Properties

Label 6080.2.a.bp.1.2
Level $6080$
Weight $2$
Character 6080.1
Self dual yes
Analytic conductor $48.549$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6080,2,Mod(1,6080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6080.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6080.a (trivial)

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: yes
Analytic conductor: \(48.5490444289\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3040)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.571993\) of defining polynomial
Character \(\chi\) \(=\) 6080.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.428007 q^{3} +1.00000 q^{5} -4.67282 q^{7} -2.81681 q^{9} +O(q^{10})\) \(q-0.428007 q^{3} +1.00000 q^{5} -4.67282 q^{7} -2.81681 q^{9} +2.67282 q^{11} +2.24482 q^{13} -0.428007 q^{15} +2.00000 q^{17} +1.00000 q^{19} +2.00000 q^{21} +0.672824 q^{23} +1.00000 q^{25} +2.48963 q^{27} -1.14399 q^{29} -1.14399 q^{31} -1.14399 q^{33} -4.67282 q^{35} -4.24482 q^{37} -0.960797 q^{39} +4.85601 q^{41} -5.52884 q^{43} -2.81681 q^{45} -4.67282 q^{47} +14.8353 q^{49} -0.856013 q^{51} +0.244817 q^{53} +2.67282 q^{55} -0.428007 q^{57} +0.287973 q^{59} +8.87448 q^{61} +13.1625 q^{63} +2.24482 q^{65} -9.20561 q^{67} -0.287973 q^{69} -9.14399 q^{71} +1.14399 q^{73} -0.428007 q^{75} -12.4896 q^{77} +2.00000 q^{79} +7.38485 q^{81} -0.183190 q^{83} +2.00000 q^{85} +0.489634 q^{87} +9.14399 q^{89} -10.4896 q^{91} +0.489634 q^{93} +1.00000 q^{95} +9.10083 q^{97} -7.52884 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 2 q^{11} - 4 q^{13} - 2 q^{15} + 6 q^{17} + 3 q^{19} + 6 q^{21} - 8 q^{23} + 3 q^{25} - 14 q^{27} - 2 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{35} - 2 q^{37} + 10 q^{39} + 16 q^{41} - 8 q^{43} + 3 q^{45} - 4 q^{47} + 3 q^{49} - 4 q^{51} - 10 q^{53} - 2 q^{55} - 2 q^{57} - 2 q^{59} - 2 q^{61} + 8 q^{63} - 4 q^{65} - 4 q^{67} + 2 q^{69} - 26 q^{71} + 2 q^{73} - 2 q^{75} - 16 q^{77} + 6 q^{79} + 15 q^{81} - 12 q^{83} + 6 q^{85} - 20 q^{87} + 26 q^{89} - 10 q^{91} - 20 q^{93} + 3 q^{95} + 18 q^{97} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −0.428007 −0.247110 −0.123555 0.992338i \(-0.539430\pi\)
−0.123555 + 0.992338i \(0.539430\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.67282 −1.76616 −0.883081 0.469221i \(-0.844535\pi\)
−0.883081 + 0.469221i \(0.844535\pi\)
\(8\) 0 0
\(9\) −2.81681 −0.938937
\(10\) 0 0
\(11\) 2.67282 0.805887 0.402943 0.915225i \(-0.367987\pi\)
0.402943 + 0.915225i \(0.367987\pi\)
\(12\) 0 0
\(13\) 2.24482 0.622600 0.311300 0.950312i \(-0.399236\pi\)
0.311300 + 0.950312i \(0.399236\pi\)
\(14\) 0 0
\(15\) −0.428007 −0.110511
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 0.672824 0.140293 0.0701467 0.997537i \(-0.477653\pi\)
0.0701467 + 0.997537i \(0.477653\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 2.48963 0.479130
\(28\) 0 0
\(29\) −1.14399 −0.212433 −0.106216 0.994343i \(-0.533874\pi\)
−0.106216 + 0.994343i \(0.533874\pi\)
\(30\) 0 0
\(31\) −1.14399 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(32\) 0 0
\(33\) −1.14399 −0.199142
\(34\) 0 0
\(35\) −4.67282 −0.789851
\(36\) 0 0
\(37\) −4.24482 −0.697844 −0.348922 0.937152i \(-0.613452\pi\)
−0.348922 + 0.937152i \(0.613452\pi\)
\(38\) 0 0
\(39\) −0.960797 −0.153851
\(40\) 0 0
\(41\) 4.85601 0.758382 0.379191 0.925318i \(-0.376202\pi\)
0.379191 + 0.925318i \(0.376202\pi\)
\(42\) 0 0
\(43\) −5.52884 −0.843140 −0.421570 0.906796i \(-0.638521\pi\)
−0.421570 + 0.906796i \(0.638521\pi\)
\(44\) 0 0
\(45\) −2.81681 −0.419905
\(46\) 0 0
\(47\) −4.67282 −0.681601 −0.340801 0.940136i \(-0.610698\pi\)
−0.340801 + 0.940136i \(0.610698\pi\)
\(48\) 0 0
\(49\) 14.8353 2.11933
\(50\) 0 0
\(51\) −0.856013 −0.119866
\(52\) 0 0
\(53\) 0.244817 0.0336282 0.0168141 0.999859i \(-0.494648\pi\)
0.0168141 + 0.999859i \(0.494648\pi\)
\(54\) 0 0
\(55\) 2.67282 0.360403
\(56\) 0 0
\(57\) −0.428007 −0.0566909
\(58\) 0 0
\(59\) 0.287973 0.0374909 0.0187455 0.999824i \(-0.494033\pi\)
0.0187455 + 0.999824i \(0.494033\pi\)
\(60\) 0 0
\(61\) 8.87448 1.13626 0.568131 0.822938i \(-0.307667\pi\)
0.568131 + 0.822938i \(0.307667\pi\)
\(62\) 0 0
\(63\) 13.1625 1.65831
\(64\) 0 0
\(65\) 2.24482 0.278435
\(66\) 0 0
\(67\) −9.20561 −1.12464 −0.562322 0.826918i \(-0.690092\pi\)
−0.562322 + 0.826918i \(0.690092\pi\)
\(68\) 0 0
\(69\) −0.287973 −0.0346679
\(70\) 0 0
\(71\) −9.14399 −1.08519 −0.542596 0.839994i \(-0.682558\pi\)
−0.542596 + 0.839994i \(0.682558\pi\)
\(72\) 0 0
\(73\) 1.14399 0.133893 0.0669467 0.997757i \(-0.478674\pi\)
0.0669467 + 0.997757i \(0.478674\pi\)
\(74\) 0 0
\(75\) −0.428007 −0.0494220
\(76\) 0 0
\(77\) −12.4896 −1.42333
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 7.38485 0.820539
\(82\) 0 0
\(83\) −0.183190 −0.0201077 −0.0100538 0.999949i \(-0.503200\pi\)
−0.0100538 + 0.999949i \(0.503200\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0.489634 0.0524943
\(88\) 0 0
\(89\) 9.14399 0.969261 0.484630 0.874719i \(-0.338954\pi\)
0.484630 + 0.874719i \(0.338954\pi\)
\(90\) 0 0
\(91\) −10.4896 −1.09961
\(92\) 0 0
\(93\) 0.489634 0.0507727
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 9.10083 0.924049 0.462025 0.886867i \(-0.347123\pi\)
0.462025 + 0.886867i \(0.347123\pi\)
\(98\) 0 0
\(99\) −7.52884 −0.756677
\(100\) 0 0
\(101\) −2.67282 −0.265956 −0.132978 0.991119i \(-0.542454\pi\)
−0.132978 + 0.991119i \(0.542454\pi\)
\(102\) 0 0
\(103\) 6.91764 0.681615 0.340808 0.940133i \(-0.389299\pi\)
0.340808 + 0.940133i \(0.389299\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) 8.62967 0.834261 0.417131 0.908846i \(-0.363036\pi\)
0.417131 + 0.908846i \(0.363036\pi\)
\(108\) 0 0
\(109\) −11.6336 −1.11430 −0.557149 0.830412i \(-0.688105\pi\)
−0.557149 + 0.830412i \(0.688105\pi\)
\(110\) 0 0
\(111\) 1.81681 0.172444
\(112\) 0 0
\(113\) 6.61120 0.621929 0.310965 0.950422i \(-0.399348\pi\)
0.310965 + 0.950422i \(0.399348\pi\)
\(114\) 0 0
\(115\) 0.672824 0.0627411
\(116\) 0 0
\(117\) −6.32322 −0.584582
\(118\) 0 0
\(119\) −9.34565 −0.856714
\(120\) 0 0
\(121\) −3.85601 −0.350547
\(122\) 0 0
\(123\) −2.07841 −0.187404
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.91764 −0.791313 −0.395656 0.918399i \(-0.629483\pi\)
−0.395656 + 0.918399i \(0.629483\pi\)
\(128\) 0 0
\(129\) 2.36638 0.208348
\(130\) 0 0
\(131\) −6.77761 −0.592162 −0.296081 0.955163i \(-0.595680\pi\)
−0.296081 + 0.955163i \(0.595680\pi\)
\(132\) 0 0
\(133\) −4.67282 −0.405185
\(134\) 0 0
\(135\) 2.48963 0.214274
\(136\) 0 0
\(137\) −14.0369 −1.19926 −0.599628 0.800279i \(-0.704685\pi\)
−0.599628 + 0.800279i \(0.704685\pi\)
\(138\) 0 0
\(139\) −4.96080 −0.420769 −0.210385 0.977619i \(-0.567472\pi\)
−0.210385 + 0.977619i \(0.567472\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) −1.14399 −0.0950029
\(146\) 0 0
\(147\) −6.34960 −0.523706
\(148\) 0 0
\(149\) 0.874485 0.0716406 0.0358203 0.999358i \(-0.488596\pi\)
0.0358203 + 0.999358i \(0.488596\pi\)
\(150\) 0 0
\(151\) −21.4689 −1.74711 −0.873557 0.486721i \(-0.838193\pi\)
−0.873557 + 0.486721i \(0.838193\pi\)
\(152\) 0 0
\(153\) −5.63362 −0.455451
\(154\) 0 0
\(155\) −1.14399 −0.0918872
\(156\) 0 0
\(157\) −11.9216 −0.951447 −0.475723 0.879595i \(-0.657814\pi\)
−0.475723 + 0.879595i \(0.657814\pi\)
\(158\) 0 0
\(159\) −0.104783 −0.00830986
\(160\) 0 0
\(161\) −3.14399 −0.247781
\(162\) 0 0
\(163\) −2.47116 −0.193556 −0.0967782 0.995306i \(-0.530854\pi\)
−0.0967782 + 0.995306i \(0.530854\pi\)
\(164\) 0 0
\(165\) −1.14399 −0.0890592
\(166\) 0 0
\(167\) −19.4857 −1.50785 −0.753924 0.656962i \(-0.771841\pi\)
−0.753924 + 0.656962i \(0.771841\pi\)
\(168\) 0 0
\(169\) −7.96080 −0.612369
\(170\) 0 0
\(171\) −2.81681 −0.215407
\(172\) 0 0
\(173\) −21.3025 −1.61960 −0.809799 0.586707i \(-0.800424\pi\)
−0.809799 + 0.586707i \(0.800424\pi\)
\(174\) 0 0
\(175\) −4.67282 −0.353232
\(176\) 0 0
\(177\) −0.123254 −0.00926437
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) 21.2672 1.58078 0.790391 0.612603i \(-0.209878\pi\)
0.790391 + 0.612603i \(0.209878\pi\)
\(182\) 0 0
\(183\) −3.79834 −0.280781
\(184\) 0 0
\(185\) −4.24482 −0.312085
\(186\) 0 0
\(187\) 5.34565 0.390912
\(188\) 0 0
\(189\) −11.6336 −0.846221
\(190\) 0 0
\(191\) −24.0369 −1.73925 −0.869626 0.493711i \(-0.835640\pi\)
−0.869626 + 0.493711i \(0.835640\pi\)
\(192\) 0 0
\(193\) 22.0801 1.58936 0.794680 0.607028i \(-0.207639\pi\)
0.794680 + 0.607028i \(0.207639\pi\)
\(194\) 0 0
\(195\) −0.960797 −0.0688041
\(196\) 0 0
\(197\) −25.5473 −1.82017 −0.910085 0.414421i \(-0.863984\pi\)
−0.910085 + 0.414421i \(0.863984\pi\)
\(198\) 0 0
\(199\) −16.9793 −1.20363 −0.601814 0.798636i \(-0.705555\pi\)
−0.601814 + 0.798636i \(0.705555\pi\)
\(200\) 0 0
\(201\) 3.94006 0.277911
\(202\) 0 0
\(203\) 5.34565 0.375191
\(204\) 0 0
\(205\) 4.85601 0.339159
\(206\) 0 0
\(207\) −1.89522 −0.131727
\(208\) 0 0
\(209\) 2.67282 0.184883
\(210\) 0 0
\(211\) 8.48963 0.584451 0.292225 0.956350i \(-0.405604\pi\)
0.292225 + 0.956350i \(0.405604\pi\)
\(212\) 0 0
\(213\) 3.91369 0.268161
\(214\) 0 0
\(215\) −5.52884 −0.377064
\(216\) 0 0
\(217\) 5.34565 0.362886
\(218\) 0 0
\(219\) −0.489634 −0.0330864
\(220\) 0 0
\(221\) 4.48963 0.302005
\(222\) 0 0
\(223\) 8.55126 0.572635 0.286317 0.958135i \(-0.407569\pi\)
0.286317 + 0.958135i \(0.407569\pi\)
\(224\) 0 0
\(225\) −2.81681 −0.187787
\(226\) 0 0
\(227\) −19.9832 −1.32633 −0.663166 0.748472i \(-0.730788\pi\)
−0.663166 + 0.748472i \(0.730788\pi\)
\(228\) 0 0
\(229\) −0.751230 −0.0496427 −0.0248213 0.999692i \(-0.507902\pi\)
−0.0248213 + 0.999692i \(0.507902\pi\)
\(230\) 0 0
\(231\) 5.34565 0.351718
\(232\) 0 0
\(233\) −15.2593 −0.999672 −0.499836 0.866120i \(-0.666606\pi\)
−0.499836 + 0.866120i \(0.666606\pi\)
\(234\) 0 0
\(235\) −4.67282 −0.304821
\(236\) 0 0
\(237\) −0.856013 −0.0556040
\(238\) 0 0
\(239\) 16.5266 1.06902 0.534508 0.845164i \(-0.320497\pi\)
0.534508 + 0.845164i \(0.320497\pi\)
\(240\) 0 0
\(241\) 4.48963 0.289203 0.144601 0.989490i \(-0.453810\pi\)
0.144601 + 0.989490i \(0.453810\pi\)
\(242\) 0 0
\(243\) −10.6297 −0.681893
\(244\) 0 0
\(245\) 14.8353 0.947791
\(246\) 0 0
\(247\) 2.24482 0.142834
\(248\) 0 0
\(249\) 0.0784065 0.00496881
\(250\) 0 0
\(251\) 20.0369 1.26472 0.632360 0.774674i \(-0.282086\pi\)
0.632360 + 0.774674i \(0.282086\pi\)
\(252\) 0 0
\(253\) 1.79834 0.113061
\(254\) 0 0
\(255\) −0.856013 −0.0536056
\(256\) 0 0
\(257\) −4.15850 −0.259400 −0.129700 0.991553i \(-0.541401\pi\)
−0.129700 + 0.991553i \(0.541401\pi\)
\(258\) 0 0
\(259\) 19.8353 1.23250
\(260\) 0 0
\(261\) 3.22239 0.199461
\(262\) 0 0
\(263\) −17.0761 −1.05296 −0.526480 0.850187i \(-0.676489\pi\)
−0.526480 + 0.850187i \(0.676489\pi\)
\(264\) 0 0
\(265\) 0.244817 0.0150390
\(266\) 0 0
\(267\) −3.91369 −0.239514
\(268\) 0 0
\(269\) −29.3826 −1.79149 −0.895744 0.444570i \(-0.853356\pi\)
−0.895744 + 0.444570i \(0.853356\pi\)
\(270\) 0 0
\(271\) 16.5944 1.00804 0.504020 0.863692i \(-0.331854\pi\)
0.504020 + 0.863692i \(0.331854\pi\)
\(272\) 0 0
\(273\) 4.48963 0.271725
\(274\) 0 0
\(275\) 2.67282 0.161177
\(276\) 0 0
\(277\) −6.36638 −0.382519 −0.191259 0.981540i \(-0.561257\pi\)
−0.191259 + 0.981540i \(0.561257\pi\)
\(278\) 0 0
\(279\) 3.22239 0.192920
\(280\) 0 0
\(281\) −0.280067 −0.0167074 −0.00835371 0.999965i \(-0.502659\pi\)
−0.00835371 + 0.999965i \(0.502659\pi\)
\(282\) 0 0
\(283\) 12.1832 0.724215 0.362108 0.932136i \(-0.382057\pi\)
0.362108 + 0.932136i \(0.382057\pi\)
\(284\) 0 0
\(285\) −0.428007 −0.0253529
\(286\) 0 0
\(287\) −22.6913 −1.33942
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −3.89522 −0.228342
\(292\) 0 0
\(293\) −25.5042 −1.48997 −0.744984 0.667082i \(-0.767543\pi\)
−0.744984 + 0.667082i \(0.767543\pi\)
\(294\) 0 0
\(295\) 0.287973 0.0167664
\(296\) 0 0
\(297\) 6.65435 0.386125
\(298\) 0 0
\(299\) 1.51037 0.0873467
\(300\) 0 0
\(301\) 25.8353 1.48912
\(302\) 0 0
\(303\) 1.14399 0.0657203
\(304\) 0 0
\(305\) 8.87448 0.508152
\(306\) 0 0
\(307\) −16.3496 −0.933121 −0.466560 0.884489i \(-0.654507\pi\)
−0.466560 + 0.884489i \(0.654507\pi\)
\(308\) 0 0
\(309\) −2.96080 −0.168434
\(310\) 0 0
\(311\) 10.3064 0.584425 0.292212 0.956353i \(-0.405609\pi\)
0.292212 + 0.956353i \(0.405609\pi\)
\(312\) 0 0
\(313\) −16.9714 −0.959278 −0.479639 0.877466i \(-0.659232\pi\)
−0.479639 + 0.877466i \(0.659232\pi\)
\(314\) 0 0
\(315\) 13.1625 0.741620
\(316\) 0 0
\(317\) −31.3473 −1.76064 −0.880321 0.474378i \(-0.842673\pi\)
−0.880321 + 0.474378i \(0.842673\pi\)
\(318\) 0 0
\(319\) −3.05767 −0.171197
\(320\) 0 0
\(321\) −3.69356 −0.206154
\(322\) 0 0
\(323\) 2.00000 0.111283
\(324\) 0 0
\(325\) 2.24482 0.124520
\(326\) 0 0
\(327\) 4.97927 0.275354
\(328\) 0 0
\(329\) 21.8353 1.20382
\(330\) 0 0
\(331\) −26.4112 −1.45169 −0.725846 0.687857i \(-0.758552\pi\)
−0.725846 + 0.687857i \(0.758552\pi\)
\(332\) 0 0
\(333\) 11.9568 0.655231
\(334\) 0 0
\(335\) −9.20561 −0.502956
\(336\) 0 0
\(337\) −23.7137 −1.29177 −0.645884 0.763435i \(-0.723511\pi\)
−0.645884 + 0.763435i \(0.723511\pi\)
\(338\) 0 0
\(339\) −2.82964 −0.153685
\(340\) 0 0
\(341\) −3.05767 −0.165582
\(342\) 0 0
\(343\) −36.6129 −1.97691
\(344\) 0 0
\(345\) −0.287973 −0.0155039
\(346\) 0 0
\(347\) −3.69356 −0.198280 −0.0991402 0.995073i \(-0.531609\pi\)
−0.0991402 + 0.995073i \(0.531609\pi\)
\(348\) 0 0
\(349\) 24.9714 1.33669 0.668343 0.743853i \(-0.267004\pi\)
0.668343 + 0.743853i \(0.267004\pi\)
\(350\) 0 0
\(351\) 5.58877 0.298307
\(352\) 0 0
\(353\) 12.8639 0.684677 0.342339 0.939577i \(-0.388781\pi\)
0.342339 + 0.939577i \(0.388781\pi\)
\(354\) 0 0
\(355\) −9.14399 −0.485312
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) 32.0554 1.69182 0.845910 0.533326i \(-0.179058\pi\)
0.845910 + 0.533326i \(0.179058\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 1.65040 0.0866235
\(364\) 0 0
\(365\) 1.14399 0.0598790
\(366\) 0 0
\(367\) 20.4297 1.06642 0.533211 0.845982i \(-0.320985\pi\)
0.533211 + 0.845982i \(0.320985\pi\)
\(368\) 0 0
\(369\) −13.6785 −0.712073
\(370\) 0 0
\(371\) −1.14399 −0.0593928
\(372\) 0 0
\(373\) −5.50415 −0.284994 −0.142497 0.989795i \(-0.545513\pi\)
−0.142497 + 0.989795i \(0.545513\pi\)
\(374\) 0 0
\(375\) −0.428007 −0.0221022
\(376\) 0 0
\(377\) −2.56804 −0.132261
\(378\) 0 0
\(379\) −26.6913 −1.37104 −0.685520 0.728054i \(-0.740425\pi\)
−0.685520 + 0.728054i \(0.740425\pi\)
\(380\) 0 0
\(381\) 3.81681 0.195541
\(382\) 0 0
\(383\) 7.77365 0.397215 0.198608 0.980079i \(-0.436358\pi\)
0.198608 + 0.980079i \(0.436358\pi\)
\(384\) 0 0
\(385\) −12.4896 −0.636531
\(386\) 0 0
\(387\) 15.5737 0.791655
\(388\) 0 0
\(389\) 24.8145 1.25815 0.629074 0.777346i \(-0.283434\pi\)
0.629074 + 0.777346i \(0.283434\pi\)
\(390\) 0 0
\(391\) 1.34565 0.0680523
\(392\) 0 0
\(393\) 2.90086 0.146329
\(394\) 0 0
\(395\) 2.00000 0.100631
\(396\) 0 0
\(397\) 32.4482 1.62853 0.814263 0.580495i \(-0.197141\pi\)
0.814263 + 0.580495i \(0.197141\pi\)
\(398\) 0 0
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) 0.481728 0.0240564 0.0120282 0.999928i \(-0.496171\pi\)
0.0120282 + 0.999928i \(0.496171\pi\)
\(402\) 0 0
\(403\) −2.56804 −0.127923
\(404\) 0 0
\(405\) 7.38485 0.366956
\(406\) 0 0
\(407\) −11.3456 −0.562383
\(408\) 0 0
\(409\) 16.9793 0.839571 0.419785 0.907623i \(-0.362105\pi\)
0.419785 + 0.907623i \(0.362105\pi\)
\(410\) 0 0
\(411\) 6.00791 0.296348
\(412\) 0 0
\(413\) −1.34565 −0.0662150
\(414\) 0 0
\(415\) −0.183190 −0.00899243
\(416\) 0 0
\(417\) 2.12325 0.103976
\(418\) 0 0
\(419\) 9.43196 0.460781 0.230391 0.973098i \(-0.426000\pi\)
0.230391 + 0.973098i \(0.426000\pi\)
\(420\) 0 0
\(421\) −10.6050 −0.516855 −0.258428 0.966031i \(-0.583204\pi\)
−0.258428 + 0.966031i \(0.583204\pi\)
\(422\) 0 0
\(423\) 13.1625 0.639981
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −41.4689 −2.00682
\(428\) 0 0
\(429\) −2.56804 −0.123986
\(430\) 0 0
\(431\) 13.2593 0.638680 0.319340 0.947640i \(-0.396539\pi\)
0.319340 + 0.947640i \(0.396539\pi\)
\(432\) 0 0
\(433\) 20.5697 0.988518 0.494259 0.869315i \(-0.335439\pi\)
0.494259 + 0.869315i \(0.335439\pi\)
\(434\) 0 0
\(435\) 0.489634 0.0234762
\(436\) 0 0
\(437\) 0.672824 0.0321855
\(438\) 0 0
\(439\) −12.0784 −0.576471 −0.288235 0.957560i \(-0.593069\pi\)
−0.288235 + 0.957560i \(0.593069\pi\)
\(440\) 0 0
\(441\) −41.7882 −1.98991
\(442\) 0 0
\(443\) 2.01847 0.0959005 0.0479502 0.998850i \(-0.484731\pi\)
0.0479502 + 0.998850i \(0.484731\pi\)
\(444\) 0 0
\(445\) 9.14399 0.433467
\(446\) 0 0
\(447\) −0.374285 −0.0177031
\(448\) 0 0
\(449\) −12.0369 −0.568058 −0.284029 0.958816i \(-0.591671\pi\)
−0.284029 + 0.958816i \(0.591671\pi\)
\(450\) 0 0
\(451\) 12.9793 0.611170
\(452\) 0 0
\(453\) 9.18883 0.431729
\(454\) 0 0
\(455\) −10.4896 −0.491762
\(456\) 0 0
\(457\) 38.6498 1.80796 0.903981 0.427572i \(-0.140631\pi\)
0.903981 + 0.427572i \(0.140631\pi\)
\(458\) 0 0
\(459\) 4.97927 0.232412
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) −12.8824 −0.598695 −0.299348 0.954144i \(-0.596769\pi\)
−0.299348 + 0.954144i \(0.596769\pi\)
\(464\) 0 0
\(465\) 0.489634 0.0227062
\(466\) 0 0
\(467\) −13.2488 −0.613080 −0.306540 0.951858i \(-0.599171\pi\)
−0.306540 + 0.951858i \(0.599171\pi\)
\(468\) 0 0
\(469\) 43.0162 1.98630
\(470\) 0 0
\(471\) 5.10252 0.235112
\(472\) 0 0
\(473\) −14.7776 −0.679475
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −0.689603 −0.0315747
\(478\) 0 0
\(479\) 24.9608 1.14049 0.570244 0.821475i \(-0.306849\pi\)
0.570244 + 0.821475i \(0.306849\pi\)
\(480\) 0 0
\(481\) −9.52884 −0.434478
\(482\) 0 0
\(483\) 1.34565 0.0612291
\(484\) 0 0
\(485\) 9.10083 0.413247
\(486\) 0 0
\(487\) −3.73671 −0.169327 −0.0846633 0.996410i \(-0.526981\pi\)
−0.0846633 + 0.996410i \(0.526981\pi\)
\(488\) 0 0
\(489\) 1.05767 0.0478297
\(490\) 0 0
\(491\) −41.9955 −1.89523 −0.947615 0.319416i \(-0.896513\pi\)
−0.947615 + 0.319416i \(0.896513\pi\)
\(492\) 0 0
\(493\) −2.28797 −0.103045
\(494\) 0 0
\(495\) −7.52884 −0.338396
\(496\) 0 0
\(497\) 42.7282 1.91662
\(498\) 0 0
\(499\) −12.0185 −0.538021 −0.269010 0.963137i \(-0.586697\pi\)
−0.269010 + 0.963137i \(0.586697\pi\)
\(500\) 0 0
\(501\) 8.34000 0.372604
\(502\) 0 0
\(503\) −14.9977 −0.668716 −0.334358 0.942446i \(-0.608519\pi\)
−0.334358 + 0.942446i \(0.608519\pi\)
\(504\) 0 0
\(505\) −2.67282 −0.118939
\(506\) 0 0
\(507\) 3.40727 0.151322
\(508\) 0 0
\(509\) −7.79834 −0.345655 −0.172828 0.984952i \(-0.555290\pi\)
−0.172828 + 0.984952i \(0.555290\pi\)
\(510\) 0 0
\(511\) −5.34565 −0.236478
\(512\) 0 0
\(513\) 2.48963 0.109920
\(514\) 0 0
\(515\) 6.91764 0.304828
\(516\) 0 0
\(517\) −12.4896 −0.549293
\(518\) 0 0
\(519\) 9.11761 0.400219
\(520\) 0 0
\(521\) −19.3456 −0.847548 −0.423774 0.905768i \(-0.639295\pi\)
−0.423774 + 0.905768i \(0.639295\pi\)
\(522\) 0 0
\(523\) 8.42801 0.368531 0.184266 0.982877i \(-0.441009\pi\)
0.184266 + 0.982877i \(0.441009\pi\)
\(524\) 0 0
\(525\) 2.00000 0.0872872
\(526\) 0 0
\(527\) −2.28797 −0.0996657
\(528\) 0 0
\(529\) −22.5473 −0.980318
\(530\) 0 0
\(531\) −0.811166 −0.0352016
\(532\) 0 0
\(533\) 10.9009 0.472169
\(534\) 0 0
\(535\) 8.62967 0.373093
\(536\) 0 0
\(537\) 2.56804 0.110819
\(538\) 0 0
\(539\) 39.6521 1.70794
\(540\) 0 0
\(541\) 40.6314 1.74688 0.873439 0.486933i \(-0.161884\pi\)
0.873439 + 0.486933i \(0.161884\pi\)
\(542\) 0 0
\(543\) −9.10252 −0.390627
\(544\) 0 0
\(545\) −11.6336 −0.498330
\(546\) 0 0
\(547\) −35.5226 −1.51884 −0.759419 0.650602i \(-0.774517\pi\)
−0.759419 + 0.650602i \(0.774517\pi\)
\(548\) 0 0
\(549\) −24.9977 −1.06688
\(550\) 0 0
\(551\) −1.14399 −0.0487355
\(552\) 0 0
\(553\) −9.34565 −0.397417
\(554\) 0 0
\(555\) 1.81681 0.0771193
\(556\) 0 0
\(557\) −3.67508 −0.155718 −0.0778592 0.996964i \(-0.524808\pi\)
−0.0778592 + 0.996964i \(0.524808\pi\)
\(558\) 0 0
\(559\) −12.4112 −0.524939
\(560\) 0 0
\(561\) −2.28797 −0.0965983
\(562\) 0 0
\(563\) 10.3865 0.437741 0.218870 0.975754i \(-0.429763\pi\)
0.218870 + 0.975754i \(0.429763\pi\)
\(564\) 0 0
\(565\) 6.61120 0.278135
\(566\) 0 0
\(567\) −34.5081 −1.44920
\(568\) 0 0
\(569\) 32.4033 1.35842 0.679209 0.733945i \(-0.262323\pi\)
0.679209 + 0.733945i \(0.262323\pi\)
\(570\) 0 0
\(571\) −6.70977 −0.280795 −0.140397 0.990095i \(-0.544838\pi\)
−0.140397 + 0.990095i \(0.544838\pi\)
\(572\) 0 0
\(573\) 10.2880 0.429786
\(574\) 0 0
\(575\) 0.672824 0.0280587
\(576\) 0 0
\(577\) 13.7199 0.571168 0.285584 0.958354i \(-0.407812\pi\)
0.285584 + 0.958354i \(0.407812\pi\)
\(578\) 0 0
\(579\) −9.45043 −0.392746
\(580\) 0 0
\(581\) 0.856013 0.0355134
\(582\) 0 0
\(583\) 0.654353 0.0271005
\(584\) 0 0
\(585\) −6.32322 −0.261433
\(586\) 0 0
\(587\) −6.10478 −0.251971 −0.125986 0.992032i \(-0.540209\pi\)
−0.125986 + 0.992032i \(0.540209\pi\)
\(588\) 0 0
\(589\) −1.14399 −0.0471371
\(590\) 0 0
\(591\) 10.9344 0.449782
\(592\) 0 0
\(593\) −26.4033 −1.08425 −0.542127 0.840296i \(-0.682381\pi\)
−0.542127 + 0.840296i \(0.682381\pi\)
\(594\) 0 0
\(595\) −9.34565 −0.383134
\(596\) 0 0
\(597\) 7.26724 0.297428
\(598\) 0 0
\(599\) 7.34565 0.300135 0.150068 0.988676i \(-0.452051\pi\)
0.150068 + 0.988676i \(0.452051\pi\)
\(600\) 0 0
\(601\) −18.7776 −0.765955 −0.382977 0.923758i \(-0.625101\pi\)
−0.382977 + 0.923758i \(0.625101\pi\)
\(602\) 0 0
\(603\) 25.9305 1.05597
\(604\) 0 0
\(605\) −3.85601 −0.156769
\(606\) 0 0
\(607\) 5.20561 0.211289 0.105645 0.994404i \(-0.466309\pi\)
0.105645 + 0.994404i \(0.466309\pi\)
\(608\) 0 0
\(609\) −2.28797 −0.0927133
\(610\) 0 0
\(611\) −10.4896 −0.424365
\(612\) 0 0
\(613\) −28.8145 −1.16381 −0.581904 0.813257i \(-0.697692\pi\)
−0.581904 + 0.813257i \(0.697692\pi\)
\(614\) 0 0
\(615\) −2.07841 −0.0838094
\(616\) 0 0
\(617\) 38.5266 1.55102 0.775511 0.631334i \(-0.217492\pi\)
0.775511 + 0.631334i \(0.217492\pi\)
\(618\) 0 0
\(619\) 36.4218 1.46392 0.731958 0.681350i \(-0.238607\pi\)
0.731958 + 0.681350i \(0.238607\pi\)
\(620\) 0 0
\(621\) 1.67508 0.0672188
\(622\) 0 0
\(623\) −42.7282 −1.71187
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.14399 −0.0456864
\(628\) 0 0
\(629\) −8.48963 −0.338504
\(630\) 0 0
\(631\) 10.7098 0.426349 0.213175 0.977014i \(-0.431620\pi\)
0.213175 + 0.977014i \(0.431620\pi\)
\(632\) 0 0
\(633\) −3.63362 −0.144423
\(634\) 0 0
\(635\) −8.91764 −0.353886
\(636\) 0 0
\(637\) 33.3025 1.31949
\(638\) 0 0
\(639\) 25.7569 1.01893
\(640\) 0 0
\(641\) −30.6050 −1.20882 −0.604412 0.796672i \(-0.706592\pi\)
−0.604412 + 0.796672i \(0.706592\pi\)
\(642\) 0 0
\(643\) −13.4425 −0.530121 −0.265061 0.964232i \(-0.585392\pi\)
−0.265061 + 0.964232i \(0.585392\pi\)
\(644\) 0 0
\(645\) 2.36638 0.0931761
\(646\) 0 0
\(647\) −33.5288 −1.31815 −0.659077 0.752075i \(-0.729053\pi\)
−0.659077 + 0.752075i \(0.729053\pi\)
\(648\) 0 0
\(649\) 0.769701 0.0302134
\(650\) 0 0
\(651\) −2.28797 −0.0896727
\(652\) 0 0
\(653\) 23.2593 0.910208 0.455104 0.890438i \(-0.349602\pi\)
0.455104 + 0.890438i \(0.349602\pi\)
\(654\) 0 0
\(655\) −6.77761 −0.264823
\(656\) 0 0
\(657\) −3.22239 −0.125718
\(658\) 0 0
\(659\) −30.9793 −1.20678 −0.603390 0.797446i \(-0.706184\pi\)
−0.603390 + 0.797446i \(0.706184\pi\)
\(660\) 0 0
\(661\) 10.3743 0.403513 0.201756 0.979436i \(-0.435335\pi\)
0.201756 + 0.979436i \(0.435335\pi\)
\(662\) 0 0
\(663\) −1.92159 −0.0746285
\(664\) 0 0
\(665\) −4.67282 −0.181204
\(666\) 0 0
\(667\) −0.769701 −0.0298030
\(668\) 0 0
\(669\) −3.66000 −0.141504
\(670\) 0 0
\(671\) 23.7199 0.915698
\(672\) 0 0
\(673\) −26.0880 −1.00562 −0.502809 0.864397i \(-0.667700\pi\)
−0.502809 + 0.864397i \(0.667700\pi\)
\(674\) 0 0
\(675\) 2.48963 0.0958261
\(676\) 0 0
\(677\) −12.5249 −0.481370 −0.240685 0.970603i \(-0.577372\pi\)
−0.240685 + 0.970603i \(0.577372\pi\)
\(678\) 0 0
\(679\) −42.5266 −1.63202
\(680\) 0 0
\(681\) 8.55295 0.327750
\(682\) 0 0
\(683\) −25.9339 −0.992331 −0.496166 0.868228i \(-0.665259\pi\)
−0.496166 + 0.868228i \(0.665259\pi\)
\(684\) 0 0
\(685\) −14.0369 −0.536324
\(686\) 0 0
\(687\) 0.321532 0.0122672
\(688\) 0 0
\(689\) 0.549569 0.0209369
\(690\) 0 0
\(691\) 5.52093 0.210026 0.105013 0.994471i \(-0.466512\pi\)
0.105013 + 0.994471i \(0.466512\pi\)
\(692\) 0 0
\(693\) 35.1809 1.33641
\(694\) 0 0
\(695\) −4.96080 −0.188174
\(696\) 0 0
\(697\) 9.71203 0.367869
\(698\) 0 0
\(699\) 6.53110 0.247029
\(700\) 0 0
\(701\) 22.9687 0.867516 0.433758 0.901029i \(-0.357187\pi\)
0.433758 + 0.901029i \(0.357187\pi\)
\(702\) 0 0
\(703\) −4.24482 −0.160096
\(704\) 0 0
\(705\) 2.00000 0.0753244
\(706\) 0 0
\(707\) 12.4896 0.469721
\(708\) 0 0
\(709\) −16.2386 −0.609854 −0.304927 0.952376i \(-0.598632\pi\)
−0.304927 + 0.952376i \(0.598632\pi\)
\(710\) 0 0
\(711\) −5.63362 −0.211277
\(712\) 0 0
\(713\) −0.769701 −0.0288255
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) 0 0
\(717\) −7.07349 −0.264164
\(718\) 0 0
\(719\) 13.1704 0.491172 0.245586 0.969375i \(-0.421020\pi\)
0.245586 + 0.969375i \(0.421020\pi\)
\(720\) 0 0
\(721\) −32.3249 −1.20384
\(722\) 0 0
\(723\) −1.92159 −0.0714648
\(724\) 0 0
\(725\) −1.14399 −0.0424866
\(726\) 0 0
\(727\) −40.7961 −1.51304 −0.756521 0.653969i \(-0.773103\pi\)
−0.756521 + 0.653969i \(0.773103\pi\)
\(728\) 0 0
\(729\) −17.6050 −0.652036
\(730\) 0 0
\(731\) −11.0577 −0.408983
\(732\) 0 0
\(733\) 27.8722 1.02948 0.514742 0.857345i \(-0.327888\pi\)
0.514742 + 0.857345i \(0.327888\pi\)
\(734\) 0 0
\(735\) −6.34960 −0.234209
\(736\) 0 0
\(737\) −24.6050 −0.906336
\(738\) 0 0
\(739\) 36.0739 1.32700 0.663500 0.748177i \(-0.269070\pi\)
0.663500 + 0.748177i \(0.269070\pi\)
\(740\) 0 0
\(741\) −0.960797 −0.0352958
\(742\) 0 0
\(743\) 19.8890 0.729657 0.364828 0.931075i \(-0.381128\pi\)
0.364828 + 0.931075i \(0.381128\pi\)
\(744\) 0 0
\(745\) 0.874485 0.0320386
\(746\) 0 0
\(747\) 0.516011 0.0188798
\(748\) 0 0
\(749\) −40.3249 −1.47344
\(750\) 0 0
\(751\) 17.1025 0.624080 0.312040 0.950069i \(-0.398988\pi\)
0.312040 + 0.950069i \(0.398988\pi\)
\(752\) 0 0
\(753\) −8.57595 −0.312525
\(754\) 0 0
\(755\) −21.4689 −0.781333
\(756\) 0 0
\(757\) −10.6992 −0.388869 −0.194435 0.980915i \(-0.562287\pi\)
−0.194435 + 0.980915i \(0.562287\pi\)
\(758\) 0 0
\(759\) −0.769701 −0.0279384
\(760\) 0 0
\(761\) −48.4218 −1.75529 −0.877644 0.479312i \(-0.840886\pi\)
−0.877644 + 0.479312i \(0.840886\pi\)
\(762\) 0 0
\(763\) 54.3619 1.96803
\(764\) 0 0
\(765\) −5.63362 −0.203684
\(766\) 0 0
\(767\) 0.646447 0.0233418
\(768\) 0 0
\(769\) −49.4011 −1.78145 −0.890724 0.454545i \(-0.849802\pi\)
−0.890724 + 0.454545i \(0.849802\pi\)
\(770\) 0 0
\(771\) 1.77987 0.0641004
\(772\) 0 0
\(773\) −32.8129 −1.18020 −0.590098 0.807331i \(-0.700911\pi\)
−0.590098 + 0.807331i \(0.700911\pi\)
\(774\) 0 0
\(775\) −1.14399 −0.0410932
\(776\) 0 0
\(777\) −8.48963 −0.304564
\(778\) 0 0
\(779\) 4.85601 0.173985
\(780\) 0 0
\(781\) −24.4403 −0.874541
\(782\) 0 0
\(783\) −2.84811 −0.101783
\(784\) 0 0
\(785\) −11.9216 −0.425500
\(786\) 0 0
\(787\) −30.1400 −1.07438 −0.537188 0.843462i \(-0.680513\pi\)
−0.537188 + 0.843462i \(0.680513\pi\)
\(788\) 0 0
\(789\) 7.30871 0.260197
\(790\) 0 0
\(791\) −30.8930 −1.09843
\(792\) 0 0
\(793\) 19.9216 0.707437
\(794\) 0 0
\(795\) −0.104783 −0.00371628
\(796\) 0 0
\(797\) −1.31040 −0.0464166 −0.0232083 0.999731i \(-0.507388\pi\)
−0.0232083 + 0.999731i \(0.507388\pi\)
\(798\) 0 0
\(799\) −9.34565 −0.330625
\(800\) 0 0
\(801\) −25.7569 −0.910074
\(802\) 0 0
\(803\) 3.05767 0.107903
\(804\) 0 0
\(805\) −3.14399 −0.110811
\(806\) 0 0
\(807\) 12.5759 0.442694
\(808\) 0 0
\(809\) 35.5058 1.24832 0.624159 0.781297i \(-0.285442\pi\)
0.624159 + 0.781297i \(0.285442\pi\)
\(810\) 0 0
\(811\) 26.9299 0.945637 0.472818 0.881160i \(-0.343237\pi\)
0.472818 + 0.881160i \(0.343237\pi\)
\(812\) 0 0
\(813\) −7.10252 −0.249096
\(814\) 0 0
\(815\) −2.47116 −0.0865611
\(816\) 0 0
\(817\) −5.52884 −0.193430
\(818\) 0 0
\(819\) 29.5473 1.03247
\(820\) 0 0
\(821\) −13.1440 −0.458728 −0.229364 0.973341i \(-0.573665\pi\)
−0.229364 + 0.973341i \(0.573665\pi\)
\(822\) 0 0
\(823\) 40.3928 1.40800 0.704001 0.710198i \(-0.251395\pi\)
0.704001 + 0.710198i \(0.251395\pi\)
\(824\) 0 0
\(825\) −1.14399 −0.0398285
\(826\) 0 0
\(827\) 27.3658 0.951602 0.475801 0.879553i \(-0.342158\pi\)
0.475801 + 0.879553i \(0.342158\pi\)
\(828\) 0 0
\(829\) 29.3456 1.01922 0.509608 0.860407i \(-0.329790\pi\)
0.509608 + 0.860407i \(0.329790\pi\)
\(830\) 0 0
\(831\) 2.72485 0.0945241
\(832\) 0 0
\(833\) 29.6706 1.02802
\(834\) 0 0
\(835\) −19.4857 −0.674330
\(836\) 0 0
\(837\) −2.84811 −0.0984450
\(838\) 0 0
\(839\) 49.6336 1.71354 0.856771 0.515696i \(-0.172467\pi\)
0.856771 + 0.515696i \(0.172467\pi\)
\(840\) 0 0
\(841\) −27.6913 −0.954872
\(842\) 0 0
\(843\) 0.119871 0.00412857
\(844\) 0 0
\(845\) −7.96080 −0.273860
\(846\) 0 0
\(847\) 18.0185 0.619122
\(848\) 0 0
\(849\) −5.21449 −0.178961
\(850\) 0 0
\(851\) −2.85601 −0.0979029
\(852\) 0 0
\(853\) −31.1730 −1.06734 −0.533672 0.845692i \(-0.679188\pi\)
−0.533672 + 0.845692i \(0.679188\pi\)
\(854\) 0 0
\(855\) −2.81681 −0.0963329
\(856\) 0 0
\(857\) 48.4755 1.65589 0.827946 0.560808i \(-0.189509\pi\)
0.827946 + 0.560808i \(0.189509\pi\)
\(858\) 0 0
\(859\) 25.8353 0.881488 0.440744 0.897633i \(-0.354715\pi\)
0.440744 + 0.897633i \(0.354715\pi\)
\(860\) 0 0
\(861\) 9.71203 0.330985
\(862\) 0 0
\(863\) 38.3865 1.30669 0.653347 0.757059i \(-0.273364\pi\)
0.653347 + 0.757059i \(0.273364\pi\)
\(864\) 0 0
\(865\) −21.3025 −0.724306
\(866\) 0 0
\(867\) 5.56409 0.188966
\(868\) 0 0
\(869\) 5.34565 0.181339
\(870\) 0 0
\(871\) −20.6649 −0.700204
\(872\) 0 0
\(873\) −25.6353 −0.867624
\(874\) 0 0
\(875\) −4.67282 −0.157970
\(876\) 0 0
\(877\) 17.2610 0.582863 0.291432 0.956592i \(-0.405868\pi\)
0.291432 + 0.956592i \(0.405868\pi\)
\(878\) 0 0
\(879\) 10.9159 0.368186
\(880\) 0 0
\(881\) −13.8168 −0.465500 −0.232750 0.972537i \(-0.574772\pi\)
−0.232750 + 0.972537i \(0.574772\pi\)
\(882\) 0 0
\(883\) −51.2073 −1.72326 −0.861632 0.507534i \(-0.830557\pi\)
−0.861632 + 0.507534i \(0.830557\pi\)
\(884\) 0 0
\(885\) −0.123254 −0.00414315
\(886\) 0 0
\(887\) 39.6538 1.33144 0.665722 0.746200i \(-0.268124\pi\)
0.665722 + 0.746200i \(0.268124\pi\)
\(888\) 0 0
\(889\) 41.6706 1.39759
\(890\) 0 0
\(891\) 19.7384 0.661261
\(892\) 0 0
\(893\) −4.67282 −0.156370
\(894\) 0 0
\(895\) −6.00000 −0.200558
\(896\) 0 0
\(897\) −0.646447 −0.0215842
\(898\) 0 0
\(899\) 1.30871 0.0436478
\(900\) 0 0
\(901\) 0.489634 0.0163121
\(902\) 0 0
\(903\) −11.0577 −0.367976
\(904\) 0 0
\(905\) 21.2672 0.706947
\(906\) 0 0
\(907\) 4.62967 0.153726 0.0768628 0.997042i \(-0.475510\pi\)
0.0768628 + 0.997042i \(0.475510\pi\)
\(908\) 0 0
\(909\) 7.52884 0.249716
\(910\) 0 0
\(911\) −15.9506 −0.528468 −0.264234 0.964459i \(-0.585119\pi\)
−0.264234 + 0.964459i \(0.585119\pi\)
\(912\) 0 0
\(913\) −0.489634 −0.0162045
\(914\) 0 0
\(915\) −3.79834 −0.125569
\(916\) 0 0
\(917\) 31.6706 1.04585
\(918\) 0 0
\(919\) −32.1233 −1.05965 −0.529824 0.848107i \(-0.677742\pi\)
−0.529824 + 0.848107i \(0.677742\pi\)
\(920\) 0 0
\(921\) 6.99774 0.230583
\(922\) 0 0
\(923\) −20.5266 −0.675640
\(924\) 0 0
\(925\) −4.24482 −0.139569
\(926\) 0 0
\(927\) −19.4857 −0.639994
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 14.8353 0.486207
\(932\) 0 0
\(933\) −4.41123 −0.144417
\(934\) 0 0
\(935\) 5.34565 0.174821
\(936\) 0 0
\(937\) −20.9872 −0.685621 −0.342811 0.939405i \(-0.611379\pi\)
−0.342811 + 0.939405i \(0.611379\pi\)
\(938\) 0 0
\(939\) 7.26386 0.237047
\(940\) 0 0
\(941\) 3.51827 0.114692 0.0573462 0.998354i \(-0.481736\pi\)
0.0573462 + 0.998354i \(0.481736\pi\)
\(942\) 0 0
\(943\) 3.26724 0.106396
\(944\) 0 0
\(945\) −11.6336 −0.378442
\(946\) 0 0
\(947\) 0.112689 0.00366190 0.00183095 0.999998i \(-0.499417\pi\)
0.00183095 + 0.999998i \(0.499417\pi\)
\(948\) 0 0
\(949\) 2.56804 0.0833621
\(950\) 0 0
\(951\) 13.4169 0.435072
\(952\) 0 0
\(953\) −31.6353 −1.02477 −0.512384 0.858756i \(-0.671238\pi\)
−0.512384 + 0.858756i \(0.671238\pi\)
\(954\) 0 0
\(955\) −24.0369 −0.777817
\(956\) 0 0
\(957\) 1.30871 0.0423044
\(958\) 0 0
\(959\) 65.5922 2.11808
\(960\) 0 0
\(961\) −29.6913 −0.957784
\(962\) 0 0
\(963\) −24.3081 −0.783319
\(964\) 0 0
\(965\) 22.0801 0.710784
\(966\) 0 0
\(967\) 31.6442 1.01761 0.508804 0.860882i \(-0.330088\pi\)
0.508804 + 0.860882i \(0.330088\pi\)
\(968\) 0 0
\(969\) −0.856013 −0.0274991
\(970\) 0 0
\(971\) 55.4689 1.78008 0.890041 0.455881i \(-0.150676\pi\)
0.890041 + 0.455881i \(0.150676\pi\)
\(972\) 0 0
\(973\) 23.1809 0.743146
\(974\) 0 0
\(975\) −0.960797 −0.0307701
\(976\) 0 0
\(977\) −2.82076 −0.0902442 −0.0451221 0.998981i \(-0.514368\pi\)
−0.0451221 + 0.998981i \(0.514368\pi\)
\(978\) 0 0
\(979\) 24.4403 0.781114
\(980\) 0 0
\(981\) 32.7697 1.04626
\(982\) 0 0
\(983\) 24.9467 0.795675 0.397838 0.917456i \(-0.369761\pi\)
0.397838 + 0.917456i \(0.369761\pi\)
\(984\) 0 0
\(985\) −25.5473 −0.814005
\(986\) 0 0
\(987\) −9.34565 −0.297475
\(988\) 0 0
\(989\) −3.71993 −0.118287
\(990\) 0 0
\(991\) 31.0241 0.985514 0.492757 0.870167i \(-0.335989\pi\)
0.492757 + 0.870167i \(0.335989\pi\)
\(992\) 0 0
\(993\) 11.3042 0.358727
\(994\) 0 0
\(995\) −16.9793 −0.538279
\(996\) 0 0
\(997\) −55.7859 −1.76676 −0.883379 0.468660i \(-0.844737\pi\)
−0.883379 + 0.468660i \(0.844737\pi\)
\(998\) 0 0
\(999\) −10.5680 −0.334358
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6080.2.a.bp.1.2 3
4.3 odd 2 6080.2.a.bz.1.2 3
8.3 odd 2 3040.2.a.k.1.2 3
8.5 even 2 3040.2.a.n.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.k.1.2 3 8.3 odd 2
3040.2.a.n.1.2 yes 3 8.5 even 2
6080.2.a.bp.1.2 3 1.1 even 1 trivial
6080.2.a.bz.1.2 3 4.3 odd 2