Properties

Label 2760.3.g.a.2161.2
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 2161.2
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.47

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} -10.7897i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} -10.7897i q^{7} +3.00000 q^{9} +11.4935i q^{11} +14.7362 q^{13} +3.87298i q^{15} -15.3979i q^{17} +10.6333i q^{19} +18.6883i q^{21} +(4.78633 + 22.4965i) q^{23} -5.00000 q^{25} -5.19615 q^{27} -4.52280 q^{29} +56.9464 q^{31} -19.9073i q^{33} -24.1265 q^{35} +58.3932i q^{37} -25.5238 q^{39} -73.8140 q^{41} +31.6819i q^{43} -6.70820i q^{45} +76.0242 q^{47} -67.4172 q^{49} +26.6699i q^{51} -11.5025i q^{53} +25.7002 q^{55} -18.4174i q^{57} -42.5561 q^{59} +92.6940i q^{61} -32.3690i q^{63} -32.9511i q^{65} +129.520i q^{67} +(-8.29017 - 38.9650i) q^{69} +0.281247 q^{71} -29.9031 q^{73} +8.66025 q^{75} +124.011 q^{77} -97.2130i q^{79} +9.00000 q^{81} +19.7885i q^{83} -34.4307 q^{85} +7.83371 q^{87} +12.7665i q^{89} -158.999i q^{91} -98.6341 q^{93} +23.7768 q^{95} +153.639i q^{97} +34.4804i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 96 q + 288 q^{9} - 16 q^{23} - 480 q^{25} - 80 q^{31} + 80 q^{35} + 48 q^{39} + 112 q^{41} + 32 q^{47} - 688 q^{49} - 80 q^{55} - 496 q^{59} - 96 q^{69} - 416 q^{71} - 320 q^{73} + 864 q^{81} + 192 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1657\) \(1841\) \(2071\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.73205 −0.577350
\(4\) 0 0
\(5\) 2.23607i 0.447214i
\(6\) 0 0
\(7\) 10.7897i 1.54138i −0.637208 0.770692i \(-0.719911\pi\)
0.637208 0.770692i \(-0.280089\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 11.4935i 1.04486i 0.852682 + 0.522430i \(0.174975\pi\)
−0.852682 + 0.522430i \(0.825025\pi\)
\(12\) 0 0
\(13\) 14.7362 1.13355 0.566776 0.823872i \(-0.308191\pi\)
0.566776 + 0.823872i \(0.308191\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 15.3979i 0.905757i −0.891572 0.452879i \(-0.850397\pi\)
0.891572 0.452879i \(-0.149603\pi\)
\(18\) 0 0
\(19\) 10.6333i 0.559647i 0.960052 + 0.279823i \(0.0902759\pi\)
−0.960052 + 0.279823i \(0.909724\pi\)
\(20\) 0 0
\(21\) 18.6883i 0.889918i
\(22\) 0 0
\(23\) 4.78633 + 22.4965i 0.208101 + 0.978107i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) −4.52280 −0.155958 −0.0779792 0.996955i \(-0.524847\pi\)
−0.0779792 + 0.996955i \(0.524847\pi\)
\(30\) 0 0
\(31\) 56.9464 1.83698 0.918491 0.395442i \(-0.129408\pi\)
0.918491 + 0.395442i \(0.129408\pi\)
\(32\) 0 0
\(33\) 19.9073i 0.603251i
\(34\) 0 0
\(35\) −24.1265 −0.689327
\(36\) 0 0
\(37\) 58.3932i 1.57819i 0.614268 + 0.789097i \(0.289451\pi\)
−0.614268 + 0.789097i \(0.710549\pi\)
\(38\) 0 0
\(39\) −25.5238 −0.654457
\(40\) 0 0
\(41\) −73.8140 −1.80034 −0.900170 0.435538i \(-0.856558\pi\)
−0.900170 + 0.435538i \(0.856558\pi\)
\(42\) 0 0
\(43\) 31.6819i 0.736788i 0.929670 + 0.368394i \(0.120092\pi\)
−0.929670 + 0.368394i \(0.879908\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) 76.0242 1.61754 0.808768 0.588128i \(-0.200135\pi\)
0.808768 + 0.588128i \(0.200135\pi\)
\(48\) 0 0
\(49\) −67.4172 −1.37586
\(50\) 0 0
\(51\) 26.6699i 0.522939i
\(52\) 0 0
\(53\) 11.5025i 0.217029i −0.994095 0.108515i \(-0.965391\pi\)
0.994095 0.108515i \(-0.0346094\pi\)
\(54\) 0 0
\(55\) 25.7002 0.467276
\(56\) 0 0
\(57\) 18.4174i 0.323112i
\(58\) 0 0
\(59\) −42.5561 −0.721290 −0.360645 0.932703i \(-0.617443\pi\)
−0.360645 + 0.932703i \(0.617443\pi\)
\(60\) 0 0
\(61\) 92.6940i 1.51957i 0.650172 + 0.759787i \(0.274697\pi\)
−0.650172 + 0.759787i \(0.725303\pi\)
\(62\) 0 0
\(63\) 32.3690i 0.513794i
\(64\) 0 0
\(65\) 32.9511i 0.506940i
\(66\) 0 0
\(67\) 129.520i 1.93314i 0.256411 + 0.966568i \(0.417460\pi\)
−0.256411 + 0.966568i \(0.582540\pi\)
\(68\) 0 0
\(69\) −8.29017 38.9650i −0.120147 0.564711i
\(70\) 0 0
\(71\) 0.281247 0.00396122 0.00198061 0.999998i \(-0.499370\pi\)
0.00198061 + 0.999998i \(0.499370\pi\)
\(72\) 0 0
\(73\) −29.9031 −0.409631 −0.204816 0.978801i \(-0.565659\pi\)
−0.204816 + 0.978801i \(0.565659\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) 124.011 1.61053
\(78\) 0 0
\(79\) 97.2130i 1.23054i −0.788315 0.615272i \(-0.789046\pi\)
0.788315 0.615272i \(-0.210954\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 19.7885i 0.238415i 0.992869 + 0.119208i \(0.0380354\pi\)
−0.992869 + 0.119208i \(0.961965\pi\)
\(84\) 0 0
\(85\) −34.4307 −0.405067
\(86\) 0 0
\(87\) 7.83371 0.0900427
\(88\) 0 0
\(89\) 12.7665i 0.143444i 0.997425 + 0.0717219i \(0.0228494\pi\)
−0.997425 + 0.0717219i \(0.977151\pi\)
\(90\) 0 0
\(91\) 158.999i 1.74724i
\(92\) 0 0
\(93\) −98.6341 −1.06058
\(94\) 0 0
\(95\) 23.7768 0.250282
\(96\) 0 0
\(97\) 153.639i 1.58391i 0.610578 + 0.791956i \(0.290937\pi\)
−0.610578 + 0.791956i \(0.709063\pi\)
\(98\) 0 0
\(99\) 34.4804i 0.348287i
\(100\) 0 0
\(101\) −20.2771 −0.200764 −0.100382 0.994949i \(-0.532006\pi\)
−0.100382 + 0.994949i \(0.532006\pi\)
\(102\) 0 0
\(103\) 48.4339i 0.470232i −0.971967 0.235116i \(-0.924453\pi\)
0.971967 0.235116i \(-0.0755471\pi\)
\(104\) 0 0
\(105\) 41.7883 0.397983
\(106\) 0 0
\(107\) 74.2355i 0.693790i −0.937904 0.346895i \(-0.887236\pi\)
0.937904 0.346895i \(-0.112764\pi\)
\(108\) 0 0
\(109\) 30.8271i 0.282817i 0.989951 + 0.141409i \(0.0451631\pi\)
−0.989951 + 0.141409i \(0.954837\pi\)
\(110\) 0 0
\(111\) 101.140i 0.911171i
\(112\) 0 0
\(113\) 50.5566i 0.447403i −0.974658 0.223702i \(-0.928186\pi\)
0.974658 0.223702i \(-0.0718142\pi\)
\(114\) 0 0
\(115\) 50.3036 10.7026i 0.437423 0.0930657i
\(116\) 0 0
\(117\) 44.2085 0.377851
\(118\) 0 0
\(119\) −166.138 −1.39612
\(120\) 0 0
\(121\) −11.0998 −0.0917341
\(122\) 0 0
\(123\) 127.850 1.03943
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) 132.524 1.04349 0.521747 0.853101i \(-0.325281\pi\)
0.521747 + 0.853101i \(0.325281\pi\)
\(128\) 0 0
\(129\) 54.8746i 0.425385i
\(130\) 0 0
\(131\) 131.397 1.00303 0.501514 0.865149i \(-0.332776\pi\)
0.501514 + 0.865149i \(0.332776\pi\)
\(132\) 0 0
\(133\) 114.730 0.862630
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 88.4658i 0.645736i −0.946444 0.322868i \(-0.895353\pi\)
0.946444 0.322868i \(-0.104647\pi\)
\(138\) 0 0
\(139\) −171.779 −1.23582 −0.617911 0.786248i \(-0.712021\pi\)
−0.617911 + 0.786248i \(0.712021\pi\)
\(140\) 0 0
\(141\) −131.678 −0.933884
\(142\) 0 0
\(143\) 169.370i 1.18440i
\(144\) 0 0
\(145\) 10.1133i 0.0697468i
\(146\) 0 0
\(147\) 116.770 0.794354
\(148\) 0 0
\(149\) 119.944i 0.804996i −0.915421 0.402498i \(-0.868142\pi\)
0.915421 0.402498i \(-0.131858\pi\)
\(150\) 0 0
\(151\) 211.908 1.40336 0.701682 0.712490i \(-0.252433\pi\)
0.701682 + 0.712490i \(0.252433\pi\)
\(152\) 0 0
\(153\) 46.1936i 0.301919i
\(154\) 0 0
\(155\) 127.336i 0.821523i
\(156\) 0 0
\(157\) 9.52690i 0.0606809i −0.999540 0.0303405i \(-0.990341\pi\)
0.999540 0.0303405i \(-0.00965915\pi\)
\(158\) 0 0
\(159\) 19.9230i 0.125302i
\(160\) 0 0
\(161\) 242.730 51.6430i 1.50764 0.320764i
\(162\) 0 0
\(163\) 196.552 1.20584 0.602921 0.797801i \(-0.294003\pi\)
0.602921 + 0.797801i \(0.294003\pi\)
\(164\) 0 0
\(165\) −44.5140 −0.269782
\(166\) 0 0
\(167\) 25.5543 0.153020 0.0765100 0.997069i \(-0.475622\pi\)
0.0765100 + 0.997069i \(0.475622\pi\)
\(168\) 0 0
\(169\) 48.1549 0.284940
\(170\) 0 0
\(171\) 31.8999i 0.186549i
\(172\) 0 0
\(173\) −104.292 −0.602843 −0.301422 0.953491i \(-0.597461\pi\)
−0.301422 + 0.953491i \(0.597461\pi\)
\(174\) 0 0
\(175\) 53.9484i 0.308277i
\(176\) 0 0
\(177\) 73.7094 0.416437
\(178\) 0 0
\(179\) 71.8422 0.401353 0.200676 0.979658i \(-0.435686\pi\)
0.200676 + 0.979658i \(0.435686\pi\)
\(180\) 0 0
\(181\) 240.851i 1.33067i 0.746545 + 0.665335i \(0.231711\pi\)
−0.746545 + 0.665335i \(0.768289\pi\)
\(182\) 0 0
\(183\) 160.551i 0.877326i
\(184\) 0 0
\(185\) 130.571 0.705790
\(186\) 0 0
\(187\) 176.975 0.946390
\(188\) 0 0
\(189\) 56.0648i 0.296639i
\(190\) 0 0
\(191\) 118.234i 0.619024i 0.950896 + 0.309512i \(0.100166\pi\)
−0.950896 + 0.309512i \(0.899834\pi\)
\(192\) 0 0
\(193\) 178.198 0.923305 0.461653 0.887061i \(-0.347257\pi\)
0.461653 + 0.887061i \(0.347257\pi\)
\(194\) 0 0
\(195\) 57.0730i 0.292682i
\(196\) 0 0
\(197\) 204.923 1.04022 0.520109 0.854100i \(-0.325891\pi\)
0.520109 + 0.854100i \(0.325891\pi\)
\(198\) 0 0
\(199\) 24.9515i 0.125385i 0.998033 + 0.0626923i \(0.0199687\pi\)
−0.998033 + 0.0626923i \(0.980031\pi\)
\(200\) 0 0
\(201\) 224.335i 1.11610i
\(202\) 0 0
\(203\) 48.7995i 0.240392i
\(204\) 0 0
\(205\) 165.053i 0.805137i
\(206\) 0 0
\(207\) 14.3590 + 67.4894i 0.0693671 + 0.326036i
\(208\) 0 0
\(209\) −122.213 −0.584753
\(210\) 0 0
\(211\) −115.242 −0.546169 −0.273084 0.961990i \(-0.588044\pi\)
−0.273084 + 0.961990i \(0.588044\pi\)
\(212\) 0 0
\(213\) −0.487133 −0.00228701
\(214\) 0 0
\(215\) 70.8428 0.329501
\(216\) 0 0
\(217\) 614.434i 2.83149i
\(218\) 0 0
\(219\) 51.7936 0.236501
\(220\) 0 0
\(221\) 226.906i 1.02672i
\(222\) 0 0
\(223\) 206.691 0.926864 0.463432 0.886133i \(-0.346618\pi\)
0.463432 + 0.886133i \(0.346618\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 53.1648i 0.234206i −0.993120 0.117103i \(-0.962639\pi\)
0.993120 0.117103i \(-0.0373608\pi\)
\(228\) 0 0
\(229\) 49.7814i 0.217386i 0.994075 + 0.108693i \(0.0346665\pi\)
−0.994075 + 0.108693i \(0.965333\pi\)
\(230\) 0 0
\(231\) −214.793 −0.929840
\(232\) 0 0
\(233\) −259.481 −1.11365 −0.556825 0.830630i \(-0.687981\pi\)
−0.556825 + 0.830630i \(0.687981\pi\)
\(234\) 0 0
\(235\) 169.995i 0.723384i
\(236\) 0 0
\(237\) 168.378i 0.710455i
\(238\) 0 0
\(239\) 31.0729 0.130012 0.0650061 0.997885i \(-0.479293\pi\)
0.0650061 + 0.997885i \(0.479293\pi\)
\(240\) 0 0
\(241\) 187.036i 0.776084i −0.921642 0.388042i \(-0.873152\pi\)
0.921642 0.388042i \(-0.126848\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 150.749i 0.615304i
\(246\) 0 0
\(247\) 156.694i 0.634389i
\(248\) 0 0
\(249\) 34.2746i 0.137649i
\(250\) 0 0
\(251\) 339.867i 1.35405i 0.735960 + 0.677025i \(0.236731\pi\)
−0.735960 + 0.677025i \(0.763269\pi\)
\(252\) 0 0
\(253\) −258.562 + 55.0115i −1.02199 + 0.217437i
\(254\) 0 0
\(255\) 59.6357 0.233866
\(256\) 0 0
\(257\) 43.4012 0.168876 0.0844381 0.996429i \(-0.473090\pi\)
0.0844381 + 0.996429i \(0.473090\pi\)
\(258\) 0 0
\(259\) 630.044 2.43260
\(260\) 0 0
\(261\) −13.5684 −0.0519862
\(262\) 0 0
\(263\) 218.063i 0.829138i 0.910018 + 0.414569i \(0.136068\pi\)
−0.910018 + 0.414569i \(0.863932\pi\)
\(264\) 0 0
\(265\) −25.7205 −0.0970583
\(266\) 0 0
\(267\) 22.1122i 0.0828173i
\(268\) 0 0
\(269\) 239.467 0.890211 0.445105 0.895478i \(-0.353166\pi\)
0.445105 + 0.895478i \(0.353166\pi\)
\(270\) 0 0
\(271\) 171.905 0.634337 0.317168 0.948369i \(-0.397268\pi\)
0.317168 + 0.948369i \(0.397268\pi\)
\(272\) 0 0
\(273\) 275.394i 1.00877i
\(274\) 0 0
\(275\) 57.4673i 0.208972i
\(276\) 0 0
\(277\) −498.670 −1.80025 −0.900127 0.435628i \(-0.856526\pi\)
−0.900127 + 0.435628i \(0.856526\pi\)
\(278\) 0 0
\(279\) 170.839 0.612327
\(280\) 0 0
\(281\) 83.1491i 0.295904i −0.988995 0.147952i \(-0.952732\pi\)
0.988995 0.147952i \(-0.0472682\pi\)
\(282\) 0 0
\(283\) 482.766i 1.70589i 0.522004 + 0.852943i \(0.325185\pi\)
−0.522004 + 0.852943i \(0.674815\pi\)
\(284\) 0 0
\(285\) −41.1825 −0.144500
\(286\) 0 0
\(287\) 796.429i 2.77501i
\(288\) 0 0
\(289\) 51.9055 0.179604
\(290\) 0 0
\(291\) 266.111i 0.914472i
\(292\) 0 0
\(293\) 457.339i 1.56088i 0.625229 + 0.780442i \(0.285006\pi\)
−0.625229 + 0.780442i \(0.714994\pi\)
\(294\) 0 0
\(295\) 95.1584i 0.322571i
\(296\) 0 0
\(297\) 59.7218i 0.201084i
\(298\) 0 0
\(299\) 70.5322 + 331.512i 0.235894 + 1.10874i
\(300\) 0 0
\(301\) 341.837 1.13567
\(302\) 0 0
\(303\) 35.1210 0.115911
\(304\) 0 0
\(305\) 207.270 0.679574
\(306\) 0 0
\(307\) 331.731 1.08056 0.540279 0.841486i \(-0.318319\pi\)
0.540279 + 0.841486i \(0.318319\pi\)
\(308\) 0 0
\(309\) 83.8901i 0.271489i
\(310\) 0 0
\(311\) 596.980 1.91955 0.959775 0.280772i \(-0.0905904\pi\)
0.959775 + 0.280772i \(0.0905904\pi\)
\(312\) 0 0
\(313\) 455.304i 1.45464i 0.686296 + 0.727322i \(0.259235\pi\)
−0.686296 + 0.727322i \(0.740765\pi\)
\(314\) 0 0
\(315\) −72.3794 −0.229776
\(316\) 0 0
\(317\) −121.440 −0.383090 −0.191545 0.981484i \(-0.561350\pi\)
−0.191545 + 0.981484i \(0.561350\pi\)
\(318\) 0 0
\(319\) 51.9826i 0.162955i
\(320\) 0 0
\(321\) 128.580i 0.400560i
\(322\) 0 0
\(323\) 163.730 0.506904
\(324\) 0 0
\(325\) −73.6809 −0.226710
\(326\) 0 0
\(327\) 53.3941i 0.163285i
\(328\) 0 0
\(329\) 820.276i 2.49324i
\(330\) 0 0
\(331\) −289.476 −0.874549 −0.437275 0.899328i \(-0.644056\pi\)
−0.437275 + 0.899328i \(0.644056\pi\)
\(332\) 0 0
\(333\) 175.180i 0.526065i
\(334\) 0 0
\(335\) 289.616 0.864525
\(336\) 0 0
\(337\) 116.256i 0.344974i 0.985012 + 0.172487i \(0.0551803\pi\)
−0.985012 + 0.172487i \(0.944820\pi\)
\(338\) 0 0
\(339\) 87.5666i 0.258309i
\(340\) 0 0
\(341\) 654.512i 1.91939i
\(342\) 0 0
\(343\) 198.716i 0.579347i
\(344\) 0 0
\(345\) −87.1284 + 18.5374i −0.252546 + 0.0537315i
\(346\) 0 0
\(347\) −62.2654 −0.179439 −0.0897196 0.995967i \(-0.528597\pi\)
−0.0897196 + 0.995967i \(0.528597\pi\)
\(348\) 0 0
\(349\) 366.444 1.04998 0.524991 0.851108i \(-0.324069\pi\)
0.524991 + 0.851108i \(0.324069\pi\)
\(350\) 0 0
\(351\) −76.5714 −0.218152
\(352\) 0 0
\(353\) −555.325 −1.57316 −0.786580 0.617489i \(-0.788150\pi\)
−0.786580 + 0.617489i \(0.788150\pi\)
\(354\) 0 0
\(355\) 0.628887i 0.00177151i
\(356\) 0 0
\(357\) 287.760 0.806050
\(358\) 0 0
\(359\) 130.657i 0.363946i 0.983303 + 0.181973i \(0.0582484\pi\)
−0.983303 + 0.181973i \(0.941752\pi\)
\(360\) 0 0
\(361\) 247.933 0.686796
\(362\) 0 0
\(363\) 19.2255 0.0529627
\(364\) 0 0
\(365\) 66.8653i 0.183193i
\(366\) 0 0
\(367\) 477.734i 1.30173i −0.759194 0.650864i \(-0.774407\pi\)
0.759194 0.650864i \(-0.225593\pi\)
\(368\) 0 0
\(369\) −221.442 −0.600114
\(370\) 0 0
\(371\) −124.109 −0.334525
\(372\) 0 0
\(373\) 528.557i 1.41704i −0.705689 0.708522i \(-0.749362\pi\)
0.705689 0.708522i \(-0.250638\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) −66.6487 −0.176787
\(378\) 0 0
\(379\) 357.539i 0.943375i −0.881766 0.471688i \(-0.843645\pi\)
0.881766 0.471688i \(-0.156355\pi\)
\(380\) 0 0
\(381\) −229.538 −0.602461
\(382\) 0 0
\(383\) 276.904i 0.722987i 0.932375 + 0.361493i \(0.117733\pi\)
−0.932375 + 0.361493i \(0.882267\pi\)
\(384\) 0 0
\(385\) 277.297i 0.720251i
\(386\) 0 0
\(387\) 95.0456i 0.245596i
\(388\) 0 0
\(389\) 419.905i 1.07945i −0.841842 0.539724i \(-0.818529\pi\)
0.841842 0.539724i \(-0.181471\pi\)
\(390\) 0 0
\(391\) 346.398 73.6993i 0.885928 0.188489i
\(392\) 0 0
\(393\) −227.586 −0.579099
\(394\) 0 0
\(395\) −217.375 −0.550316
\(396\) 0 0
\(397\) 211.846 0.533617 0.266808 0.963750i \(-0.414031\pi\)
0.266808 + 0.963750i \(0.414031\pi\)
\(398\) 0 0
\(399\) −198.718 −0.498040
\(400\) 0 0
\(401\) 198.458i 0.494909i −0.968900 0.247454i \(-0.920406\pi\)
0.968900 0.247454i \(-0.0795940\pi\)
\(402\) 0 0
\(403\) 839.173 2.08231
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) −671.140 −1.64899
\(408\) 0 0
\(409\) 778.670 1.90384 0.951919 0.306349i \(-0.0991073\pi\)
0.951919 + 0.306349i \(0.0991073\pi\)
\(410\) 0 0
\(411\) 153.227i 0.372816i
\(412\) 0 0
\(413\) 459.167i 1.11178i
\(414\) 0 0
\(415\) 44.2484 0.106623
\(416\) 0 0
\(417\) 297.530 0.713502
\(418\) 0 0
\(419\) 747.867i 1.78489i −0.451160 0.892443i \(-0.648990\pi\)
0.451160 0.892443i \(-0.351010\pi\)
\(420\) 0 0
\(421\) 521.513i 1.23875i −0.785096 0.619374i \(-0.787387\pi\)
0.785096 0.619374i \(-0.212613\pi\)
\(422\) 0 0
\(423\) 228.072 0.539178
\(424\) 0 0
\(425\) 76.9894i 0.181151i
\(426\) 0 0
\(427\) 1000.14 2.34224
\(428\) 0 0
\(429\) 293.357i 0.683816i
\(430\) 0 0
\(431\) 355.459i 0.824730i −0.911019 0.412365i \(-0.864703\pi\)
0.911019 0.412365i \(-0.135297\pi\)
\(432\) 0 0
\(433\) 44.0527i 0.101738i 0.998705 + 0.0508691i \(0.0161991\pi\)
−0.998705 + 0.0508691i \(0.983801\pi\)
\(434\) 0 0
\(435\) 17.5167i 0.0402683i
\(436\) 0 0
\(437\) −239.211 + 50.8944i −0.547394 + 0.116463i
\(438\) 0 0
\(439\) −717.173 −1.63365 −0.816826 0.576884i \(-0.804269\pi\)
−0.816826 + 0.576884i \(0.804269\pi\)
\(440\) 0 0
\(441\) −202.252 −0.458621
\(442\) 0 0
\(443\) 633.931 1.43100 0.715498 0.698615i \(-0.246200\pi\)
0.715498 + 0.698615i \(0.246200\pi\)
\(444\) 0 0
\(445\) 28.5467 0.0641500
\(446\) 0 0
\(447\) 207.750i 0.464765i
\(448\) 0 0
\(449\) −518.240 −1.15421 −0.577104 0.816670i \(-0.695817\pi\)
−0.577104 + 0.816670i \(0.695817\pi\)
\(450\) 0 0
\(451\) 848.379i 1.88111i
\(452\) 0 0
\(453\) −367.035 −0.810232
\(454\) 0 0
\(455\) −355.532 −0.781388
\(456\) 0 0
\(457\) 574.358i 1.25680i −0.777890 0.628400i \(-0.783710\pi\)
0.777890 0.628400i \(-0.216290\pi\)
\(458\) 0 0
\(459\) 80.0097i 0.174313i
\(460\) 0 0
\(461\) −297.239 −0.644770 −0.322385 0.946609i \(-0.604485\pi\)
−0.322385 + 0.946609i \(0.604485\pi\)
\(462\) 0 0
\(463\) −47.9910 −0.103652 −0.0518261 0.998656i \(-0.516504\pi\)
−0.0518261 + 0.998656i \(0.516504\pi\)
\(464\) 0 0
\(465\) 220.553i 0.474307i
\(466\) 0 0
\(467\) 389.865i 0.834828i −0.908716 0.417414i \(-0.862936\pi\)
0.908716 0.417414i \(-0.137064\pi\)
\(468\) 0 0
\(469\) 1397.48 2.97970
\(470\) 0 0
\(471\) 16.5011i 0.0350341i
\(472\) 0 0
\(473\) −364.135 −0.769841
\(474\) 0 0
\(475\) 53.1664i 0.111929i
\(476\) 0 0
\(477\) 34.5076i 0.0723430i
\(478\) 0 0
\(479\) 40.4531i 0.0844533i −0.999108 0.0422266i \(-0.986555\pi\)
0.999108 0.0422266i \(-0.0134452\pi\)
\(480\) 0 0
\(481\) 860.492i 1.78897i
\(482\) 0 0
\(483\) −420.420 + 89.4482i −0.870435 + 0.185193i
\(484\) 0 0
\(485\) 343.548 0.708347
\(486\) 0 0
\(487\) 689.576 1.41597 0.707984 0.706229i \(-0.249605\pi\)
0.707984 + 0.706229i \(0.249605\pi\)
\(488\) 0 0
\(489\) −340.439 −0.696194
\(490\) 0 0
\(491\) 707.866 1.44168 0.720841 0.693101i \(-0.243756\pi\)
0.720841 + 0.693101i \(0.243756\pi\)
\(492\) 0 0
\(493\) 69.6414i 0.141261i
\(494\) 0 0
\(495\) 77.1005 0.155759
\(496\) 0 0
\(497\) 3.03456i 0.00610576i
\(498\) 0 0
\(499\) 315.622 0.632509 0.316254 0.948674i \(-0.397575\pi\)
0.316254 + 0.948674i \(0.397575\pi\)
\(500\) 0 0
\(501\) −44.2614 −0.0883461
\(502\) 0 0
\(503\) 470.530i 0.935446i −0.883875 0.467723i \(-0.845074\pi\)
0.883875 0.467723i \(-0.154926\pi\)
\(504\) 0 0
\(505\) 45.3411i 0.0897843i
\(506\) 0 0
\(507\) −83.4067 −0.164510
\(508\) 0 0
\(509\) 396.564 0.779104 0.389552 0.921004i \(-0.372630\pi\)
0.389552 + 0.921004i \(0.372630\pi\)
\(510\) 0 0
\(511\) 322.645i 0.631399i
\(512\) 0 0
\(513\) 55.2522i 0.107704i
\(514\) 0 0
\(515\) −108.302 −0.210294
\(516\) 0 0
\(517\) 873.781i 1.69010i
\(518\) 0 0
\(519\) 180.639 0.348052
\(520\) 0 0
\(521\) 204.608i 0.392721i −0.980532 0.196360i \(-0.937088\pi\)
0.980532 0.196360i \(-0.0629123\pi\)
\(522\) 0 0
\(523\) 550.203i 1.05201i 0.850481 + 0.526006i \(0.176311\pi\)
−0.850481 + 0.526006i \(0.823689\pi\)
\(524\) 0 0
\(525\) 93.4414i 0.177984i
\(526\) 0 0
\(527\) 876.854i 1.66386i
\(528\) 0 0
\(529\) −483.182 + 215.351i −0.913388 + 0.407091i
\(530\) 0 0
\(531\) −127.668 −0.240430
\(532\) 0 0
\(533\) −1087.74 −2.04078
\(534\) 0 0
\(535\) −165.996 −0.310272
\(536\) 0 0
\(537\) −124.434 −0.231721
\(538\) 0 0
\(539\) 774.858i 1.43758i
\(540\) 0 0
\(541\) −597.569 −1.10456 −0.552282 0.833658i \(-0.686243\pi\)
−0.552282 + 0.833658i \(0.686243\pi\)
\(542\) 0 0
\(543\) 417.167i 0.768263i
\(544\) 0 0
\(545\) 68.9314 0.126480
\(546\) 0 0
\(547\) −79.3357 −0.145038 −0.0725189 0.997367i \(-0.523104\pi\)
−0.0725189 + 0.997367i \(0.523104\pi\)
\(548\) 0 0
\(549\) 278.082i 0.506525i
\(550\) 0 0
\(551\) 48.0922i 0.0872816i
\(552\) 0 0
\(553\) −1048.90 −1.89674
\(554\) 0 0
\(555\) −226.156 −0.407488
\(556\) 0 0
\(557\) 748.441i 1.34370i −0.740687 0.671850i \(-0.765500\pi\)
0.740687 0.671850i \(-0.234500\pi\)
\(558\) 0 0
\(559\) 466.870i 0.835187i
\(560\) 0 0
\(561\) −306.530 −0.546399
\(562\) 0 0
\(563\) 155.587i 0.276354i 0.990408 + 0.138177i \(0.0441243\pi\)
−0.990408 + 0.138177i \(0.955876\pi\)
\(564\) 0 0
\(565\) −113.048 −0.200085
\(566\) 0 0
\(567\) 97.1071i 0.171265i
\(568\) 0 0
\(569\) 886.249i 1.55756i 0.627300 + 0.778778i \(0.284160\pi\)
−0.627300 + 0.778778i \(0.715840\pi\)
\(570\) 0 0
\(571\) 349.302i 0.611737i 0.952074 + 0.305869i \(0.0989469\pi\)
−0.952074 + 0.305869i \(0.901053\pi\)
\(572\) 0 0
\(573\) 204.787i 0.357394i
\(574\) 0 0
\(575\) −23.9316 112.482i −0.0416203 0.195621i
\(576\) 0 0
\(577\) −113.202 −0.196191 −0.0980956 0.995177i \(-0.531275\pi\)
−0.0980956 + 0.995177i \(0.531275\pi\)
\(578\) 0 0
\(579\) −308.648 −0.533071
\(580\) 0 0
\(581\) 213.511 0.367489
\(582\) 0 0
\(583\) 132.204 0.226765
\(584\) 0 0
\(585\) 98.8533i 0.168980i
\(586\) 0 0
\(587\) −842.829 −1.43582 −0.717912 0.696134i \(-0.754902\pi\)
−0.717912 + 0.696134i \(0.754902\pi\)
\(588\) 0 0
\(589\) 605.528i 1.02806i
\(590\) 0 0
\(591\) −354.937 −0.600571
\(592\) 0 0
\(593\) 426.718 0.719591 0.359796 0.933031i \(-0.382846\pi\)
0.359796 + 0.933031i \(0.382846\pi\)
\(594\) 0 0
\(595\) 371.496i 0.624363i
\(596\) 0 0
\(597\) 43.2173i 0.0723909i
\(598\) 0 0
\(599\) −1129.00 −1.88481 −0.942406 0.334471i \(-0.891442\pi\)
−0.942406 + 0.334471i \(0.891442\pi\)
\(600\) 0 0
\(601\) −376.320 −0.626156 −0.313078 0.949727i \(-0.601360\pi\)
−0.313078 + 0.949727i \(0.601360\pi\)
\(602\) 0 0
\(603\) 388.560i 0.644379i
\(604\) 0 0
\(605\) 24.8200i 0.0410247i
\(606\) 0 0
\(607\) −835.834 −1.37699 −0.688496 0.725240i \(-0.741729\pi\)
−0.688496 + 0.725240i \(0.741729\pi\)
\(608\) 0 0
\(609\) 84.5233i 0.138790i
\(610\) 0 0
\(611\) 1120.31 1.83356
\(612\) 0 0
\(613\) 130.322i 0.212597i −0.994334 0.106299i \(-0.966100\pi\)
0.994334 0.106299i \(-0.0339000\pi\)
\(614\) 0 0
\(615\) 285.880i 0.464846i
\(616\) 0 0
\(617\) 256.390i 0.415543i −0.978177 0.207772i \(-0.933379\pi\)
0.978177 0.207772i \(-0.0666211\pi\)
\(618\) 0 0
\(619\) 46.6594i 0.0753786i −0.999290 0.0376893i \(-0.988000\pi\)
0.999290 0.0376893i \(-0.0119997\pi\)
\(620\) 0 0
\(621\) −24.8705 116.895i −0.0400491 0.188237i
\(622\) 0 0
\(623\) 137.746 0.221102
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 211.680 0.337607
\(628\) 0 0
\(629\) 899.131 1.42946
\(630\) 0 0
\(631\) 99.7585i 0.158096i 0.996871 + 0.0790480i \(0.0251880\pi\)
−0.996871 + 0.0790480i \(0.974812\pi\)
\(632\) 0 0
\(633\) 199.604 0.315331
\(634\) 0 0
\(635\) 296.332i 0.466664i
\(636\) 0 0
\(637\) −993.472 −1.55961
\(638\) 0 0
\(639\) 0.843740 0.00132041
\(640\) 0 0
\(641\) 42.7619i 0.0667112i −0.999444 0.0333556i \(-0.989381\pi\)
0.999444 0.0333556i \(-0.0106194\pi\)
\(642\) 0 0
\(643\) 268.507i 0.417585i 0.977960 + 0.208793i \(0.0669534\pi\)
−0.977960 + 0.208793i \(0.933047\pi\)
\(644\) 0 0
\(645\) −122.703 −0.190238
\(646\) 0 0
\(647\) −759.120 −1.17329 −0.586646 0.809843i \(-0.699552\pi\)
−0.586646 + 0.809843i \(0.699552\pi\)
\(648\) 0 0
\(649\) 489.118i 0.753648i
\(650\) 0 0
\(651\) 1064.23i 1.63476i
\(652\) 0 0
\(653\) 658.226 1.00800 0.504001 0.863703i \(-0.331861\pi\)
0.504001 + 0.863703i \(0.331861\pi\)
\(654\) 0 0
\(655\) 293.812i 0.448568i
\(656\) 0 0
\(657\) −89.7092 −0.136544
\(658\) 0 0
\(659\) 485.635i 0.736927i −0.929642 0.368464i \(-0.879884\pi\)
0.929642 0.368464i \(-0.120116\pi\)
\(660\) 0 0
\(661\) 342.321i 0.517883i −0.965893 0.258941i \(-0.916626\pi\)
0.965893 0.258941i \(-0.0833737\pi\)
\(662\) 0 0
\(663\) 393.012i 0.592779i
\(664\) 0 0
\(665\) 256.544i 0.385780i
\(666\) 0 0
\(667\) −21.6476 101.747i −0.0324552 0.152544i
\(668\) 0 0
\(669\) −357.999 −0.535125
\(670\) 0 0
\(671\) −1065.38 −1.58774
\(672\) 0 0
\(673\) 984.166 1.46236 0.731178 0.682187i \(-0.238971\pi\)
0.731178 + 0.682187i \(0.238971\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 164.649i 0.243203i 0.992579 + 0.121602i \(0.0388030\pi\)
−0.992579 + 0.121602i \(0.961197\pi\)
\(678\) 0 0
\(679\) 1657.72 2.44142
\(680\) 0 0
\(681\) 92.0841i 0.135219i
\(682\) 0 0
\(683\) −461.309 −0.675415 −0.337708 0.941251i \(-0.609652\pi\)
−0.337708 + 0.941251i \(0.609652\pi\)
\(684\) 0 0
\(685\) −197.816 −0.288782
\(686\) 0 0
\(687\) 86.2239i 0.125508i
\(688\) 0 0
\(689\) 169.503i 0.246014i
\(690\) 0 0
\(691\) −539.917 −0.781356 −0.390678 0.920527i \(-0.627759\pi\)
−0.390678 + 0.920527i \(0.627759\pi\)
\(692\) 0 0
\(693\) 372.033 0.536844
\(694\) 0 0
\(695\) 384.110i 0.552676i
\(696\) 0 0
\(697\) 1136.58i 1.63067i
\(698\) 0 0
\(699\) 449.433 0.642966
\(700\) 0 0
\(701\) 510.672i 0.728490i 0.931303 + 0.364245i \(0.118673\pi\)
−0.931303 + 0.364245i \(0.881327\pi\)
\(702\) 0 0
\(703\) −620.911 −0.883231
\(704\) 0 0
\(705\) 294.440i 0.417646i
\(706\) 0 0
\(707\) 218.784i 0.309454i
\(708\) 0 0
\(709\) 1093.73i 1.54263i −0.636451 0.771317i \(-0.719598\pi\)
0.636451 0.771317i \(-0.280402\pi\)
\(710\) 0 0
\(711\) 291.639i 0.410182i
\(712\) 0 0
\(713\) 272.564 + 1281.09i 0.382278 + 1.79677i
\(714\) 0 0
\(715\) 378.722 0.529682
\(716\) 0 0
\(717\) −53.8198 −0.0750625
\(718\) 0 0
\(719\) −396.671 −0.551698 −0.275849 0.961201i \(-0.588959\pi\)
−0.275849 + 0.961201i \(0.588959\pi\)
\(720\) 0 0
\(721\) −522.587 −0.724808
\(722\) 0 0
\(723\) 323.956i 0.448072i
\(724\) 0 0
\(725\) 22.6140 0.0311917
\(726\) 0 0
\(727\) 700.150i 0.963068i 0.876427 + 0.481534i \(0.159920\pi\)
−0.876427 + 0.481534i \(0.840080\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 487.833 0.667351
\(732\) 0 0
\(733\) 78.4236i 0.106990i −0.998568 0.0534949i \(-0.982964\pi\)
0.998568 0.0534949i \(-0.0170361\pi\)
\(734\) 0 0
\(735\) 261.106i 0.355246i
\(736\) 0 0
\(737\) −1488.64 −2.01986
\(738\) 0 0
\(739\) 1250.86 1.69264 0.846319 0.532676i \(-0.178813\pi\)
0.846319 + 0.532676i \(0.178813\pi\)
\(740\) 0 0
\(741\) 271.402i 0.366264i
\(742\) 0 0
\(743\) 75.7488i 0.101950i −0.998700 0.0509750i \(-0.983767\pi\)
0.998700 0.0509750i \(-0.0162329\pi\)
\(744\) 0 0
\(745\) −268.204 −0.360005
\(746\) 0 0
\(747\) 59.3654i 0.0794718i
\(748\) 0 0
\(749\) −800.978 −1.06940
\(750\) 0 0
\(751\) 475.572i 0.633252i 0.948550 + 0.316626i \(0.102550\pi\)
−0.948550 + 0.316626i \(0.897450\pi\)
\(752\) 0 0
\(753\) 588.666i 0.781761i
\(754\) 0 0
\(755\) 473.840i 0.627603i
\(756\) 0 0
\(757\) 283.248i 0.374172i −0.982344 0.187086i \(-0.940096\pi\)
0.982344 0.187086i \(-0.0599043\pi\)
\(758\) 0 0
\(759\) 447.843 95.2828i 0.590044 0.125537i
\(760\) 0 0
\(761\) −303.162 −0.398373 −0.199186 0.979962i \(-0.563830\pi\)
−0.199186 + 0.979962i \(0.563830\pi\)
\(762\) 0 0
\(763\) 332.614 0.435930
\(764\) 0 0
\(765\) −103.292 −0.135022
\(766\) 0 0
\(767\) −627.115 −0.817620
\(768\) 0 0
\(769\) 1104.26i 1.43597i −0.696058 0.717985i \(-0.745065\pi\)
0.696058 0.717985i \(-0.254935\pi\)
\(770\) 0 0
\(771\) −75.1731 −0.0975008
\(772\) 0 0
\(773\) 98.5755i 0.127523i −0.997965 0.0637617i \(-0.979690\pi\)
0.997965 0.0637617i \(-0.0203097\pi\)
\(774\) 0 0
\(775\) −284.732 −0.367396
\(776\) 0 0
\(777\) −1091.27 −1.40446
\(778\) 0 0
\(779\) 784.885i 1.00755i
\(780\) 0 0
\(781\) 3.23250i 0.00413892i
\(782\) 0 0
\(783\) 23.5011 0.0300142
\(784\) 0 0
\(785\) −21.3028 −0.0271373
\(786\) 0 0
\(787\) 1154.73i 1.46725i 0.679555 + 0.733625i \(0.262173\pi\)
−0.679555 + 0.733625i \(0.737827\pi\)
\(788\) 0 0
\(789\) 377.697i 0.478703i
\(790\) 0 0
\(791\) −545.490 −0.689620
\(792\) 0 0
\(793\) 1365.95i 1.72252i
\(794\) 0 0
\(795\) 44.5491 0.0560367
\(796\) 0 0
\(797\) 905.937i 1.13668i −0.822793 0.568342i \(-0.807585\pi\)
0.822793 0.568342i \(-0.192415\pi\)
\(798\) 0 0