 + 211.621i 0.749583 + 0.278816i
\(760\) 0 0
\(761\) −1278.07 −1.67946 −0.839731 0.543003i \(-0.817287\pi\)
−0.839731 + 0.543003i \(0.817287\pi\)
\(762\) 0 0
\(763\) −0.0647682 −8.48862e−5
\(764\) 0 0
\(765\) −201.034 −0.262789
\(766\) 0 0
\(767\) 1935.12 2.52297
\(768\) 0 0
\(769\) 1344.09i 1.74784i 0.486067 + 0.873921i \(0.338431\pi\)
−0.486067 + 0.873921i \(0.661569\pi\)
\(770\) 0 0
\(771\) 89.3647 0.115908
\(772\) 0 0
\(773\) 169.656i 0.219477i −0.993960 0.109739i \(-0.964999\pi\)
0.993960 0.109739i \(-0.0350014\pi\)
\(774\) 0 0
\(775\) −250.651 −0.323421
\(776\) 0 0
\(777\) 0.0643577 8.28285e−5
\(778\) 0 0
\(779\) 56.5106i 0.0725425i
\(780\) 0 0
\(781\) 98.2499i 0.125800i
\(782\) 0 0
\(783\) −17.2682 −0.0220539
\(784\) 0 0
\(785\) −85.9593 −0.109502
\(786\) 0 0
\(787\) 772.353i 0.981389i −0.871332 0.490694i \(-0.836743\pi\)
0.871332 0.490694i \(-0.163257\pi\)
\(788\) 0 0
\(789\) 596.396i 0.755888i
\(790\) 0 0
\(791\) 0.107980 0.000136510
\(792\) 0 0
\(793\) 879.663i 1.10929i
\(794\) 0 0
\(795\) 135.988 0.171054
\(796\) 0 0
\(797\) 446.742i 0.560530i −0.959923 0.280265i \(-0.909578\pi\)
0.959923 0.280265i \(-0.0904223\pi\)
\(798\) 0 0