Properties

Label 6016.2.a.o.1.11
Level $6016$
Weight $2$
Character 6016.1
Self dual yes
Analytic conductor $48.038$
Analytic rank $0$
Dimension $13$
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] = [6016,2,Mod(1,6016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6016 = 2^{7} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6016.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: \(48.0380018560\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} + \cdots - 832 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(2.61905\) of defining polynomial
Character \(\chi\) \(=\) 6016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.61905 q^{3} -1.63183 q^{5} +3.46539 q^{7} +3.85941 q^{9} +O(q^{10})\) \(q+2.61905 q^{3} -1.63183 q^{5} +3.46539 q^{7} +3.85941 q^{9} +1.65260 q^{11} -2.82156 q^{13} -4.27384 q^{15} +6.37109 q^{17} +3.98956 q^{19} +9.07601 q^{21} -2.04749 q^{23} -2.33713 q^{25} +2.25082 q^{27} -0.0143200 q^{29} +7.31164 q^{31} +4.32824 q^{33} -5.65493 q^{35} -4.31853 q^{37} -7.38981 q^{39} +10.1303 q^{41} -1.72811 q^{43} -6.29790 q^{45} -1.00000 q^{47} +5.00891 q^{49} +16.6862 q^{51} +5.63901 q^{53} -2.69677 q^{55} +10.4488 q^{57} -13.0598 q^{59} +1.84494 q^{61} +13.3743 q^{63} +4.60432 q^{65} +3.19593 q^{67} -5.36247 q^{69} -1.78389 q^{71} +13.3324 q^{73} -6.12104 q^{75} +5.72690 q^{77} -12.6560 q^{79} -5.68320 q^{81} +7.77661 q^{83} -10.3965 q^{85} -0.0375047 q^{87} -0.300991 q^{89} -9.77781 q^{91} +19.1495 q^{93} -6.51029 q^{95} +3.62682 q^{97} +6.37806 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9} + 10 q^{11} - 4 q^{13} + 14 q^{15} + 10 q^{17} + 8 q^{19} - 10 q^{21} + 18 q^{23} + 23 q^{25} + 16 q^{27} - 14 q^{29} + 4 q^{31} + 14 q^{33} + 14 q^{35} - 16 q^{37} + 12 q^{39} + 10 q^{41} + 12 q^{43} - 10 q^{45} - 13 q^{47} + 9 q^{49} + 22 q^{51} - 26 q^{53} - 2 q^{55} + 20 q^{57} + 30 q^{59} - 18 q^{61} + 12 q^{63} - 4 q^{65} + 4 q^{67} - 2 q^{69} + 36 q^{71} + 10 q^{73} + 38 q^{75} - 42 q^{77} + 21 q^{81} + 12 q^{83} - 4 q^{85} + 6 q^{87} + 50 q^{89} - 4 q^{91} - 52 q^{93} + 8 q^{95} - 10 q^{97} + 34 q^{99}+O(q^{100}) \) 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
\(3\) 2.61905 1.51211 0.756054 0.654510i \(-0.227125\pi\)
0.756054 + 0.654510i \(0.227125\pi\)
\(4\) 0 0
\(5\) −1.63183 −0.729777 −0.364889 0.931051i \(-0.618893\pi\)
−0.364889 + 0.931051i \(0.618893\pi\)
\(6\) 0 0
\(7\) 3.46539 1.30979 0.654897 0.755719i \(-0.272712\pi\)
0.654897 + 0.755719i \(0.272712\pi\)
\(8\) 0 0
\(9\) 3.85941 1.28647
\(10\) 0 0
\(11\) 1.65260 0.498278 0.249139 0.968468i \(-0.419852\pi\)
0.249139 + 0.968468i \(0.419852\pi\)
\(12\) 0 0
\(13\) −2.82156 −0.782561 −0.391280 0.920271i \(-0.627968\pi\)
−0.391280 + 0.920271i \(0.627968\pi\)
\(14\) 0 0
\(15\) −4.27384 −1.10350
\(16\) 0 0
\(17\) 6.37109 1.54522 0.772608 0.634883i \(-0.218952\pi\)
0.772608 + 0.634883i \(0.218952\pi\)
\(18\) 0 0
\(19\) 3.98956 0.915267 0.457634 0.889141i \(-0.348697\pi\)
0.457634 + 0.889141i \(0.348697\pi\)
\(20\) 0 0
\(21\) 9.07601 1.98055
\(22\) 0 0
\(23\) −2.04749 −0.426931 −0.213466 0.976951i \(-0.568475\pi\)
−0.213466 + 0.976951i \(0.568475\pi\)
\(24\) 0 0
\(25\) −2.33713 −0.467425
\(26\) 0 0
\(27\) 2.25082 0.433171
\(28\) 0 0
\(29\) −0.0143200 −0.00265916 −0.00132958 0.999999i \(-0.500423\pi\)
−0.00132958 + 0.999999i \(0.500423\pi\)
\(30\) 0 0
\(31\) 7.31164 1.31321 0.656605 0.754235i \(-0.271992\pi\)
0.656605 + 0.754235i \(0.271992\pi\)
\(32\) 0 0
\(33\) 4.32824 0.753450
\(34\) 0 0
\(35\) −5.65493 −0.955857
\(36\) 0 0
\(37\) −4.31853 −0.709962 −0.354981 0.934874i \(-0.615513\pi\)
−0.354981 + 0.934874i \(0.615513\pi\)
\(38\) 0 0
\(39\) −7.38981 −1.18332
\(40\) 0 0
\(41\) 10.1303 1.58208 0.791040 0.611764i \(-0.209540\pi\)
0.791040 + 0.611764i \(0.209540\pi\)
\(42\) 0 0
\(43\) −1.72811 −0.263535 −0.131767 0.991281i \(-0.542065\pi\)
−0.131767 + 0.991281i \(0.542065\pi\)
\(44\) 0 0
\(45\) −6.29790 −0.938836
\(46\) 0 0
\(47\) −1.00000 −0.145865
\(48\) 0 0
\(49\) 5.00891 0.715558
\(50\) 0 0
\(51\) 16.6862 2.33653
\(52\) 0 0
\(53\) 5.63901 0.774577 0.387289 0.921959i \(-0.373412\pi\)
0.387289 + 0.921959i \(0.373412\pi\)
\(54\) 0 0
\(55\) −2.69677 −0.363632
\(56\) 0 0
\(57\) 10.4488 1.38398
\(58\) 0 0
\(59\) −13.0598 −1.70024 −0.850118 0.526592i \(-0.823469\pi\)
−0.850118 + 0.526592i \(0.823469\pi\)
\(60\) 0 0
\(61\) 1.84494 0.236220 0.118110 0.993001i \(-0.462316\pi\)
0.118110 + 0.993001i \(0.462316\pi\)
\(62\) 0 0
\(63\) 13.3743 1.68501
\(64\) 0 0
\(65\) 4.60432 0.571095
\(66\) 0 0
\(67\) 3.19593 0.390445 0.195223 0.980759i \(-0.437457\pi\)
0.195223 + 0.980759i \(0.437457\pi\)
\(68\) 0 0
\(69\) −5.36247 −0.645566
\(70\) 0 0
\(71\) −1.78389 −0.211708 −0.105854 0.994382i \(-0.533758\pi\)
−0.105854 + 0.994382i \(0.533758\pi\)
\(72\) 0 0
\(73\) 13.3324 1.56044 0.780221 0.625504i \(-0.215107\pi\)
0.780221 + 0.625504i \(0.215107\pi\)
\(74\) 0 0
\(75\) −6.12104 −0.706797
\(76\) 0 0
\(77\) 5.72690 0.652641
\(78\) 0 0
\(79\) −12.6560 −1.42391 −0.711954 0.702226i \(-0.752189\pi\)
−0.711954 + 0.702226i \(0.752189\pi\)
\(80\) 0 0
\(81\) −5.68320 −0.631467
\(82\) 0 0
\(83\) 7.77661 0.853594 0.426797 0.904348i \(-0.359642\pi\)
0.426797 + 0.904348i \(0.359642\pi\)
\(84\) 0 0
\(85\) −10.3965 −1.12766
\(86\) 0 0
\(87\) −0.0375047 −0.00402093
\(88\) 0 0
\(89\) −0.300991 −0.0319050 −0.0159525 0.999873i \(-0.505078\pi\)
−0.0159525 + 0.999873i \(0.505078\pi\)
\(90\) 0 0
\(91\) −9.77781 −1.02499
\(92\) 0 0
\(93\) 19.1495 1.98571
\(94\) 0 0
\(95\) −6.51029 −0.667941
\(96\) 0 0
\(97\) 3.62682 0.368248 0.184124 0.982903i \(-0.441055\pi\)
0.184124 + 0.982903i \(0.441055\pi\)
\(98\) 0 0
\(99\) 6.37806 0.641019
\(100\) 0 0
\(101\) 0.0396915 0.00394946 0.00197473 0.999998i \(-0.499371\pi\)
0.00197473 + 0.999998i \(0.499371\pi\)
\(102\) 0 0
\(103\) 6.87008 0.676930 0.338465 0.940979i \(-0.390092\pi\)
0.338465 + 0.940979i \(0.390092\pi\)
\(104\) 0 0
\(105\) −14.8105 −1.44536
\(106\) 0 0
\(107\) −12.6779 −1.22562 −0.612810 0.790231i \(-0.709961\pi\)
−0.612810 + 0.790231i \(0.709961\pi\)
\(108\) 0 0
\(109\) −2.62362 −0.251297 −0.125649 0.992075i \(-0.540101\pi\)
−0.125649 + 0.992075i \(0.540101\pi\)
\(110\) 0 0
\(111\) −11.3104 −1.07354
\(112\) 0 0
\(113\) 6.67123 0.627577 0.313788 0.949493i \(-0.398402\pi\)
0.313788 + 0.949493i \(0.398402\pi\)
\(114\) 0 0
\(115\) 3.34116 0.311565
\(116\) 0 0
\(117\) −10.8896 −1.00674
\(118\) 0 0
\(119\) 22.0783 2.02391
\(120\) 0 0
\(121\) −8.26891 −0.751719
\(122\) 0 0
\(123\) 26.5316 2.39227
\(124\) 0 0
\(125\) 11.9730 1.07089
\(126\) 0 0
\(127\) 3.22819 0.286456 0.143228 0.989690i \(-0.454252\pi\)
0.143228 + 0.989690i \(0.454252\pi\)
\(128\) 0 0
\(129\) −4.52601 −0.398493
\(130\) 0 0
\(131\) 1.35330 0.118239 0.0591193 0.998251i \(-0.481171\pi\)
0.0591193 + 0.998251i \(0.481171\pi\)
\(132\) 0 0
\(133\) 13.8254 1.19881
\(134\) 0 0
\(135\) −3.67297 −0.316119
\(136\) 0 0
\(137\) 8.40162 0.717799 0.358899 0.933376i \(-0.383152\pi\)
0.358899 + 0.933376i \(0.383152\pi\)
\(138\) 0 0
\(139\) 17.1511 1.45474 0.727369 0.686246i \(-0.240743\pi\)
0.727369 + 0.686246i \(0.240743\pi\)
\(140\) 0 0
\(141\) −2.61905 −0.220564
\(142\) 0 0
\(143\) −4.66292 −0.389933
\(144\) 0 0
\(145\) 0.0233678 0.00194059
\(146\) 0 0
\(147\) 13.1186 1.08200
\(148\) 0 0
\(149\) 23.7241 1.94355 0.971777 0.235902i \(-0.0758045\pi\)
0.971777 + 0.235902i \(0.0758045\pi\)
\(150\) 0 0
\(151\) −9.68732 −0.788343 −0.394171 0.919037i \(-0.628968\pi\)
−0.394171 + 0.919037i \(0.628968\pi\)
\(152\) 0 0
\(153\) 24.5886 1.98787
\(154\) 0 0
\(155\) −11.9314 −0.958351
\(156\) 0 0
\(157\) −9.30532 −0.742645 −0.371323 0.928504i \(-0.621096\pi\)
−0.371323 + 0.928504i \(0.621096\pi\)
\(158\) 0 0
\(159\) 14.7688 1.17124
\(160\) 0 0
\(161\) −7.09534 −0.559192
\(162\) 0 0
\(163\) 0.783564 0.0613735 0.0306867 0.999529i \(-0.490231\pi\)
0.0306867 + 0.999529i \(0.490231\pi\)
\(164\) 0 0
\(165\) −7.06296 −0.549851
\(166\) 0 0
\(167\) −3.84752 −0.297730 −0.148865 0.988858i \(-0.547562\pi\)
−0.148865 + 0.988858i \(0.547562\pi\)
\(168\) 0 0
\(169\) −5.03878 −0.387599
\(170\) 0 0
\(171\) 15.3973 1.17746
\(172\) 0 0
\(173\) 2.64476 0.201078 0.100539 0.994933i \(-0.467943\pi\)
0.100539 + 0.994933i \(0.467943\pi\)
\(174\) 0 0
\(175\) −8.09905 −0.612230
\(176\) 0 0
\(177\) −34.2041 −2.57094
\(178\) 0 0
\(179\) 21.9594 1.64132 0.820660 0.571417i \(-0.193606\pi\)
0.820660 + 0.571417i \(0.193606\pi\)
\(180\) 0 0
\(181\) −22.3190 −1.65896 −0.829479 0.558538i \(-0.811362\pi\)
−0.829479 + 0.558538i \(0.811362\pi\)
\(182\) 0 0
\(183\) 4.83198 0.357190
\(184\) 0 0
\(185\) 7.04711 0.518114
\(186\) 0 0
\(187\) 10.5289 0.769947
\(188\) 0 0
\(189\) 7.79998 0.567365
\(190\) 0 0
\(191\) −17.3650 −1.25649 −0.628244 0.778016i \(-0.716226\pi\)
−0.628244 + 0.778016i \(0.716226\pi\)
\(192\) 0 0
\(193\) −2.45220 −0.176513 −0.0882567 0.996098i \(-0.528130\pi\)
−0.0882567 + 0.996098i \(0.528130\pi\)
\(194\) 0 0
\(195\) 12.0589 0.863557
\(196\) 0 0
\(197\) −22.2939 −1.58838 −0.794189 0.607671i \(-0.792104\pi\)
−0.794189 + 0.607671i \(0.792104\pi\)
\(198\) 0 0
\(199\) 11.3127 0.801936 0.400968 0.916092i \(-0.368674\pi\)
0.400968 + 0.916092i \(0.368674\pi\)
\(200\) 0 0
\(201\) 8.37029 0.590395
\(202\) 0 0
\(203\) −0.0496243 −0.00348294
\(204\) 0 0
\(205\) −16.5309 −1.15457
\(206\) 0 0
\(207\) −7.90210 −0.549234
\(208\) 0 0
\(209\) 6.59315 0.456058
\(210\) 0 0
\(211\) 14.4994 0.998179 0.499089 0.866551i \(-0.333668\pi\)
0.499089 + 0.866551i \(0.333668\pi\)
\(212\) 0 0
\(213\) −4.67208 −0.320126
\(214\) 0 0
\(215\) 2.81999 0.192322
\(216\) 0 0
\(217\) 25.3377 1.72003
\(218\) 0 0
\(219\) 34.9182 2.35956
\(220\) 0 0
\(221\) −17.9764 −1.20923
\(222\) 0 0
\(223\) −6.57749 −0.440462 −0.220231 0.975448i \(-0.570681\pi\)
−0.220231 + 0.975448i \(0.570681\pi\)
\(224\) 0 0
\(225\) −9.01992 −0.601328
\(226\) 0 0
\(227\) −5.60362 −0.371925 −0.185963 0.982557i \(-0.559540\pi\)
−0.185963 + 0.982557i \(0.559540\pi\)
\(228\) 0 0
\(229\) 9.74446 0.643932 0.321966 0.946751i \(-0.395656\pi\)
0.321966 + 0.946751i \(0.395656\pi\)
\(230\) 0 0
\(231\) 14.9990 0.986863
\(232\) 0 0
\(233\) −18.7199 −1.22638 −0.613190 0.789935i \(-0.710114\pi\)
−0.613190 + 0.789935i \(0.710114\pi\)
\(234\) 0 0
\(235\) 1.63183 0.106449
\(236\) 0 0
\(237\) −33.1466 −2.15310
\(238\) 0 0
\(239\) 7.87605 0.509460 0.254730 0.967012i \(-0.418013\pi\)
0.254730 + 0.967012i \(0.418013\pi\)
\(240\) 0 0
\(241\) 5.54408 0.357125 0.178563 0.983929i \(-0.442855\pi\)
0.178563 + 0.983929i \(0.442855\pi\)
\(242\) 0 0
\(243\) −21.6370 −1.38802
\(244\) 0 0
\(245\) −8.17369 −0.522198
\(246\) 0 0
\(247\) −11.2568 −0.716252
\(248\) 0 0
\(249\) 20.3673 1.29073
\(250\) 0 0
\(251\) −22.0982 −1.39483 −0.697414 0.716669i \(-0.745666\pi\)
−0.697414 + 0.716669i \(0.745666\pi\)
\(252\) 0 0
\(253\) −3.38368 −0.212730
\(254\) 0 0
\(255\) −27.2290 −1.70515
\(256\) 0 0
\(257\) 19.2945 1.20356 0.601779 0.798663i \(-0.294459\pi\)
0.601779 + 0.798663i \(0.294459\pi\)
\(258\) 0 0
\(259\) −14.9654 −0.929903
\(260\) 0 0
\(261\) −0.0552667 −0.00342092
\(262\) 0 0
\(263\) 20.3983 1.25781 0.628907 0.777480i \(-0.283502\pi\)
0.628907 + 0.777480i \(0.283502\pi\)
\(264\) 0 0
\(265\) −9.20191 −0.565269
\(266\) 0 0
\(267\) −0.788310 −0.0482438
\(268\) 0 0
\(269\) −26.7330 −1.62994 −0.814969 0.579505i \(-0.803246\pi\)
−0.814969 + 0.579505i \(0.803246\pi\)
\(270\) 0 0
\(271\) 13.7183 0.833329 0.416665 0.909060i \(-0.363199\pi\)
0.416665 + 0.909060i \(0.363199\pi\)
\(272\) 0 0
\(273\) −25.6085 −1.54990
\(274\) 0 0
\(275\) −3.86234 −0.232908
\(276\) 0 0
\(277\) −1.58257 −0.0950877 −0.0475438 0.998869i \(-0.515139\pi\)
−0.0475438 + 0.998869i \(0.515139\pi\)
\(278\) 0 0
\(279\) 28.2186 1.68940
\(280\) 0 0
\(281\) 10.4464 0.623179 0.311590 0.950217i \(-0.399139\pi\)
0.311590 + 0.950217i \(0.399139\pi\)
\(282\) 0 0
\(283\) −21.2512 −1.26325 −0.631626 0.775273i \(-0.717612\pi\)
−0.631626 + 0.775273i \(0.717612\pi\)
\(284\) 0 0
\(285\) −17.0507 −1.01000
\(286\) 0 0
\(287\) 35.1052 2.07220
\(288\) 0 0
\(289\) 23.5908 1.38769
\(290\) 0 0
\(291\) 9.49881 0.556830
\(292\) 0 0
\(293\) 30.2919 1.76967 0.884835 0.465905i \(-0.154271\pi\)
0.884835 + 0.465905i \(0.154271\pi\)
\(294\) 0 0
\(295\) 21.3113 1.24079
\(296\) 0 0
\(297\) 3.71971 0.215840
\(298\) 0 0
\(299\) 5.77712 0.334100
\(300\) 0 0
\(301\) −5.98858 −0.345176
\(302\) 0 0
\(303\) 0.103954 0.00597200
\(304\) 0 0
\(305\) −3.01063 −0.172388
\(306\) 0 0
\(307\) −1.59742 −0.0911695 −0.0455847 0.998960i \(-0.514515\pi\)
−0.0455847 + 0.998960i \(0.514515\pi\)
\(308\) 0 0
\(309\) 17.9931 1.02359
\(310\) 0 0
\(311\) 20.6187 1.16918 0.584588 0.811330i \(-0.301256\pi\)
0.584588 + 0.811330i \(0.301256\pi\)
\(312\) 0 0
\(313\) 23.8852 1.35007 0.675035 0.737786i \(-0.264129\pi\)
0.675035 + 0.737786i \(0.264129\pi\)
\(314\) 0 0
\(315\) −21.8247 −1.22968
\(316\) 0 0
\(317\) −16.2961 −0.915283 −0.457641 0.889137i \(-0.651306\pi\)
−0.457641 + 0.889137i \(0.651306\pi\)
\(318\) 0 0
\(319\) −0.0236652 −0.00132500
\(320\) 0 0
\(321\) −33.2040 −1.85327
\(322\) 0 0
\(323\) 25.4178 1.41429
\(324\) 0 0
\(325\) 6.59435 0.365789
\(326\) 0 0
\(327\) −6.87139 −0.379989
\(328\) 0 0
\(329\) −3.46539 −0.191053
\(330\) 0 0
\(331\) 2.75022 0.151166 0.0755829 0.997140i \(-0.475918\pi\)
0.0755829 + 0.997140i \(0.475918\pi\)
\(332\) 0 0
\(333\) −16.6670 −0.913344
\(334\) 0 0
\(335\) −5.21522 −0.284938
\(336\) 0 0
\(337\) −32.7156 −1.78213 −0.891067 0.453871i \(-0.850043\pi\)
−0.891067 + 0.453871i \(0.850043\pi\)
\(338\) 0 0
\(339\) 17.4723 0.948964
\(340\) 0 0
\(341\) 12.0832 0.654344
\(342\) 0 0
\(343\) −6.89991 −0.372560
\(344\) 0 0
\(345\) 8.75065 0.471119
\(346\) 0 0
\(347\) −7.03353 −0.377580 −0.188790 0.982018i \(-0.560456\pi\)
−0.188790 + 0.982018i \(0.560456\pi\)
\(348\) 0 0
\(349\) 23.2479 1.24443 0.622216 0.782846i \(-0.286233\pi\)
0.622216 + 0.782846i \(0.286233\pi\)
\(350\) 0 0
\(351\) −6.35084 −0.338983
\(352\) 0 0
\(353\) 15.6215 0.831446 0.415723 0.909491i \(-0.363529\pi\)
0.415723 + 0.909491i \(0.363529\pi\)
\(354\) 0 0
\(355\) 2.91100 0.154500
\(356\) 0 0
\(357\) 57.8241 3.06037
\(358\) 0 0
\(359\) 33.7998 1.78388 0.891942 0.452150i \(-0.149343\pi\)
0.891942 + 0.452150i \(0.149343\pi\)
\(360\) 0 0
\(361\) −3.08343 −0.162286
\(362\) 0 0
\(363\) −21.6567 −1.13668
\(364\) 0 0
\(365\) −21.7563 −1.13878
\(366\) 0 0
\(367\) −23.7411 −1.23928 −0.619638 0.784887i \(-0.712721\pi\)
−0.619638 + 0.784887i \(0.712721\pi\)
\(368\) 0 0
\(369\) 39.0968 2.03530
\(370\) 0 0
\(371\) 19.5413 1.01454
\(372\) 0 0
\(373\) −2.15684 −0.111677 −0.0558385 0.998440i \(-0.517783\pi\)
−0.0558385 + 0.998440i \(0.517783\pi\)
\(374\) 0 0
\(375\) 31.3577 1.61931
\(376\) 0 0
\(377\) 0.0404048 0.00208095
\(378\) 0 0
\(379\) −12.3159 −0.632624 −0.316312 0.948655i \(-0.602445\pi\)
−0.316312 + 0.948655i \(0.602445\pi\)
\(380\) 0 0
\(381\) 8.45478 0.433152
\(382\) 0 0
\(383\) 3.48160 0.177901 0.0889506 0.996036i \(-0.471649\pi\)
0.0889506 + 0.996036i \(0.471649\pi\)
\(384\) 0 0
\(385\) −9.34534 −0.476283
\(386\) 0 0
\(387\) −6.66949 −0.339029
\(388\) 0 0
\(389\) −11.5890 −0.587583 −0.293792 0.955870i \(-0.594917\pi\)
−0.293792 + 0.955870i \(0.594917\pi\)
\(390\) 0 0
\(391\) −13.0447 −0.659701
\(392\) 0 0
\(393\) 3.54436 0.178790
\(394\) 0 0
\(395\) 20.6524 1.03914
\(396\) 0 0
\(397\) −21.5181 −1.07996 −0.539982 0.841677i \(-0.681569\pi\)
−0.539982 + 0.841677i \(0.681569\pi\)
\(398\) 0 0
\(399\) 36.2093 1.81273
\(400\) 0 0
\(401\) −12.8674 −0.642565 −0.321283 0.946983i \(-0.604114\pi\)
−0.321283 + 0.946983i \(0.604114\pi\)
\(402\) 0 0
\(403\) −20.6303 −1.02767
\(404\) 0 0
\(405\) 9.27403 0.460830
\(406\) 0 0
\(407\) −7.13681 −0.353758
\(408\) 0 0
\(409\) −26.3544 −1.30314 −0.651569 0.758589i \(-0.725889\pi\)
−0.651569 + 0.758589i \(0.725889\pi\)
\(410\) 0 0
\(411\) 22.0042 1.08539
\(412\) 0 0
\(413\) −45.2571 −2.22696
\(414\) 0 0
\(415\) −12.6901 −0.622933
\(416\) 0 0
\(417\) 44.9196 2.19972
\(418\) 0 0
\(419\) −5.30358 −0.259097 −0.129549 0.991573i \(-0.541353\pi\)
−0.129549 + 0.991573i \(0.541353\pi\)
\(420\) 0 0
\(421\) −14.6840 −0.715652 −0.357826 0.933788i \(-0.616482\pi\)
−0.357826 + 0.933788i \(0.616482\pi\)
\(422\) 0 0
\(423\) −3.85941 −0.187651
\(424\) 0 0
\(425\) −14.8900 −0.722273
\(426\) 0 0
\(427\) 6.39343 0.309400
\(428\) 0 0
\(429\) −12.2124 −0.589620
\(430\) 0 0
\(431\) 25.9251 1.24877 0.624384 0.781118i \(-0.285350\pi\)
0.624384 + 0.781118i \(0.285350\pi\)
\(432\) 0 0
\(433\) −35.8031 −1.72059 −0.860293 0.509800i \(-0.829719\pi\)
−0.860293 + 0.509800i \(0.829719\pi\)
\(434\) 0 0
\(435\) 0.0612014 0.00293438
\(436\) 0 0
\(437\) −8.16858 −0.390756
\(438\) 0 0
\(439\) 40.2511 1.92108 0.960541 0.278140i \(-0.0897178\pi\)
0.960541 + 0.278140i \(0.0897178\pi\)
\(440\) 0 0
\(441\) 19.3314 0.920543
\(442\) 0 0
\(443\) −22.1914 −1.05435 −0.527173 0.849758i \(-0.676748\pi\)
−0.527173 + 0.849758i \(0.676748\pi\)
\(444\) 0 0
\(445\) 0.491167 0.0232835
\(446\) 0 0
\(447\) 62.1345 2.93886
\(448\) 0 0
\(449\) −10.3870 −0.490191 −0.245095 0.969499i \(-0.578819\pi\)
−0.245095 + 0.969499i \(0.578819\pi\)
\(450\) 0 0
\(451\) 16.7413 0.788316
\(452\) 0 0
\(453\) −25.3715 −1.19206
\(454\) 0 0
\(455\) 15.9557 0.748016
\(456\) 0 0
\(457\) 19.1022 0.893565 0.446783 0.894643i \(-0.352570\pi\)
0.446783 + 0.894643i \(0.352570\pi\)
\(458\) 0 0
\(459\) 14.3402 0.669343
\(460\) 0 0
\(461\) 30.4515 1.41827 0.709135 0.705073i \(-0.249086\pi\)
0.709135 + 0.705073i \(0.249086\pi\)
\(462\) 0 0
\(463\) 10.2642 0.477020 0.238510 0.971140i \(-0.423341\pi\)
0.238510 + 0.971140i \(0.423341\pi\)
\(464\) 0 0
\(465\) −31.2488 −1.44913
\(466\) 0 0
\(467\) −16.5025 −0.763647 −0.381823 0.924235i \(-0.624704\pi\)
−0.381823 + 0.924235i \(0.624704\pi\)
\(468\) 0 0
\(469\) 11.0751 0.511402
\(470\) 0 0
\(471\) −24.3711 −1.12296
\(472\) 0 0
\(473\) −2.85588 −0.131314
\(474\) 0 0
\(475\) −9.32410 −0.427819
\(476\) 0 0
\(477\) 21.7632 0.996469
\(478\) 0 0
\(479\) 14.7101 0.672120 0.336060 0.941841i \(-0.390905\pi\)
0.336060 + 0.941841i \(0.390905\pi\)
\(480\) 0 0
\(481\) 12.1850 0.555588
\(482\) 0 0
\(483\) −18.5830 −0.845558
\(484\) 0 0
\(485\) −5.91836 −0.268739
\(486\) 0 0
\(487\) −29.9025 −1.35501 −0.677506 0.735517i \(-0.736939\pi\)
−0.677506 + 0.735517i \(0.736939\pi\)
\(488\) 0 0
\(489\) 2.05219 0.0928033
\(490\) 0 0
\(491\) 16.6866 0.753056 0.376528 0.926405i \(-0.377118\pi\)
0.376528 + 0.926405i \(0.377118\pi\)
\(492\) 0 0
\(493\) −0.0912340 −0.00410897
\(494\) 0 0
\(495\) −10.4079 −0.467801
\(496\) 0 0
\(497\) −6.18185 −0.277294
\(498\) 0 0
\(499\) −24.3924 −1.09195 −0.545976 0.837801i \(-0.683841\pi\)
−0.545976 + 0.837801i \(0.683841\pi\)
\(500\) 0 0
\(501\) −10.0768 −0.450200
\(502\) 0 0
\(503\) 7.84938 0.349987 0.174993 0.984570i \(-0.444010\pi\)
0.174993 + 0.984570i \(0.444010\pi\)
\(504\) 0 0
\(505\) −0.0647699 −0.00288222
\(506\) 0 0
\(507\) −13.1968 −0.586091
\(508\) 0 0
\(509\) −12.2126 −0.541314 −0.270657 0.962676i \(-0.587241\pi\)
−0.270657 + 0.962676i \(0.587241\pi\)
\(510\) 0 0
\(511\) 46.2020 2.04386
\(512\) 0 0
\(513\) 8.97979 0.396468
\(514\) 0 0
\(515\) −11.2108 −0.494008
\(516\) 0 0
\(517\) −1.65260 −0.0726813
\(518\) 0 0
\(519\) 6.92676 0.304051
\(520\) 0 0
\(521\) 6.64477 0.291112 0.145556 0.989350i \(-0.453503\pi\)
0.145556 + 0.989350i \(0.453503\pi\)
\(522\) 0 0
\(523\) 34.7936 1.52142 0.760710 0.649092i \(-0.224851\pi\)
0.760710 + 0.649092i \(0.224851\pi\)
\(524\) 0 0
\(525\) −21.2118 −0.925758
\(526\) 0 0
\(527\) 46.5831 2.02919
\(528\) 0 0
\(529\) −18.8078 −0.817730
\(530\) 0 0
\(531\) −50.4029 −2.18730
\(532\) 0 0
\(533\) −28.5832 −1.23807
\(534\) 0 0
\(535\) 20.6882 0.894429
\(536\) 0 0
\(537\) 57.5126 2.48185
\(538\) 0 0
\(539\) 8.27772 0.356547
\(540\) 0 0
\(541\) −8.37482 −0.360062 −0.180031 0.983661i \(-0.557620\pi\)
−0.180031 + 0.983661i \(0.557620\pi\)
\(542\) 0 0
\(543\) −58.4545 −2.50852
\(544\) 0 0
\(545\) 4.28131 0.183391
\(546\) 0 0
\(547\) −26.8403 −1.14761 −0.573804 0.818992i \(-0.694533\pi\)
−0.573804 + 0.818992i \(0.694533\pi\)
\(548\) 0 0
\(549\) 7.12037 0.303890
\(550\) 0 0
\(551\) −0.0571304 −0.00243384
\(552\) 0 0
\(553\) −43.8578 −1.86502
\(554\) 0 0
\(555\) 18.4567 0.783444
\(556\) 0 0
\(557\) −15.2642 −0.646765 −0.323383 0.946268i \(-0.604820\pi\)
−0.323383 + 0.946268i \(0.604820\pi\)
\(558\) 0 0
\(559\) 4.87598 0.206232
\(560\) 0 0
\(561\) 27.5756 1.16424
\(562\) 0 0
\(563\) 31.0665 1.30930 0.654649 0.755933i \(-0.272817\pi\)
0.654649 + 0.755933i \(0.272817\pi\)
\(564\) 0 0
\(565\) −10.8863 −0.457991
\(566\) 0 0
\(567\) −19.6945 −0.827091
\(568\) 0 0
\(569\) −22.5746 −0.946378 −0.473189 0.880961i \(-0.656897\pi\)
−0.473189 + 0.880961i \(0.656897\pi\)
\(570\) 0 0
\(571\) −4.84825 −0.202893 −0.101446 0.994841i \(-0.532347\pi\)
−0.101446 + 0.994841i \(0.532347\pi\)
\(572\) 0 0
\(573\) −45.4798 −1.89995
\(574\) 0 0
\(575\) 4.78524 0.199558
\(576\) 0 0
\(577\) −33.6964 −1.40280 −0.701400 0.712768i \(-0.747441\pi\)
−0.701400 + 0.712768i \(0.747441\pi\)
\(578\) 0 0
\(579\) −6.42243 −0.266907
\(580\) 0 0
\(581\) 26.9490 1.11803
\(582\) 0 0
\(583\) 9.31903 0.385955
\(584\) 0 0
\(585\) 17.7699 0.734696
\(586\) 0 0
\(587\) −9.80827 −0.404830 −0.202415 0.979300i \(-0.564879\pi\)
−0.202415 + 0.979300i \(0.564879\pi\)
\(588\) 0 0
\(589\) 29.1702 1.20194
\(590\) 0 0
\(591\) −58.3889 −2.40180
\(592\) 0 0
\(593\) 31.5006 1.29357 0.646787 0.762671i \(-0.276112\pi\)
0.646787 + 0.762671i \(0.276112\pi\)
\(594\) 0 0
\(595\) −36.0280 −1.47701
\(596\) 0 0
\(597\) 29.6285 1.21261
\(598\) 0 0
\(599\) −30.1137 −1.23041 −0.615207 0.788366i \(-0.710928\pi\)
−0.615207 + 0.788366i \(0.710928\pi\)
\(600\) 0 0
\(601\) 14.8979 0.607700 0.303850 0.952720i \(-0.401728\pi\)
0.303850 + 0.952720i \(0.401728\pi\)
\(602\) 0 0
\(603\) 12.3344 0.502295
\(604\) 0 0
\(605\) 13.4935 0.548587
\(606\) 0 0
\(607\) −16.1466 −0.655370 −0.327685 0.944787i \(-0.606268\pi\)
−0.327685 + 0.944787i \(0.606268\pi\)
\(608\) 0 0
\(609\) −0.129968 −0.00526658
\(610\) 0 0
\(611\) 2.82156 0.114148
\(612\) 0 0
\(613\) 2.28385 0.0922436 0.0461218 0.998936i \(-0.485314\pi\)
0.0461218 + 0.998936i \(0.485314\pi\)
\(614\) 0 0
\(615\) −43.2951 −1.74583
\(616\) 0 0
\(617\) −22.7258 −0.914907 −0.457453 0.889234i \(-0.651238\pi\)
−0.457453 + 0.889234i \(0.651238\pi\)
\(618\) 0 0
\(619\) −47.8116 −1.92171 −0.960856 0.277049i \(-0.910644\pi\)
−0.960856 + 0.277049i \(0.910644\pi\)
\(620\) 0 0
\(621\) −4.60854 −0.184934
\(622\) 0 0
\(623\) −1.04305 −0.0417889
\(624\) 0 0
\(625\) −7.85221 −0.314089
\(626\) 0 0
\(627\) 17.2678 0.689608
\(628\) 0 0
\(629\) −27.5137 −1.09704
\(630\) 0 0
\(631\) −2.71388 −0.108038 −0.0540190 0.998540i \(-0.517203\pi\)
−0.0540190 + 0.998540i \(0.517203\pi\)
\(632\) 0 0
\(633\) 37.9746 1.50935
\(634\) 0 0
\(635\) −5.26786 −0.209049
\(636\) 0 0
\(637\) −14.1329 −0.559968
\(638\) 0 0
\(639\) −6.88474 −0.272356
\(640\) 0 0
\(641\) −12.9355 −0.510920 −0.255460 0.966820i \(-0.582227\pi\)
−0.255460 + 0.966820i \(0.582227\pi\)
\(642\) 0 0
\(643\) −8.84592 −0.348849 −0.174424 0.984671i \(-0.555806\pi\)
−0.174424 + 0.984671i \(0.555806\pi\)
\(644\) 0 0
\(645\) 7.38568 0.290811
\(646\) 0 0
\(647\) 39.0899 1.53678 0.768391 0.639981i \(-0.221058\pi\)
0.768391 + 0.639981i \(0.221058\pi\)
\(648\) 0 0
\(649\) −21.5826 −0.847190
\(650\) 0 0
\(651\) 66.3606 2.60088
\(652\) 0 0
\(653\) 33.0186 1.29212 0.646058 0.763288i \(-0.276416\pi\)
0.646058 + 0.763288i \(0.276416\pi\)
\(654\) 0 0
\(655\) −2.20836 −0.0862879
\(656\) 0 0
\(657\) 51.4552 2.00746
\(658\) 0 0
\(659\) −29.1223 −1.13444 −0.567222 0.823565i \(-0.691982\pi\)
−0.567222 + 0.823565i \(0.691982\pi\)
\(660\) 0 0
\(661\) −43.5149 −1.69253 −0.846267 0.532759i \(-0.821155\pi\)
−0.846267 + 0.532759i \(0.821155\pi\)
\(662\) 0 0
\(663\) −47.0811 −1.82848
\(664\) 0 0
\(665\) −22.5607 −0.874865
\(666\) 0 0
\(667\) 0.0293200 0.00113528
\(668\) 0 0
\(669\) −17.2268 −0.666025
\(670\) 0 0
\(671\) 3.04895 0.117703
\(672\) 0 0
\(673\) −33.2537 −1.28184 −0.640918 0.767609i \(-0.721446\pi\)
−0.640918 + 0.767609i \(0.721446\pi\)
\(674\) 0 0
\(675\) −5.26046 −0.202475
\(676\) 0 0
\(677\) 13.4478 0.516840 0.258420 0.966033i \(-0.416798\pi\)
0.258420 + 0.966033i \(0.416798\pi\)
\(678\) 0 0
\(679\) 12.5683 0.482328
\(680\) 0 0
\(681\) −14.6761 −0.562391
\(682\) 0 0
\(683\) −41.3546 −1.58239 −0.791196 0.611563i \(-0.790541\pi\)
−0.791196 + 0.611563i \(0.790541\pi\)
\(684\) 0 0
\(685\) −13.7100 −0.523833
\(686\) 0 0
\(687\) 25.5212 0.973694
\(688\) 0 0
\(689\) −15.9108 −0.606154
\(690\) 0 0
\(691\) −38.0453 −1.44731 −0.723656 0.690161i \(-0.757540\pi\)
−0.723656 + 0.690161i \(0.757540\pi\)
\(692\) 0 0
\(693\) 22.1024 0.839602
\(694\) 0 0
\(695\) −27.9877 −1.06164
\(696\) 0 0
\(697\) 64.5408 2.44466
\(698\) 0 0
\(699\) −49.0283 −1.85442
\(700\) 0 0
\(701\) 8.60173 0.324883 0.162441 0.986718i \(-0.448063\pi\)
0.162441 + 0.986718i \(0.448063\pi\)
\(702\) 0 0
\(703\) −17.2290 −0.649805
\(704\) 0 0
\(705\) 4.27384 0.160962
\(706\) 0 0
\(707\) 0.137547 0.00517297
\(708\) 0 0
\(709\) −9.55583 −0.358877 −0.179438 0.983769i \(-0.557428\pi\)
−0.179438 + 0.983769i \(0.557428\pi\)
\(710\) 0 0
\(711\) −48.8445 −1.83181
\(712\) 0 0
\(713\) −14.9705 −0.560650
\(714\) 0 0
\(715\) 7.60910 0.284564
\(716\) 0 0
\(717\) 20.6278 0.770358
\(718\) 0 0
\(719\) 27.2538 1.01639 0.508197 0.861241i \(-0.330312\pi\)
0.508197 + 0.861241i \(0.330312\pi\)
\(720\) 0 0
\(721\) 23.8075 0.886638
\(722\) 0 0
\(723\) 14.5202 0.540012
\(724\) 0 0
\(725\) 0.0334676 0.00124296
\(726\) 0 0
\(727\) −32.1115 −1.19095 −0.595475 0.803374i \(-0.703036\pi\)
−0.595475 + 0.803374i \(0.703036\pi\)
\(728\) 0 0
\(729\) −39.6188 −1.46736
\(730\) 0 0
\(731\) −11.0100 −0.407218
\(732\) 0 0
\(733\) 26.1534 0.965999 0.483000 0.875621i \(-0.339547\pi\)
0.483000 + 0.875621i \(0.339547\pi\)
\(734\) 0 0
\(735\) −21.4073 −0.789619
\(736\) 0 0
\(737\) 5.28160 0.194550
\(738\) 0 0
\(739\) −26.3389 −0.968891 −0.484445 0.874821i \(-0.660979\pi\)
−0.484445 + 0.874821i \(0.660979\pi\)
\(740\) 0 0
\(741\) −29.4821 −1.08305
\(742\) 0 0
\(743\) −33.7923 −1.23972 −0.619860 0.784713i \(-0.712811\pi\)
−0.619860 + 0.784713i \(0.712811\pi\)
\(744\) 0 0
\(745\) −38.7137 −1.41836
\(746\) 0 0
\(747\) 30.0131 1.09812
\(748\) 0 0
\(749\) −43.9338 −1.60531
\(750\) 0 0
\(751\) −45.8704 −1.67384 −0.836918 0.547329i \(-0.815645\pi\)
−0.836918 + 0.547329i \(0.815645\pi\)
\(752\) 0 0
\(753\) −57.8763 −2.10913
\(754\) 0 0
\(755\) 15.8081 0.575315
\(756\) 0 0
\(757\) −7.91756 −0.287768 −0.143884 0.989595i \(-0.545959\pi\)
−0.143884 + 0.989595i \(0.545959\pi\)
\(758\) 0 0
\(759\) −8.86203 −0.321671
\(760\) 0 0
\(761\) −6.22789 −0.225761 −0.112881 0.993609i \(-0.536008\pi\)
−0.112881 + 0.993609i \(0.536008\pi\)
\(762\) 0 0
\(763\) −9.09186 −0.329148
\(764\) 0 0
\(765\) −40.1245 −1.45070
\(766\) 0 0
\(767\) 36.8489 1.33054
\(768\) 0 0
\(769\) 0.267101 0.00963192 0.00481596 0.999988i \(-0.498467\pi\)
0.00481596 + 0.999988i \(0.498467\pi\)
\(770\) 0 0
\(771\) 50.5332 1.81991
\(772\) 0 0
\(773\) 15.0193 0.540206 0.270103 0.962831i \(-0.412942\pi\)
0.270103 + 0.962831i \(0.412942\pi\)
\(774\) 0 0
\(775\) −17.0882 −0.613827
\(776\) 0 0
\(777\) −39.1950 −1.40611
\(778\) 0 0
\(779\) 40.4152 1.44803
\(780\) 0 0
\(781\) −2.94805 −0.105490
\(782\) 0 0
\(783\) −0.0322318 −0.00115187
\(784\) 0 0
\(785\) 15.1847 0.541966
\(786\) 0 0
\(787\) −2.17662 −0.0775882 −0.0387941 0.999247i \(-0.512352\pi\)
−0.0387941 + 0.999247i \(0.512352\pi\)
\(788\) 0 0
\(789\) 53.4242 1.90195
\(790\) 0 0
\(791\) 23.1184 0.821996
\(792\) 0 0
\(793\) −5.20561 −0.184857
\(794\) 0 0
\(795\) −24.1002 −0.854747
\(796\) 0 0
\(797\) 17.3168 0.613394 0.306697 0.951807i \(-0.400776\pi\)
0.306697 + 0.951807i \(0.400776\pi\)
\(798\) 0 0
\(799\) −6.37109 −0.225393
\(800\) 0 0
\(801\) −1.16165 −0.0410448
\(802\) 0 0
\(803\) 22.0332 0.777534
\(804\) 0 0
\(805\) 11.5784 0.408085
\(806\) 0 0
\(807\) −70.0149 −2.46464
\(808\) 0 0
\(809\) −15.9862 −0.562046 −0.281023 0.959701i \(-0.590674\pi\)
−0.281023 + 0.959701i \(0.590674\pi\)
\(810\) 0 0
\(811\) 31.9350 1.12139 0.560695 0.828022i \(-0.310534\pi\)
0.560695 + 0.828022i \(0.310534\pi\)
\(812\) 0 0
\(813\) 35.9290 1.26008
\(814\) 0 0
\(815\) −1.27865 −0.0447890
\(816\) 0 0
\(817\) −6.89441 −0.241205
\(818\) 0 0
\(819\) −37.7365 −1.31862
\(820\) 0 0
\(821\) −24.0453 −0.839186 −0.419593 0.907712i \(-0.637827\pi\)
−0.419593 + 0.907712i \(0.637827\pi\)
\(822\) 0 0
\(823\) 38.2905 1.33472 0.667362 0.744734i \(-0.267424\pi\)
0.667362 + 0.744734i \(0.267424\pi\)
\(824\) 0 0
\(825\) −10.1156 −0.352181
\(826\) 0 0
\(827\) −42.4132 −1.47485 −0.737425 0.675429i \(-0.763959\pi\)
−0.737425 + 0.675429i \(0.763959\pi\)
\(828\) 0 0
\(829\) −55.0385 −1.91157 −0.955783 0.294072i \(-0.904990\pi\)
−0.955783 + 0.294072i \(0.904990\pi\)
\(830\) 0 0
\(831\) −4.14484 −0.143783
\(832\) 0 0
\(833\) 31.9122 1.10569
\(834\) 0 0
\(835\) 6.27851 0.217277
\(836\) 0 0
\(837\) 16.4572 0.568845
\(838\) 0 0
\(839\) 12.1435 0.419241 0.209621 0.977783i \(-0.432777\pi\)
0.209621 + 0.977783i \(0.432777\pi\)
\(840\) 0 0
\(841\) −28.9998 −0.999993
\(842\) 0 0
\(843\) 27.3596 0.942314
\(844\) 0 0
\(845\) 8.22244 0.282861
\(846\) 0 0
\(847\) −28.6550 −0.984596
\(848\) 0 0
\(849\) −55.6578 −1.91017
\(850\) 0 0
\(851\) 8.84214 0.303105
\(852\) 0 0
\(853\) 27.8185 0.952487 0.476244 0.879313i \(-0.341998\pi\)
0.476244 + 0.879313i \(0.341998\pi\)
\(854\) 0 0
\(855\) −25.1258 −0.859285
\(856\) 0 0
\(857\) 43.0034 1.46897 0.734484 0.678626i \(-0.237424\pi\)
0.734484 + 0.678626i \(0.237424\pi\)
\(858\) 0 0
\(859\) 17.8086 0.607620 0.303810 0.952733i \(-0.401741\pi\)
0.303810 + 0.952733i \(0.401741\pi\)
\(860\) 0 0
\(861\) 91.9423 3.13338
\(862\) 0 0
\(863\) 20.0611 0.682887 0.341444 0.939902i \(-0.389084\pi\)
0.341444 + 0.939902i \(0.389084\pi\)
\(864\) 0 0
\(865\) −4.31581 −0.146742
\(866\) 0 0
\(867\) 61.7854 2.09834
\(868\) 0 0
\(869\) −20.9153 −0.709502
\(870\) 0 0
\(871\) −9.01752 −0.305547
\(872\) 0 0
\(873\) 13.9974 0.473739
\(874\) 0 0
\(875\) 41.4909 1.40265
\(876\) 0 0
\(877\) 27.3471 0.923446 0.461723 0.887024i \(-0.347231\pi\)
0.461723 + 0.887024i \(0.347231\pi\)
\(878\) 0 0
\(879\) 79.3358 2.67593
\(880\) 0 0
\(881\) 35.9501 1.21119 0.605595 0.795773i \(-0.292935\pi\)
0.605595 + 0.795773i \(0.292935\pi\)
\(882\) 0 0
\(883\) 36.9962 1.24502 0.622511 0.782611i \(-0.286112\pi\)
0.622511 + 0.782611i \(0.286112\pi\)
\(884\) 0 0
\(885\) 55.8154 1.87621
\(886\) 0 0
\(887\) 31.5813 1.06040 0.530198 0.847874i \(-0.322118\pi\)
0.530198 + 0.847874i \(0.322118\pi\)
\(888\) 0 0
\(889\) 11.1869 0.375198
\(890\) 0 0
\(891\) −9.39207 −0.314646
\(892\) 0 0
\(893\) −3.98956 −0.133505
\(894\) 0 0
\(895\) −35.8340 −1.19780
\(896\) 0 0
\(897\) 15.1306 0.505195
\(898\) 0 0
\(899\) −0.104703 −0.00349203
\(900\) 0 0
\(901\) 35.9266 1.19689
\(902\) 0 0
\(903\) −15.6844 −0.521943
\(904\) 0 0
\(905\) 36.4208 1.21067
\(906\) 0 0
\(907\) 15.3465 0.509571 0.254786 0.966998i \(-0.417995\pi\)
0.254786 + 0.966998i \(0.417995\pi\)
\(908\) 0 0
\(909\) 0.153186 0.00508085
\(910\) 0 0
\(911\) −39.5434 −1.31013 −0.655066 0.755572i \(-0.727359\pi\)
−0.655066 + 0.755572i \(0.727359\pi\)
\(912\) 0 0
\(913\) 12.8516 0.425327
\(914\) 0 0
\(915\) −7.88498 −0.260669
\(916\) 0 0
\(917\) 4.68972 0.154868
\(918\) 0 0
\(919\) 15.4055 0.508179 0.254090 0.967181i \(-0.418224\pi\)
0.254090 + 0.967181i \(0.418224\pi\)
\(920\) 0 0
\(921\) −4.18371 −0.137858
\(922\) 0 0
\(923\) 5.03335 0.165675
\(924\) 0 0
\(925\) 10.0929 0.331854
\(926\) 0 0
\(927\) 26.5144 0.870849
\(928\) 0 0
\(929\) 10.4517 0.342908 0.171454 0.985192i \(-0.445154\pi\)
0.171454 + 0.985192i \(0.445154\pi\)
\(930\) 0 0
\(931\) 19.9833 0.654927
\(932\) 0 0
\(933\) 54.0012 1.76792
\(934\) 0 0
\(935\) −17.1813 −0.561890
\(936\) 0 0
\(937\) 12.9830 0.424136 0.212068 0.977255i \(-0.431980\pi\)
0.212068 + 0.977255i \(0.431980\pi\)
\(938\) 0 0
\(939\) 62.5564 2.04145
\(940\) 0 0
\(941\) −31.4045 −1.02376 −0.511879 0.859058i \(-0.671050\pi\)
−0.511879 + 0.859058i \(0.671050\pi\)
\(942\) 0 0
\(943\) −20.7416 −0.675439
\(944\) 0 0
\(945\) −12.7282 −0.414050
\(946\) 0 0
\(947\) −43.1004 −1.40058 −0.700288 0.713860i \(-0.746945\pi\)
−0.700288 + 0.713860i \(0.746945\pi\)
\(948\) 0 0
\(949\) −37.6183 −1.22114
\(950\) 0 0
\(951\) −42.6804 −1.38401
\(952\) 0 0
\(953\) 21.4745 0.695626 0.347813 0.937564i \(-0.386924\pi\)
0.347813 + 0.937564i \(0.386924\pi\)
\(954\) 0 0
\(955\) 28.3368 0.916957
\(956\) 0 0
\(957\) −0.0619804 −0.00200354
\(958\) 0 0
\(959\) 29.1149 0.940168
\(960\) 0 0
\(961\) 22.4601 0.724521
\(962\) 0 0
\(963\) −48.9292 −1.57672
\(964\) 0 0
\(965\) 4.00158 0.128815
\(966\) 0 0
\(967\) 11.8062 0.379662 0.189831 0.981817i \(-0.439206\pi\)
0.189831 + 0.981817i \(0.439206\pi\)
\(968\) 0 0
\(969\) 66.5705 2.13855
\(970\) 0 0
\(971\) −4.32111 −0.138671 −0.0693354 0.997593i \(-0.522088\pi\)
−0.0693354 + 0.997593i \(0.522088\pi\)
\(972\) 0 0
\(973\) 59.4353 1.90541
\(974\) 0 0
\(975\) 17.2709 0.553112
\(976\) 0 0
\(977\) −56.8679 −1.81936 −0.909682 0.415305i \(-0.863675\pi\)
−0.909682 + 0.415305i \(0.863675\pi\)
\(978\) 0 0
\(979\) −0.497418 −0.0158976
\(980\) 0 0
\(981\) −10.1256 −0.323286
\(982\) 0 0
\(983\) −12.6165 −0.402404 −0.201202 0.979550i \(-0.564485\pi\)
−0.201202 + 0.979550i \(0.564485\pi\)
\(984\) 0 0
\(985\) 36.3800 1.15916
\(986\) 0 0
\(987\) −9.07601 −0.288893
\(988\) 0 0
\(989\) 3.53829 0.112511
\(990\) 0 0
\(991\) −17.0364 −0.541180 −0.270590 0.962695i \(-0.587219\pi\)
−0.270590 + 0.962695i \(0.587219\pi\)
\(992\) 0 0
\(993\) 7.20296 0.228579
\(994\) 0 0
\(995\) −18.4604 −0.585235
\(996\) 0 0
\(997\) −13.1632 −0.416882 −0.208441 0.978035i \(-0.566839\pi\)
−0.208441 + 0.978035i \(0.566839\pi\)
\(998\) 0 0
\(999\) −9.72025 −0.307535
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 6016.2.a.o.1.11 yes 13
4.3 odd 2 6016.2.a.m.1.3 13
8.3 odd 2 6016.2.a.p.1.11 yes 13
8.5 even 2 6016.2.a.n.1.3 yes 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6016.2.a.m.1.3 13 4.3 odd 2
6016.2.a.n.1.3 yes 13 8.5 even 2
6016.2.a.o.1.11 yes 13 1.1 even 1 trivial
6016.2.a.p.1.11 yes 13 8.3 odd 2