Properties

Label 2760.3.g.a.2161.9
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 2161.9
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.40

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} -4.66176i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} -4.66176i q^{7} +3.00000 q^{9} -4.25334i q^{11} +13.7492 q^{13} +3.87298i q^{15} -24.0967i q^{17} -25.5990i q^{19} +8.07441i q^{21} +(-10.9359 - 20.2338i) q^{23} -5.00000 q^{25} -5.19615 q^{27} +39.4767 q^{29} -35.4816 q^{31} +7.36700i q^{33} -10.4240 q^{35} -12.1117i q^{37} -23.8142 q^{39} -23.6271 q^{41} +47.0808i q^{43} -6.70820i q^{45} +24.3062 q^{47} +27.2680 q^{49} +41.7368i q^{51} +24.7444i q^{53} -9.51076 q^{55} +44.3389i q^{57} +83.7286 q^{59} -51.1738i q^{61} -13.9853i q^{63} -30.7441i q^{65} -20.8505i q^{67} +(18.9416 + 35.0459i) q^{69} +35.0018 q^{71} -13.4481 q^{73} +8.66025 q^{75} -19.8281 q^{77} -50.1144i q^{79} +9.00000 q^{81} -72.3206i q^{83} -53.8819 q^{85} -68.3757 q^{87} +153.289i q^{89} -64.0953i q^{91} +61.4560 q^{93} -57.2412 q^{95} -102.710i q^{97} -12.7600i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 4.66176i 0.665966i −0.942933 0.332983i \(-0.891945\pi\)
0.942933 0.332983i \(-0.108055\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 4.25334i 0.386667i −0.981133 0.193334i \(-0.938070\pi\)
0.981133 0.193334i \(-0.0619300\pi\)
\(12\) 0 0
\(13\) 13.7492 1.05763 0.528814 0.848738i \(-0.322637\pi\)
0.528814 + 0.848738i \(0.322637\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 24.0967i 1.41745i −0.705483 0.708727i \(-0.749270\pi\)
0.705483 0.708727i \(-0.250730\pi\)
\(18\) 0 0
\(19\) 25.5990i 1.34732i −0.739042 0.673659i \(-0.764722\pi\)
0.739042 0.673659i \(-0.235278\pi\)
\(20\) 0 0
\(21\) 8.07441i 0.384496i
\(22\) 0 0
\(23\) −10.9359 20.2338i −0.475475 0.879729i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 39.4767 1.36127 0.680633 0.732624i \(-0.261705\pi\)
0.680633 + 0.732624i \(0.261705\pi\)
\(30\) 0 0
\(31\) −35.4816 −1.14457 −0.572284 0.820055i \(-0.693943\pi\)
−0.572284 + 0.820055i \(0.693943\pi\)
\(32\) 0 0
\(33\) 7.36700i 0.223242i
\(34\) 0 0
\(35\) −10.4240 −0.297829
\(36\) 0 0
\(37\) 12.1117i 0.327344i −0.986515 0.163672i \(-0.947666\pi\)
0.986515 0.163672i \(-0.0523338\pi\)
\(38\) 0 0
\(39\) −23.8142 −0.610622
\(40\) 0 0
\(41\) −23.6271 −0.576270 −0.288135 0.957590i \(-0.593035\pi\)
−0.288135 + 0.957590i \(0.593035\pi\)
\(42\) 0 0
\(43\) 47.0808i 1.09490i 0.836837 + 0.547452i \(0.184402\pi\)
−0.836837 + 0.547452i \(0.815598\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) 24.3062 0.517152 0.258576 0.965991i \(-0.416747\pi\)
0.258576 + 0.965991i \(0.416747\pi\)
\(48\) 0 0
\(49\) 27.2680 0.556489
\(50\) 0 0
\(51\) 41.7368i 0.818368i
\(52\) 0 0
\(53\) 24.7444i 0.466875i 0.972372 + 0.233438i \(0.0749975\pi\)
−0.972372 + 0.233438i \(0.925003\pi\)
\(54\) 0 0
\(55\) −9.51076 −0.172923
\(56\) 0 0
\(57\) 44.3389i 0.777875i
\(58\) 0 0
\(59\) 83.7286 1.41913 0.709565 0.704640i \(-0.248892\pi\)
0.709565 + 0.704640i \(0.248892\pi\)
\(60\) 0 0
\(61\) 51.1738i 0.838915i −0.907775 0.419457i \(-0.862220\pi\)
0.907775 0.419457i \(-0.137780\pi\)
\(62\) 0 0
\(63\) 13.9853i 0.221989i
\(64\) 0 0
\(65\) 30.7441i 0.472986i
\(66\) 0 0
\(67\) 20.8505i 0.311201i −0.987820 0.155601i \(-0.950269\pi\)
0.987820 0.155601i \(-0.0497313\pi\)
\(68\) 0 0
\(69\) 18.9416 + 35.0459i 0.274516 + 0.507912i
\(70\) 0 0
\(71\) 35.0018 0.492983 0.246491 0.969145i \(-0.420722\pi\)
0.246491 + 0.969145i \(0.420722\pi\)
\(72\) 0 0
\(73\) −13.4481 −0.184220 −0.0921102 0.995749i \(-0.529361\pi\)
−0.0921102 + 0.995749i \(0.529361\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) −19.8281 −0.257507
\(78\) 0 0
\(79\) 50.1144i 0.634360i −0.948365 0.317180i \(-0.897264\pi\)
0.948365 0.317180i \(-0.102736\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 72.3206i 0.871332i −0.900108 0.435666i \(-0.856513\pi\)
0.900108 0.435666i \(-0.143487\pi\)
\(84\) 0 0
\(85\) −53.8819 −0.633905
\(86\) 0 0
\(87\) −68.3757 −0.785928
\(88\) 0 0
\(89\) 153.289i 1.72235i 0.508309 + 0.861175i \(0.330271\pi\)
−0.508309 + 0.861175i \(0.669729\pi\)
\(90\) 0 0
\(91\) 64.0953i 0.704344i
\(92\) 0 0
\(93\) 61.4560 0.660817
\(94\) 0 0
\(95\) −57.2412 −0.602539
\(96\) 0 0
\(97\) 102.710i 1.05887i −0.848351 0.529435i \(-0.822404\pi\)
0.848351 0.529435i \(-0.177596\pi\)
\(98\) 0 0
\(99\) 12.7600i 0.128889i
\(100\) 0 0
\(101\) 47.7174 0.472449 0.236225 0.971698i \(-0.424090\pi\)
0.236225 + 0.971698i \(0.424090\pi\)
\(102\) 0 0
\(103\) 97.0273i 0.942013i 0.882130 + 0.471007i \(0.156109\pi\)
−0.882130 + 0.471007i \(0.843891\pi\)
\(104\) 0 0
\(105\) 18.0549 0.171952
\(106\) 0 0
\(107\) 77.8133i 0.727227i −0.931550 0.363614i \(-0.881543\pi\)
0.931550 0.363614i \(-0.118457\pi\)
\(108\) 0 0
\(109\) 62.5850i 0.574174i 0.957905 + 0.287087i \(0.0926869\pi\)
−0.957905 + 0.287087i \(0.907313\pi\)
\(110\) 0 0
\(111\) 20.9781i 0.188992i
\(112\) 0 0
\(113\) 36.8574i 0.326172i −0.986612 0.163086i \(-0.947855\pi\)
0.986612 0.163086i \(-0.0521448\pi\)
\(114\) 0 0
\(115\) −45.2441 + 24.4535i −0.393427 + 0.212639i
\(116\) 0 0
\(117\) 41.2475 0.352543
\(118\) 0 0
\(119\) −112.333 −0.943977
\(120\) 0 0
\(121\) 102.909 0.850488
\(122\) 0 0
\(123\) 40.9233 0.332710
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −78.7276 −0.619903 −0.309951 0.950752i \(-0.600313\pi\)
−0.309951 + 0.950752i \(0.600313\pi\)
\(128\) 0 0
\(129\) 81.5464i 0.632143i
\(130\) 0 0
\(131\) 42.8980 0.327466 0.163733 0.986505i \(-0.447646\pi\)
0.163733 + 0.986505i \(0.447646\pi\)
\(132\) 0 0
\(133\) −119.337 −0.897268
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 12.8820i 0.0940294i 0.998894 + 0.0470147i \(0.0149708\pi\)
−0.998894 + 0.0470147i \(0.985029\pi\)
\(138\) 0 0
\(139\) −94.4408 −0.679430 −0.339715 0.940528i \(-0.610331\pi\)
−0.339715 + 0.940528i \(0.610331\pi\)
\(140\) 0 0
\(141\) −42.0995 −0.298578
\(142\) 0 0
\(143\) 58.4799i 0.408950i
\(144\) 0 0
\(145\) 88.2727i 0.608777i
\(146\) 0 0
\(147\) −47.2295 −0.321289
\(148\) 0 0
\(149\) 106.924i 0.717609i 0.933413 + 0.358805i \(0.116816\pi\)
−0.933413 + 0.358805i \(0.883184\pi\)
\(150\) 0 0
\(151\) −272.102 −1.80200 −0.901002 0.433816i \(-0.857167\pi\)
−0.901002 + 0.433816i \(0.857167\pi\)
\(152\) 0 0
\(153\) 72.2902i 0.472485i
\(154\) 0 0
\(155\) 79.3394i 0.511867i
\(156\) 0 0
\(157\) 52.1305i 0.332041i 0.986122 + 0.166021i \(0.0530918\pi\)
−0.986122 + 0.166021i \(0.946908\pi\)
\(158\) 0 0
\(159\) 42.8586i 0.269551i
\(160\) 0 0
\(161\) −94.3250 + 50.9807i −0.585870 + 0.316650i
\(162\) 0 0
\(163\) 31.8795 0.195580 0.0977899 0.995207i \(-0.468823\pi\)
0.0977899 + 0.995207i \(0.468823\pi\)
\(164\) 0 0
\(165\) 16.4731 0.0998371
\(166\) 0 0
\(167\) 42.9231 0.257025 0.128512 0.991708i \(-0.458980\pi\)
0.128512 + 0.991708i \(0.458980\pi\)
\(168\) 0 0
\(169\) 20.0395 0.118577
\(170\) 0 0
\(171\) 76.7971i 0.449106i
\(172\) 0 0
\(173\) −203.590 −1.17682 −0.588410 0.808563i \(-0.700246\pi\)
−0.588410 + 0.808563i \(0.700246\pi\)
\(174\) 0 0
\(175\) 23.3088i 0.133193i
\(176\) 0 0
\(177\) −145.022 −0.819335
\(178\) 0 0
\(179\) −319.518 −1.78502 −0.892509 0.451030i \(-0.851057\pi\)
−0.892509 + 0.451030i \(0.851057\pi\)
\(180\) 0 0
\(181\) 34.4654i 0.190417i 0.995457 + 0.0952084i \(0.0303518\pi\)
−0.995457 + 0.0952084i \(0.969648\pi\)
\(182\) 0 0
\(183\) 88.6356i 0.484348i
\(184\) 0 0
\(185\) −27.0826 −0.146393
\(186\) 0 0
\(187\) −102.492 −0.548083
\(188\) 0 0
\(189\) 24.2232i 0.128165i
\(190\) 0 0
\(191\) 93.9125i 0.491688i 0.969309 + 0.245844i \(0.0790652\pi\)
−0.969309 + 0.245844i \(0.920935\pi\)
\(192\) 0 0
\(193\) 264.208 1.36895 0.684476 0.729035i \(-0.260031\pi\)
0.684476 + 0.729035i \(0.260031\pi\)
\(194\) 0 0
\(195\) 53.2503i 0.273078i
\(196\) 0 0
\(197\) −248.093 −1.25936 −0.629679 0.776856i \(-0.716813\pi\)
−0.629679 + 0.776856i \(0.716813\pi\)
\(198\) 0 0
\(199\) 156.904i 0.788460i −0.919012 0.394230i \(-0.871011\pi\)
0.919012 0.394230i \(-0.128989\pi\)
\(200\) 0 0
\(201\) 36.1141i 0.179672i
\(202\) 0 0
\(203\) 184.031i 0.906557i
\(204\) 0 0
\(205\) 52.8318i 0.257716i
\(206\) 0 0
\(207\) −32.8078 60.7013i −0.158492 0.293243i
\(208\) 0 0
\(209\) −108.881 −0.520964
\(210\) 0 0
\(211\) −168.706 −0.799556 −0.399778 0.916612i \(-0.630913\pi\)
−0.399778 + 0.916612i \(0.630913\pi\)
\(212\) 0 0
\(213\) −60.6248 −0.284624
\(214\) 0 0
\(215\) 105.276 0.489656
\(216\) 0 0
\(217\) 165.407i 0.762244i
\(218\) 0 0
\(219\) 23.2928 0.106360
\(220\) 0 0
\(221\) 331.310i 1.49914i
\(222\) 0 0
\(223\) 5.34896 0.0239864 0.0119932 0.999928i \(-0.496182\pi\)
0.0119932 + 0.999928i \(0.496182\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 45.7195i 0.201407i −0.994916 0.100704i \(-0.967891\pi\)
0.994916 0.100704i \(-0.0321094\pi\)
\(228\) 0 0
\(229\) 74.5583i 0.325582i −0.986661 0.162791i \(-0.947950\pi\)
0.986661 0.162791i \(-0.0520496\pi\)
\(230\) 0 0
\(231\) 34.3432 0.148672
\(232\) 0 0
\(233\) 264.901 1.13691 0.568456 0.822714i \(-0.307541\pi\)
0.568456 + 0.822714i \(0.307541\pi\)
\(234\) 0 0
\(235\) 54.3502i 0.231278i
\(236\) 0 0
\(237\) 86.8008i 0.366248i
\(238\) 0 0
\(239\) −15.8209 −0.0661964 −0.0330982 0.999452i \(-0.510537\pi\)
−0.0330982 + 0.999452i \(0.510537\pi\)
\(240\) 0 0
\(241\) 136.798i 0.567627i −0.958880 0.283813i \(-0.908400\pi\)
0.958880 0.283813i \(-0.0915996\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 60.9731i 0.248870i
\(246\) 0 0
\(247\) 351.965i 1.42496i
\(248\) 0 0
\(249\) 125.263i 0.503064i
\(250\) 0 0
\(251\) 76.4348i 0.304521i −0.988340 0.152261i \(-0.951345\pi\)
0.988340 0.152261i \(-0.0486553\pi\)
\(252\) 0 0
\(253\) −86.0611 + 46.5142i −0.340162 + 0.183851i
\(254\) 0 0
\(255\) 93.3262 0.365985
\(256\) 0 0
\(257\) −136.428 −0.530848 −0.265424 0.964132i \(-0.585512\pi\)
−0.265424 + 0.964132i \(0.585512\pi\)
\(258\) 0 0
\(259\) −56.4619 −0.218000
\(260\) 0 0
\(261\) 118.430 0.453756
\(262\) 0 0
\(263\) 252.546i 0.960249i −0.877200 0.480125i \(-0.840591\pi\)
0.877200 0.480125i \(-0.159409\pi\)
\(264\) 0 0
\(265\) 55.3302 0.208793
\(266\) 0 0
\(267\) 265.505i 0.994399i
\(268\) 0 0
\(269\) 227.313 0.845030 0.422515 0.906356i \(-0.361147\pi\)
0.422515 + 0.906356i \(0.361147\pi\)
\(270\) 0 0
\(271\) −161.253 −0.595030 −0.297515 0.954717i \(-0.596158\pi\)
−0.297515 + 0.954717i \(0.596158\pi\)
\(272\) 0 0
\(273\) 111.016i 0.406653i
\(274\) 0 0
\(275\) 21.2667i 0.0773335i
\(276\) 0 0
\(277\) −164.517 −0.593924 −0.296962 0.954889i \(-0.595973\pi\)
−0.296962 + 0.954889i \(0.595973\pi\)
\(278\) 0 0
\(279\) −106.445 −0.381523
\(280\) 0 0
\(281\) 363.810i 1.29470i 0.762194 + 0.647348i \(0.224122\pi\)
−0.762194 + 0.647348i \(0.775878\pi\)
\(282\) 0 0
\(283\) 271.078i 0.957871i −0.877850 0.478936i \(-0.841023\pi\)
0.877850 0.478936i \(-0.158977\pi\)
\(284\) 0 0
\(285\) 99.1447 0.347876
\(286\) 0 0
\(287\) 110.144i 0.383776i
\(288\) 0 0
\(289\) −291.652 −1.00918
\(290\) 0 0
\(291\) 177.900i 0.611339i
\(292\) 0 0
\(293\) 342.140i 1.16771i −0.811857 0.583856i \(-0.801543\pi\)
0.811857 0.583856i \(-0.198457\pi\)
\(294\) 0 0
\(295\) 187.223i 0.634654i
\(296\) 0 0
\(297\) 22.1010i 0.0744142i
\(298\) 0 0
\(299\) −150.360 278.197i −0.502876 0.930426i
\(300\) 0 0
\(301\) 219.480 0.729168
\(302\) 0 0
\(303\) −82.6489 −0.272769
\(304\) 0 0
\(305\) −114.428 −0.375174
\(306\) 0 0
\(307\) −437.617 −1.42546 −0.712732 0.701436i \(-0.752542\pi\)
−0.712732 + 0.701436i \(0.752542\pi\)
\(308\) 0 0
\(309\) 168.056i 0.543871i
\(310\) 0 0
\(311\) 38.4361 0.123589 0.0617943 0.998089i \(-0.480318\pi\)
0.0617943 + 0.998089i \(0.480318\pi\)
\(312\) 0 0
\(313\) 18.5679i 0.0593222i −0.999560 0.0296611i \(-0.990557\pi\)
0.999560 0.0296611i \(-0.00944281\pi\)
\(314\) 0 0
\(315\) −31.2720 −0.0992763
\(316\) 0 0
\(317\) 298.683 0.942217 0.471108 0.882075i \(-0.343854\pi\)
0.471108 + 0.882075i \(0.343854\pi\)
\(318\) 0 0
\(319\) 167.908i 0.526357i
\(320\) 0 0
\(321\) 134.777i 0.419865i
\(322\) 0 0
\(323\) −616.853 −1.90976
\(324\) 0 0
\(325\) −68.7458 −0.211526
\(326\) 0 0
\(327\) 108.400i 0.331500i
\(328\) 0 0
\(329\) 113.310i 0.344406i
\(330\) 0 0
\(331\) −516.170 −1.55943 −0.779713 0.626137i \(-0.784635\pi\)
−0.779713 + 0.626137i \(0.784635\pi\)
\(332\) 0 0
\(333\) 36.3351i 0.109115i
\(334\) 0 0
\(335\) −46.6231 −0.139174
\(336\) 0 0
\(337\) 105.196i 0.312155i 0.987745 + 0.156078i \(0.0498850\pi\)
−0.987745 + 0.156078i \(0.950115\pi\)
\(338\) 0 0
\(339\) 63.8390i 0.188316i
\(340\) 0 0
\(341\) 150.915i 0.442567i
\(342\) 0 0
\(343\) 355.543i 1.03657i
\(344\) 0 0
\(345\) 78.3650 42.3547i 0.227145 0.122767i
\(346\) 0 0
\(347\) 57.0577 0.164431 0.0822157 0.996615i \(-0.473800\pi\)
0.0822157 + 0.996615i \(0.473800\pi\)
\(348\) 0 0
\(349\) −560.826 −1.60695 −0.803476 0.595337i \(-0.797018\pi\)
−0.803476 + 0.595337i \(0.797018\pi\)
\(350\) 0 0
\(351\) −71.4427 −0.203541
\(352\) 0 0
\(353\) 537.587 1.52291 0.761455 0.648218i \(-0.224485\pi\)
0.761455 + 0.648218i \(0.224485\pi\)
\(354\) 0 0
\(355\) 78.2663i 0.220469i
\(356\) 0 0
\(357\) 194.567 0.545005
\(358\) 0 0
\(359\) 555.758i 1.54807i 0.633141 + 0.774037i \(0.281765\pi\)
−0.633141 + 0.774037i \(0.718235\pi\)
\(360\) 0 0
\(361\) −294.311 −0.815267
\(362\) 0 0
\(363\) −178.244 −0.491030
\(364\) 0 0
\(365\) 30.0708i 0.0823858i
\(366\) 0 0
\(367\) 67.9377i 0.185116i 0.995707 + 0.0925582i \(0.0295044\pi\)
−0.995707 + 0.0925582i \(0.970496\pi\)
\(368\) 0 0
\(369\) −70.8813 −0.192090
\(370\) 0 0
\(371\) 115.352 0.310923
\(372\) 0 0
\(373\) 567.325i 1.52098i −0.649350 0.760490i \(-0.724959\pi\)
0.649350 0.760490i \(-0.275041\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) 542.772 1.43971
\(378\) 0 0
\(379\) 433.317i 1.14332i 0.820491 + 0.571659i \(0.193700\pi\)
−0.820491 + 0.571659i \(0.806300\pi\)
\(380\) 0 0
\(381\) 136.360 0.357901
\(382\) 0 0
\(383\) 661.407i 1.72691i −0.504425 0.863456i \(-0.668295\pi\)
0.504425 0.863456i \(-0.331705\pi\)
\(384\) 0 0
\(385\) 44.3369i 0.115161i
\(386\) 0 0
\(387\) 141.243i 0.364968i
\(388\) 0 0
\(389\) 100.973i 0.259572i 0.991542 + 0.129786i \(0.0414290\pi\)
−0.991542 + 0.129786i \(0.958571\pi\)
\(390\) 0 0
\(391\) −487.568 + 263.520i −1.24698 + 0.673965i
\(392\) 0 0
\(393\) −74.3016 −0.189062
\(394\) 0 0
\(395\) −112.059 −0.283694
\(396\) 0 0
\(397\) −123.035 −0.309912 −0.154956 0.987921i \(-0.549524\pi\)
−0.154956 + 0.987921i \(0.549524\pi\)
\(398\) 0 0
\(399\) 206.697 0.518038
\(400\) 0 0
\(401\) 60.9863i 0.152086i 0.997105 + 0.0760428i \(0.0242286\pi\)
−0.997105 + 0.0760428i \(0.975771\pi\)
\(402\) 0 0
\(403\) −487.843 −1.21053
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) −51.5152 −0.126573
\(408\) 0 0
\(409\) −23.7887 −0.0581631 −0.0290816 0.999577i \(-0.509258\pi\)
−0.0290816 + 0.999577i \(0.509258\pi\)
\(410\) 0 0
\(411\) 22.3123i 0.0542879i
\(412\) 0 0
\(413\) 390.323i 0.945092i
\(414\) 0 0
\(415\) −161.714 −0.389672
\(416\) 0 0
\(417\) 163.576 0.392269
\(418\) 0 0
\(419\) 301.074i 0.718554i 0.933231 + 0.359277i \(0.116977\pi\)
−0.933231 + 0.359277i \(0.883023\pi\)
\(420\) 0 0
\(421\) 12.0466i 0.0286143i 0.999898 + 0.0143072i \(0.00455427\pi\)
−0.999898 + 0.0143072i \(0.995446\pi\)
\(422\) 0 0
\(423\) 72.9185 0.172384
\(424\) 0 0
\(425\) 120.484i 0.283491i
\(426\) 0 0
\(427\) −238.560 −0.558689
\(428\) 0 0
\(429\) 101.290i 0.236107i
\(430\) 0 0
\(431\) 165.871i 0.384852i −0.981311 0.192426i \(-0.938364\pi\)
0.981311 0.192426i \(-0.0616356\pi\)
\(432\) 0 0
\(433\) 160.034i 0.369594i 0.982777 + 0.184797i \(0.0591627\pi\)
−0.982777 + 0.184797i \(0.940837\pi\)
\(434\) 0 0
\(435\) 152.893i 0.351478i
\(436\) 0 0
\(437\) −517.965 + 279.949i −1.18528 + 0.640617i
\(438\) 0 0
\(439\) 614.944 1.40078 0.700392 0.713759i \(-0.253009\pi\)
0.700392 + 0.713759i \(0.253009\pi\)
\(440\) 0 0
\(441\) 81.8039 0.185496
\(442\) 0 0
\(443\) 610.421 1.37793 0.688963 0.724796i \(-0.258066\pi\)
0.688963 + 0.724796i \(0.258066\pi\)
\(444\) 0 0
\(445\) 342.765 0.770258
\(446\) 0 0
\(447\) 185.197i 0.414312i
\(448\) 0 0
\(449\) −566.327 −1.26131 −0.630653 0.776065i \(-0.717213\pi\)
−0.630653 + 0.776065i \(0.717213\pi\)
\(450\) 0 0
\(451\) 100.494i 0.222825i
\(452\) 0 0
\(453\) 471.295 1.04039
\(454\) 0 0
\(455\) −143.321 −0.314992
\(456\) 0 0
\(457\) 221.765i 0.485262i −0.970119 0.242631i \(-0.921990\pi\)
0.970119 0.242631i \(-0.0780105\pi\)
\(458\) 0 0
\(459\) 125.210i 0.272789i
\(460\) 0 0
\(461\) −12.0521 −0.0261434 −0.0130717 0.999915i \(-0.504161\pi\)
−0.0130717 + 0.999915i \(0.504161\pi\)
\(462\) 0 0
\(463\) −258.140 −0.557538 −0.278769 0.960358i \(-0.589926\pi\)
−0.278769 + 0.960358i \(0.589926\pi\)
\(464\) 0 0
\(465\) 137.420i 0.295526i
\(466\) 0 0
\(467\) 216.479i 0.463552i 0.972769 + 0.231776i \(0.0744537\pi\)
−0.972769 + 0.231776i \(0.925546\pi\)
\(468\) 0 0
\(469\) −97.2000 −0.207250
\(470\) 0 0
\(471\) 90.2926i 0.191704i
\(472\) 0 0
\(473\) 200.251 0.423363
\(474\) 0 0
\(475\) 127.995i 0.269464i
\(476\) 0 0
\(477\) 74.2332i 0.155625i
\(478\) 0 0
\(479\) 94.3734i 0.197022i −0.995136 0.0985108i \(-0.968592\pi\)
0.995136 0.0985108i \(-0.0314079\pi\)
\(480\) 0 0
\(481\) 166.526i 0.346208i
\(482\) 0 0
\(483\) 163.376 88.3012i 0.338252 0.182818i
\(484\) 0 0
\(485\) −229.667 −0.473541
\(486\) 0 0
\(487\) 442.700 0.909035 0.454518 0.890738i \(-0.349812\pi\)
0.454518 + 0.890738i \(0.349812\pi\)
\(488\) 0 0
\(489\) −55.2169 −0.112918
\(490\) 0 0
\(491\) −622.766 −1.26836 −0.634182 0.773184i \(-0.718663\pi\)
−0.634182 + 0.773184i \(0.718663\pi\)
\(492\) 0 0
\(493\) 951.260i 1.92953i
\(494\) 0 0
\(495\) −28.5323 −0.0576410
\(496\) 0 0
\(497\) 163.170i 0.328310i
\(498\) 0 0
\(499\) −235.999 −0.472944 −0.236472 0.971638i \(-0.575991\pi\)
−0.236472 + 0.971638i \(0.575991\pi\)
\(500\) 0 0
\(501\) −74.3450 −0.148393
\(502\) 0 0
\(503\) 232.171i 0.461572i 0.973004 + 0.230786i \(0.0741298\pi\)
−0.973004 + 0.230786i \(0.925870\pi\)
\(504\) 0 0
\(505\) 106.699i 0.211286i
\(506\) 0 0
\(507\) −34.7094 −0.0684603
\(508\) 0 0
\(509\) 806.081 1.58366 0.791828 0.610744i \(-0.209130\pi\)
0.791828 + 0.610744i \(0.209130\pi\)
\(510\) 0 0
\(511\) 62.6918i 0.122684i
\(512\) 0 0
\(513\) 133.017i 0.259292i
\(514\) 0 0
\(515\) 216.960 0.421281
\(516\) 0 0
\(517\) 103.382i 0.199966i
\(518\) 0 0
\(519\) 352.628 0.679437
\(520\) 0 0
\(521\) 29.3804i 0.0563923i −0.999602 0.0281961i \(-0.991024\pi\)
0.999602 0.0281961i \(-0.00897630\pi\)
\(522\) 0 0
\(523\) 315.284i 0.602837i −0.953492 0.301418i \(-0.902540\pi\)
0.953492 0.301418i \(-0.0974601\pi\)
\(524\) 0 0
\(525\) 40.3720i 0.0768991i
\(526\) 0 0
\(527\) 854.991i 1.62237i
\(528\) 0 0
\(529\) −289.811 + 442.550i −0.547846 + 0.836579i
\(530\) 0 0
\(531\) 251.186 0.473043
\(532\) 0 0
\(533\) −324.853 −0.609480
\(534\) 0 0
\(535\) −173.996 −0.325226
\(536\) 0 0
\(537\) 553.422 1.03058
\(538\) 0 0
\(539\) 115.980i 0.215176i
\(540\) 0 0
\(541\) 865.567 1.59994 0.799969 0.600041i \(-0.204849\pi\)
0.799969 + 0.600041i \(0.204849\pi\)
\(542\) 0 0
\(543\) 59.6959i 0.109937i
\(544\) 0 0
\(545\) 139.944 0.256778
\(546\) 0 0
\(547\) −474.301 −0.867095 −0.433548 0.901131i \(-0.642738\pi\)
−0.433548 + 0.901131i \(0.642738\pi\)
\(548\) 0 0
\(549\) 153.521i 0.279638i
\(550\) 0 0
\(551\) 1010.57i 1.83406i
\(552\) 0 0
\(553\) −233.622 −0.422462
\(554\) 0 0
\(555\) 46.9085 0.0845198
\(556\) 0 0
\(557\) 8.09742i 0.0145376i 0.999974 + 0.00726878i \(0.00231374\pi\)
−0.999974 + 0.00726878i \(0.997686\pi\)
\(558\) 0 0
\(559\) 647.322i 1.15800i
\(560\) 0 0
\(561\) 177.521 0.316436
\(562\) 0 0
\(563\) 568.644i 1.01002i 0.863112 + 0.505012i \(0.168512\pi\)
−0.863112 + 0.505012i \(0.831488\pi\)
\(564\) 0 0
\(565\) −82.4157 −0.145869
\(566\) 0 0
\(567\) 41.9559i 0.0739962i
\(568\) 0 0
\(569\) 878.881i 1.54461i 0.635254 + 0.772303i \(0.280895\pi\)
−0.635254 + 0.772303i \(0.719105\pi\)
\(570\) 0 0
\(571\) 518.757i 0.908506i −0.890873 0.454253i \(-0.849906\pi\)
0.890873 0.454253i \(-0.150094\pi\)
\(572\) 0 0
\(573\) 162.661i 0.283876i
\(574\) 0 0
\(575\) 54.6797 + 101.169i 0.0950951 + 0.175946i
\(576\) 0 0
\(577\) −43.5483 −0.0754737 −0.0377369 0.999288i \(-0.512015\pi\)
−0.0377369 + 0.999288i \(0.512015\pi\)
\(578\) 0 0
\(579\) −457.621 −0.790365
\(580\) 0 0
\(581\) −337.141 −0.580278
\(582\) 0 0
\(583\) 105.246 0.180525
\(584\) 0 0
\(585\) 92.2322i 0.157662i
\(586\) 0 0
\(587\) 251.693 0.428778 0.214389 0.976748i \(-0.431224\pi\)
0.214389 + 0.976748i \(0.431224\pi\)
\(588\) 0 0
\(589\) 908.296i 1.54210i
\(590\) 0 0
\(591\) 429.710 0.727090
\(592\) 0 0
\(593\) 926.581 1.56253 0.781265 0.624199i \(-0.214575\pi\)
0.781265 + 0.624199i \(0.214575\pi\)
\(594\) 0 0
\(595\) 251.185i 0.422159i
\(596\) 0 0
\(597\) 271.765i 0.455218i
\(598\) 0 0
\(599\) 513.994 0.858087 0.429044 0.903284i \(-0.358851\pi\)
0.429044 + 0.903284i \(0.358851\pi\)
\(600\) 0 0
\(601\) −356.434 −0.593068 −0.296534 0.955022i \(-0.595831\pi\)
−0.296534 + 0.955022i \(0.595831\pi\)
\(602\) 0 0
\(603\) 62.5515i 0.103734i
\(604\) 0 0
\(605\) 230.112i 0.380350i
\(606\) 0 0
\(607\) −533.311 −0.878602 −0.439301 0.898340i \(-0.644774\pi\)
−0.439301 + 0.898340i \(0.644774\pi\)
\(608\) 0 0
\(609\) 318.751i 0.523401i
\(610\) 0 0
\(611\) 334.189 0.546955
\(612\) 0 0
\(613\) 560.532i 0.914408i 0.889362 + 0.457204i \(0.151149\pi\)
−0.889362 + 0.457204i \(0.848851\pi\)
\(614\) 0 0
\(615\) 91.5073i 0.148792i
\(616\) 0 0
\(617\) 607.192i 0.984104i 0.870566 + 0.492052i \(0.163753\pi\)
−0.870566 + 0.492052i \(0.836247\pi\)
\(618\) 0 0
\(619\) 580.955i 0.938539i 0.883055 + 0.469269i \(0.155483\pi\)
−0.883055 + 0.469269i \(0.844517\pi\)
\(620\) 0 0
\(621\) 56.8248 + 105.138i 0.0915053 + 0.169304i
\(622\) 0 0
\(623\) 714.597 1.14703
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 188.588 0.300779
\(628\) 0 0
\(629\) −291.853 −0.463995
\(630\) 0 0
\(631\) 484.582i 0.767958i −0.923342 0.383979i \(-0.874553\pi\)
0.923342 0.383979i \(-0.125447\pi\)
\(632\) 0 0
\(633\) 292.208 0.461624
\(634\) 0 0
\(635\) 176.040i 0.277229i
\(636\) 0 0
\(637\) 374.912 0.588559
\(638\) 0 0
\(639\) 105.005 0.164328
\(640\) 0 0
\(641\) 647.562i 1.01024i −0.863050 0.505119i \(-0.831449\pi\)
0.863050 0.505119i \(-0.168551\pi\)
\(642\) 0 0
\(643\) 1145.10i 1.78087i −0.455115 0.890433i \(-0.650402\pi\)
0.455115 0.890433i \(-0.349598\pi\)
\(644\) 0 0
\(645\) −182.343 −0.282703
\(646\) 0 0
\(647\) 193.934 0.299744 0.149872 0.988705i \(-0.452114\pi\)
0.149872 + 0.988705i \(0.452114\pi\)
\(648\) 0 0
\(649\) 356.126i 0.548731i
\(650\) 0 0
\(651\) 286.493i 0.440082i
\(652\) 0 0
\(653\) 581.744 0.890879 0.445440 0.895312i \(-0.353047\pi\)
0.445440 + 0.895312i \(0.353047\pi\)
\(654\) 0 0
\(655\) 95.9229i 0.146447i
\(656\) 0 0
\(657\) −40.3443 −0.0614068
\(658\) 0 0
\(659\) 121.497i 0.184366i −0.995742 0.0921832i \(-0.970615\pi\)
0.995742 0.0921832i \(-0.0293845\pi\)
\(660\) 0 0
\(661\) 466.856i 0.706287i 0.935569 + 0.353143i \(0.114887\pi\)
−0.935569 + 0.353143i \(0.885113\pi\)
\(662\) 0 0
\(663\) 573.845i 0.865529i
\(664\) 0 0
\(665\) 266.845i 0.401270i
\(666\) 0 0
\(667\) −431.715 798.763i −0.647249 1.19755i
\(668\) 0 0
\(669\) −9.26468 −0.0138485
\(670\) 0 0
\(671\) −217.660 −0.324381
\(672\) 0 0
\(673\) 301.067 0.447351 0.223676 0.974664i \(-0.428194\pi\)
0.223676 + 0.974664i \(0.428194\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 941.302i 1.39040i 0.718815 + 0.695201i \(0.244685\pi\)
−0.718815 + 0.695201i \(0.755315\pi\)
\(678\) 0 0
\(679\) −478.811 −0.705171
\(680\) 0 0
\(681\) 79.1885i 0.116283i
\(682\) 0 0
\(683\) −357.401 −0.523281 −0.261640 0.965165i \(-0.584263\pi\)
−0.261640 + 0.965165i \(0.584263\pi\)
\(684\) 0 0
\(685\) 28.8051 0.0420512
\(686\) 0 0
\(687\) 129.139i 0.187975i
\(688\) 0 0
\(689\) 340.215i 0.493780i
\(690\) 0 0
\(691\) 635.656 0.919907 0.459954 0.887943i \(-0.347866\pi\)
0.459954 + 0.887943i \(0.347866\pi\)
\(692\) 0 0
\(693\) −59.4842 −0.0858357
\(694\) 0 0
\(695\) 211.176i 0.303850i
\(696\) 0 0
\(697\) 569.336i 0.816837i
\(698\) 0 0
\(699\) −458.821 −0.656397
\(700\) 0 0
\(701\) 185.616i 0.264788i 0.991197 + 0.132394i \(0.0422663\pi\)
−0.991197 + 0.132394i \(0.957734\pi\)
\(702\) 0 0
\(703\) −310.048 −0.441036
\(704\) 0 0
\(705\) 94.1374i 0.133528i
\(706\) 0 0
\(707\) 222.447i 0.314635i
\(708\) 0 0
\(709\) 134.870i 0.190225i −0.995467 0.0951125i \(-0.969679\pi\)
0.995467 0.0951125i \(-0.0303211\pi\)
\(710\) 0 0
\(711\) 150.343i 0.211453i
\(712\) 0 0
\(713\) 388.025 + 717.927i 0.544214 + 1.00691i
\(714\) 0 0
\(715\) −130.765 −0.182888
\(716\) 0 0
\(717\) 27.4027 0.0382185
\(718\) 0 0
\(719\) 632.645 0.879896 0.439948 0.898023i \(-0.354997\pi\)
0.439948 + 0.898023i \(0.354997\pi\)
\(720\) 0 0
\(721\) 452.318 0.627349
\(722\) 0 0
\(723\) 236.941i 0.327719i
\(724\) 0 0
\(725\) −197.384 −0.272253
\(726\) 0 0
\(727\) 314.817i 0.433036i 0.976279 + 0.216518i \(0.0694700\pi\)
−0.976279 + 0.216518i \(0.930530\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 1134.49 1.55198
\(732\) 0 0
\(733\) 323.931i 0.441925i 0.975282 + 0.220963i \(0.0709199\pi\)
−0.975282 + 0.220963i \(0.929080\pi\)
\(734\) 0 0
\(735\) 105.608i 0.143685i
\(736\) 0 0
\(737\) −88.6843 −0.120331
\(738\) 0 0
\(739\) −607.960 −0.822680 −0.411340 0.911482i \(-0.634939\pi\)
−0.411340 + 0.911482i \(0.634939\pi\)
\(740\) 0 0
\(741\) 609.622i 0.822702i
\(742\) 0 0
\(743\) 882.588i 1.18787i 0.804513 + 0.593935i \(0.202427\pi\)
−0.804513 + 0.593935i \(0.797573\pi\)
\(744\) 0 0
\(745\) 239.089 0.320925
\(746\) 0 0
\(747\) 216.962i 0.290444i
\(748\) 0 0
\(749\) −362.747 −0.484308
\(750\) 0 0
\(751\) 804.965i 1.07186i 0.844263 + 0.535929i \(0.180038\pi\)
−0.844263 + 0.535929i \(0.819962\pi\)
\(752\) 0 0
\(753\) 132.389i 0.175815i
\(754\) 0 0
\(755\) 608.440i 0.805880i
\(756\) 0 0
\(757\) 1169.26i 1.54460i −0.635261 0.772298i \(-0.719107\pi\)
0.635261 0.772298i \(-0.280893\pi\)
\(758\) 0 0
\(759\) 149.062 80.5650i 0.196393 0.106146i
\(760\) 0 0
\(761\) −899.032 −1.18138 −0.590691 0.806898i \(-0.701145\pi\)
−0.590691 + 0.806898i \(0.701145\pi\)
\(762\) 0 0
\(763\) 291.756 0.382380
\(764\) 0 0
\(765\) −161.646 −0.211302
\(766\) 0 0
\(767\) 1151.20 1.50091
\(768\) 0 0
\(769\) 1000.49i 1.30103i −0.759494 0.650514i \(-0.774553\pi\)
0.759494 0.650514i \(-0.225447\pi\)
\(770\) 0 0
\(771\) 236.300 0.306485
\(772\) 0 0
\(773\) 145.852i 0.188683i −0.995540 0.0943415i \(-0.969925\pi\)
0.995540 0.0943415i \(-0.0300746\pi\)
\(774\) 0 0
\(775\) 177.408 0.228914
\(776\) 0 0
\(777\) 97.7949 0.125862
\(778\) 0 0
\(779\) 604.831i 0.776420i
\(780\) 0 0
\(781\) 148.874i 0.190620i
\(782\) 0 0
\(783\) −205.127 −0.261976
\(784\) 0 0
\(785\) 116.567 0.148493
\(786\) 0 0
\(787\) 1052.12i 1.33688i −0.743766 0.668440i \(-0.766962\pi\)
0.743766 0.668440i \(-0.233038\pi\)
\(788\) 0 0
\(789\) 437.422i 0.554400i
\(790\) 0 0
\(791\) −171.821 −0.217219
\(792\) 0 0
\(793\) 703.597i 0.887260i
\(794\) 0 0
\(795\) −95.8346 −0.120547
\(796\) 0 0
\(797\) 575.129i 0.721618i −0.932640 0.360809i \(-0.882501\pi\)
0.932640 0.360809i \(-0.117499\pi\)
\(798\) 0 0