Properties

Label 2760.3.g.a.2161.8
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 2161.8
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.41

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} -5.01595i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} -5.01595i q^{7} +3.00000 q^{9} +14.3210i q^{11} -4.09528 q^{13} +3.87298i q^{15} +27.0428i q^{17} -20.8125i q^{19} +8.68788i q^{21} +(2.05935 + 22.9076i) q^{23} -5.00000 q^{25} -5.19615 q^{27} +13.9171 q^{29} +10.3779 q^{31} -24.8047i q^{33} -11.2160 q^{35} -57.4834i q^{37} +7.09324 q^{39} -64.5214 q^{41} -40.8745i q^{43} -6.70820i q^{45} +40.1864 q^{47} +23.8402 q^{49} -46.8394i q^{51} -12.2488i q^{53} +32.0228 q^{55} +36.0483i q^{57} +10.7490 q^{59} -7.90396i q^{61} -15.0478i q^{63} +9.15733i q^{65} -102.594i q^{67} +(-3.56690 - 39.6772i) q^{69} -54.4839 q^{71} -107.868 q^{73} +8.66025 q^{75} +71.8335 q^{77} +89.7931i q^{79} +9.00000 q^{81} +62.5763i q^{83} +60.4695 q^{85} -24.1051 q^{87} -46.1734i q^{89} +20.5417i q^{91} -17.9751 q^{93} -46.5382 q^{95} -1.14500i q^{97} +42.9631i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 5.01595i 0.716564i −0.933613 0.358282i \(-0.883363\pi\)
0.933613 0.358282i \(-0.116637\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 14.3210i 1.30191i 0.759116 + 0.650956i \(0.225632\pi\)
−0.759116 + 0.650956i \(0.774368\pi\)
\(12\) 0 0
\(13\) −4.09528 −0.315022 −0.157511 0.987517i \(-0.550347\pi\)
−0.157511 + 0.987517i \(0.550347\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 27.0428i 1.59075i 0.606117 + 0.795376i \(0.292726\pi\)
−0.606117 + 0.795376i \(0.707274\pi\)
\(18\) 0 0
\(19\) 20.8125i 1.09540i −0.836676 0.547698i \(-0.815504\pi\)
0.836676 0.547698i \(-0.184496\pi\)
\(20\) 0 0
\(21\) 8.68788i 0.413709i
\(22\) 0 0
\(23\) 2.05935 + 22.9076i 0.0895370 + 0.995983i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 13.9171 0.479899 0.239949 0.970785i \(-0.422869\pi\)
0.239949 + 0.970785i \(0.422869\pi\)
\(30\) 0 0
\(31\) 10.3779 0.334771 0.167386 0.985891i \(-0.446467\pi\)
0.167386 + 0.985891i \(0.446467\pi\)
\(32\) 0 0
\(33\) 24.8047i 0.751659i
\(34\) 0 0
\(35\) −11.2160 −0.320457
\(36\) 0 0
\(37\) 57.4834i 1.55361i −0.629744 0.776803i \(-0.716840\pi\)
0.629744 0.776803i \(-0.283160\pi\)
\(38\) 0 0
\(39\) 7.09324 0.181878
\(40\) 0 0
\(41\) −64.5214 −1.57369 −0.786846 0.617149i \(-0.788288\pi\)
−0.786846 + 0.617149i \(0.788288\pi\)
\(42\) 0 0
\(43\) 40.8745i 0.950571i −0.879832 0.475285i \(-0.842345\pi\)
0.879832 0.475285i \(-0.157655\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) 40.1864 0.855030 0.427515 0.904008i \(-0.359389\pi\)
0.427515 + 0.904008i \(0.359389\pi\)
\(48\) 0 0
\(49\) 23.8402 0.486536
\(50\) 0 0
\(51\) 46.8394i 0.918421i
\(52\) 0 0
\(53\) 12.2488i 0.231109i −0.993301 0.115555i \(-0.963135\pi\)
0.993301 0.115555i \(-0.0368645\pi\)
\(54\) 0 0
\(55\) 32.0228 0.582232
\(56\) 0 0
\(57\) 36.0483i 0.632427i
\(58\) 0 0
\(59\) 10.7490 0.182187 0.0910936 0.995842i \(-0.470964\pi\)
0.0910936 + 0.995842i \(0.470964\pi\)
\(60\) 0 0
\(61\) 7.90396i 0.129573i −0.997899 0.0647866i \(-0.979363\pi\)
0.997899 0.0647866i \(-0.0206367\pi\)
\(62\) 0 0
\(63\) 15.0478i 0.238855i
\(64\) 0 0
\(65\) 9.15733i 0.140882i
\(66\) 0 0
\(67\) 102.594i 1.53125i −0.643288 0.765624i \(-0.722430\pi\)
0.643288 0.765624i \(-0.277570\pi\)
\(68\) 0 0
\(69\) −3.56690 39.6772i −0.0516942 0.575031i
\(70\) 0 0
\(71\) −54.4839 −0.767379 −0.383690 0.923462i \(-0.625347\pi\)
−0.383690 + 0.923462i \(0.625347\pi\)
\(72\) 0 0
\(73\) −107.868 −1.47764 −0.738819 0.673904i \(-0.764616\pi\)
−0.738819 + 0.673904i \(0.764616\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) 71.8335 0.932903
\(78\) 0 0
\(79\) 89.7931i 1.13662i 0.822814 + 0.568311i \(0.192403\pi\)
−0.822814 + 0.568311i \(0.807597\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 62.5763i 0.753931i 0.926227 + 0.376966i \(0.123032\pi\)
−0.926227 + 0.376966i \(0.876968\pi\)
\(84\) 0 0
\(85\) 60.4695 0.711405
\(86\) 0 0
\(87\) −24.1051 −0.277070
\(88\) 0 0
\(89\) 46.1734i 0.518802i −0.965770 0.259401i \(-0.916475\pi\)
0.965770 0.259401i \(-0.0835252\pi\)
\(90\) 0 0
\(91\) 20.5417i 0.225733i
\(92\) 0 0
\(93\) −17.9751 −0.193280
\(94\) 0 0
\(95\) −46.5382 −0.489876
\(96\) 0 0
\(97\) 1.14500i 0.0118042i −0.999983 0.00590208i \(-0.998121\pi\)
0.999983 0.00590208i \(-0.00187870\pi\)
\(98\) 0 0
\(99\) 42.9631i 0.433970i
\(100\) 0 0
\(101\) −154.327 −1.52799 −0.763996 0.645221i \(-0.776765\pi\)
−0.763996 + 0.645221i \(0.776765\pi\)
\(102\) 0 0
\(103\) 25.9515i 0.251956i −0.992033 0.125978i \(-0.959793\pi\)
0.992033 0.125978i \(-0.0402069\pi\)
\(104\) 0 0
\(105\) 19.4267 0.185016
\(106\) 0 0
\(107\) 135.819i 1.26933i 0.772786 + 0.634667i \(0.218863\pi\)
−0.772786 + 0.634667i \(0.781137\pi\)
\(108\) 0 0
\(109\) 53.8547i 0.494080i 0.969005 + 0.247040i \(0.0794579\pi\)
−0.969005 + 0.247040i \(0.920542\pi\)
\(110\) 0 0
\(111\) 99.5642i 0.896975i
\(112\) 0 0
\(113\) 187.288i 1.65742i −0.559679 0.828710i \(-0.689075\pi\)
0.559679 0.828710i \(-0.310925\pi\)
\(114\) 0 0
\(115\) 51.2230 4.60485i 0.445417 0.0400422i
\(116\) 0 0
\(117\) −12.2858 −0.105007
\(118\) 0 0
\(119\) 135.645 1.13988
\(120\) 0 0
\(121\) −84.0918 −0.694973
\(122\) 0 0
\(123\) 111.754 0.908572
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −100.128 −0.788412 −0.394206 0.919022i \(-0.628980\pi\)
−0.394206 + 0.919022i \(0.628980\pi\)
\(128\) 0 0
\(129\) 70.7968i 0.548812i
\(130\) 0 0
\(131\) 143.876 1.09829 0.549145 0.835727i \(-0.314953\pi\)
0.549145 + 0.835727i \(0.314953\pi\)
\(132\) 0 0
\(133\) −104.395 −0.784922
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 63.6725i 0.464763i −0.972625 0.232381i \(-0.925348\pi\)
0.972625 0.232381i \(-0.0746518\pi\)
\(138\) 0 0
\(139\) −16.2501 −0.116907 −0.0584537 0.998290i \(-0.518617\pi\)
−0.0584537 + 0.998290i \(0.518617\pi\)
\(140\) 0 0
\(141\) −69.6049 −0.493652
\(142\) 0 0
\(143\) 58.6486i 0.410130i
\(144\) 0 0
\(145\) 31.1195i 0.214617i
\(146\) 0 0
\(147\) −41.2925 −0.280901
\(148\) 0 0
\(149\) 47.7678i 0.320589i 0.987069 + 0.160295i \(0.0512444\pi\)
−0.987069 + 0.160295i \(0.948756\pi\)
\(150\) 0 0
\(151\) −178.814 −1.18420 −0.592101 0.805864i \(-0.701701\pi\)
−0.592101 + 0.805864i \(0.701701\pi\)
\(152\) 0 0
\(153\) 81.1283i 0.530250i
\(154\) 0 0
\(155\) 23.2057i 0.149714i
\(156\) 0 0
\(157\) 236.434i 1.50595i 0.658051 + 0.752974i \(0.271381\pi\)
−0.658051 + 0.752974i \(0.728619\pi\)
\(158\) 0 0
\(159\) 21.2155i 0.133431i
\(160\) 0 0
\(161\) 114.903 10.3296i 0.713686 0.0641590i
\(162\) 0 0
\(163\) 140.046 0.859178 0.429589 0.903025i \(-0.358659\pi\)
0.429589 + 0.903025i \(0.358659\pi\)
\(164\) 0 0
\(165\) −55.4651 −0.336152
\(166\) 0 0
\(167\) 53.2885 0.319093 0.159547 0.987190i \(-0.448997\pi\)
0.159547 + 0.987190i \(0.448997\pi\)
\(168\) 0 0
\(169\) −152.229 −0.900761
\(170\) 0 0
\(171\) 62.4376i 0.365132i
\(172\) 0 0
\(173\) −229.196 −1.32484 −0.662418 0.749135i \(-0.730470\pi\)
−0.662418 + 0.749135i \(0.730470\pi\)
\(174\) 0 0
\(175\) 25.0797i 0.143313i
\(176\) 0 0
\(177\) −18.6179 −0.105186
\(178\) 0 0
\(179\) −175.528 −0.980602 −0.490301 0.871553i \(-0.663113\pi\)
−0.490301 + 0.871553i \(0.663113\pi\)
\(180\) 0 0
\(181\) 227.004i 1.25417i 0.778953 + 0.627083i \(0.215751\pi\)
−0.778953 + 0.627083i \(0.784249\pi\)
\(182\) 0 0
\(183\) 13.6901i 0.0748091i
\(184\) 0 0
\(185\) −128.537 −0.694794
\(186\) 0 0
\(187\) −387.280 −2.07102
\(188\) 0 0
\(189\) 26.0636i 0.137903i
\(190\) 0 0
\(191\) 52.8640i 0.276775i 0.990378 + 0.138387i \(0.0441919\pi\)
−0.990378 + 0.138387i \(0.955808\pi\)
\(192\) 0 0
\(193\) −351.731 −1.82244 −0.911220 0.411919i \(-0.864859\pi\)
−0.911220 + 0.411919i \(0.864859\pi\)
\(194\) 0 0
\(195\) 15.8610i 0.0813382i
\(196\) 0 0
\(197\) −244.948 −1.24339 −0.621695 0.783259i \(-0.713556\pi\)
−0.621695 + 0.783259i \(0.713556\pi\)
\(198\) 0 0
\(199\) 160.660i 0.807337i −0.914905 0.403669i \(-0.867735\pi\)
0.914905 0.403669i \(-0.132265\pi\)
\(200\) 0 0
\(201\) 177.697i 0.884067i
\(202\) 0 0
\(203\) 69.8073i 0.343878i
\(204\) 0 0
\(205\) 144.274i 0.703777i
\(206\) 0 0
\(207\) 6.17805 + 68.7229i 0.0298457 + 0.331994i
\(208\) 0 0
\(209\) 298.057 1.42611
\(210\) 0 0
\(211\) 394.621 1.87024 0.935121 0.354329i \(-0.115291\pi\)
0.935121 + 0.354329i \(0.115291\pi\)
\(212\) 0 0
\(213\) 94.3689 0.443047
\(214\) 0 0
\(215\) −91.3983 −0.425108
\(216\) 0 0
\(217\) 52.0551i 0.239885i
\(218\) 0 0
\(219\) 186.832 0.853114
\(220\) 0 0
\(221\) 110.748i 0.501121i
\(222\) 0 0
\(223\) 127.704 0.572662 0.286331 0.958131i \(-0.407564\pi\)
0.286331 + 0.958131i \(0.407564\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 417.523i 1.83931i −0.392729 0.919654i \(-0.628469\pi\)
0.392729 0.919654i \(-0.371531\pi\)
\(228\) 0 0
\(229\) 93.6775i 0.409072i 0.978859 + 0.204536i \(0.0655685\pi\)
−0.978859 + 0.204536i \(0.934431\pi\)
\(230\) 0 0
\(231\) −124.419 −0.538612
\(232\) 0 0
\(233\) −177.697 −0.762649 −0.381324 0.924441i \(-0.624532\pi\)
−0.381324 + 0.924441i \(0.624532\pi\)
\(234\) 0 0
\(235\) 89.8595i 0.382381i
\(236\) 0 0
\(237\) 155.526i 0.656228i
\(238\) 0 0
\(239\) −320.958 −1.34292 −0.671461 0.741040i \(-0.734333\pi\)
−0.671461 + 0.741040i \(0.734333\pi\)
\(240\) 0 0
\(241\) 323.249i 1.34128i −0.741782 0.670641i \(-0.766019\pi\)
0.741782 0.670641i \(-0.233981\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 53.3084i 0.217585i
\(246\) 0 0
\(247\) 85.2331i 0.345073i
\(248\) 0 0
\(249\) 108.385i 0.435282i
\(250\) 0 0
\(251\) 333.280i 1.32781i 0.747818 + 0.663904i \(0.231102\pi\)
−0.747818 + 0.663904i \(0.768898\pi\)
\(252\) 0 0
\(253\) −328.061 + 29.4920i −1.29668 + 0.116569i
\(254\) 0 0
\(255\) −104.736 −0.410730
\(256\) 0 0
\(257\) −297.458 −1.15743 −0.578713 0.815532i \(-0.696445\pi\)
−0.578713 + 0.815532i \(0.696445\pi\)
\(258\) 0 0
\(259\) −288.334 −1.11326
\(260\) 0 0
\(261\) 41.7512 0.159966
\(262\) 0 0
\(263\) 184.541i 0.701675i −0.936436 0.350838i \(-0.885897\pi\)
0.936436 0.350838i \(-0.114103\pi\)
\(264\) 0 0
\(265\) −27.3891 −0.103355
\(266\) 0 0
\(267\) 79.9747i 0.299531i
\(268\) 0 0
\(269\) 487.231 1.81127 0.905634 0.424060i \(-0.139395\pi\)
0.905634 + 0.424060i \(0.139395\pi\)
\(270\) 0 0
\(271\) −46.8344 −0.172821 −0.0864104 0.996260i \(-0.527540\pi\)
−0.0864104 + 0.996260i \(0.527540\pi\)
\(272\) 0 0
\(273\) 35.5793i 0.130327i
\(274\) 0 0
\(275\) 71.6051i 0.260382i
\(276\) 0 0
\(277\) −517.507 −1.86826 −0.934129 0.356937i \(-0.883821\pi\)
−0.934129 + 0.356937i \(0.883821\pi\)
\(278\) 0 0
\(279\) 31.1337 0.111590
\(280\) 0 0
\(281\) 242.086i 0.861516i −0.902468 0.430758i \(-0.858246\pi\)
0.902468 0.430758i \(-0.141754\pi\)
\(282\) 0 0
\(283\) 240.836i 0.851012i −0.904956 0.425506i \(-0.860096\pi\)
0.904956 0.425506i \(-0.139904\pi\)
\(284\) 0 0
\(285\) 80.6066 0.282830
\(286\) 0 0
\(287\) 323.636i 1.12765i
\(288\) 0 0
\(289\) −442.311 −1.53049
\(290\) 0 0
\(291\) 1.98320i 0.00681513i
\(292\) 0 0
\(293\) 144.224i 0.492232i −0.969240 0.246116i \(-0.920846\pi\)
0.969240 0.246116i \(-0.0791544\pi\)
\(294\) 0 0
\(295\) 24.0356i 0.0814766i
\(296\) 0 0
\(297\) 74.4142i 0.250553i
\(298\) 0 0
\(299\) −8.43362 93.8132i −0.0282061 0.313756i
\(300\) 0 0
\(301\) −205.025 −0.681145
\(302\) 0 0
\(303\) 267.303 0.882187
\(304\) 0 0
\(305\) −17.6738 −0.0579469
\(306\) 0 0
\(307\) −213.289 −0.694751 −0.347376 0.937726i \(-0.612927\pi\)
−0.347376 + 0.937726i \(0.612927\pi\)
\(308\) 0 0
\(309\) 44.9493i 0.145467i
\(310\) 0 0
\(311\) −280.257 −0.901149 −0.450574 0.892739i \(-0.648781\pi\)
−0.450574 + 0.892739i \(0.648781\pi\)
\(312\) 0 0
\(313\) 44.1529i 0.141064i −0.997510 0.0705318i \(-0.977530\pi\)
0.997510 0.0705318i \(-0.0224696\pi\)
\(314\) 0 0
\(315\) −33.6480 −0.106819
\(316\) 0 0
\(317\) −457.676 −1.44377 −0.721886 0.692012i \(-0.756725\pi\)
−0.721886 + 0.692012i \(0.756725\pi\)
\(318\) 0 0
\(319\) 199.307i 0.624786i
\(320\) 0 0
\(321\) 235.245i 0.732850i
\(322\) 0 0
\(323\) 562.828 1.74250
\(324\) 0 0
\(325\) 20.4764 0.0630043
\(326\) 0 0
\(327\) 93.2791i 0.285257i
\(328\) 0 0
\(329\) 201.573i 0.612684i
\(330\) 0 0
\(331\) 261.847 0.791078 0.395539 0.918449i \(-0.370558\pi\)
0.395539 + 0.918449i \(0.370558\pi\)
\(332\) 0 0
\(333\) 172.450i 0.517869i
\(334\) 0 0
\(335\) −229.406 −0.684795
\(336\) 0 0
\(337\) 458.566i 1.36073i −0.732874 0.680365i \(-0.761821\pi\)
0.732874 0.680365i \(-0.238179\pi\)
\(338\) 0 0
\(339\) 324.393i 0.956911i
\(340\) 0 0
\(341\) 148.622i 0.435843i
\(342\) 0 0
\(343\) 365.363i 1.06520i
\(344\) 0 0
\(345\) −88.7208 + 7.97583i −0.257162 + 0.0231184i
\(346\) 0 0
\(347\) −18.7902 −0.0541505 −0.0270753 0.999633i \(-0.508619\pi\)
−0.0270753 + 0.999633i \(0.508619\pi\)
\(348\) 0 0
\(349\) 64.9794 0.186187 0.0930936 0.995657i \(-0.470324\pi\)
0.0930936 + 0.995657i \(0.470324\pi\)
\(350\) 0 0
\(351\) 21.2797 0.0606259
\(352\) 0 0
\(353\) 9.31904 0.0263995 0.0131998 0.999913i \(-0.495798\pi\)
0.0131998 + 0.999913i \(0.495798\pi\)
\(354\) 0 0
\(355\) 121.830i 0.343182i
\(356\) 0 0
\(357\) −234.944 −0.658107
\(358\) 0 0
\(359\) 255.330i 0.711226i 0.934633 + 0.355613i \(0.115728\pi\)
−0.934633 + 0.355613i \(0.884272\pi\)
\(360\) 0 0
\(361\) −72.1611 −0.199892
\(362\) 0 0
\(363\) 145.651 0.401243
\(364\) 0 0
\(365\) 241.199i 0.660819i
\(366\) 0 0
\(367\) 203.091i 0.553380i 0.960959 + 0.276690i \(0.0892375\pi\)
−0.960959 + 0.276690i \(0.910762\pi\)
\(368\) 0 0
\(369\) −193.564 −0.524564
\(370\) 0 0
\(371\) −61.4393 −0.165605
\(372\) 0 0
\(373\) 116.897i 0.313397i −0.987646 0.156698i \(-0.949915\pi\)
0.987646 0.156698i \(-0.0500851\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) −56.9943 −0.151179
\(378\) 0 0
\(379\) 377.534i 0.996133i 0.867139 + 0.498067i \(0.165956\pi\)
−0.867139 + 0.498067i \(0.834044\pi\)
\(380\) 0 0
\(381\) 173.427 0.455190
\(382\) 0 0
\(383\) 123.784i 0.323196i 0.986857 + 0.161598i \(0.0516647\pi\)
−0.986857 + 0.161598i \(0.948335\pi\)
\(384\) 0 0
\(385\) 160.625i 0.417207i
\(386\) 0 0
\(387\) 122.624i 0.316857i
\(388\) 0 0
\(389\) 260.381i 0.669360i 0.942332 + 0.334680i \(0.108628\pi\)
−0.942332 + 0.334680i \(0.891372\pi\)
\(390\) 0 0
\(391\) −619.485 + 55.6905i −1.58436 + 0.142431i
\(392\) 0 0
\(393\) −249.201 −0.634098
\(394\) 0 0
\(395\) 200.783 0.508312
\(396\) 0 0
\(397\) 505.931 1.27438 0.637192 0.770705i \(-0.280096\pi\)
0.637192 + 0.770705i \(0.280096\pi\)
\(398\) 0 0
\(399\) 180.817 0.453175
\(400\) 0 0
\(401\) 186.821i 0.465888i 0.972490 + 0.232944i \(0.0748358\pi\)
−0.972490 + 0.232944i \(0.925164\pi\)
\(402\) 0 0
\(403\) −42.5005 −0.105460
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) 823.221 2.02266
\(408\) 0 0
\(409\) −624.495 −1.52688 −0.763441 0.645878i \(-0.776492\pi\)
−0.763441 + 0.645878i \(0.776492\pi\)
\(410\) 0 0
\(411\) 110.284i 0.268331i
\(412\) 0 0
\(413\) 53.9167i 0.130549i
\(414\) 0 0
\(415\) 139.925 0.337168
\(416\) 0 0
\(417\) 28.1460 0.0674965
\(418\) 0 0
\(419\) 391.008i 0.933192i 0.884470 + 0.466596i \(0.154520\pi\)
−0.884470 + 0.466596i \(0.845480\pi\)
\(420\) 0 0
\(421\) 480.994i 1.14250i 0.820775 + 0.571252i \(0.193542\pi\)
−0.820775 + 0.571252i \(0.806458\pi\)
\(422\) 0 0
\(423\) 120.559 0.285010
\(424\) 0 0
\(425\) 135.214i 0.318150i
\(426\) 0 0
\(427\) −39.6459 −0.0928475
\(428\) 0 0
\(429\) 101.582i 0.236789i
\(430\) 0 0
\(431\) 471.625i 1.09426i −0.837049 0.547128i \(-0.815721\pi\)
0.837049 0.547128i \(-0.184279\pi\)
\(432\) 0 0
\(433\) 617.256i 1.42553i −0.701401 0.712767i \(-0.747442\pi\)
0.701401 0.712767i \(-0.252558\pi\)
\(434\) 0 0
\(435\) 53.9006i 0.123909i
\(436\) 0 0
\(437\) 476.765 42.8603i 1.09100 0.0980785i
\(438\) 0 0
\(439\) −214.067 −0.487624 −0.243812 0.969823i \(-0.578398\pi\)
−0.243812 + 0.969823i \(0.578398\pi\)
\(440\) 0 0
\(441\) 71.5207 0.162179
\(442\) 0 0
\(443\) −235.723 −0.532107 −0.266053 0.963958i \(-0.585720\pi\)
−0.266053 + 0.963958i \(0.585720\pi\)
\(444\) 0 0
\(445\) −103.247 −0.232015
\(446\) 0 0
\(447\) 82.7363i 0.185092i
\(448\) 0 0
\(449\) −566.769 −1.26229 −0.631146 0.775664i \(-0.717415\pi\)
−0.631146 + 0.775664i \(0.717415\pi\)
\(450\) 0 0
\(451\) 924.013i 2.04881i
\(452\) 0 0
\(453\) 309.716 0.683699
\(454\) 0 0
\(455\) 45.9327 0.100951
\(456\) 0 0
\(457\) 374.677i 0.819863i 0.912116 + 0.409931i \(0.134447\pi\)
−0.912116 + 0.409931i \(0.865553\pi\)
\(458\) 0 0
\(459\) 140.518i 0.306140i
\(460\) 0 0
\(461\) −424.356 −0.920512 −0.460256 0.887786i \(-0.652242\pi\)
−0.460256 + 0.887786i \(0.652242\pi\)
\(462\) 0 0
\(463\) −330.778 −0.714423 −0.357212 0.934023i \(-0.616273\pi\)
−0.357212 + 0.934023i \(0.616273\pi\)
\(464\) 0 0
\(465\) 40.1935i 0.0864376i
\(466\) 0 0
\(467\) 348.886i 0.747079i 0.927614 + 0.373539i \(0.121856\pi\)
−0.927614 + 0.373539i \(0.878144\pi\)
\(468\) 0 0
\(469\) −514.605 −1.09724
\(470\) 0 0
\(471\) 409.515i 0.869459i
\(472\) 0 0
\(473\) 585.365 1.23756
\(474\) 0 0
\(475\) 104.063i 0.219079i
\(476\) 0 0
\(477\) 36.7463i 0.0770364i
\(478\) 0 0
\(479\) 320.316i 0.668717i −0.942446 0.334359i \(-0.891480\pi\)
0.942446 0.334359i \(-0.108520\pi\)
\(480\) 0 0
\(481\) 235.411i 0.489419i
\(482\) 0 0
\(483\) −199.019 + 17.8914i −0.412047 + 0.0370422i
\(484\) 0 0
\(485\) −2.56030 −0.00527898
\(486\) 0 0
\(487\) 306.331 0.629017 0.314508 0.949255i \(-0.398160\pi\)
0.314508 + 0.949255i \(0.398160\pi\)
\(488\) 0 0
\(489\) −242.567 −0.496046
\(490\) 0 0
\(491\) −78.1853 −0.159237 −0.0796184 0.996825i \(-0.525370\pi\)
−0.0796184 + 0.996825i \(0.525370\pi\)
\(492\) 0 0
\(493\) 376.356i 0.763399i
\(494\) 0 0
\(495\) 96.0684 0.194077
\(496\) 0 0
\(497\) 273.289i 0.549876i
\(498\) 0 0
\(499\) −492.042 −0.986057 −0.493028 0.870013i \(-0.664110\pi\)
−0.493028 + 0.870013i \(0.664110\pi\)
\(500\) 0 0
\(501\) −92.2985 −0.184228
\(502\) 0 0
\(503\) 170.410i 0.338788i 0.985548 + 0.169394i \(0.0541810\pi\)
−0.985548 + 0.169394i \(0.945819\pi\)
\(504\) 0 0
\(505\) 345.086i 0.683339i
\(506\) 0 0
\(507\) 263.668 0.520055
\(508\) 0 0
\(509\) −90.6462 −0.178087 −0.0890434 0.996028i \(-0.528381\pi\)
−0.0890434 + 0.996028i \(0.528381\pi\)
\(510\) 0 0
\(511\) 541.058i 1.05882i
\(512\) 0 0
\(513\) 108.145i 0.210809i
\(514\) 0 0
\(515\) −58.0293 −0.112678
\(516\) 0 0
\(517\) 575.511i 1.11317i
\(518\) 0 0
\(519\) 396.980 0.764894
\(520\) 0 0
\(521\) 155.726i 0.298898i 0.988769 + 0.149449i \(0.0477500\pi\)
−0.988769 + 0.149449i \(0.952250\pi\)
\(522\) 0 0
\(523\) 0.939326i 0.00179604i 1.00000 0.000898018i \(0.000285848\pi\)
−1.00000 0.000898018i \(0.999714\pi\)
\(524\) 0 0
\(525\) 43.4394i 0.0827417i
\(526\) 0 0
\(527\) 280.647i 0.532538i
\(528\) 0 0
\(529\) −520.518 + 94.3496i −0.983966 + 0.178355i
\(530\) 0 0
\(531\) 32.2471 0.0607291
\(532\) 0 0
\(533\) 264.233 0.495747
\(534\) 0 0
\(535\) 303.700 0.567663
\(536\) 0 0
\(537\) 304.023 0.566151
\(538\) 0 0
\(539\) 341.417i 0.633426i
\(540\) 0 0
\(541\) −845.264 −1.56241 −0.781205 0.624275i \(-0.785395\pi\)
−0.781205 + 0.624275i \(0.785395\pi\)
\(542\) 0 0
\(543\) 393.182i 0.724093i
\(544\) 0 0
\(545\) 120.423 0.220959
\(546\) 0 0
\(547\) 1029.41 1.88191 0.940956 0.338528i \(-0.109929\pi\)
0.940956 + 0.338528i \(0.109929\pi\)
\(548\) 0 0
\(549\) 23.7119i 0.0431911i
\(550\) 0 0
\(551\) 289.649i 0.525679i
\(552\) 0 0
\(553\) 450.398 0.814462
\(554\) 0 0
\(555\) 222.632 0.401139
\(556\) 0 0
\(557\) 996.228i 1.78856i −0.447507 0.894280i \(-0.647688\pi\)
0.447507 0.894280i \(-0.352312\pi\)
\(558\) 0 0
\(559\) 167.393i 0.299450i
\(560\) 0 0
\(561\) 670.789 1.19570
\(562\) 0 0
\(563\) 1008.08i 1.79056i −0.445505 0.895279i \(-0.646976\pi\)
0.445505 0.895279i \(-0.353024\pi\)
\(564\) 0 0
\(565\) −418.790 −0.741220
\(566\) 0 0
\(567\) 45.1435i 0.0796183i
\(568\) 0 0
\(569\) 805.065i 1.41488i 0.706775 + 0.707439i \(0.250149\pi\)
−0.706775 + 0.707439i \(0.749851\pi\)
\(570\) 0 0
\(571\) 493.693i 0.864612i 0.901727 + 0.432306i \(0.142300\pi\)
−0.901727 + 0.432306i \(0.857700\pi\)
\(572\) 0 0
\(573\) 91.5631i 0.159796i
\(574\) 0 0
\(575\) −10.2968 114.538i −0.0179074 0.199197i
\(576\) 0 0
\(577\) −83.4667 −0.144656 −0.0723282 0.997381i \(-0.523043\pi\)
−0.0723282 + 0.997381i \(0.523043\pi\)
\(578\) 0 0
\(579\) 609.216 1.05219
\(580\) 0 0
\(581\) 313.879 0.540240
\(582\) 0 0
\(583\) 175.415 0.300884
\(584\) 0 0
\(585\) 27.4720i 0.0469607i
\(586\) 0 0
\(587\) 119.506 0.203588 0.101794 0.994805i \(-0.467542\pi\)
0.101794 + 0.994805i \(0.467542\pi\)
\(588\) 0 0
\(589\) 215.991i 0.366707i
\(590\) 0 0
\(591\) 424.262 0.717872
\(592\) 0 0
\(593\) −632.239 −1.06617 −0.533085 0.846062i \(-0.678968\pi\)
−0.533085 + 0.846062i \(0.678968\pi\)
\(594\) 0 0
\(595\) 303.312i 0.509768i
\(596\) 0 0
\(597\) 278.271i 0.466116i
\(598\) 0 0
\(599\) −122.676 −0.204801 −0.102401 0.994743i \(-0.532652\pi\)
−0.102401 + 0.994743i \(0.532652\pi\)
\(600\) 0 0
\(601\) 153.679 0.255706 0.127853 0.991793i \(-0.459191\pi\)
0.127853 + 0.991793i \(0.459191\pi\)
\(602\) 0 0
\(603\) 307.781i 0.510416i
\(604\) 0 0
\(605\) 188.035i 0.310802i
\(606\) 0 0
\(607\) 126.114 0.207767 0.103883 0.994589i \(-0.466873\pi\)
0.103883 + 0.994589i \(0.466873\pi\)
\(608\) 0 0
\(609\) 120.910i 0.198538i
\(610\) 0 0
\(611\) −164.575 −0.269353
\(612\) 0 0
\(613\) 63.0090i 0.102788i −0.998678 0.0513940i \(-0.983634\pi\)
0.998678 0.0513940i \(-0.0163664\pi\)
\(614\) 0 0
\(615\) 249.890i 0.406326i
\(616\) 0 0
\(617\) 719.648i 1.16637i 0.812341 + 0.583183i \(0.198193\pi\)
−0.812341 + 0.583183i \(0.801807\pi\)
\(618\) 0 0
\(619\) 383.857i 0.620124i 0.950716 + 0.310062i \(0.100350\pi\)
−0.950716 + 0.310062i \(0.899650\pi\)
\(620\) 0 0
\(621\) −10.7007 119.031i −0.0172314 0.191677i
\(622\) 0 0
\(623\) −231.604 −0.371755
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −516.249 −0.823364
\(628\) 0 0
\(629\) 1554.51 2.47140
\(630\) 0 0
\(631\) 406.691i 0.644518i −0.946652 0.322259i \(-0.895558\pi\)
0.946652 0.322259i \(-0.104442\pi\)
\(632\) 0 0
\(633\) −683.504 −1.07978
\(634\) 0 0
\(635\) 223.894i 0.352589i
\(636\) 0 0
\(637\) −97.6325 −0.153269
\(638\) 0 0
\(639\) −163.452 −0.255793
\(640\) 0 0
\(641\) 601.561i 0.938472i 0.883073 + 0.469236i \(0.155471\pi\)
−0.883073 + 0.469236i \(0.844529\pi\)
\(642\) 0 0
\(643\) 921.609i 1.43330i −0.697435 0.716648i \(-0.745676\pi\)
0.697435 0.716648i \(-0.254324\pi\)
\(644\) 0 0
\(645\) 158.306 0.245436
\(646\) 0 0
\(647\) 673.319 1.04068 0.520339 0.853960i \(-0.325806\pi\)
0.520339 + 0.853960i \(0.325806\pi\)
\(648\) 0 0
\(649\) 153.937i 0.237192i
\(650\) 0 0
\(651\) 90.1621i 0.138498i
\(652\) 0 0
\(653\) 961.797 1.47289 0.736445 0.676497i \(-0.236503\pi\)
0.736445 + 0.676497i \(0.236503\pi\)
\(654\) 0 0
\(655\) 321.717i 0.491171i
\(656\) 0 0
\(657\) −323.603 −0.492546
\(658\) 0 0
\(659\) 839.994i 1.27465i −0.770596 0.637324i \(-0.780041\pi\)
0.770596 0.637324i \(-0.219959\pi\)
\(660\) 0 0
\(661\) 1067.98i 1.61571i −0.589382 0.807855i \(-0.700629\pi\)
0.589382 0.807855i \(-0.299371\pi\)
\(662\) 0 0
\(663\) 191.821i 0.289322i
\(664\) 0 0
\(665\) 233.433i 0.351028i
\(666\) 0 0
\(667\) 28.6601 + 318.807i 0.0429687 + 0.477971i
\(668\) 0 0
\(669\) −221.189 −0.330626
\(670\) 0 0
\(671\) 113.193 0.168693
\(672\) 0 0
\(673\) −425.208 −0.631809 −0.315905 0.948791i \(-0.602308\pi\)
−0.315905 + 0.948791i \(0.602308\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 1037.54i 1.53256i 0.642507 + 0.766280i \(0.277894\pi\)
−0.642507 + 0.766280i \(0.722106\pi\)
\(678\) 0 0
\(679\) −5.74328 −0.00845844
\(680\) 0 0
\(681\) 723.171i 1.06193i
\(682\) 0 0
\(683\) 486.938 0.712940 0.356470 0.934307i \(-0.383980\pi\)
0.356470 + 0.934307i \(0.383980\pi\)
\(684\) 0 0
\(685\) −142.376 −0.207848
\(686\) 0 0
\(687\) 162.254i 0.236178i
\(688\) 0 0
\(689\) 50.1622i 0.0728044i
\(690\) 0 0
\(691\) 1356.87 1.96364 0.981819 0.189821i \(-0.0607907\pi\)
0.981819 + 0.189821i \(0.0607907\pi\)
\(692\) 0 0
\(693\) 215.501 0.310968
\(694\) 0 0
\(695\) 36.3364i 0.0522826i
\(696\) 0 0
\(697\) 1744.84i 2.50335i
\(698\) 0 0
\(699\) 307.780 0.440315
\(700\) 0 0
\(701\) 831.928i 1.18677i −0.804918 0.593386i \(-0.797791\pi\)
0.804918 0.593386i \(-0.202209\pi\)
\(702\) 0 0
\(703\) −1196.37 −1.70181
\(704\) 0 0
\(705\) 155.641i 0.220768i
\(706\) 0 0
\(707\) 774.098i 1.09490i
\(708\) 0 0
\(709\) 706.558i 0.996556i 0.867017 + 0.498278i \(0.166034\pi\)
−0.867017 + 0.498278i \(0.833966\pi\)
\(710\) 0 0
\(711\) 269.379i 0.378874i
\(712\) 0 0
\(713\) 21.3718 + 237.733i 0.0299744 + 0.333427i
\(714\) 0 0
\(715\) −131.142 −0.183416
\(716\) 0 0
\(717\) 555.916 0.775336
\(718\) 0 0
\(719\) −179.512 −0.249669 −0.124835 0.992178i \(-0.539840\pi\)
−0.124835 + 0.992178i \(0.539840\pi\)
\(720\) 0 0
\(721\) −130.171 −0.180543
\(722\) 0 0
\(723\) 559.883i 0.774389i
\(724\) 0 0
\(725\) −69.5853 −0.0959798
\(726\) 0 0
\(727\) 262.547i 0.361137i −0.983562 0.180569i \(-0.942206\pi\)
0.983562 0.180569i \(-0.0577938\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 1105.36 1.51212
\(732\) 0 0
\(733\) 627.829i 0.856520i −0.903656 0.428260i \(-0.859127\pi\)
0.903656 0.428260i \(-0.140873\pi\)
\(734\) 0 0
\(735\) 92.3329i 0.125623i
\(736\) 0 0
\(737\) 1469.25 1.99355
\(738\) 0 0
\(739\) 337.605 0.456841 0.228420 0.973563i \(-0.426644\pi\)
0.228420 + 0.973563i \(0.426644\pi\)
\(740\) 0 0
\(741\) 147.628i 0.199228i
\(742\) 0 0
\(743\) 436.216i 0.587101i −0.955944 0.293550i \(-0.905163\pi\)
0.955944 0.293550i \(-0.0948368\pi\)
\(744\) 0 0
\(745\) 106.812 0.143372
\(746\) 0 0
\(747\) 187.729i 0.251310i
\(748\) 0 0
\(749\) 681.260 0.909559
\(750\) 0 0
\(751\) 881.633i 1.17395i 0.809607 + 0.586973i \(0.199680\pi\)
−0.809607 + 0.586973i \(0.800320\pi\)
\(752\) 0 0
\(753\) 577.257i 0.766610i
\(754\) 0 0
\(755\) 399.841i 0.529591i
\(756\) 0 0
\(757\) 1452.78i 1.91913i −0.281495 0.959563i \(-0.590830\pi\)
0.281495 0.959563i \(-0.409170\pi\)
\(758\) 0 0
\(759\) 568.218 51.0817i 0.748640 0.0673013i
\(760\) 0 0
\(761\) 205.662 0.270252 0.135126 0.990828i \(-0.456856\pi\)
0.135126 + 0.990828i \(0.456856\pi\)
\(762\) 0 0
\(763\) 270.132 0.354040
\(764\) 0 0
\(765\) 181.408 0.237135
\(766\) 0 0
\(767\) −44.0204 −0.0573929
\(768\) 0 0
\(769\) 371.221i 0.482732i −0.970434 0.241366i \(-0.922405\pi\)
0.970434 0.241366i \(-0.0775954\pi\)
\(770\) 0 0
\(771\) 515.213 0.668240
\(772\) 0 0
\(773\) 885.458i 1.14548i −0.819736 0.572741i \(-0.805880\pi\)
0.819736 0.572741i \(-0.194120\pi\)
\(774\) 0 0
\(775\) −51.8896 −0.0669543
\(776\) 0 0
\(777\) 499.409 0.642740
\(778\) 0 0
\(779\) 1342.85i 1.72382i
\(780\) 0 0
\(781\) 780.266i 0.999060i
\(782\) 0 0
\(783\) −72.3152 −0.0923566
\(784\) 0 0
\(785\) 528.682 0.673480
\(786\) 0 0
\(787\) 200.375i 0.254606i −0.991864 0.127303i \(-0.959368\pi\)
0.991864 0.127303i \(-0.0406321\pi\)
\(788\) 0 0
\(789\) 319.634i 0.405112i
\(790\) 0 0
\(791\) −939.429 −1.18765
\(792\) 0 0
\(793\) 32.3690i 0.0408183i
\(794\) 0 0
\(795\) 47.4393 0.0596721
\(796\) 0 0
\(797\) 805.665i 1.01087i −0.862864 0.505436i \(-0.831332\pi\)
0.862864 0.505436i \(-0.168668\pi\)
\(798\) 0 0