Properties

Label 2760.3.g.a.2161.1
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 2161.1
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.48

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} -12.2824i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} -12.2824i q^{7} +3.00000 q^{9} -2.61742i q^{11} +10.1017 q^{13} +3.87298i q^{15} +24.1412i q^{17} +13.4990i q^{19} +21.2737i q^{21} +(-21.6769 - 7.68853i) q^{23} -5.00000 q^{25} -5.19615 q^{27} +20.0216 q^{29} -51.6425 q^{31} +4.53350i q^{33} -27.4642 q^{35} +12.4541i q^{37} -17.4967 q^{39} +18.3729 q^{41} +66.3980i q^{43} -6.70820i q^{45} -24.1786 q^{47} -101.856 q^{49} -41.8138i q^{51} -15.4135i q^{53} -5.85272 q^{55} -23.3810i q^{57} -54.7222 q^{59} +30.4427i q^{61} -36.8471i q^{63} -22.5881i q^{65} +50.5504i q^{67} +(37.5454 + 13.3169i) q^{69} +69.7461 q^{71} -43.1315 q^{73} +8.66025 q^{75} -32.1480 q^{77} +30.4115i q^{79} +9.00000 q^{81} +23.6084i q^{83} +53.9814 q^{85} -34.6784 q^{87} -157.292i q^{89} -124.073i q^{91} +89.4474 q^{93} +30.1847 q^{95} +103.724i q^{97} -7.85225i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 12.2824i 1.75462i −0.479923 0.877311i \(-0.659335\pi\)
0.479923 0.877311i \(-0.340665\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 2.61742i 0.237947i −0.992897 0.118974i \(-0.962040\pi\)
0.992897 0.118974i \(-0.0379604\pi\)
\(12\) 0 0
\(13\) 10.1017 0.777054 0.388527 0.921437i \(-0.372984\pi\)
0.388527 + 0.921437i \(0.372984\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 24.1412i 1.42007i 0.704166 + 0.710036i \(0.251321\pi\)
−0.704166 + 0.710036i \(0.748679\pi\)
\(18\) 0 0
\(19\) 13.4990i 0.710475i 0.934776 + 0.355238i \(0.115600\pi\)
−0.934776 + 0.355238i \(0.884400\pi\)
\(20\) 0 0
\(21\) 21.2737i 1.01303i
\(22\) 0 0
\(23\) −21.6769 7.68853i −0.942472 0.334284i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 20.0216 0.690400 0.345200 0.938529i \(-0.387811\pi\)
0.345200 + 0.938529i \(0.387811\pi\)
\(30\) 0 0
\(31\) −51.6425 −1.66589 −0.832943 0.553358i \(-0.813346\pi\)
−0.832943 + 0.553358i \(0.813346\pi\)
\(32\) 0 0
\(33\) 4.53350i 0.137379i
\(34\) 0 0
\(35\) −27.4642 −0.784691
\(36\) 0 0
\(37\) 12.4541i 0.336597i 0.985736 + 0.168299i \(0.0538273\pi\)
−0.985736 + 0.168299i \(0.946173\pi\)
\(38\) 0 0
\(39\) −17.4967 −0.448633
\(40\) 0 0
\(41\) 18.3729 0.448119 0.224060 0.974575i \(-0.428069\pi\)
0.224060 + 0.974575i \(0.428069\pi\)
\(42\) 0 0
\(43\) 66.3980i 1.54414i 0.635537 + 0.772070i \(0.280779\pi\)
−0.635537 + 0.772070i \(0.719221\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) −24.1786 −0.514439 −0.257220 0.966353i \(-0.582806\pi\)
−0.257220 + 0.966353i \(0.582806\pi\)
\(48\) 0 0
\(49\) −101.856 −2.07870
\(50\) 0 0
\(51\) 41.8138i 0.819879i
\(52\) 0 0
\(53\) 15.4135i 0.290821i −0.989371 0.145410i \(-0.953550\pi\)
0.989371 0.145410i \(-0.0464503\pi\)
\(54\) 0 0
\(55\) −5.85272 −0.106413
\(56\) 0 0
\(57\) 23.3810i 0.410193i
\(58\) 0 0
\(59\) −54.7222 −0.927496 −0.463748 0.885967i \(-0.653496\pi\)
−0.463748 + 0.885967i \(0.653496\pi\)
\(60\) 0 0
\(61\) 30.4427i 0.499060i 0.968367 + 0.249530i \(0.0802762\pi\)
−0.968367 + 0.249530i \(0.919724\pi\)
\(62\) 0 0
\(63\) 36.8471i 0.584874i
\(64\) 0 0
\(65\) 22.5881i 0.347509i
\(66\) 0 0
\(67\) 50.5504i 0.754484i 0.926115 + 0.377242i \(0.123127\pi\)
−0.926115 + 0.377242i \(0.876873\pi\)
\(68\) 0 0
\(69\) 37.5454 + 13.3169i 0.544137 + 0.192999i
\(70\) 0 0
\(71\) 69.7461 0.982340 0.491170 0.871064i \(-0.336569\pi\)
0.491170 + 0.871064i \(0.336569\pi\)
\(72\) 0 0
\(73\) −43.1315 −0.590843 −0.295421 0.955367i \(-0.595460\pi\)
−0.295421 + 0.955367i \(0.595460\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) −32.1480 −0.417507
\(78\) 0 0
\(79\) 30.4115i 0.384956i 0.981301 + 0.192478i \(0.0616524\pi\)
−0.981301 + 0.192478i \(0.938348\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 23.6084i 0.284438i 0.989835 + 0.142219i \(0.0454238\pi\)
−0.989835 + 0.142219i \(0.954576\pi\)
\(84\) 0 0
\(85\) 53.9814 0.635075
\(86\) 0 0
\(87\) −34.6784 −0.398603
\(88\) 0 0
\(89\) 157.292i 1.76732i −0.468127 0.883661i \(-0.655071\pi\)
0.468127 0.883661i \(-0.344929\pi\)
\(90\) 0 0
\(91\) 124.073i 1.36344i
\(92\) 0 0
\(93\) 89.4474 0.961800
\(94\) 0 0
\(95\) 30.1847 0.317734
\(96\) 0 0
\(97\) 103.724i 1.06931i 0.845069 + 0.534657i \(0.179559\pi\)
−0.845069 + 0.534657i \(0.820441\pi\)
\(98\) 0 0
\(99\) 7.85225i 0.0793157i
\(100\) 0 0
\(101\) 52.2898 0.517721 0.258860 0.965915i \(-0.416653\pi\)
0.258860 + 0.965915i \(0.416653\pi\)
\(102\) 0 0
\(103\) 114.755i 1.11413i −0.830469 0.557065i \(-0.811927\pi\)
0.830469 0.557065i \(-0.188073\pi\)
\(104\) 0 0
\(105\) 47.5693 0.453041
\(106\) 0 0
\(107\) 80.5500i 0.752804i −0.926456 0.376402i \(-0.877161\pi\)
0.926456 0.376402i \(-0.122839\pi\)
\(108\) 0 0
\(109\) 136.462i 1.25194i 0.779846 + 0.625972i \(0.215298\pi\)
−0.779846 + 0.625972i \(0.784702\pi\)
\(110\) 0 0
\(111\) 21.5711i 0.194334i
\(112\) 0 0
\(113\) 181.271i 1.60417i 0.597211 + 0.802084i \(0.296275\pi\)
−0.597211 + 0.802084i \(0.703725\pi\)
\(114\) 0 0
\(115\) −17.1921 + 48.4709i −0.149496 + 0.421486i
\(116\) 0 0
\(117\) 30.3051 0.259018
\(118\) 0 0
\(119\) 296.511 2.49169
\(120\) 0 0
\(121\) 114.149 0.943381
\(122\) 0 0
\(123\) −31.8228 −0.258722
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) 132.399 1.04251 0.521257 0.853400i \(-0.325463\pi\)
0.521257 + 0.853400i \(0.325463\pi\)
\(128\) 0 0
\(129\) 115.005i 0.891510i
\(130\) 0 0
\(131\) 12.1681 0.0928860 0.0464430 0.998921i \(-0.485211\pi\)
0.0464430 + 0.998921i \(0.485211\pi\)
\(132\) 0 0
\(133\) 165.800 1.24662
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 175.314i 1.27966i 0.768516 + 0.639830i \(0.220995\pi\)
−0.768516 + 0.639830i \(0.779005\pi\)
\(138\) 0 0
\(139\) 81.4330 0.585849 0.292924 0.956136i \(-0.405372\pi\)
0.292924 + 0.956136i \(0.405372\pi\)
\(140\) 0 0
\(141\) 41.8786 0.297012
\(142\) 0 0
\(143\) 26.4404i 0.184898i
\(144\) 0 0
\(145\) 44.7697i 0.308756i
\(146\) 0 0
\(147\) 176.420 1.20014
\(148\) 0 0
\(149\) 171.763i 1.15277i −0.817178 0.576386i \(-0.804463\pi\)
0.817178 0.576386i \(-0.195537\pi\)
\(150\) 0 0
\(151\) −251.558 −1.66595 −0.832974 0.553313i \(-0.813363\pi\)
−0.832974 + 0.553313i \(0.813363\pi\)
\(152\) 0 0
\(153\) 72.4236i 0.473357i
\(154\) 0 0
\(155\) 115.476i 0.745007i
\(156\) 0 0
\(157\) 220.214i 1.40264i 0.712849 + 0.701318i \(0.247405\pi\)
−0.712849 + 0.701318i \(0.752595\pi\)
\(158\) 0 0
\(159\) 26.6970i 0.167906i
\(160\) 0 0
\(161\) −94.4333 + 266.243i −0.586542 + 1.65368i
\(162\) 0 0
\(163\) −239.735 −1.47076 −0.735382 0.677652i \(-0.762997\pi\)
−0.735382 + 0.677652i \(0.762997\pi\)
\(164\) 0 0
\(165\) 10.1372 0.0614377
\(166\) 0 0
\(167\) −53.6350 −0.321168 −0.160584 0.987022i \(-0.551338\pi\)
−0.160584 + 0.987022i \(0.551338\pi\)
\(168\) 0 0
\(169\) −66.9555 −0.396186
\(170\) 0 0
\(171\) 40.4971i 0.236825i
\(172\) 0 0
\(173\) −134.909 −0.779821 −0.389910 0.920853i \(-0.627494\pi\)
−0.389910 + 0.920853i \(0.627494\pi\)
\(174\) 0 0
\(175\) 61.4118i 0.350924i
\(176\) 0 0
\(177\) 94.7817 0.535490
\(178\) 0 0
\(179\) −88.0374 −0.491829 −0.245914 0.969292i \(-0.579088\pi\)
−0.245914 + 0.969292i \(0.579088\pi\)
\(180\) 0 0
\(181\) 106.980i 0.591049i −0.955335 0.295524i \(-0.904506\pi\)
0.955335 0.295524i \(-0.0954943\pi\)
\(182\) 0 0
\(183\) 52.7283i 0.288133i
\(184\) 0 0
\(185\) 27.8482 0.150531
\(186\) 0 0
\(187\) 63.1876 0.337902
\(188\) 0 0
\(189\) 63.8210i 0.337677i
\(190\) 0 0
\(191\) 208.748i 1.09292i 0.837485 + 0.546460i \(0.184025\pi\)
−0.837485 + 0.546460i \(0.815975\pi\)
\(192\) 0 0
\(193\) 77.2636 0.400329 0.200165 0.979762i \(-0.435852\pi\)
0.200165 + 0.979762i \(0.435852\pi\)
\(194\) 0 0
\(195\) 39.1237i 0.200635i
\(196\) 0 0
\(197\) 61.4348 0.311852 0.155926 0.987769i \(-0.450164\pi\)
0.155926 + 0.987769i \(0.450164\pi\)
\(198\) 0 0
\(199\) 105.523i 0.530266i 0.964212 + 0.265133i \(0.0854159\pi\)
−0.964212 + 0.265133i \(0.914584\pi\)
\(200\) 0 0
\(201\) 87.5559i 0.435601i
\(202\) 0 0
\(203\) 245.912i 1.21139i
\(204\) 0 0
\(205\) 41.0830i 0.200405i
\(206\) 0 0
\(207\) −65.0306 23.0656i −0.314157 0.111428i
\(208\) 0 0
\(209\) 35.3326 0.169055
\(210\) 0 0
\(211\) 12.7456 0.0604058 0.0302029 0.999544i \(-0.490385\pi\)
0.0302029 + 0.999544i \(0.490385\pi\)
\(212\) 0 0
\(213\) −120.804 −0.567154
\(214\) 0 0
\(215\) 148.470 0.690560
\(216\) 0 0
\(217\) 634.291i 2.92300i
\(218\) 0 0
\(219\) 74.7060 0.341123
\(220\) 0 0
\(221\) 243.868i 1.10347i
\(222\) 0 0
\(223\) 226.612 1.01620 0.508099 0.861299i \(-0.330348\pi\)
0.508099 + 0.861299i \(0.330348\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 187.673i 0.826755i 0.910560 + 0.413378i \(0.135651\pi\)
−0.910560 + 0.413378i \(0.864349\pi\)
\(228\) 0 0
\(229\) 126.045i 0.550415i −0.961385 0.275208i \(-0.911253\pi\)
0.961385 0.275208i \(-0.0887466\pi\)
\(230\) 0 0
\(231\) 55.6820 0.241048
\(232\) 0 0
\(233\) −414.940 −1.78086 −0.890429 0.455122i \(-0.849595\pi\)
−0.890429 + 0.455122i \(0.849595\pi\)
\(234\) 0 0
\(235\) 54.0651i 0.230064i
\(236\) 0 0
\(237\) 52.6743i 0.222255i
\(238\) 0 0
\(239\) 310.421 1.29883 0.649417 0.760432i \(-0.275013\pi\)
0.649417 + 0.760432i \(0.275013\pi\)
\(240\) 0 0
\(241\) 45.3011i 0.187971i 0.995574 + 0.0939857i \(0.0299608\pi\)
−0.995574 + 0.0939857i \(0.970039\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 227.757i 0.929621i
\(246\) 0 0
\(247\) 136.363i 0.552078i
\(248\) 0 0
\(249\) 40.8909i 0.164221i
\(250\) 0 0
\(251\) 37.6728i 0.150091i −0.997180 0.0750455i \(-0.976090\pi\)
0.997180 0.0750455i \(-0.0239102\pi\)
\(252\) 0 0
\(253\) −20.1241 + 56.7374i −0.0795419 + 0.224259i
\(254\) 0 0
\(255\) −93.4985 −0.366661
\(256\) 0 0
\(257\) 128.614 0.500443 0.250221 0.968189i \(-0.419497\pi\)
0.250221 + 0.968189i \(0.419497\pi\)
\(258\) 0 0
\(259\) 152.966 0.590600
\(260\) 0 0
\(261\) 60.0648 0.230133
\(262\) 0 0
\(263\) 316.334i 1.20279i −0.798952 0.601395i \(-0.794612\pi\)
0.798952 0.601395i \(-0.205388\pi\)
\(264\) 0 0
\(265\) −34.4657 −0.130059
\(266\) 0 0
\(267\) 272.437i 1.02036i
\(268\) 0 0
\(269\) −167.129 −0.621297 −0.310649 0.950525i \(-0.600546\pi\)
−0.310649 + 0.950525i \(0.600546\pi\)
\(270\) 0 0
\(271\) −283.584 −1.04643 −0.523217 0.852199i \(-0.675268\pi\)
−0.523217 + 0.852199i \(0.675268\pi\)
\(272\) 0 0
\(273\) 214.900i 0.787180i
\(274\) 0 0
\(275\) 13.0871i 0.0475894i
\(276\) 0 0
\(277\) 460.666 1.66306 0.831528 0.555483i \(-0.187467\pi\)
0.831528 + 0.555483i \(0.187467\pi\)
\(278\) 0 0
\(279\) −154.927 −0.555295
\(280\) 0 0
\(281\) 338.211i 1.20360i 0.798648 + 0.601798i \(0.205549\pi\)
−0.798648 + 0.601798i \(0.794451\pi\)
\(282\) 0 0
\(283\) 162.159i 0.572999i −0.958080 0.286500i \(-0.907508\pi\)
0.958080 0.286500i \(-0.0924917\pi\)
\(284\) 0 0
\(285\) −52.2815 −0.183444
\(286\) 0 0
\(287\) 225.662i 0.786280i
\(288\) 0 0
\(289\) −293.798 −1.01660
\(290\) 0 0
\(291\) 179.654i 0.617369i
\(292\) 0 0
\(293\) 26.8813i 0.0917451i −0.998947 0.0458726i \(-0.985393\pi\)
0.998947 0.0458726i \(-0.0146068\pi\)
\(294\) 0 0
\(295\) 122.363i 0.414789i
\(296\) 0 0
\(297\) 13.6005i 0.0457929i
\(298\) 0 0
\(299\) −218.973 77.6673i −0.732352 0.259757i
\(300\) 0 0
\(301\) 815.524 2.70938
\(302\) 0 0
\(303\) −90.5686 −0.298906
\(304\) 0 0
\(305\) 68.0719 0.223187
\(306\) 0 0
\(307\) −329.938 −1.07472 −0.537358 0.843354i \(-0.680577\pi\)
−0.537358 + 0.843354i \(0.680577\pi\)
\(308\) 0 0
\(309\) 198.762i 0.643244i
\(310\) 0 0
\(311\) 401.321 1.29042 0.645210 0.764005i \(-0.276770\pi\)
0.645210 + 0.764005i \(0.276770\pi\)
\(312\) 0 0
\(313\) 615.573i 1.96669i 0.181760 + 0.983343i \(0.441820\pi\)
−0.181760 + 0.983343i \(0.558180\pi\)
\(314\) 0 0
\(315\) −82.3925 −0.261564
\(316\) 0 0
\(317\) −580.768 −1.83208 −0.916038 0.401090i \(-0.868631\pi\)
−0.916038 + 0.401090i \(0.868631\pi\)
\(318\) 0 0
\(319\) 52.4049i 0.164279i
\(320\) 0 0
\(321\) 139.517i 0.434632i
\(322\) 0 0
\(323\) −325.883 −1.00893
\(324\) 0 0
\(325\) −50.5085 −0.155411
\(326\) 0 0
\(327\) 236.359i 0.722810i
\(328\) 0 0
\(329\) 296.971i 0.902646i
\(330\) 0 0
\(331\) 249.686 0.754340 0.377170 0.926144i \(-0.376897\pi\)
0.377170 + 0.926144i \(0.376897\pi\)
\(332\) 0 0
\(333\) 37.3623i 0.112199i
\(334\) 0 0
\(335\) 113.034 0.337415
\(336\) 0 0
\(337\) 476.822i 1.41490i 0.706762 + 0.707451i \(0.250155\pi\)
−0.706762 + 0.707451i \(0.749845\pi\)
\(338\) 0 0
\(339\) 313.971i 0.926167i
\(340\) 0 0
\(341\) 135.170i 0.396393i
\(342\) 0 0
\(343\) 649.198i 1.89270i
\(344\) 0 0
\(345\) 29.7776 83.9541i 0.0863118 0.243345i
\(346\) 0 0
\(347\) 541.004 1.55909 0.779544 0.626347i \(-0.215451\pi\)
0.779544 + 0.626347i \(0.215451\pi\)
\(348\) 0 0
\(349\) 398.231 1.14106 0.570532 0.821276i \(-0.306737\pi\)
0.570532 + 0.821276i \(0.306737\pi\)
\(350\) 0 0
\(351\) −52.4900 −0.149544
\(352\) 0 0
\(353\) −163.029 −0.461840 −0.230920 0.972973i \(-0.574174\pi\)
−0.230920 + 0.972973i \(0.574174\pi\)
\(354\) 0 0
\(355\) 155.957i 0.439316i
\(356\) 0 0
\(357\) −513.572 −1.43858
\(358\) 0 0
\(359\) 82.8993i 0.230917i 0.993312 + 0.115459i \(0.0368338\pi\)
−0.993312 + 0.115459i \(0.963166\pi\)
\(360\) 0 0
\(361\) 178.776 0.495225
\(362\) 0 0
\(363\) −197.712 −0.544661
\(364\) 0 0
\(365\) 96.4450i 0.264233i
\(366\) 0 0
\(367\) 258.720i 0.704959i 0.935819 + 0.352480i \(0.114661\pi\)
−0.935819 + 0.352480i \(0.885339\pi\)
\(368\) 0 0
\(369\) 55.1187 0.149373
\(370\) 0 0
\(371\) −189.314 −0.510281
\(372\) 0 0
\(373\) 259.475i 0.695643i 0.937561 + 0.347822i \(0.113079\pi\)
−0.937561 + 0.347822i \(0.886921\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) 202.252 0.536478
\(378\) 0 0
\(379\) 374.651i 0.988525i 0.869313 + 0.494262i \(0.164562\pi\)
−0.869313 + 0.494262i \(0.835438\pi\)
\(380\) 0 0
\(381\) −229.322 −0.601896
\(382\) 0 0
\(383\) 289.150i 0.754961i 0.926017 + 0.377481i \(0.123210\pi\)
−0.926017 + 0.377481i \(0.876790\pi\)
\(384\) 0 0
\(385\) 71.8852i 0.186715i
\(386\) 0 0
\(387\) 199.194i 0.514713i
\(388\) 0 0
\(389\) 666.363i 1.71301i −0.516135 0.856507i \(-0.672630\pi\)
0.516135 0.856507i \(-0.327370\pi\)
\(390\) 0 0
\(391\) 185.611 523.306i 0.474707 1.33838i
\(392\) 0 0
\(393\) −21.0757 −0.0536278
\(394\) 0 0
\(395\) 68.0023 0.172158
\(396\) 0 0
\(397\) 401.838 1.01219 0.506093 0.862479i \(-0.331089\pi\)
0.506093 + 0.862479i \(0.331089\pi\)
\(398\) 0 0
\(399\) −287.174 −0.719734
\(400\) 0 0
\(401\) 260.004i 0.648388i −0.945991 0.324194i \(-0.894907\pi\)
0.945991 0.324194i \(-0.105093\pi\)
\(402\) 0 0
\(403\) −521.677 −1.29448
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) 32.5976 0.0800923
\(408\) 0 0
\(409\) 252.495 0.617346 0.308673 0.951168i \(-0.400115\pi\)
0.308673 + 0.951168i \(0.400115\pi\)
\(410\) 0 0
\(411\) 303.652i 0.738812i
\(412\) 0 0
\(413\) 672.118i 1.62740i
\(414\) 0 0
\(415\) 52.7900 0.127205
\(416\) 0 0
\(417\) −141.046 −0.338240
\(418\) 0 0
\(419\) 43.9500i 0.104893i 0.998624 + 0.0524463i \(0.0167018\pi\)
−0.998624 + 0.0524463i \(0.983298\pi\)
\(420\) 0 0
\(421\) 1.79264i 0.00425806i −0.999998 0.00212903i \(-0.999322\pi\)
0.999998 0.00212903i \(-0.000677692\pi\)
\(422\) 0 0
\(423\) −72.5359 −0.171480
\(424\) 0 0
\(425\) 120.706i 0.284014i
\(426\) 0 0
\(427\) 373.908 0.875662
\(428\) 0 0
\(429\) 45.7961i 0.106751i
\(430\) 0 0
\(431\) 339.722i 0.788217i 0.919064 + 0.394109i \(0.128947\pi\)
−0.919064 + 0.394109i \(0.871053\pi\)
\(432\) 0 0
\(433\) 205.296i 0.474124i 0.971495 + 0.237062i \(0.0761844\pi\)
−0.971495 + 0.237062i \(0.923816\pi\)
\(434\) 0 0
\(435\) 77.5433i 0.178261i
\(436\) 0 0
\(437\) 103.788 292.617i 0.237501 0.669603i
\(438\) 0 0
\(439\) 464.621 1.05836 0.529181 0.848509i \(-0.322499\pi\)
0.529181 + 0.848509i \(0.322499\pi\)
\(440\) 0 0
\(441\) −305.568 −0.692899
\(442\) 0 0
\(443\) −276.866 −0.624980 −0.312490 0.949921i \(-0.601163\pi\)
−0.312490 + 0.949921i \(0.601163\pi\)
\(444\) 0 0
\(445\) −351.715 −0.790371
\(446\) 0 0
\(447\) 297.502i 0.665553i
\(448\) 0 0
\(449\) −269.011 −0.599133 −0.299566 0.954075i \(-0.596842\pi\)
−0.299566 + 0.954075i \(0.596842\pi\)
\(450\) 0 0
\(451\) 48.0895i 0.106629i
\(452\) 0 0
\(453\) 435.711 0.961835
\(454\) 0 0
\(455\) −277.435 −0.609747
\(456\) 0 0
\(457\) 360.775i 0.789443i −0.918801 0.394721i \(-0.870841\pi\)
0.918801 0.394721i \(-0.129159\pi\)
\(458\) 0 0
\(459\) 125.441i 0.273293i
\(460\) 0 0
\(461\) 335.080 0.726855 0.363427 0.931623i \(-0.381606\pi\)
0.363427 + 0.931623i \(0.381606\pi\)
\(462\) 0 0
\(463\) −651.856 −1.40790 −0.703948 0.710252i \(-0.748581\pi\)
−0.703948 + 0.710252i \(0.748581\pi\)
\(464\) 0 0
\(465\) 200.010i 0.430130i
\(466\) 0 0
\(467\) 552.138i 1.18231i −0.806558 0.591154i \(-0.798672\pi\)
0.806558 0.591154i \(-0.201328\pi\)
\(468\) 0 0
\(469\) 620.878 1.32383
\(470\) 0 0
\(471\) 381.421i 0.809812i
\(472\) 0 0
\(473\) 173.791 0.367424
\(474\) 0 0
\(475\) 67.4951i 0.142095i
\(476\) 0 0
\(477\) 46.2405i 0.0969403i
\(478\) 0 0
\(479\) 224.318i 0.468306i −0.972200 0.234153i \(-0.924768\pi\)
0.972200 0.234153i \(-0.0752316\pi\)
\(480\) 0 0
\(481\) 125.808i 0.261554i
\(482\) 0 0
\(483\) 163.563 461.146i 0.338640 0.954754i
\(484\) 0 0
\(485\) 231.933 0.478212
\(486\) 0 0
\(487\) 196.147 0.402766 0.201383 0.979513i \(-0.435456\pi\)
0.201383 + 0.979513i \(0.435456\pi\)
\(488\) 0 0
\(489\) 415.233 0.849146
\(490\) 0 0
\(491\) −273.172 −0.556359 −0.278179 0.960529i \(-0.589731\pi\)
−0.278179 + 0.960529i \(0.589731\pi\)
\(492\) 0 0
\(493\) 483.346i 0.980417i
\(494\) 0 0
\(495\) −17.5582 −0.0354711
\(496\) 0 0
\(497\) 856.647i 1.72364i
\(498\) 0 0
\(499\) 697.703 1.39820 0.699101 0.715023i \(-0.253584\pi\)
0.699101 + 0.715023i \(0.253584\pi\)
\(500\) 0 0
\(501\) 92.8985 0.185426
\(502\) 0 0
\(503\) 159.605i 0.317307i 0.987334 + 0.158653i \(0.0507152\pi\)
−0.987334 + 0.158653i \(0.949285\pi\)
\(504\) 0 0
\(505\) 116.924i 0.231532i
\(506\) 0 0
\(507\) 115.970 0.228738
\(508\) 0 0
\(509\) 808.833 1.58906 0.794531 0.607223i \(-0.207717\pi\)
0.794531 + 0.607223i \(0.207717\pi\)
\(510\) 0 0
\(511\) 529.757i 1.03671i
\(512\) 0 0
\(513\) 70.1430i 0.136731i
\(514\) 0 0
\(515\) −256.601 −0.498255
\(516\) 0 0
\(517\) 63.2856i 0.122409i
\(518\) 0 0
\(519\) 233.669 0.450230
\(520\) 0 0
\(521\) 558.655i 1.07227i 0.844131 + 0.536137i \(0.180117\pi\)
−0.844131 + 0.536137i \(0.819883\pi\)
\(522\) 0 0
\(523\) 393.177i 0.751772i −0.926666 0.375886i \(-0.877338\pi\)
0.926666 0.375886i \(-0.122662\pi\)
\(524\) 0 0
\(525\) 106.368i 0.202606i
\(526\) 0 0
\(527\) 1246.71i 2.36568i
\(528\) 0 0
\(529\) 410.773 + 333.327i 0.776508 + 0.630107i
\(530\) 0 0
\(531\) −164.167 −0.309165
\(532\) 0 0
\(533\) 185.598 0.348213
\(534\) 0 0
\(535\) −180.115 −0.336664
\(536\) 0 0
\(537\) 152.485 0.283957
\(538\) 0 0
\(539\) 266.600i 0.494620i
\(540\) 0 0
\(541\) −922.762 −1.70566 −0.852830 0.522189i \(-0.825115\pi\)
−0.852830 + 0.522189i \(0.825115\pi\)
\(542\) 0 0
\(543\) 185.294i 0.341242i
\(544\) 0 0
\(545\) 305.138 0.559886
\(546\) 0 0
\(547\) −128.168 −0.234311 −0.117155 0.993114i \(-0.537378\pi\)
−0.117155 + 0.993114i \(0.537378\pi\)
\(548\) 0 0
\(549\) 91.3281i 0.166353i
\(550\) 0 0
\(551\) 270.272i 0.490512i
\(552\) 0 0
\(553\) 373.525 0.675452
\(554\) 0 0
\(555\) −48.2345 −0.0869090
\(556\) 0 0
\(557\) 299.230i 0.537217i −0.963249 0.268608i \(-0.913436\pi\)
0.963249 0.268608i \(-0.0865638\pi\)
\(558\) 0 0
\(559\) 670.733i 1.19988i
\(560\) 0 0
\(561\) −109.444 −0.195088
\(562\) 0 0
\(563\) 375.056i 0.666173i 0.942896 + 0.333087i \(0.108090\pi\)
−0.942896 + 0.333087i \(0.891910\pi\)
\(564\) 0 0
\(565\) 405.334 0.717406
\(566\) 0 0
\(567\) 110.541i 0.194958i
\(568\) 0 0
\(569\) 136.669i 0.240191i 0.992762 + 0.120095i \(0.0383201\pi\)
−0.992762 + 0.120095i \(0.961680\pi\)
\(570\) 0 0
\(571\) 86.2562i 0.151062i −0.997143 0.0755308i \(-0.975935\pi\)
0.997143 0.0755308i \(-0.0240651\pi\)
\(572\) 0 0
\(573\) 361.562i 0.630998i
\(574\) 0 0
\(575\) 108.384 + 38.4427i 0.188494 + 0.0668568i
\(576\) 0 0
\(577\) 595.470 1.03201 0.516005 0.856585i \(-0.327418\pi\)
0.516005 + 0.856585i \(0.327418\pi\)
\(578\) 0 0
\(579\) −133.824 −0.231130
\(580\) 0 0
\(581\) 289.966 0.499082
\(582\) 0 0
\(583\) −40.3436 −0.0692000
\(584\) 0 0
\(585\) 67.7643i 0.115836i
\(586\) 0 0
\(587\) −579.280 −0.986849 −0.493425 0.869789i \(-0.664255\pi\)
−0.493425 + 0.869789i \(0.664255\pi\)
\(588\) 0 0
\(589\) 697.123i 1.18357i
\(590\) 0 0
\(591\) −106.408 −0.180048
\(592\) 0 0
\(593\) −785.739 −1.32502 −0.662511 0.749052i \(-0.730509\pi\)
−0.662511 + 0.749052i \(0.730509\pi\)
\(594\) 0 0
\(595\) 663.018i 1.11432i
\(596\) 0 0
\(597\) 182.771i 0.306149i
\(598\) 0 0
\(599\) 898.045 1.49924 0.749620 0.661869i \(-0.230236\pi\)
0.749620 + 0.661869i \(0.230236\pi\)
\(600\) 0 0
\(601\) −291.411 −0.484877 −0.242439 0.970167i \(-0.577947\pi\)
−0.242439 + 0.970167i \(0.577947\pi\)
\(602\) 0 0
\(603\) 151.651i 0.251495i
\(604\) 0 0
\(605\) 255.245i 0.421893i
\(606\) 0 0
\(607\) −736.652 −1.21359 −0.606797 0.794857i \(-0.707546\pi\)
−0.606797 + 0.794857i \(0.707546\pi\)
\(608\) 0 0
\(609\) 425.933i 0.699397i
\(610\) 0 0
\(611\) −244.246 −0.399747
\(612\) 0 0
\(613\) 770.317i 1.25663i −0.777957 0.628317i \(-0.783744\pi\)
0.777957 0.628317i \(-0.216256\pi\)
\(614\) 0 0
\(615\) 71.1579i 0.115704i
\(616\) 0 0
\(617\) 191.122i 0.309760i 0.987933 + 0.154880i \(0.0494991\pi\)
−0.987933 + 0.154880i \(0.950501\pi\)
\(618\) 0 0
\(619\) 184.597i 0.298219i −0.988821 0.149109i \(-0.952359\pi\)
0.988821 0.149109i \(-0.0476406\pi\)
\(620\) 0 0
\(621\) 112.636 + 39.9508i 0.181379 + 0.0643330i
\(622\) 0 0
\(623\) −1931.91 −3.10098
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −61.1978 −0.0976042
\(628\) 0 0
\(629\) −300.657 −0.477992
\(630\) 0 0
\(631\) 250.458i 0.396923i 0.980109 + 0.198461i \(0.0635944\pi\)
−0.980109 + 0.198461i \(0.936406\pi\)
\(632\) 0 0
\(633\) −22.0761 −0.0348753
\(634\) 0 0
\(635\) 296.054i 0.466227i
\(636\) 0 0
\(637\) −1028.92 −1.61526
\(638\) 0 0
\(639\) 209.238 0.327447
\(640\) 0 0
\(641\) 404.622i 0.631235i −0.948887 0.315617i \(-0.897788\pi\)
0.948887 0.315617i \(-0.102212\pi\)
\(642\) 0 0
\(643\) 305.472i 0.475073i 0.971379 + 0.237537i \(0.0763400\pi\)
−0.971379 + 0.237537i \(0.923660\pi\)
\(644\) 0 0
\(645\) −257.158 −0.398695
\(646\) 0 0
\(647\) 163.887 0.253303 0.126651 0.991947i \(-0.459577\pi\)
0.126651 + 0.991947i \(0.459577\pi\)
\(648\) 0 0
\(649\) 143.231i 0.220695i
\(650\) 0 0
\(651\) 1098.62i 1.68759i
\(652\) 0 0
\(653\) −54.7661 −0.0838685 −0.0419343 0.999120i \(-0.513352\pi\)
−0.0419343 + 0.999120i \(0.513352\pi\)
\(654\) 0 0
\(655\) 27.2086i 0.0415399i
\(656\) 0 0
\(657\) −129.395 −0.196948
\(658\) 0 0
\(659\) 575.629i 0.873489i −0.899586 0.436744i \(-0.856131\pi\)
0.899586 0.436744i \(-0.143869\pi\)
\(660\) 0 0
\(661\) 887.745i 1.34303i 0.740990 + 0.671516i \(0.234357\pi\)
−0.740990 + 0.671516i \(0.765643\pi\)
\(662\) 0 0
\(663\) 422.391i 0.637090i
\(664\) 0 0
\(665\) 370.740i 0.557503i
\(666\) 0 0
\(667\) −434.005 153.937i −0.650683 0.230790i
\(668\) 0 0
\(669\) −392.504 −0.586703
\(670\) 0 0
\(671\) 79.6812 0.118750
\(672\) 0 0
\(673\) −881.325 −1.30955 −0.654773 0.755826i \(-0.727236\pi\)
−0.654773 + 0.755826i \(0.727236\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 1159.61i 1.71286i −0.516263 0.856430i \(-0.672677\pi\)
0.516263 0.856430i \(-0.327323\pi\)
\(678\) 0 0
\(679\) 1273.97 1.87624
\(680\) 0 0
\(681\) 325.060i 0.477327i
\(682\) 0 0
\(683\) −41.3337 −0.0605179 −0.0302589 0.999542i \(-0.509633\pi\)
−0.0302589 + 0.999542i \(0.509633\pi\)
\(684\) 0 0
\(685\) 392.013 0.572282
\(686\) 0 0
\(687\) 218.316i 0.317782i
\(688\) 0 0
\(689\) 155.703i 0.225984i
\(690\) 0 0
\(691\) 1086.08 1.57175 0.785873 0.618388i \(-0.212214\pi\)
0.785873 + 0.618388i \(0.212214\pi\)
\(692\) 0 0
\(693\) −96.4441 −0.139169
\(694\) 0 0
\(695\) 182.090i 0.262000i
\(696\) 0 0
\(697\) 443.544i 0.636361i
\(698\) 0 0
\(699\) 718.697 1.02818
\(700\) 0 0
\(701\) 745.281i 1.06317i −0.847006 0.531584i \(-0.821597\pi\)
0.847006 0.531584i \(-0.178403\pi\)
\(702\) 0 0
\(703\) −168.118 −0.239144
\(704\) 0 0
\(705\) 93.6435i 0.132828i
\(706\) 0 0
\(707\) 642.242i 0.908404i
\(708\) 0 0
\(709\) 911.582i 1.28573i 0.765980 + 0.642865i \(0.222254\pi\)
−0.765980 + 0.642865i \(0.777746\pi\)
\(710\) 0 0
\(711\) 91.2346i 0.128319i
\(712\) 0 0
\(713\) 1119.45 + 397.055i 1.57005 + 0.556879i
\(714\) 0 0
\(715\) −59.1225 −0.0826888
\(716\) 0 0
\(717\) −537.666 −0.749882
\(718\) 0 0
\(719\) −831.013 −1.15579 −0.577895 0.816111i \(-0.696126\pi\)
−0.577895 + 0.816111i \(0.696126\pi\)
\(720\) 0 0
\(721\) −1409.47 −1.95488
\(722\) 0 0
\(723\) 78.4638i 0.108525i
\(724\) 0 0
\(725\) −100.108 −0.138080
\(726\) 0 0
\(727\) 1131.77i 1.55677i −0.627785 0.778387i \(-0.716038\pi\)
0.627785 0.778387i \(-0.283962\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) −1602.93 −2.19279
\(732\) 0 0
\(733\) 1230.83i 1.67916i 0.543234 + 0.839581i \(0.317200\pi\)
−0.543234 + 0.839581i \(0.682800\pi\)
\(734\) 0 0
\(735\) 394.487i 0.536717i
\(736\) 0 0
\(737\) 132.312 0.179527
\(738\) 0 0
\(739\) −1319.55 −1.78559 −0.892794 0.450464i \(-0.851258\pi\)
−0.892794 + 0.450464i \(0.851258\pi\)
\(740\) 0 0
\(741\) 236.188i 0.318742i
\(742\) 0 0
\(743\) 80.2284i 0.107979i −0.998542 0.0539895i \(-0.982806\pi\)
0.998542 0.0539895i \(-0.0171938\pi\)
\(744\) 0 0
\(745\) −384.074 −0.515535
\(746\) 0 0
\(747\) 70.8252i 0.0948128i
\(748\) 0 0
\(749\) −989.344 −1.32089
\(750\) 0 0
\(751\) 836.857i 1.11432i 0.830404 + 0.557161i \(0.188110\pi\)
−0.830404 + 0.557161i \(0.811890\pi\)
\(752\) 0 0
\(753\) 65.2512i 0.0866550i
\(754\) 0 0
\(755\) 562.501i 0.745034i
\(756\) 0 0
\(757\) 588.692i 0.777665i 0.921309 + 0.388832i \(0.127121\pi\)
−0.921309 + 0.388832i \(0.872879\pi\)
\(758\) 0 0
\(759\) 34.8560 98.2721i 0.0459235 0.129476i
\(760\) 0 0
\(761\) 365.766 0.480638 0.240319 0.970694i \(-0.422748\pi\)
0.240319 + 0.970694i \(0.422748\pi\)
\(762\) 0 0
\(763\) 1676.07 2.19669
\(764\) 0 0
\(765\) 161.944 0.211692
\(766\) 0 0
\(767\) −552.788 −0.720715
\(768\) 0 0
\(769\) 1113.26i 1.44767i 0.689974 + 0.723835i \(0.257622\pi\)
−0.689974 + 0.723835i \(0.742378\pi\)
\(770\) 0 0
\(771\) −222.766 −0.288931
\(772\) 0 0
\(773\) 598.085i 0.773720i −0.922139 0.386860i \(-0.873560\pi\)
0.922139 0.386860i \(-0.126440\pi\)
\(774\) 0 0
\(775\) 258.212 0.333177
\(776\) 0 0
\(777\) −264.944 −0.340983
\(778\) 0 0
\(779\) 248.016i 0.318378i
\(780\) 0 0
\(781\) 182.555i 0.233745i
\(782\) 0 0
\(783\) −104.035 −0.132868
\(784\) 0 0
\(785\) 492.413 0.627278
\(786\) 0 0
\(787\) 1031.54i 1.31073i −0.755314 0.655363i \(-0.772516\pi\)
0.755314 0.655363i \(-0.227484\pi\)
\(788\) 0 0
\(789\) 547.906i 0.694431i
\(790\) 0 0
\(791\) 2226.43 2.81471
\(792\) 0 0
\(793\) 307.523i 0.387797i
\(794\) 0 0
\(795\) 59.6963 0.0750896
\(796\) 0 0
\(797\) 845.388i 1.06071i −0.847775 0.530356i \(-0.822058\pi\)
0.847775 0.530356i \(-0.177942\pi\)
\(798\) 0 0
\(799\)