Properties

Label 2760.3.g.a.2161.10
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 2161.10
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.39

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} -2.69051i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} -2.69051i q^{7} +3.00000 q^{9} -21.3735i q^{11} -18.3139 q^{13} +3.87298i q^{15} +26.4299i q^{17} +14.5414i q^{19} +4.66010i q^{21} +(-8.29751 + 21.4511i) q^{23} -5.00000 q^{25} -5.19615 q^{27} +48.8802 q^{29} -8.01485 q^{31} +37.0200i q^{33} -6.01617 q^{35} +36.9406i q^{37} +31.7205 q^{39} -12.0057 q^{41} +33.1839i q^{43} -6.70820i q^{45} +48.1910 q^{47} +41.7611 q^{49} -45.7779i q^{51} +63.1698i q^{53} -47.7926 q^{55} -25.1864i q^{57} +21.5809 q^{59} -82.0890i q^{61} -8.07154i q^{63} +40.9510i q^{65} -27.9275i q^{67} +(14.3717 - 37.1545i) q^{69} +20.6113 q^{71} +20.4485 q^{73} +8.66025 q^{75} -57.5056 q^{77} -6.01518i q^{79} +9.00000 q^{81} -45.2575i q^{83} +59.0990 q^{85} -84.6631 q^{87} -3.60325i q^{89} +49.2737i q^{91} +13.8821 q^{93} +32.5156 q^{95} +35.7362i q^{97} -64.1204i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 2.69051i 0.384359i −0.981360 0.192179i \(-0.938444\pi\)
0.981360 0.192179i \(-0.0615556\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 21.3735i 1.94304i −0.236950 0.971522i \(-0.576148\pi\)
0.236950 0.971522i \(-0.423852\pi\)
\(12\) 0 0
\(13\) −18.3139 −1.40876 −0.704379 0.709824i \(-0.748775\pi\)
−0.704379 + 0.709824i \(0.748775\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 26.4299i 1.55470i 0.629070 + 0.777349i \(0.283436\pi\)
−0.629070 + 0.777349i \(0.716564\pi\)
\(18\) 0 0
\(19\) 14.5414i 0.765337i 0.923886 + 0.382668i \(0.124995\pi\)
−0.923886 + 0.382668i \(0.875005\pi\)
\(20\) 0 0
\(21\) 4.66010i 0.221910i
\(22\) 0 0
\(23\) −8.29751 + 21.4511i −0.360761 + 0.932658i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 48.8802 1.68553 0.842763 0.538285i \(-0.180928\pi\)
0.842763 + 0.538285i \(0.180928\pi\)
\(30\) 0 0
\(31\) −8.01485 −0.258543 −0.129272 0.991609i \(-0.541264\pi\)
−0.129272 + 0.991609i \(0.541264\pi\)
\(32\) 0 0
\(33\) 37.0200i 1.12182i
\(34\) 0 0
\(35\) −6.01617 −0.171891
\(36\) 0 0
\(37\) 36.9406i 0.998393i 0.866489 + 0.499197i \(0.166372\pi\)
−0.866489 + 0.499197i \(0.833628\pi\)
\(38\) 0 0
\(39\) 31.7205 0.813347
\(40\) 0 0
\(41\) −12.0057 −0.292822 −0.146411 0.989224i \(-0.546772\pi\)
−0.146411 + 0.989224i \(0.546772\pi\)
\(42\) 0 0
\(43\) 33.1839i 0.771719i 0.922558 + 0.385860i \(0.126095\pi\)
−0.922558 + 0.385860i \(0.873905\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) 48.1910 1.02534 0.512671 0.858585i \(-0.328656\pi\)
0.512671 + 0.858585i \(0.328656\pi\)
\(48\) 0 0
\(49\) 41.7611 0.852268
\(50\) 0 0
\(51\) 45.7779i 0.897605i
\(52\) 0 0
\(53\) 63.1698i 1.19188i 0.803028 + 0.595942i \(0.203221\pi\)
−0.803028 + 0.595942i \(0.796779\pi\)
\(54\) 0 0
\(55\) −47.7926 −0.868956
\(56\) 0 0
\(57\) 25.1864i 0.441867i
\(58\) 0 0
\(59\) 21.5809 0.365779 0.182889 0.983133i \(-0.441455\pi\)
0.182889 + 0.983133i \(0.441455\pi\)
\(60\) 0 0
\(61\) 82.0890i 1.34572i −0.739769 0.672861i \(-0.765065\pi\)
0.739769 0.672861i \(-0.234935\pi\)
\(62\) 0 0
\(63\) 8.07154i 0.128120i
\(64\) 0 0
\(65\) 40.9510i 0.630016i
\(66\) 0 0
\(67\) 27.9275i 0.416828i −0.978041 0.208414i \(-0.933170\pi\)
0.978041 0.208414i \(-0.0668302\pi\)
\(68\) 0 0
\(69\) 14.3717 37.1545i 0.208286 0.538470i
\(70\) 0 0
\(71\) 20.6113 0.290300 0.145150 0.989410i \(-0.453634\pi\)
0.145150 + 0.989410i \(0.453634\pi\)
\(72\) 0 0
\(73\) 20.4485 0.280116 0.140058 0.990143i \(-0.455271\pi\)
0.140058 + 0.990143i \(0.455271\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) −57.5056 −0.746826
\(78\) 0 0
\(79\) 6.01518i 0.0761415i −0.999275 0.0380707i \(-0.987879\pi\)
0.999275 0.0380707i \(-0.0121212\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 45.2575i 0.545271i −0.962117 0.272635i \(-0.912105\pi\)
0.962117 0.272635i \(-0.0878953\pi\)
\(84\) 0 0
\(85\) 59.0990 0.695282
\(86\) 0 0
\(87\) −84.6631 −0.973139
\(88\) 0 0
\(89\) 3.60325i 0.0404860i −0.999795 0.0202430i \(-0.993556\pi\)
0.999795 0.0202430i \(-0.00644398\pi\)
\(90\) 0 0
\(91\) 49.2737i 0.541469i
\(92\) 0 0
\(93\) 13.8821 0.149270
\(94\) 0 0
\(95\) 32.5156 0.342269
\(96\) 0 0
\(97\) 35.7362i 0.368414i 0.982887 + 0.184207i \(0.0589717\pi\)
−0.982887 + 0.184207i \(0.941028\pi\)
\(98\) 0 0
\(99\) 64.1204i 0.647681i
\(100\) 0 0
\(101\) −139.643 −1.38260 −0.691301 0.722567i \(-0.742962\pi\)
−0.691301 + 0.722567i \(0.742962\pi\)
\(102\) 0 0
\(103\) 31.8584i 0.309305i −0.987969 0.154653i \(-0.950574\pi\)
0.987969 0.154653i \(-0.0494258\pi\)
\(104\) 0 0
\(105\) 10.4203 0.0992410
\(106\) 0 0
\(107\) 186.172i 1.73992i −0.493120 0.869962i \(-0.664144\pi\)
0.493120 0.869962i \(-0.335856\pi\)
\(108\) 0 0
\(109\) 184.591i 1.69350i −0.531994 0.846748i \(-0.678557\pi\)
0.531994 0.846748i \(-0.321443\pi\)
\(110\) 0 0
\(111\) 63.9829i 0.576423i
\(112\) 0 0
\(113\) 42.5561i 0.376603i −0.982111 0.188301i \(-0.939702\pi\)
0.982111 0.188301i \(-0.0602982\pi\)
\(114\) 0 0
\(115\) 47.9662 + 18.5538i 0.417097 + 0.161337i
\(116\) 0 0
\(117\) −54.9416 −0.469586
\(118\) 0 0
\(119\) 71.1099 0.597562
\(120\) 0 0
\(121\) −335.826 −2.77542
\(122\) 0 0
\(123\) 20.7945 0.169061
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) 205.828 1.62069 0.810345 0.585953i \(-0.199280\pi\)
0.810345 + 0.585953i \(0.199280\pi\)
\(128\) 0 0
\(129\) 57.4762i 0.445552i
\(130\) 0 0
\(131\) 204.886 1.56401 0.782006 0.623271i \(-0.214197\pi\)
0.782006 + 0.623271i \(0.214197\pi\)
\(132\) 0 0
\(133\) 39.1238 0.294164
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 189.756i 1.38508i 0.721381 + 0.692539i \(0.243508\pi\)
−0.721381 + 0.692539i \(0.756492\pi\)
\(138\) 0 0
\(139\) −123.679 −0.889780 −0.444890 0.895585i \(-0.646757\pi\)
−0.444890 + 0.895585i \(0.646757\pi\)
\(140\) 0 0
\(141\) −83.4693 −0.591981
\(142\) 0 0
\(143\) 391.431i 2.73728i
\(144\) 0 0
\(145\) 109.300i 0.753790i
\(146\) 0 0
\(147\) −72.3324 −0.492057
\(148\) 0 0
\(149\) 34.4336i 0.231098i 0.993302 + 0.115549i \(0.0368627\pi\)
−0.993302 + 0.115549i \(0.963137\pi\)
\(150\) 0 0
\(151\) 184.051 1.21888 0.609440 0.792832i \(-0.291394\pi\)
0.609440 + 0.792832i \(0.291394\pi\)
\(152\) 0 0
\(153\) 79.2896i 0.518232i
\(154\) 0 0
\(155\) 17.9217i 0.115624i
\(156\) 0 0
\(157\) 274.951i 1.75128i −0.482966 0.875639i \(-0.660441\pi\)
0.482966 0.875639i \(-0.339559\pi\)
\(158\) 0 0
\(159\) 109.413i 0.688134i
\(160\) 0 0
\(161\) 57.7145 + 22.3246i 0.358475 + 0.138662i
\(162\) 0 0
\(163\) 137.650 0.844480 0.422240 0.906484i \(-0.361244\pi\)
0.422240 + 0.906484i \(0.361244\pi\)
\(164\) 0 0
\(165\) 82.7791 0.501692
\(166\) 0 0
\(167\) −115.388 −0.690943 −0.345472 0.938429i \(-0.612281\pi\)
−0.345472 + 0.938429i \(0.612281\pi\)
\(168\) 0 0
\(169\) 166.398 0.984601
\(170\) 0 0
\(171\) 43.6242i 0.255112i
\(172\) 0 0
\(173\) −246.911 −1.42723 −0.713615 0.700539i \(-0.752943\pi\)
−0.713615 + 0.700539i \(0.752943\pi\)
\(174\) 0 0
\(175\) 13.4526i 0.0768718i
\(176\) 0 0
\(177\) −37.3793 −0.211182
\(178\) 0 0
\(179\) 266.937 1.49127 0.745633 0.666357i \(-0.232147\pi\)
0.745633 + 0.666357i \(0.232147\pi\)
\(180\) 0 0
\(181\) 258.616i 1.42882i 0.699728 + 0.714409i \(0.253304\pi\)
−0.699728 + 0.714409i \(0.746696\pi\)
\(182\) 0 0
\(183\) 142.182i 0.776953i
\(184\) 0 0
\(185\) 82.6016 0.446495
\(186\) 0 0
\(187\) 564.898 3.02084
\(188\) 0 0
\(189\) 13.9803i 0.0739699i
\(190\) 0 0
\(191\) 22.1454i 0.115944i 0.998318 + 0.0579722i \(0.0184635\pi\)
−0.998318 + 0.0579722i \(0.981537\pi\)
\(192\) 0 0
\(193\) −96.4573 −0.499779 −0.249889 0.968274i \(-0.580394\pi\)
−0.249889 + 0.968274i \(0.580394\pi\)
\(194\) 0 0
\(195\) 70.9293i 0.363740i
\(196\) 0 0
\(197\) 13.6447 0.0692625 0.0346312 0.999400i \(-0.488974\pi\)
0.0346312 + 0.999400i \(0.488974\pi\)
\(198\) 0 0
\(199\) 230.917i 1.16039i 0.814479 + 0.580193i \(0.197023\pi\)
−0.814479 + 0.580193i \(0.802977\pi\)
\(200\) 0 0
\(201\) 48.3718i 0.240656i
\(202\) 0 0
\(203\) 131.513i 0.647847i
\(204\) 0 0
\(205\) 26.8456i 0.130954i
\(206\) 0 0
\(207\) −24.8925 + 64.3534i −0.120254 + 0.310886i
\(208\) 0 0
\(209\) 310.800 1.48708
\(210\) 0 0
\(211\) 208.485 0.988082 0.494041 0.869438i \(-0.335519\pi\)
0.494041 + 0.869438i \(0.335519\pi\)
\(212\) 0 0
\(213\) −35.6998 −0.167605
\(214\) 0 0
\(215\) 74.2015 0.345123
\(216\) 0 0
\(217\) 21.5640i 0.0993734i
\(218\) 0 0
\(219\) −35.4178 −0.161725
\(220\) 0 0
\(221\) 484.033i 2.19019i
\(222\) 0 0
\(223\) −9.98893 −0.0447934 −0.0223967 0.999749i \(-0.507130\pi\)
−0.0223967 + 0.999749i \(0.507130\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 108.272i 0.476969i −0.971146 0.238484i \(-0.923349\pi\)
0.971146 0.238484i \(-0.0766506\pi\)
\(228\) 0 0
\(229\) 9.41714i 0.0411229i 0.999789 + 0.0205614i \(0.00654537\pi\)
−0.999789 + 0.0205614i \(0.993455\pi\)
\(230\) 0 0
\(231\) 99.6026 0.431180
\(232\) 0 0
\(233\) 361.719 1.55244 0.776221 0.630461i \(-0.217134\pi\)
0.776221 + 0.630461i \(0.217134\pi\)
\(234\) 0 0
\(235\) 107.758i 0.458547i
\(236\) 0 0
\(237\) 10.4186i 0.0439603i
\(238\) 0 0
\(239\) −290.677 −1.21622 −0.608111 0.793852i \(-0.708072\pi\)
−0.608111 + 0.793852i \(0.708072\pi\)
\(240\) 0 0
\(241\) 200.333i 0.831259i −0.909534 0.415629i \(-0.863561\pi\)
0.909534 0.415629i \(-0.136439\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 93.3808i 0.381146i
\(246\) 0 0
\(247\) 266.309i 1.07817i
\(248\) 0 0
\(249\) 78.3883i 0.314812i
\(250\) 0 0
\(251\) 219.568i 0.874771i −0.899274 0.437386i \(-0.855904\pi\)
0.899274 0.437386i \(-0.144096\pi\)
\(252\) 0 0
\(253\) 458.485 + 177.347i 1.81220 + 0.700975i
\(254\) 0 0
\(255\) −102.362 −0.401421
\(256\) 0 0
\(257\) −181.700 −0.707004 −0.353502 0.935434i \(-0.615009\pi\)
−0.353502 + 0.935434i \(0.615009\pi\)
\(258\) 0 0
\(259\) 99.3890 0.383741
\(260\) 0 0
\(261\) 146.641 0.561842
\(262\) 0 0
\(263\) 299.898i 1.14030i −0.821542 0.570149i \(-0.806886\pi\)
0.821542 0.570149i \(-0.193114\pi\)
\(264\) 0 0
\(265\) 141.252 0.533027
\(266\) 0 0
\(267\) 6.24101i 0.0233746i
\(268\) 0 0
\(269\) −359.382 −1.33599 −0.667996 0.744165i \(-0.732848\pi\)
−0.667996 + 0.744165i \(0.732848\pi\)
\(270\) 0 0
\(271\) −23.9982 −0.0885543 −0.0442771 0.999019i \(-0.514098\pi\)
−0.0442771 + 0.999019i \(0.514098\pi\)
\(272\) 0 0
\(273\) 85.3445i 0.312617i
\(274\) 0 0
\(275\) 106.867i 0.388609i
\(276\) 0 0
\(277\) 532.611 1.92278 0.961391 0.275186i \(-0.0887395\pi\)
0.961391 + 0.275186i \(0.0887395\pi\)
\(278\) 0 0
\(279\) −24.0445 −0.0861811
\(280\) 0 0
\(281\) 152.634i 0.543181i −0.962413 0.271590i \(-0.912450\pi\)
0.962413 0.271590i \(-0.0875495\pi\)
\(282\) 0 0
\(283\) 227.841i 0.805092i −0.915400 0.402546i \(-0.868125\pi\)
0.915400 0.402546i \(-0.131875\pi\)
\(284\) 0 0
\(285\) −56.3186 −0.197609
\(286\) 0 0
\(287\) 32.3015i 0.112549i
\(288\) 0 0
\(289\) −409.537 −1.41708
\(290\) 0 0
\(291\) 61.8969i 0.212704i
\(292\) 0 0
\(293\) 364.685i 1.24466i −0.782755 0.622330i \(-0.786186\pi\)
0.782755 0.622330i \(-0.213814\pi\)
\(294\) 0 0
\(295\) 48.2565i 0.163581i
\(296\) 0 0
\(297\) 111.060i 0.373939i
\(298\) 0 0
\(299\) 151.959 392.853i 0.508226 1.31389i
\(300\) 0 0
\(301\) 89.2818 0.296617
\(302\) 0 0
\(303\) 241.868 0.798246
\(304\) 0 0
\(305\) −183.557 −0.601825
\(306\) 0 0
\(307\) 608.582 1.98235 0.991176 0.132556i \(-0.0423183\pi\)
0.991176 + 0.132556i \(0.0423183\pi\)
\(308\) 0 0
\(309\) 55.1804i 0.178577i
\(310\) 0 0
\(311\) 393.737 1.26603 0.633017 0.774138i \(-0.281816\pi\)
0.633017 + 0.774138i \(0.281816\pi\)
\(312\) 0 0
\(313\) 368.782i 1.17822i 0.808054 + 0.589109i \(0.200521\pi\)
−0.808054 + 0.589109i \(0.799479\pi\)
\(314\) 0 0
\(315\) −18.0485 −0.0572968
\(316\) 0 0
\(317\) 330.351 1.04212 0.521059 0.853521i \(-0.325537\pi\)
0.521059 + 0.853521i \(0.325537\pi\)
\(318\) 0 0
\(319\) 1044.74i 3.27505i
\(320\) 0 0
\(321\) 322.459i 1.00455i
\(322\) 0 0
\(323\) −384.327 −1.18987
\(324\) 0 0
\(325\) 91.5693 0.281752
\(326\) 0 0
\(327\) 319.721i 0.977740i
\(328\) 0 0
\(329\) 129.659i 0.394099i
\(330\) 0 0
\(331\) 328.089 0.991206 0.495603 0.868549i \(-0.334947\pi\)
0.495603 + 0.868549i \(0.334947\pi\)
\(332\) 0 0
\(333\) 110.822i 0.332798i
\(334\) 0 0
\(335\) −62.4478 −0.186411
\(336\) 0 0
\(337\) 142.686i 0.423401i 0.977335 + 0.211701i \(0.0679002\pi\)
−0.977335 + 0.211701i \(0.932100\pi\)
\(338\) 0 0
\(339\) 73.7094i 0.217432i
\(340\) 0 0
\(341\) 171.305i 0.502361i
\(342\) 0 0
\(343\) 244.194i 0.711936i
\(344\) 0 0
\(345\) −83.0799 32.1361i −0.240811 0.0931482i
\(346\) 0 0
\(347\) −240.165 −0.692118 −0.346059 0.938213i \(-0.612480\pi\)
−0.346059 + 0.938213i \(0.612480\pi\)
\(348\) 0 0
\(349\) −234.803 −0.672788 −0.336394 0.941721i \(-0.609207\pi\)
−0.336394 + 0.941721i \(0.609207\pi\)
\(350\) 0 0
\(351\) 95.1616 0.271116
\(352\) 0 0
\(353\) 467.622 1.32471 0.662355 0.749190i \(-0.269557\pi\)
0.662355 + 0.749190i \(0.269557\pi\)
\(354\) 0 0
\(355\) 46.0882i 0.129826i
\(356\) 0 0
\(357\) −123.166 −0.345002
\(358\) 0 0
\(359\) 571.031i 1.59062i 0.606206 + 0.795308i \(0.292691\pi\)
−0.606206 + 0.795308i \(0.707309\pi\)
\(360\) 0 0
\(361\) 149.548 0.414260
\(362\) 0 0
\(363\) 581.667 1.60239
\(364\) 0 0
\(365\) 45.7242i 0.125272i
\(366\) 0 0
\(367\) 396.719i 1.08098i −0.841351 0.540489i \(-0.818239\pi\)
0.841351 0.540489i \(-0.181761\pi\)
\(368\) 0 0
\(369\) −36.0171 −0.0976074
\(370\) 0 0
\(371\) 169.959 0.458111
\(372\) 0 0
\(373\) 395.967i 1.06157i 0.847505 + 0.530787i \(0.178104\pi\)
−0.847505 + 0.530787i \(0.821896\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) −895.186 −2.37450
\(378\) 0 0
\(379\) 335.982i 0.886497i 0.896399 + 0.443248i \(0.146174\pi\)
−0.896399 + 0.443248i \(0.853826\pi\)
\(380\) 0 0
\(381\) −356.504 −0.935706
\(382\) 0 0
\(383\) 597.598i 1.56031i 0.625587 + 0.780155i \(0.284860\pi\)
−0.625587 + 0.780155i \(0.715140\pi\)
\(384\) 0 0
\(385\) 128.586i 0.333991i
\(386\) 0 0
\(387\) 99.5518i 0.257240i
\(388\) 0 0
\(389\) 457.226i 1.17539i 0.809083 + 0.587694i \(0.199964\pi\)
−0.809083 + 0.587694i \(0.800036\pi\)
\(390\) 0 0
\(391\) −566.950 219.302i −1.45000 0.560875i
\(392\) 0 0
\(393\) −354.872 −0.902983
\(394\) 0 0
\(395\) −13.4503 −0.0340515
\(396\) 0 0
\(397\) 554.607 1.39699 0.698497 0.715613i \(-0.253853\pi\)
0.698497 + 0.715613i \(0.253853\pi\)
\(398\) 0 0
\(399\) −67.7644 −0.169836
\(400\) 0 0
\(401\) 217.722i 0.542947i 0.962446 + 0.271473i \(0.0875108\pi\)
−0.962446 + 0.271473i \(0.912489\pi\)
\(402\) 0 0
\(403\) 146.783 0.364225
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) 789.548 1.93992
\(408\) 0 0
\(409\) −82.7621 −0.202352 −0.101176 0.994869i \(-0.532261\pi\)
−0.101176 + 0.994869i \(0.532261\pi\)
\(410\) 0 0
\(411\) 328.666i 0.799675i
\(412\) 0 0
\(413\) 58.0638i 0.140590i
\(414\) 0 0
\(415\) −101.199 −0.243853
\(416\) 0 0
\(417\) 214.219 0.513714
\(418\) 0 0
\(419\) 509.789i 1.21668i 0.793676 + 0.608340i \(0.208164\pi\)
−0.793676 + 0.608340i \(0.791836\pi\)
\(420\) 0 0
\(421\) 625.639i 1.48608i 0.669248 + 0.743039i \(0.266616\pi\)
−0.669248 + 0.743039i \(0.733384\pi\)
\(422\) 0 0
\(423\) 144.573 0.341780
\(424\) 0 0
\(425\) 132.149i 0.310939i
\(426\) 0 0
\(427\) −220.861 −0.517240
\(428\) 0 0
\(429\) 677.978i 1.58037i
\(430\) 0 0
\(431\) 197.410i 0.458029i −0.973423 0.229014i \(-0.926450\pi\)
0.973423 0.229014i \(-0.0735503\pi\)
\(432\) 0 0
\(433\) 309.110i 0.713880i −0.934127 0.356940i \(-0.883820\pi\)
0.934127 0.356940i \(-0.116180\pi\)
\(434\) 0 0
\(435\) 189.312i 0.435201i
\(436\) 0 0
\(437\) −311.930 120.657i −0.713798 0.276104i
\(438\) 0 0
\(439\) −280.440 −0.638815 −0.319407 0.947617i \(-0.603484\pi\)
−0.319407 + 0.947617i \(0.603484\pi\)
\(440\) 0 0
\(441\) 125.283 0.284089
\(442\) 0 0
\(443\) −392.013 −0.884906 −0.442453 0.896792i \(-0.645892\pi\)
−0.442453 + 0.896792i \(0.645892\pi\)
\(444\) 0 0
\(445\) −8.05711 −0.0181059
\(446\) 0 0
\(447\) 59.6407i 0.133424i
\(448\) 0 0
\(449\) −517.345 −1.15222 −0.576108 0.817374i \(-0.695429\pi\)
−0.576108 + 0.817374i \(0.695429\pi\)
\(450\) 0 0
\(451\) 256.604i 0.568966i
\(452\) 0 0
\(453\) −318.786 −0.703721
\(454\) 0 0
\(455\) 110.179 0.242152
\(456\) 0 0
\(457\) 23.0890i 0.0505229i 0.999681 + 0.0252614i \(0.00804182\pi\)
−0.999681 + 0.0252614i \(0.991958\pi\)
\(458\) 0 0
\(459\) 137.334i 0.299202i
\(460\) 0 0
\(461\) 508.964 1.10404 0.552022 0.833830i \(-0.313857\pi\)
0.552022 + 0.833830i \(0.313857\pi\)
\(462\) 0 0
\(463\) 701.424 1.51496 0.757478 0.652861i \(-0.226432\pi\)
0.757478 + 0.652861i \(0.226432\pi\)
\(464\) 0 0
\(465\) 31.0414i 0.0667556i
\(466\) 0 0
\(467\) 337.646i 0.723011i 0.932370 + 0.361506i \(0.117737\pi\)
−0.932370 + 0.361506i \(0.882263\pi\)
\(468\) 0 0
\(469\) −75.1393 −0.160212
\(470\) 0 0
\(471\) 476.229i 1.01110i
\(472\) 0 0
\(473\) 709.256 1.49948
\(474\) 0 0
\(475\) 72.7070i 0.153067i
\(476\) 0 0
\(477\) 189.509i 0.397295i
\(478\) 0 0
\(479\) 514.482i 1.07408i −0.843558 0.537038i \(-0.819543\pi\)
0.843558 0.537038i \(-0.180457\pi\)
\(480\) 0 0
\(481\) 676.524i 1.40650i
\(482\) 0 0
\(483\) −99.9645 38.6673i −0.206966 0.0800564i
\(484\) 0 0
\(485\) 79.9085 0.164760
\(486\) 0 0
\(487\) 460.379 0.945336 0.472668 0.881241i \(-0.343291\pi\)
0.472668 + 0.881241i \(0.343291\pi\)
\(488\) 0 0
\(489\) −238.417 −0.487561
\(490\) 0 0
\(491\) 572.919 1.16684 0.583420 0.812171i \(-0.301714\pi\)
0.583420 + 0.812171i \(0.301714\pi\)
\(492\) 0 0
\(493\) 1291.90i 2.62048i
\(494\) 0 0
\(495\) −143.378 −0.289652
\(496\) 0 0
\(497\) 55.4549i 0.111579i
\(498\) 0 0
\(499\) 531.929 1.06599 0.532995 0.846119i \(-0.321067\pi\)
0.532995 + 0.846119i \(0.321067\pi\)
\(500\) 0 0
\(501\) 199.857 0.398916
\(502\) 0 0
\(503\) 438.073i 0.870920i 0.900208 + 0.435460i \(0.143414\pi\)
−0.900208 + 0.435460i \(0.856586\pi\)
\(504\) 0 0
\(505\) 312.251i 0.618318i
\(506\) 0 0
\(507\) −288.209 −0.568460
\(508\) 0 0
\(509\) 462.713 0.909063 0.454532 0.890731i \(-0.349807\pi\)
0.454532 + 0.890731i \(0.349807\pi\)
\(510\) 0 0
\(511\) 55.0168i 0.107665i
\(512\) 0 0
\(513\) 75.5593i 0.147289i
\(514\) 0 0
\(515\) −71.2376 −0.138325
\(516\) 0 0
\(517\) 1030.01i 1.99228i
\(518\) 0 0
\(519\) 427.662 0.824011
\(520\) 0 0
\(521\) 489.696i 0.939916i 0.882689 + 0.469958i \(0.155731\pi\)
−0.882689 + 0.469958i \(0.844269\pi\)
\(522\) 0 0
\(523\) 133.092i 0.254479i −0.991872 0.127239i \(-0.959388\pi\)
0.991872 0.127239i \(-0.0406117\pi\)
\(524\) 0 0
\(525\) 23.3005i 0.0443819i
\(526\) 0 0
\(527\) 211.831i 0.401957i
\(528\) 0 0
\(529\) −391.303 355.982i −0.739703 0.672934i
\(530\) 0 0
\(531\) 64.7428 0.121926
\(532\) 0 0
\(533\) 219.871 0.412516
\(534\) 0 0
\(535\) −416.293 −0.778117
\(536\) 0 0
\(537\) −462.348 −0.860982
\(538\) 0 0
\(539\) 892.581i 1.65599i
\(540\) 0 0
\(541\) −249.915 −0.461951 −0.230975 0.972960i \(-0.574192\pi\)
−0.230975 + 0.972960i \(0.574192\pi\)
\(542\) 0 0
\(543\) 447.936i 0.824928i
\(544\) 0 0
\(545\) −412.758 −0.757355
\(546\) 0 0
\(547\) −282.177 −0.515862 −0.257931 0.966163i \(-0.583041\pi\)
−0.257931 + 0.966163i \(0.583041\pi\)
\(548\) 0 0
\(549\) 246.267i 0.448574i
\(550\) 0 0
\(551\) 710.787i 1.28999i
\(552\) 0 0
\(553\) −16.1839 −0.0292657
\(554\) 0 0
\(555\) −143.070 −0.257784
\(556\) 0 0
\(557\) 148.494i 0.266597i 0.991076 + 0.133298i \(0.0425568\pi\)
−0.991076 + 0.133298i \(0.957443\pi\)
\(558\) 0 0
\(559\) 607.726i 1.08717i
\(560\) 0 0
\(561\) −978.432 −1.74409
\(562\) 0 0
\(563\) 92.7944i 0.164821i −0.996598 0.0824107i \(-0.973738\pi\)
0.996598 0.0824107i \(-0.0262619\pi\)
\(564\) 0 0
\(565\) −95.1584 −0.168422
\(566\) 0 0
\(567\) 24.2146i 0.0427065i
\(568\) 0 0
\(569\) 96.4311i 0.169475i 0.996403 + 0.0847373i \(0.0270051\pi\)
−0.996403 + 0.0847373i \(0.972995\pi\)
\(570\) 0 0
\(571\) 400.496i 0.701394i −0.936489 0.350697i \(-0.885945\pi\)
0.936489 0.350697i \(-0.114055\pi\)
\(572\) 0 0
\(573\) 38.3569i 0.0669406i
\(574\) 0 0
\(575\) 41.4876 107.256i 0.0721523 0.186532i
\(576\) 0 0
\(577\) −692.314 −1.19985 −0.599926 0.800056i \(-0.704803\pi\)
−0.599926 + 0.800056i \(0.704803\pi\)
\(578\) 0 0
\(579\) 167.069 0.288547
\(580\) 0 0
\(581\) −121.766 −0.209580
\(582\) 0 0
\(583\) 1350.16 2.31588
\(584\) 0 0
\(585\) 122.853i 0.210005i
\(586\) 0 0
\(587\) −889.479 −1.51530 −0.757648 0.652663i \(-0.773652\pi\)
−0.757648 + 0.652663i \(0.773652\pi\)
\(588\) 0 0
\(589\) 116.547i 0.197873i
\(590\) 0 0
\(591\) −23.6333 −0.0399887
\(592\) 0 0
\(593\) 795.028 1.34069 0.670344 0.742050i \(-0.266146\pi\)
0.670344 + 0.742050i \(0.266146\pi\)
\(594\) 0 0
\(595\) 159.006i 0.267238i
\(596\) 0 0
\(597\) 399.960i 0.669949i
\(598\) 0 0
\(599\) −4.37359 −0.00730148 −0.00365074 0.999993i \(-0.501162\pi\)
−0.00365074 + 0.999993i \(0.501162\pi\)
\(600\) 0 0
\(601\) 912.225 1.51784 0.758922 0.651181i \(-0.225726\pi\)
0.758922 + 0.651181i \(0.225726\pi\)
\(602\) 0 0
\(603\) 83.7825i 0.138943i
\(604\) 0 0
\(605\) 750.929i 1.24121i
\(606\) 0 0
\(607\) 611.171 1.00687 0.503436 0.864032i \(-0.332069\pi\)
0.503436 + 0.864032i \(0.332069\pi\)
\(608\) 0 0
\(609\) 227.787i 0.374034i
\(610\) 0 0
\(611\) −882.564 −1.44446
\(612\) 0 0
\(613\) 639.202i 1.04274i 0.853330 + 0.521372i \(0.174580\pi\)
−0.853330 + 0.521372i \(0.825420\pi\)
\(614\) 0 0
\(615\) 46.4979i 0.0756064i
\(616\) 0 0
\(617\) 234.395i 0.379895i −0.981794 0.189948i \(-0.939168\pi\)
0.981794 0.189948i \(-0.0608319\pi\)
\(618\) 0 0
\(619\) 191.656i 0.309622i −0.987944 0.154811i \(-0.950523\pi\)
0.987944 0.154811i \(-0.0494769\pi\)
\(620\) 0 0
\(621\) 43.1151 111.463i 0.0694285 0.179490i
\(622\) 0 0
\(623\) −9.69459 −0.0155611
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −538.322 −0.858568
\(628\) 0 0
\(629\) −976.334 −1.55220
\(630\) 0 0
\(631\) 507.948i 0.804988i −0.915423 0.402494i \(-0.868143\pi\)
0.915423 0.402494i \(-0.131857\pi\)
\(632\) 0 0
\(633\) −361.107 −0.570470
\(634\) 0 0
\(635\) 460.245i 0.724795i
\(636\) 0 0
\(637\) −764.808 −1.20064
\(638\) 0 0
\(639\) 61.8338 0.0967665
\(640\) 0 0
\(641\) 154.509i 0.241043i −0.992711 0.120522i \(-0.961543\pi\)
0.992711 0.120522i \(-0.0384567\pi\)
\(642\) 0 0
\(643\) 208.249i 0.323870i 0.986801 + 0.161935i \(0.0517736\pi\)
−0.986801 + 0.161935i \(0.948226\pi\)
\(644\) 0 0
\(645\) −128.521 −0.199257
\(646\) 0 0
\(647\) −789.328 −1.21998 −0.609991 0.792408i \(-0.708827\pi\)
−0.609991 + 0.792408i \(0.708827\pi\)
\(648\) 0 0
\(649\) 461.260i 0.710724i
\(650\) 0 0
\(651\) 37.3500i 0.0573733i
\(652\) 0 0
\(653\) −1119.38 −1.71422 −0.857108 0.515136i \(-0.827741\pi\)
−0.857108 + 0.515136i \(0.827741\pi\)
\(654\) 0 0
\(655\) 458.138i 0.699447i
\(656\) 0 0
\(657\) 61.3454 0.0933720
\(658\) 0 0
\(659\) 862.648i 1.30903i 0.756051 + 0.654513i \(0.227126\pi\)
−0.756051 + 0.654513i \(0.772874\pi\)
\(660\) 0 0
\(661\) 401.095i 0.606801i −0.952863 0.303400i \(-0.901878\pi\)
0.952863 0.303400i \(-0.0981219\pi\)
\(662\) 0 0
\(663\) 838.369i 1.26451i
\(664\) 0 0
\(665\) 87.4835i 0.131554i
\(666\) 0 0
\(667\) −405.584 + 1048.54i −0.608072 + 1.57202i
\(668\) 0 0
\(669\) 17.3013 0.0258615
\(670\) 0 0
\(671\) −1754.53 −2.61480
\(672\) 0 0
\(673\) −849.026 −1.26155 −0.630777 0.775964i \(-0.717264\pi\)
−0.630777 + 0.775964i \(0.717264\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 49.1031i 0.0725304i 0.999342 + 0.0362652i \(0.0115461\pi\)
−0.999342 + 0.0362652i \(0.988454\pi\)
\(678\) 0 0
\(679\) 96.1486 0.141603
\(680\) 0 0
\(681\) 187.532i 0.275378i
\(682\) 0 0
\(683\) 589.071 0.862476 0.431238 0.902238i \(-0.358077\pi\)
0.431238 + 0.902238i \(0.358077\pi\)
\(684\) 0 0
\(685\) 424.306 0.619425
\(686\) 0 0
\(687\) 16.3110i 0.0237423i
\(688\) 0 0
\(689\) 1156.88i 1.67908i
\(690\) 0 0
\(691\) 695.256 1.00616 0.503079 0.864240i \(-0.332200\pi\)
0.503079 + 0.864240i \(0.332200\pi\)
\(692\) 0 0
\(693\) −172.517 −0.248942
\(694\) 0 0
\(695\) 276.555i 0.397922i
\(696\) 0 0
\(697\) 317.309i 0.455250i
\(698\) 0 0
\(699\) −626.516 −0.896303
\(700\) 0 0
\(701\) 636.812i 0.908434i 0.890891 + 0.454217i \(0.150081\pi\)
−0.890891 + 0.454217i \(0.849919\pi\)
\(702\) 0 0
\(703\) −537.167 −0.764107
\(704\) 0 0
\(705\) 186.643i 0.264742i
\(706\) 0 0
\(707\) 375.711i 0.531415i
\(708\) 0 0
\(709\) 694.217i 0.979149i −0.871961 0.489575i \(-0.837152\pi\)
0.871961 0.489575i \(-0.162848\pi\)
\(710\) 0 0
\(711\) 18.0455i 0.0253805i
\(712\) 0 0
\(713\) 66.5033 171.928i 0.0932725 0.241133i
\(714\) 0 0
\(715\) 875.266 1.22415
\(716\) 0 0
\(717\) 503.467 0.702186
\(718\) 0 0
\(719\) −23.9123 −0.0332577 −0.0166289 0.999862i \(-0.505293\pi\)
−0.0166289 + 0.999862i \(0.505293\pi\)
\(720\) 0 0
\(721\) −85.7155 −0.118884
\(722\) 0 0
\(723\) 346.988i 0.479928i
\(724\) 0 0
\(725\) −244.401 −0.337105
\(726\) 0 0
\(727\) 1028.96i 1.41535i 0.706539 + 0.707674i \(0.250256\pi\)
−0.706539 + 0.707674i \(0.749744\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) −877.046 −1.19979
\(732\) 0 0
\(733\) 360.850i 0.492292i −0.969233 0.246146i \(-0.920836\pi\)
0.969233 0.246146i \(-0.0791643\pi\)
\(734\) 0 0
\(735\) 161.740i 0.220055i
\(736\) 0 0
\(737\) −596.908 −0.809915
\(738\) 0 0
\(739\) 793.801 1.07416 0.537078 0.843533i \(-0.319528\pi\)
0.537078 + 0.843533i \(0.319528\pi\)
\(740\) 0 0
\(741\) 461.261i 0.622484i
\(742\) 0 0
\(743\) 1466.08i 1.97319i 0.163181 + 0.986596i \(0.447824\pi\)
−0.163181 + 0.986596i \(0.552176\pi\)
\(744\) 0 0
\(745\) 76.9958 0.103350
\(746\) 0 0
\(747\) 135.772i 0.181757i
\(748\) 0 0
\(749\) −500.897 −0.668755
\(750\) 0 0
\(751\) 442.587i 0.589330i 0.955601 + 0.294665i \(0.0952081\pi\)
−0.955601 + 0.294665i \(0.904792\pi\)
\(752\) 0 0
\(753\) 380.302i 0.505049i
\(754\) 0 0
\(755\) 411.551i 0.545100i
\(756\) 0 0
\(757\) 1032.76i 1.36428i 0.731221 + 0.682141i \(0.238951\pi\)
−0.731221 + 0.682141i \(0.761049\pi\)
\(758\) 0 0
\(759\) −794.120 307.173i −1.04627 0.404708i
\(760\) 0 0
\(761\) −845.420 −1.11093 −0.555467 0.831539i \(-0.687460\pi\)
−0.555467 + 0.831539i \(0.687460\pi\)
\(762\) 0 0
\(763\) −496.645 −0.650910
\(764\) 0 0
\(765\) 177.297 0.231761
\(766\) 0 0
\(767\) −395.230 −0.515294
\(768\) 0 0
\(769\) 1238.15i 1.61008i −0.593220 0.805041i \(-0.702143\pi\)
0.593220 0.805041i \(-0.297857\pi\)
\(770\) 0 0
\(771\) 314.714 0.408189
\(772\) 0 0
\(773\) 1133.33i 1.46615i 0.680148 + 0.733075i \(0.261916\pi\)
−0.680148 + 0.733075i \(0.738084\pi\)
\(774\) 0 0
\(775\) 40.0742 0.0517087
\(776\) 0 0
\(777\) −172.147 −0.221553
\(778\) 0 0
\(779\) 174.580i 0.224108i
\(780\) 0 0
\(781\) 440.535i 0.564065i
\(782\) 0 0
\(783\) −253.989 −0.324380
\(784\) 0 0
\(785\) −614.808 −0.783195
\(786\) 0 0
\(787\) 536.686i 0.681939i −0.940074 0.340969i \(-0.889245\pi\)
0.940074 0.340969i \(-0.110755\pi\)
\(788\) 0 0
\(789\) 519.439i 0.658351i
\(790\) 0 0
\(791\) −114.498 −0.144751
\(792\) 0 0
\(793\) 1503.37i 1.89580i
\(794\) 0 0
\(795\) −244.656 −0.307743
\(796\) 0 0
\(797\) 1184.96i 1.48677i −0.668861 0.743387i \(-0.733218\pi\)
0.668861 0.743387i \(-0.266782\pi\)
\(798\) 0 0