Properties

Label 6080.2.a.cl.1.4
Level $6080$
Weight $2$
Character 6080.1
Self dual yes
Analytic conductor $48.549$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.387268.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3040)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.50372\) 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)
 
\(f(q)\) \(=\) \(q+1.76491 q^{3} +1.00000 q^{5} -3.89255 q^{7} +0.114895 q^{9} +O(q^{10})\) \(q+1.76491 q^{3} +1.00000 q^{5} -3.89255 q^{7} +0.114895 q^{9} -4.80767 q^{11} -0.0353128 q^{13} +1.76491 q^{15} -4.39275 q^{17} -1.00000 q^{19} -6.86999 q^{21} +9.12195 q^{23} +1.00000 q^{25} -5.09194 q^{27} +6.21514 q^{29} +9.71448 q^{31} -8.48508 q^{33} -3.89255 q^{35} +0.749158 q^{37} -0.0623238 q^{39} -3.52981 q^{41} +11.4625 q^{43} +0.114895 q^{45} +5.57827 q^{47} +8.15196 q^{49} -7.75279 q^{51} -8.67877 q^{53} -4.80767 q^{55} -1.76491 q^{57} +2.56252 q^{59} -6.50725 q^{61} -0.447233 q^{63} -0.0353128 q^{65} +13.2496 q^{67} +16.0994 q^{69} +1.67106 q^{71} -11.3217 q^{73} +1.76491 q^{75} +18.7141 q^{77} +11.4141 q^{79} -9.33148 q^{81} +5.64889 q^{83} -4.39275 q^{85} +10.9691 q^{87} -1.24079 q^{89} +0.137457 q^{91} +17.1451 q^{93} -1.00000 q^{95} +9.31976 q^{97} -0.552375 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 7 q^{9} + 2 q^{11} + 4 q^{13} + 4 q^{15} - 12 q^{17} - 5 q^{19} + 10 q^{21} + 8 q^{23} + 5 q^{25} + 16 q^{27} + 6 q^{29} + 10 q^{31} - 18 q^{33} + 4 q^{35} + 6 q^{37} + 18 q^{39} - 8 q^{41} + 12 q^{43} + 7 q^{45} + 16 q^{47} + 7 q^{49} - 14 q^{51} + 18 q^{53} + 2 q^{55} - 4 q^{57} + 8 q^{59} - 2 q^{61} + 36 q^{63} + 4 q^{65} + 10 q^{67} + 22 q^{69} - 18 q^{71} - 28 q^{73} + 4 q^{75} + 28 q^{77} + 14 q^{79} + 25 q^{81} + 8 q^{83} - 12 q^{85} + 24 q^{87} - 30 q^{89} + 28 q^{91} + 24 q^{93} - 5 q^{95} - 18 q^{97} - 14 q^{99}+O(q^{100}) \) 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
\(3\) 1.76491 1.01897 0.509485 0.860480i \(-0.329836\pi\)
0.509485 + 0.860480i \(0.329836\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.89255 −1.47125 −0.735623 0.677391i \(-0.763111\pi\)
−0.735623 + 0.677391i \(0.763111\pi\)
\(8\) 0 0
\(9\) 0.114895 0.0382982
\(10\) 0 0
\(11\) −4.80767 −1.44957 −0.724783 0.688977i \(-0.758060\pi\)
−0.724783 + 0.688977i \(0.758060\pi\)
\(12\) 0 0
\(13\) −0.0353128 −0.00979401 −0.00489701 0.999988i \(-0.501559\pi\)
−0.00489701 + 0.999988i \(0.501559\pi\)
\(14\) 0 0
\(15\) 1.76491 0.455697
\(16\) 0 0
\(17\) −4.39275 −1.06540 −0.532699 0.846305i \(-0.678822\pi\)
−0.532699 + 0.846305i \(0.678822\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −6.86999 −1.49915
\(22\) 0 0
\(23\) 9.12195 1.90206 0.951029 0.309102i \(-0.100028\pi\)
0.951029 + 0.309102i \(0.100028\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.09194 −0.979944
\(28\) 0 0
\(29\) 6.21514 1.15412 0.577061 0.816701i \(-0.304199\pi\)
0.577061 + 0.816701i \(0.304199\pi\)
\(30\) 0 0
\(31\) 9.71448 1.74477 0.872386 0.488818i \(-0.162572\pi\)
0.872386 + 0.488818i \(0.162572\pi\)
\(32\) 0 0
\(33\) −8.48508 −1.47706
\(34\) 0 0
\(35\) −3.89255 −0.657961
\(36\) 0 0
\(37\) 0.749158 0.123161 0.0615804 0.998102i \(-0.480386\pi\)
0.0615804 + 0.998102i \(0.480386\pi\)
\(38\) 0 0
\(39\) −0.0623238 −0.00997980
\(40\) 0 0
\(41\) −3.52981 −0.551264 −0.275632 0.961263i \(-0.588887\pi\)
−0.275632 + 0.961263i \(0.588887\pi\)
\(42\) 0 0
\(43\) 11.4625 1.74802 0.874009 0.485910i \(-0.161512\pi\)
0.874009 + 0.485910i \(0.161512\pi\)
\(44\) 0 0
\(45\) 0.114895 0.0171275
\(46\) 0 0
\(47\) 5.57827 0.813674 0.406837 0.913501i \(-0.366632\pi\)
0.406837 + 0.913501i \(0.366632\pi\)
\(48\) 0 0
\(49\) 8.15196 1.16457
\(50\) 0 0
\(51\) −7.75279 −1.08561
\(52\) 0 0
\(53\) −8.67877 −1.19212 −0.596061 0.802939i \(-0.703268\pi\)
−0.596061 + 0.802939i \(0.703268\pi\)
\(54\) 0 0
\(55\) −4.80767 −0.648265
\(56\) 0 0
\(57\) −1.76491 −0.233768
\(58\) 0 0
\(59\) 2.56252 0.333612 0.166806 0.985990i \(-0.446655\pi\)
0.166806 + 0.985990i \(0.446655\pi\)
\(60\) 0 0
\(61\) −6.50725 −0.833168 −0.416584 0.909097i \(-0.636773\pi\)
−0.416584 + 0.909097i \(0.636773\pi\)
\(62\) 0 0
\(63\) −0.447233 −0.0563461
\(64\) 0 0
\(65\) −0.0353128 −0.00438002
\(66\) 0 0
\(67\) 13.2496 1.61870 0.809348 0.587330i \(-0.199821\pi\)
0.809348 + 0.587330i \(0.199821\pi\)
\(68\) 0 0
\(69\) 16.0994 1.93814
\(70\) 0 0
\(71\) 1.67106 0.198319 0.0991594 0.995072i \(-0.468385\pi\)
0.0991594 + 0.995072i \(0.468385\pi\)
\(72\) 0 0
\(73\) −11.3217 −1.32511 −0.662554 0.749014i \(-0.730528\pi\)
−0.662554 + 0.749014i \(0.730528\pi\)
\(74\) 0 0
\(75\) 1.76491 0.203794
\(76\) 0 0
\(77\) 18.7141 2.13267
\(78\) 0 0
\(79\) 11.4141 1.28418 0.642091 0.766628i \(-0.278067\pi\)
0.642091 + 0.766628i \(0.278067\pi\)
\(80\) 0 0
\(81\) −9.33148 −1.03683
\(82\) 0 0
\(83\) 5.64889 0.620047 0.310023 0.950729i \(-0.399663\pi\)
0.310023 + 0.950729i \(0.399663\pi\)
\(84\) 0 0
\(85\) −4.39275 −0.476460
\(86\) 0 0
\(87\) 10.9691 1.17601
\(88\) 0 0
\(89\) −1.24079 −0.131523 −0.0657617 0.997835i \(-0.520948\pi\)
−0.0657617 + 0.997835i \(0.520948\pi\)
\(90\) 0 0
\(91\) 0.137457 0.0144094
\(92\) 0 0
\(93\) 17.1451 1.77787
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 9.31976 0.946278 0.473139 0.880988i \(-0.343121\pi\)
0.473139 + 0.880988i \(0.343121\pi\)
\(98\) 0 0
\(99\) −0.552375 −0.0555158
\(100\) 0 0
\(101\) −0.807666 −0.0803657 −0.0401829 0.999192i \(-0.512794\pi\)
−0.0401829 + 0.999192i \(0.512794\pi\)
\(102\) 0 0
\(103\) −17.0564 −1.68062 −0.840309 0.542108i \(-0.817626\pi\)
−0.840309 + 0.542108i \(0.817626\pi\)
\(104\) 0 0
\(105\) −6.86999 −0.670442
\(106\) 0 0
\(107\) 8.41008 0.813033 0.406516 0.913643i \(-0.366743\pi\)
0.406516 + 0.913643i \(0.366743\pi\)
\(108\) 0 0
\(109\) 16.1309 1.54506 0.772529 0.634979i \(-0.218991\pi\)
0.772529 + 0.634979i \(0.218991\pi\)
\(110\) 0 0
\(111\) 1.32219 0.125497
\(112\) 0 0
\(113\) 13.0794 1.23040 0.615201 0.788370i \(-0.289075\pi\)
0.615201 + 0.788370i \(0.289075\pi\)
\(114\) 0 0
\(115\) 9.12195 0.850626
\(116\) 0 0
\(117\) −0.00405725 −0.000375093 0
\(118\) 0 0
\(119\) 17.0990 1.56746
\(120\) 0 0
\(121\) 12.1136 1.10124
\(122\) 0 0
\(123\) −6.22979 −0.561721
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −15.0024 −1.33125 −0.665623 0.746288i \(-0.731834\pi\)
−0.665623 + 0.746288i \(0.731834\pi\)
\(128\) 0 0
\(129\) 20.2303 1.78118
\(130\) 0 0
\(131\) −4.11404 −0.359445 −0.179723 0.983717i \(-0.557520\pi\)
−0.179723 + 0.983717i \(0.557520\pi\)
\(132\) 0 0
\(133\) 3.89255 0.337527
\(134\) 0 0
\(135\) −5.09194 −0.438244
\(136\) 0 0
\(137\) −2.77742 −0.237291 −0.118645 0.992937i \(-0.537855\pi\)
−0.118645 + 0.992937i \(0.537855\pi\)
\(138\) 0 0
\(139\) 5.40850 0.458743 0.229371 0.973339i \(-0.426333\pi\)
0.229371 + 0.973339i \(0.426333\pi\)
\(140\) 0 0
\(141\) 9.84512 0.829109
\(142\) 0 0
\(143\) 0.169772 0.0141971
\(144\) 0 0
\(145\) 6.21514 0.516139
\(146\) 0 0
\(147\) 14.3874 1.18666
\(148\) 0 0
\(149\) 0.291090 0.0238470 0.0119235 0.999929i \(-0.496205\pi\)
0.0119235 + 0.999929i \(0.496205\pi\)
\(150\) 0 0
\(151\) 15.0259 1.22279 0.611395 0.791326i \(-0.290609\pi\)
0.611395 + 0.791326i \(0.290609\pi\)
\(152\) 0 0
\(153\) −0.504703 −0.0408028
\(154\) 0 0
\(155\) 9.71448 0.780286
\(156\) 0 0
\(157\) 9.60044 0.766198 0.383099 0.923707i \(-0.374857\pi\)
0.383099 + 0.923707i \(0.374857\pi\)
\(158\) 0 0
\(159\) −15.3172 −1.21473
\(160\) 0 0
\(161\) −35.5077 −2.79840
\(162\) 0 0
\(163\) 4.90681 0.384331 0.192166 0.981363i \(-0.438449\pi\)
0.192166 + 0.981363i \(0.438449\pi\)
\(164\) 0 0
\(165\) −8.48508 −0.660563
\(166\) 0 0
\(167\) −6.54582 −0.506531 −0.253266 0.967397i \(-0.581505\pi\)
−0.253266 + 0.967397i \(0.581505\pi\)
\(168\) 0 0
\(169\) −12.9988 −0.999904
\(170\) 0 0
\(171\) −0.114895 −0.00878621
\(172\) 0 0
\(173\) 23.9483 1.82076 0.910379 0.413776i \(-0.135790\pi\)
0.910379 + 0.413776i \(0.135790\pi\)
\(174\) 0 0
\(175\) −3.89255 −0.294249
\(176\) 0 0
\(177\) 4.52261 0.339940
\(178\) 0 0
\(179\) −9.98637 −0.746416 −0.373208 0.927748i \(-0.621742\pi\)
−0.373208 + 0.927748i \(0.621742\pi\)
\(180\) 0 0
\(181\) 7.41406 0.551083 0.275541 0.961289i \(-0.411143\pi\)
0.275541 + 0.961289i \(0.411143\pi\)
\(182\) 0 0
\(183\) −11.4847 −0.848973
\(184\) 0 0
\(185\) 0.749158 0.0550792
\(186\) 0 0
\(187\) 21.1189 1.54436
\(188\) 0 0
\(189\) 19.8206 1.44174
\(190\) 0 0
\(191\) −22.3007 −1.61362 −0.806809 0.590812i \(-0.798808\pi\)
−0.806809 + 0.590812i \(0.798808\pi\)
\(192\) 0 0
\(193\) −3.15393 −0.227025 −0.113512 0.993537i \(-0.536210\pi\)
−0.113512 + 0.993537i \(0.536210\pi\)
\(194\) 0 0
\(195\) −0.0623238 −0.00446310
\(196\) 0 0
\(197\) −5.21801 −0.371768 −0.185884 0.982572i \(-0.559515\pi\)
−0.185884 + 0.982572i \(0.559515\pi\)
\(198\) 0 0
\(199\) 22.2841 1.57967 0.789837 0.613316i \(-0.210165\pi\)
0.789837 + 0.613316i \(0.210165\pi\)
\(200\) 0 0
\(201\) 23.3843 1.64940
\(202\) 0 0
\(203\) −24.1927 −1.69800
\(204\) 0 0
\(205\) −3.52981 −0.246533
\(206\) 0 0
\(207\) 1.04806 0.0728454
\(208\) 0 0
\(209\) 4.80767 0.332553
\(210\) 0 0
\(211\) 2.27871 0.156873 0.0784364 0.996919i \(-0.475007\pi\)
0.0784364 + 0.996919i \(0.475007\pi\)
\(212\) 0 0
\(213\) 2.94927 0.202081
\(214\) 0 0
\(215\) 11.4625 0.781737
\(216\) 0 0
\(217\) −37.8141 −2.56699
\(218\) 0 0
\(219\) −19.9818 −1.35024
\(220\) 0 0
\(221\) 0.155120 0.0104345
\(222\) 0 0
\(223\) 20.1108 1.34672 0.673360 0.739315i \(-0.264850\pi\)
0.673360 + 0.739315i \(0.264850\pi\)
\(224\) 0 0
\(225\) 0.114895 0.00765964
\(226\) 0 0
\(227\) −1.16184 −0.0771140 −0.0385570 0.999256i \(-0.512276\pi\)
−0.0385570 + 0.999256i \(0.512276\pi\)
\(228\) 0 0
\(229\) −2.77617 −0.183454 −0.0917272 0.995784i \(-0.529239\pi\)
−0.0917272 + 0.995784i \(0.529239\pi\)
\(230\) 0 0
\(231\) 33.0286 2.17312
\(232\) 0 0
\(233\) 25.2847 1.65645 0.828227 0.560392i \(-0.189350\pi\)
0.828227 + 0.560392i \(0.189350\pi\)
\(234\) 0 0
\(235\) 5.57827 0.363886
\(236\) 0 0
\(237\) 20.1448 1.30854
\(238\) 0 0
\(239\) −0.228766 −0.0147976 −0.00739881 0.999973i \(-0.502355\pi\)
−0.00739881 + 0.999973i \(0.502355\pi\)
\(240\) 0 0
\(241\) −3.21056 −0.206810 −0.103405 0.994639i \(-0.532974\pi\)
−0.103405 + 0.994639i \(0.532974\pi\)
\(242\) 0 0
\(243\) −1.19337 −0.0765548
\(244\) 0 0
\(245\) 8.15196 0.520809
\(246\) 0 0
\(247\) 0.0353128 0.00224690
\(248\) 0 0
\(249\) 9.96977 0.631808
\(250\) 0 0
\(251\) 26.5329 1.67474 0.837372 0.546634i \(-0.184091\pi\)
0.837372 + 0.546634i \(0.184091\pi\)
\(252\) 0 0
\(253\) −43.8553 −2.75716
\(254\) 0 0
\(255\) −7.75279 −0.485498
\(256\) 0 0
\(257\) −28.5807 −1.78281 −0.891406 0.453206i \(-0.850280\pi\)
−0.891406 + 0.453206i \(0.850280\pi\)
\(258\) 0 0
\(259\) −2.91614 −0.181200
\(260\) 0 0
\(261\) 0.714086 0.0442008
\(262\) 0 0
\(263\) 18.4959 1.14050 0.570252 0.821470i \(-0.306846\pi\)
0.570252 + 0.821470i \(0.306846\pi\)
\(264\) 0 0
\(265\) −8.67877 −0.533133
\(266\) 0 0
\(267\) −2.18988 −0.134018
\(268\) 0 0
\(269\) 15.6882 0.956526 0.478263 0.878217i \(-0.341267\pi\)
0.478263 + 0.878217i \(0.341267\pi\)
\(270\) 0 0
\(271\) −27.7803 −1.68753 −0.843767 0.536710i \(-0.819667\pi\)
−0.843767 + 0.536710i \(0.819667\pi\)
\(272\) 0 0
\(273\) 0.242599 0.0146827
\(274\) 0 0
\(275\) −4.80767 −0.289913
\(276\) 0 0
\(277\) −19.9044 −1.19594 −0.597968 0.801520i \(-0.704025\pi\)
−0.597968 + 0.801520i \(0.704025\pi\)
\(278\) 0 0
\(279\) 1.11614 0.0668217
\(280\) 0 0
\(281\) 10.5220 0.627688 0.313844 0.949475i \(-0.398383\pi\)
0.313844 + 0.949475i \(0.398383\pi\)
\(282\) 0 0
\(283\) 4.26164 0.253328 0.126664 0.991946i \(-0.459573\pi\)
0.126664 + 0.991946i \(0.459573\pi\)
\(284\) 0 0
\(285\) −1.76491 −0.104544
\(286\) 0 0
\(287\) 13.7400 0.811045
\(288\) 0 0
\(289\) 2.29623 0.135072
\(290\) 0 0
\(291\) 16.4485 0.964228
\(292\) 0 0
\(293\) 2.16596 0.126536 0.0632682 0.997997i \(-0.479848\pi\)
0.0632682 + 0.997997i \(0.479848\pi\)
\(294\) 0 0
\(295\) 2.56252 0.149196
\(296\) 0 0
\(297\) 24.4803 1.42049
\(298\) 0 0
\(299\) −0.322122 −0.0186288
\(300\) 0 0
\(301\) −44.6184 −2.57176
\(302\) 0 0
\(303\) −1.42545 −0.0818902
\(304\) 0 0
\(305\) −6.50725 −0.372604
\(306\) 0 0
\(307\) 15.1287 0.863443 0.431721 0.902007i \(-0.357906\pi\)
0.431721 + 0.902007i \(0.357906\pi\)
\(308\) 0 0
\(309\) −30.1030 −1.71250
\(310\) 0 0
\(311\) 12.0180 0.681477 0.340739 0.940158i \(-0.389323\pi\)
0.340739 + 0.940158i \(0.389323\pi\)
\(312\) 0 0
\(313\) −23.1532 −1.30870 −0.654349 0.756193i \(-0.727057\pi\)
−0.654349 + 0.756193i \(0.727057\pi\)
\(314\) 0 0
\(315\) −0.447233 −0.0251987
\(316\) 0 0
\(317\) −6.88004 −0.386422 −0.193211 0.981157i \(-0.561890\pi\)
−0.193211 + 0.981157i \(0.561890\pi\)
\(318\) 0 0
\(319\) −29.8803 −1.67298
\(320\) 0 0
\(321\) 14.8430 0.828456
\(322\) 0 0
\(323\) 4.39275 0.244419
\(324\) 0 0
\(325\) −0.0353128 −0.00195880
\(326\) 0 0
\(327\) 28.4695 1.57437
\(328\) 0 0
\(329\) −21.7137 −1.19711
\(330\) 0 0
\(331\) −10.6924 −0.587706 −0.293853 0.955851i \(-0.594938\pi\)
−0.293853 + 0.955851i \(0.594938\pi\)
\(332\) 0 0
\(333\) 0.0860743 0.00471684
\(334\) 0 0
\(335\) 13.2496 0.723903
\(336\) 0 0
\(337\) −29.7049 −1.61813 −0.809065 0.587720i \(-0.800026\pi\)
−0.809065 + 0.587720i \(0.800026\pi\)
\(338\) 0 0
\(339\) 23.0838 1.25374
\(340\) 0 0
\(341\) −46.7040 −2.52916
\(342\) 0 0
\(343\) −4.48406 −0.242116
\(344\) 0 0
\(345\) 16.0994 0.866762
\(346\) 0 0
\(347\) 3.25157 0.174553 0.0872767 0.996184i \(-0.472184\pi\)
0.0872767 + 0.996184i \(0.472184\pi\)
\(348\) 0 0
\(349\) 31.4979 1.68604 0.843021 0.537880i \(-0.180775\pi\)
0.843021 + 0.537880i \(0.180775\pi\)
\(350\) 0 0
\(351\) 0.179811 0.00959759
\(352\) 0 0
\(353\) −23.0362 −1.22609 −0.613047 0.790047i \(-0.710056\pi\)
−0.613047 + 0.790047i \(0.710056\pi\)
\(354\) 0 0
\(355\) 1.67106 0.0886909
\(356\) 0 0
\(357\) 30.1781 1.59720
\(358\) 0 0
\(359\) −13.0768 −0.690168 −0.345084 0.938572i \(-0.612150\pi\)
−0.345084 + 0.938572i \(0.612150\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 21.3795 1.12213
\(364\) 0 0
\(365\) −11.3217 −0.592606
\(366\) 0 0
\(367\) 32.4936 1.69615 0.848077 0.529874i \(-0.177761\pi\)
0.848077 + 0.529874i \(0.177761\pi\)
\(368\) 0 0
\(369\) −0.405557 −0.0211124
\(370\) 0 0
\(371\) 33.7826 1.75390
\(372\) 0 0
\(373\) −6.75164 −0.349586 −0.174793 0.984605i \(-0.555926\pi\)
−0.174793 + 0.984605i \(0.555926\pi\)
\(374\) 0 0
\(375\) 1.76491 0.0911394
\(376\) 0 0
\(377\) −0.219474 −0.0113035
\(378\) 0 0
\(379\) −34.7328 −1.78410 −0.892051 0.451934i \(-0.850734\pi\)
−0.892051 + 0.451934i \(0.850734\pi\)
\(380\) 0 0
\(381\) −26.4778 −1.35650
\(382\) 0 0
\(383\) −0.230516 −0.0117788 −0.00588942 0.999983i \(-0.501875\pi\)
−0.00588942 + 0.999983i \(0.501875\pi\)
\(384\) 0 0
\(385\) 18.7141 0.953758
\(386\) 0 0
\(387\) 1.31698 0.0669460
\(388\) 0 0
\(389\) 26.1973 1.32826 0.664128 0.747619i \(-0.268803\pi\)
0.664128 + 0.747619i \(0.268803\pi\)
\(390\) 0 0
\(391\) −40.0704 −2.02645
\(392\) 0 0
\(393\) −7.26089 −0.366264
\(394\) 0 0
\(395\) 11.4141 0.574304
\(396\) 0 0
\(397\) 24.7005 1.23968 0.619840 0.784728i \(-0.287197\pi\)
0.619840 + 0.784728i \(0.287197\pi\)
\(398\) 0 0
\(399\) 6.86999 0.343930
\(400\) 0 0
\(401\) 2.43645 0.121671 0.0608354 0.998148i \(-0.480624\pi\)
0.0608354 + 0.998148i \(0.480624\pi\)
\(402\) 0 0
\(403\) −0.343046 −0.0170883
\(404\) 0 0
\(405\) −9.33148 −0.463685
\(406\) 0 0
\(407\) −3.60170 −0.178530
\(408\) 0 0
\(409\) −22.2293 −1.09917 −0.549585 0.835438i \(-0.685214\pi\)
−0.549585 + 0.835438i \(0.685214\pi\)
\(410\) 0 0
\(411\) −4.90188 −0.241792
\(412\) 0 0
\(413\) −9.97474 −0.490825
\(414\) 0 0
\(415\) 5.64889 0.277293
\(416\) 0 0
\(417\) 9.54549 0.467445
\(418\) 0 0
\(419\) −4.34305 −0.212172 −0.106086 0.994357i \(-0.533832\pi\)
−0.106086 + 0.994357i \(0.533832\pi\)
\(420\) 0 0
\(421\) 0.903722 0.0440447 0.0220224 0.999757i \(-0.492989\pi\)
0.0220224 + 0.999757i \(0.492989\pi\)
\(422\) 0 0
\(423\) 0.640913 0.0311623
\(424\) 0 0
\(425\) −4.39275 −0.213080
\(426\) 0 0
\(427\) 25.3298 1.22580
\(428\) 0 0
\(429\) 0.299632 0.0144664
\(430\) 0 0
\(431\) −27.6872 −1.33365 −0.666823 0.745216i \(-0.732346\pi\)
−0.666823 + 0.745216i \(0.732346\pi\)
\(432\) 0 0
\(433\) 15.7540 0.757088 0.378544 0.925583i \(-0.376425\pi\)
0.378544 + 0.925583i \(0.376425\pi\)
\(434\) 0 0
\(435\) 10.9691 0.525930
\(436\) 0 0
\(437\) −9.12195 −0.436362
\(438\) 0 0
\(439\) −2.54081 −0.121266 −0.0606332 0.998160i \(-0.519312\pi\)
−0.0606332 + 0.998160i \(0.519312\pi\)
\(440\) 0 0
\(441\) 0.936616 0.0446008
\(442\) 0 0
\(443\) −20.3166 −0.965270 −0.482635 0.875821i \(-0.660320\pi\)
−0.482635 + 0.875821i \(0.660320\pi\)
\(444\) 0 0
\(445\) −1.24079 −0.0588191
\(446\) 0 0
\(447\) 0.513746 0.0242993
\(448\) 0 0
\(449\) −20.0132 −0.944480 −0.472240 0.881470i \(-0.656555\pi\)
−0.472240 + 0.881470i \(0.656555\pi\)
\(450\) 0 0
\(451\) 16.9702 0.799094
\(452\) 0 0
\(453\) 26.5193 1.24599
\(454\) 0 0
\(455\) 0.137457 0.00644408
\(456\) 0 0
\(457\) 28.9331 1.35343 0.676717 0.736243i \(-0.263402\pi\)
0.676717 + 0.736243i \(0.263402\pi\)
\(458\) 0 0
\(459\) 22.3676 1.04403
\(460\) 0 0
\(461\) −9.04395 −0.421219 −0.210609 0.977570i \(-0.567545\pi\)
−0.210609 + 0.977570i \(0.567545\pi\)
\(462\) 0 0
\(463\) 33.2226 1.54399 0.771993 0.635632i \(-0.219260\pi\)
0.771993 + 0.635632i \(0.219260\pi\)
\(464\) 0 0
\(465\) 17.1451 0.795087
\(466\) 0 0
\(467\) 34.0888 1.57744 0.788722 0.614750i \(-0.210743\pi\)
0.788722 + 0.614750i \(0.210743\pi\)
\(468\) 0 0
\(469\) −51.5747 −2.38150
\(470\) 0 0
\(471\) 16.9439 0.780733
\(472\) 0 0
\(473\) −55.1079 −2.53387
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −0.997145 −0.0456561
\(478\) 0 0
\(479\) −28.2735 −1.29185 −0.645925 0.763401i \(-0.723528\pi\)
−0.645925 + 0.763401i \(0.723528\pi\)
\(480\) 0 0
\(481\) −0.0264549 −0.00120624
\(482\) 0 0
\(483\) −62.6677 −2.85148
\(484\) 0 0
\(485\) 9.31976 0.423188
\(486\) 0 0
\(487\) −25.5836 −1.15930 −0.579652 0.814864i \(-0.696811\pi\)
−0.579652 + 0.814864i \(0.696811\pi\)
\(488\) 0 0
\(489\) 8.66006 0.391622
\(490\) 0 0
\(491\) 3.53949 0.159735 0.0798675 0.996805i \(-0.474550\pi\)
0.0798675 + 0.996805i \(0.474550\pi\)
\(492\) 0 0
\(493\) −27.3015 −1.22960
\(494\) 0 0
\(495\) −0.552375 −0.0248274
\(496\) 0 0
\(497\) −6.50470 −0.291776
\(498\) 0 0
\(499\) −26.7046 −1.19546 −0.597730 0.801697i \(-0.703931\pi\)
−0.597730 + 0.801697i \(0.703931\pi\)
\(500\) 0 0
\(501\) −11.5528 −0.516140
\(502\) 0 0
\(503\) −24.7956 −1.10558 −0.552790 0.833320i \(-0.686437\pi\)
−0.552790 + 0.833320i \(0.686437\pi\)
\(504\) 0 0
\(505\) −0.807666 −0.0359406
\(506\) 0 0
\(507\) −22.9416 −1.01887
\(508\) 0 0
\(509\) −17.4860 −0.775051 −0.387526 0.921859i \(-0.626670\pi\)
−0.387526 + 0.921859i \(0.626670\pi\)
\(510\) 0 0
\(511\) 44.0704 1.94956
\(512\) 0 0
\(513\) 5.09194 0.224815
\(514\) 0 0
\(515\) −17.0564 −0.751595
\(516\) 0 0
\(517\) −26.8184 −1.17947
\(518\) 0 0
\(519\) 42.2666 1.85530
\(520\) 0 0
\(521\) −1.37888 −0.0604099 −0.0302049 0.999544i \(-0.509616\pi\)
−0.0302049 + 0.999544i \(0.509616\pi\)
\(522\) 0 0
\(523\) −38.5666 −1.68640 −0.843199 0.537602i \(-0.819330\pi\)
−0.843199 + 0.537602i \(0.819330\pi\)
\(524\) 0 0
\(525\) −6.86999 −0.299831
\(526\) 0 0
\(527\) −42.6732 −1.85888
\(528\) 0 0
\(529\) 60.2100 2.61782
\(530\) 0 0
\(531\) 0.294420 0.0127767
\(532\) 0 0
\(533\) 0.124648 0.00539909
\(534\) 0 0
\(535\) 8.41008 0.363599
\(536\) 0 0
\(537\) −17.6250 −0.760575
\(538\) 0 0
\(539\) −39.1919 −1.68811
\(540\) 0 0
\(541\) −8.73354 −0.375484 −0.187742 0.982218i \(-0.560117\pi\)
−0.187742 + 0.982218i \(0.560117\pi\)
\(542\) 0 0
\(543\) 13.0851 0.561537
\(544\) 0 0
\(545\) 16.1309 0.690971
\(546\) 0 0
\(547\) −31.8117 −1.36017 −0.680084 0.733134i \(-0.738057\pi\)
−0.680084 + 0.733134i \(0.738057\pi\)
\(548\) 0 0
\(549\) −0.747648 −0.0319089
\(550\) 0 0
\(551\) −6.21514 −0.264774
\(552\) 0 0
\(553\) −44.4298 −1.88935
\(554\) 0 0
\(555\) 1.32219 0.0561240
\(556\) 0 0
\(557\) 31.9044 1.35183 0.675915 0.736979i \(-0.263748\pi\)
0.675915 + 0.736979i \(0.263748\pi\)
\(558\) 0 0
\(559\) −0.404774 −0.0171201
\(560\) 0 0
\(561\) 37.2728 1.57366
\(562\) 0 0
\(563\) 40.7223 1.71624 0.858119 0.513450i \(-0.171633\pi\)
0.858119 + 0.513450i \(0.171633\pi\)
\(564\) 0 0
\(565\) 13.0794 0.550253
\(566\) 0 0
\(567\) 36.3233 1.52543
\(568\) 0 0
\(569\) 12.9176 0.541534 0.270767 0.962645i \(-0.412723\pi\)
0.270767 + 0.962645i \(0.412723\pi\)
\(570\) 0 0
\(571\) 14.3129 0.598976 0.299488 0.954100i \(-0.403184\pi\)
0.299488 + 0.954100i \(0.403184\pi\)
\(572\) 0 0
\(573\) −39.3586 −1.64423
\(574\) 0 0
\(575\) 9.12195 0.380412
\(576\) 0 0
\(577\) 36.0033 1.49884 0.749419 0.662095i \(-0.230333\pi\)
0.749419 + 0.662095i \(0.230333\pi\)
\(578\) 0 0
\(579\) −5.56639 −0.231331
\(580\) 0 0
\(581\) −21.9886 −0.912241
\(582\) 0 0
\(583\) 41.7246 1.72806
\(584\) 0 0
\(585\) −0.00405725 −0.000167747 0
\(586\) 0 0
\(587\) 6.30974 0.260431 0.130215 0.991486i \(-0.458433\pi\)
0.130215 + 0.991486i \(0.458433\pi\)
\(588\) 0 0
\(589\) −9.71448 −0.400278
\(590\) 0 0
\(591\) −9.20929 −0.378820
\(592\) 0 0
\(593\) 6.17104 0.253414 0.126707 0.991940i \(-0.459559\pi\)
0.126707 + 0.991940i \(0.459559\pi\)
\(594\) 0 0
\(595\) 17.0990 0.700990
\(596\) 0 0
\(597\) 39.3293 1.60964
\(598\) 0 0
\(599\) 23.3127 0.952530 0.476265 0.879302i \(-0.341990\pi\)
0.476265 + 0.879302i \(0.341990\pi\)
\(600\) 0 0
\(601\) −20.0776 −0.818984 −0.409492 0.912314i \(-0.634294\pi\)
−0.409492 + 0.912314i \(0.634294\pi\)
\(602\) 0 0
\(603\) 1.52231 0.0619932
\(604\) 0 0
\(605\) 12.1136 0.492490
\(606\) 0 0
\(607\) 5.12748 0.208118 0.104059 0.994571i \(-0.466817\pi\)
0.104059 + 0.994571i \(0.466817\pi\)
\(608\) 0 0
\(609\) −42.6979 −1.73021
\(610\) 0 0
\(611\) −0.196984 −0.00796913
\(612\) 0 0
\(613\) −24.2342 −0.978811 −0.489405 0.872056i \(-0.662786\pi\)
−0.489405 + 0.872056i \(0.662786\pi\)
\(614\) 0 0
\(615\) −6.22979 −0.251209
\(616\) 0 0
\(617\) 10.8591 0.437171 0.218585 0.975818i \(-0.429856\pi\)
0.218585 + 0.975818i \(0.429856\pi\)
\(618\) 0 0
\(619\) −19.2362 −0.773169 −0.386585 0.922254i \(-0.626345\pi\)
−0.386585 + 0.922254i \(0.626345\pi\)
\(620\) 0 0
\(621\) −46.4484 −1.86391
\(622\) 0 0
\(623\) 4.82984 0.193503
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 8.48508 0.338861
\(628\) 0 0
\(629\) −3.29086 −0.131215
\(630\) 0 0
\(631\) 46.0007 1.83126 0.915629 0.402025i \(-0.131694\pi\)
0.915629 + 0.402025i \(0.131694\pi\)
\(632\) 0 0
\(633\) 4.02171 0.159848
\(634\) 0 0
\(635\) −15.0024 −0.595351
\(636\) 0 0
\(637\) −0.287869 −0.0114058
\(638\) 0 0
\(639\) 0.191996 0.00759526
\(640\) 0 0
\(641\) 7.97279 0.314906 0.157453 0.987526i \(-0.449672\pi\)
0.157453 + 0.987526i \(0.449672\pi\)
\(642\) 0 0
\(643\) 42.2951 1.66795 0.833977 0.551799i \(-0.186058\pi\)
0.833977 + 0.551799i \(0.186058\pi\)
\(644\) 0 0
\(645\) 20.2303 0.796566
\(646\) 0 0
\(647\) 0.422756 0.0166202 0.00831012 0.999965i \(-0.497355\pi\)
0.00831012 + 0.999965i \(0.497355\pi\)
\(648\) 0 0
\(649\) −12.3197 −0.483592
\(650\) 0 0
\(651\) −66.7384 −2.61568
\(652\) 0 0
\(653\) −9.74222 −0.381242 −0.190621 0.981664i \(-0.561050\pi\)
−0.190621 + 0.981664i \(0.561050\pi\)
\(654\) 0 0
\(655\) −4.11404 −0.160749
\(656\) 0 0
\(657\) −1.30081 −0.0507493
\(658\) 0 0
\(659\) 4.05689 0.158034 0.0790170 0.996873i \(-0.474822\pi\)
0.0790170 + 0.996873i \(0.474822\pi\)
\(660\) 0 0
\(661\) 21.5278 0.837335 0.418668 0.908140i \(-0.362497\pi\)
0.418668 + 0.908140i \(0.362497\pi\)
\(662\) 0 0
\(663\) 0.273773 0.0106325
\(664\) 0 0
\(665\) 3.89255 0.150947
\(666\) 0 0
\(667\) 56.6942 2.19521
\(668\) 0 0
\(669\) 35.4937 1.37227
\(670\) 0 0
\(671\) 31.2847 1.20773
\(672\) 0 0
\(673\) −30.3026 −1.16808 −0.584039 0.811725i \(-0.698529\pi\)
−0.584039 + 0.811725i \(0.698529\pi\)
\(674\) 0 0
\(675\) −5.09194 −0.195989
\(676\) 0 0
\(677\) 18.3642 0.705795 0.352898 0.935662i \(-0.385196\pi\)
0.352898 + 0.935662i \(0.385196\pi\)
\(678\) 0 0
\(679\) −36.2776 −1.39221
\(680\) 0 0
\(681\) −2.05054 −0.0785768
\(682\) 0 0
\(683\) 9.35629 0.358009 0.179004 0.983848i \(-0.442712\pi\)
0.179004 + 0.983848i \(0.442712\pi\)
\(684\) 0 0
\(685\) −2.77742 −0.106120
\(686\) 0 0
\(687\) −4.89968 −0.186934
\(688\) 0 0
\(689\) 0.306472 0.0116757
\(690\) 0 0
\(691\) 24.6498 0.937723 0.468861 0.883272i \(-0.344664\pi\)
0.468861 + 0.883272i \(0.344664\pi\)
\(692\) 0 0
\(693\) 2.15015 0.0816774
\(694\) 0 0
\(695\) 5.40850 0.205156
\(696\) 0 0
\(697\) 15.5056 0.587316
\(698\) 0 0
\(699\) 44.6251 1.68788
\(700\) 0 0
\(701\) 13.2467 0.500321 0.250160 0.968204i \(-0.419517\pi\)
0.250160 + 0.968204i \(0.419517\pi\)
\(702\) 0 0
\(703\) −0.749158 −0.0282550
\(704\) 0 0
\(705\) 9.84512 0.370789
\(706\) 0 0
\(707\) 3.14388 0.118238
\(708\) 0 0
\(709\) 30.0979 1.13035 0.565175 0.824971i \(-0.308808\pi\)
0.565175 + 0.824971i \(0.308808\pi\)
\(710\) 0 0
\(711\) 1.31141 0.0491819
\(712\) 0 0
\(713\) 88.6150 3.31866
\(714\) 0 0
\(715\) 0.169772 0.00634912
\(716\) 0 0
\(717\) −0.403750 −0.0150783
\(718\) 0 0
\(719\) 25.5662 0.953458 0.476729 0.879050i \(-0.341822\pi\)
0.476729 + 0.879050i \(0.341822\pi\)
\(720\) 0 0
\(721\) 66.3929 2.47260
\(722\) 0 0
\(723\) −5.66633 −0.210733
\(724\) 0 0
\(725\) 6.21514 0.230824
\(726\) 0 0
\(727\) 46.2807 1.71646 0.858228 0.513268i \(-0.171565\pi\)
0.858228 + 0.513268i \(0.171565\pi\)
\(728\) 0 0
\(729\) 25.8883 0.958824
\(730\) 0 0
\(731\) −50.3519 −1.86233
\(732\) 0 0
\(733\) −33.1328 −1.22379 −0.611894 0.790940i \(-0.709592\pi\)
−0.611894 + 0.790940i \(0.709592\pi\)
\(734\) 0 0
\(735\) 14.3874 0.530689
\(736\) 0 0
\(737\) −63.6996 −2.34641
\(738\) 0 0
\(739\) −2.75493 −0.101342 −0.0506708 0.998715i \(-0.516136\pi\)
−0.0506708 + 0.998715i \(0.516136\pi\)
\(740\) 0 0
\(741\) 0.0623238 0.00228952
\(742\) 0 0
\(743\) −9.01027 −0.330555 −0.165277 0.986247i \(-0.552852\pi\)
−0.165277 + 0.986247i \(0.552852\pi\)
\(744\) 0 0
\(745\) 0.291090 0.0106647
\(746\) 0 0
\(747\) 0.649028 0.0237467
\(748\) 0 0
\(749\) −32.7367 −1.19617
\(750\) 0 0
\(751\) −10.7150 −0.390996 −0.195498 0.980704i \(-0.562632\pi\)
−0.195498 + 0.980704i \(0.562632\pi\)
\(752\) 0 0
\(753\) 46.8281 1.70651
\(754\) 0 0
\(755\) 15.0259 0.546848
\(756\) 0 0
\(757\) −22.4197 −0.814857 −0.407428 0.913237i \(-0.633574\pi\)
−0.407428 + 0.913237i \(0.633574\pi\)
\(758\) 0 0
\(759\) −77.4005 −2.80946
\(760\) 0 0
\(761\) −7.21311 −0.261475 −0.130738 0.991417i \(-0.541735\pi\)
−0.130738 + 0.991417i \(0.541735\pi\)
\(762\) 0 0
\(763\) −62.7903 −2.27316
\(764\) 0 0
\(765\) −0.504703 −0.0182476
\(766\) 0 0
\(767\) −0.0904898 −0.00326740
\(768\) 0 0
\(769\) 6.14157 0.221471 0.110735 0.993850i \(-0.464679\pi\)
0.110735 + 0.993850i \(0.464679\pi\)
\(770\) 0 0
\(771\) −50.4422 −1.81663
\(772\) 0 0
\(773\) 51.0612 1.83655 0.918273 0.395949i \(-0.129584\pi\)
0.918273 + 0.395949i \(0.129584\pi\)
\(774\) 0 0
\(775\) 9.71448 0.348954
\(776\) 0 0
\(777\) −5.14671 −0.184637
\(778\) 0 0
\(779\) 3.52981 0.126469
\(780\) 0 0
\(781\) −8.03392 −0.287476
\(782\) 0 0
\(783\) −31.6471 −1.13098
\(784\) 0 0
\(785\) 9.60044 0.342654
\(786\) 0 0
\(787\) 38.6448 1.37754 0.688769 0.724980i \(-0.258151\pi\)
0.688769 + 0.724980i \(0.258151\pi\)
\(788\) 0 0
\(789\) 32.6435 1.16214
\(790\) 0 0
\(791\) −50.9121 −1.81023
\(792\) 0 0
\(793\) 0.229789 0.00816006
\(794\) 0 0
\(795\) −15.3172 −0.543246
\(796\) 0 0
\(797\) 8.38007 0.296837 0.148419 0.988925i \(-0.452582\pi\)
0.148419 + 0.988925i \(0.452582\pi\)
\(798\) 0 0
\(799\) −24.5039 −0.866886
\(800\) 0 0
\(801\) −0.142560 −0.00503711
\(802\) 0 0
\(803\) 54.4311 1.92083
\(804\) 0 0
\(805\) −35.5077 −1.25148
\(806\) 0 0
\(807\) 27.6882 0.974671
\(808\) 0 0
\(809\) −20.2642 −0.712450 −0.356225 0.934400i \(-0.615936\pi\)
−0.356225 + 0.934400i \(0.615936\pi\)
\(810\) 0 0
\(811\) −15.1391 −0.531606 −0.265803 0.964027i \(-0.585637\pi\)
−0.265803 + 0.964027i \(0.585637\pi\)
\(812\) 0 0
\(813\) −49.0296 −1.71954
\(814\) 0 0
\(815\) 4.90681 0.171878
\(816\) 0 0
\(817\) −11.4625 −0.401023
\(818\) 0 0
\(819\) 0.0157931 0.000551855 0
\(820\) 0 0
\(821\) −11.6562 −0.406803 −0.203402 0.979095i \(-0.565200\pi\)
−0.203402 + 0.979095i \(0.565200\pi\)
\(822\) 0 0
\(823\) 37.6062 1.31087 0.655436 0.755251i \(-0.272485\pi\)
0.655436 + 0.755251i \(0.272485\pi\)
\(824\) 0 0
\(825\) −8.48508 −0.295413
\(826\) 0 0
\(827\) −50.0978 −1.74207 −0.871036 0.491219i \(-0.836551\pi\)
−0.871036 + 0.491219i \(0.836551\pi\)
\(828\) 0 0
\(829\) 18.7309 0.650552 0.325276 0.945619i \(-0.394543\pi\)
0.325276 + 0.945619i \(0.394543\pi\)
\(830\) 0 0
\(831\) −35.1293 −1.21862
\(832\) 0 0
\(833\) −35.8095 −1.24073
\(834\) 0 0
\(835\) −6.54582 −0.226528
\(836\) 0 0
\(837\) −49.4655 −1.70978
\(838\) 0 0
\(839\) −23.7615 −0.820339 −0.410170 0.912009i \(-0.634531\pi\)
−0.410170 + 0.912009i \(0.634531\pi\)
\(840\) 0 0
\(841\) 9.62793 0.331998
\(842\) 0 0
\(843\) 18.5703 0.639595
\(844\) 0 0
\(845\) −12.9988 −0.447171
\(846\) 0 0
\(847\) −47.1530 −1.62020
\(848\) 0 0
\(849\) 7.52140 0.258134
\(850\) 0 0
\(851\) 6.83378 0.234259
\(852\) 0 0
\(853\) 3.61980 0.123940 0.0619699 0.998078i \(-0.480262\pi\)
0.0619699 + 0.998078i \(0.480262\pi\)
\(854\) 0 0
\(855\) −0.114895 −0.00392931
\(856\) 0 0
\(857\) −36.1820 −1.23595 −0.617977 0.786196i \(-0.712048\pi\)
−0.617977 + 0.786196i \(0.712048\pi\)
\(858\) 0 0
\(859\) 52.3890 1.78749 0.893745 0.448576i \(-0.148069\pi\)
0.893745 + 0.448576i \(0.148069\pi\)
\(860\) 0 0
\(861\) 24.2498 0.826430
\(862\) 0 0
\(863\) −48.7594 −1.65979 −0.829895 0.557919i \(-0.811600\pi\)
−0.829895 + 0.557919i \(0.811600\pi\)
\(864\) 0 0
\(865\) 23.9483 0.814267
\(866\) 0 0
\(867\) 4.05263 0.137635
\(868\) 0 0
\(869\) −54.8750 −1.86151
\(870\) 0 0
\(871\) −0.467880 −0.0158535
\(872\) 0 0
\(873\) 1.07079 0.0362408
\(874\) 0 0
\(875\) −3.89255 −0.131592
\(876\) 0 0
\(877\) 9.17301 0.309751 0.154875 0.987934i \(-0.450502\pi\)
0.154875 + 0.987934i \(0.450502\pi\)
\(878\) 0 0
\(879\) 3.82271 0.128937
\(880\) 0 0
\(881\) 44.7985 1.50930 0.754649 0.656128i \(-0.227807\pi\)
0.754649 + 0.656128i \(0.227807\pi\)
\(882\) 0 0
\(883\) −40.4371 −1.36082 −0.680408 0.732833i \(-0.738198\pi\)
−0.680408 + 0.732833i \(0.738198\pi\)
\(884\) 0 0
\(885\) 4.52261 0.152026
\(886\) 0 0
\(887\) 11.6569 0.391400 0.195700 0.980664i \(-0.437302\pi\)
0.195700 + 0.980664i \(0.437302\pi\)
\(888\) 0 0
\(889\) 58.3975 1.95859
\(890\) 0 0
\(891\) 44.8627 1.50296
\(892\) 0 0
\(893\) −5.57827 −0.186670
\(894\) 0 0
\(895\) −9.98637 −0.333808
\(896\) 0 0
\(897\) −0.568515 −0.0189822
\(898\) 0 0
\(899\) 60.3768 2.01368
\(900\) 0 0
\(901\) 38.1237 1.27008
\(902\) 0 0
\(903\) −78.7474 −2.62055
\(904\) 0 0
\(905\) 7.41406 0.246452
\(906\) 0 0
\(907\) 29.7723 0.988575 0.494287 0.869299i \(-0.335429\pi\)
0.494287 + 0.869299i \(0.335429\pi\)
\(908\) 0 0
\(909\) −0.0927965 −0.00307786
\(910\) 0 0
\(911\) −7.86284 −0.260507 −0.130254 0.991481i \(-0.541579\pi\)
−0.130254 + 0.991481i \(0.541579\pi\)
\(912\) 0 0
\(913\) −27.1580 −0.898798
\(914\) 0 0
\(915\) −11.4847 −0.379672
\(916\) 0 0
\(917\) 16.0141 0.528832
\(918\) 0 0
\(919\) −11.2607 −0.371455 −0.185727 0.982601i \(-0.559464\pi\)
−0.185727 + 0.982601i \(0.559464\pi\)
\(920\) 0 0
\(921\) 26.7008 0.879822
\(922\) 0 0
\(923\) −0.0590100 −0.00194234
\(924\) 0 0
\(925\) 0.749158 0.0246322
\(926\) 0 0
\(927\) −1.95969 −0.0643647
\(928\) 0 0
\(929\) −6.12699 −0.201020 −0.100510 0.994936i \(-0.532047\pi\)
−0.100510 + 0.994936i \(0.532047\pi\)
\(930\) 0 0
\(931\) −8.15196 −0.267170
\(932\) 0 0
\(933\) 21.2106 0.694404
\(934\) 0 0
\(935\) 21.1189 0.690661
\(936\) 0 0
\(937\) 20.2530 0.661635 0.330818 0.943695i \(-0.392676\pi\)
0.330818 + 0.943695i \(0.392676\pi\)
\(938\) 0 0
\(939\) −40.8633 −1.33352
\(940\) 0 0
\(941\) −0.565626 −0.0184389 −0.00921945 0.999958i \(-0.502935\pi\)
−0.00921945 + 0.999958i \(0.502935\pi\)
\(942\) 0 0
\(943\) −32.1988 −1.04854
\(944\) 0 0
\(945\) 19.8206 0.644766
\(946\) 0 0
\(947\) −6.16157 −0.200224 −0.100112 0.994976i \(-0.531920\pi\)
−0.100112 + 0.994976i \(0.531920\pi\)
\(948\) 0 0
\(949\) 0.399802 0.0129781
\(950\) 0 0
\(951\) −12.1426 −0.393752
\(952\) 0 0
\(953\) 15.1913 0.492095 0.246047 0.969258i \(-0.420868\pi\)
0.246047 + 0.969258i \(0.420868\pi\)
\(954\) 0 0
\(955\) −22.3007 −0.721632
\(956\) 0 0
\(957\) −52.7359 −1.70471
\(958\) 0 0
\(959\) 10.8112 0.349113
\(960\) 0 0
\(961\) 63.3711 2.04423
\(962\) 0 0
\(963\) 0.966273 0.0311377
\(964\) 0 0
\(965\) −3.15393 −0.101529
\(966\) 0 0
\(967\) 26.5387 0.853429 0.426714 0.904386i \(-0.359671\pi\)
0.426714 + 0.904386i \(0.359671\pi\)
\(968\) 0 0
\(969\) 7.75279 0.249055
\(970\) 0 0
\(971\) 59.8610 1.92103 0.960516 0.278226i \(-0.0897463\pi\)
0.960516 + 0.278226i \(0.0897463\pi\)
\(972\) 0 0
\(973\) −21.0528 −0.674923
\(974\) 0 0
\(975\) −0.0623238 −0.00199596
\(976\) 0 0
\(977\) 22.5670 0.721983 0.360991 0.932569i \(-0.382438\pi\)
0.360991 + 0.932569i \(0.382438\pi\)
\(978\) 0 0
\(979\) 5.96530 0.190652
\(980\) 0 0
\(981\) 1.85335 0.0591730
\(982\) 0 0
\(983\) −5.97094 −0.190443 −0.0952217 0.995456i \(-0.530356\pi\)
−0.0952217 + 0.995456i \(0.530356\pi\)
\(984\) 0 0
\(985\) −5.21801 −0.166260
\(986\) 0 0
\(987\) −38.3226 −1.21982
\(988\) 0 0
\(989\) 104.561 3.32483
\(990\) 0 0
\(991\) 12.8794 0.409127 0.204564 0.978853i \(-0.434422\pi\)
0.204564 + 0.978853i \(0.434422\pi\)
\(992\) 0 0
\(993\) −18.8710 −0.598855
\(994\) 0 0
\(995\) 22.2841 0.706452
\(996\) 0 0
\(997\) −0.948160 −0.0300285 −0.0150143 0.999887i \(-0.504779\pi\)
−0.0150143 + 0.999887i \(0.504779\pi\)
\(998\) 0 0
\(999\) −3.81467 −0.120691
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.cl.1.4 5
4.3 odd 2 6080.2.a.ci.1.2 5
8.3 odd 2 3040.2.a.x.1.4 yes 5
8.5 even 2 3040.2.a.u.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.u.1.2 5 8.5 even 2
3040.2.a.x.1.4 yes 5 8.3 odd 2
6080.2.a.ci.1.2 5 4.3 odd 2
6080.2.a.cl.1.4 5 1.1 even 1 trivial