Properties

Label 2112.2.a.bg.1.2
Level $2112$
Weight $2$
Character 2112.1
Self dual yes
Analytic conductor $16.864$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2112,2,Mod(1,2112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2112, 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("2112.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2112.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: \(16.8644049069\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1056)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 2112.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.508203 q^{5} -3.87086 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.508203 q^{5} -3.87086 q^{7} +1.00000 q^{9} -1.00000 q^{11} -0.508203 q^{13} +0.508203 q^{15} +7.36266 q^{17} -5.36266 q^{19} +3.87086 q^{21} -2.50820 q^{23} -4.74173 q^{25} -1.00000 q^{27} -7.36266 q^{29} +10.7253 q^{31} +1.00000 q^{33} +1.96719 q^{35} -8.72532 q^{37} +0.508203 q^{39} +8.37907 q^{41} +2.37907 q^{43} -0.508203 q^{45} -5.49180 q^{47} +7.98359 q^{49} -7.36266 q^{51} +12.2499 q^{53} +0.508203 q^{55} +5.36266 q^{57} +1.01641 q^{59} +7.23353 q^{61} -3.87086 q^{63} +0.258271 q^{65} +12.7581 q^{67} +2.50820 q^{69} -10.5082 q^{71} +14.7581 q^{73} +4.74173 q^{75} +3.87086 q^{77} +1.14554 q^{79} +1.00000 q^{81} +8.00000 q^{83} -3.74173 q^{85} +7.36266 q^{87} +2.00000 q^{89} +1.96719 q^{91} -10.7253 q^{93} +2.72532 q^{95} +3.01641 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 4 q^{7} + 3 q^{9} - 3 q^{11} + 8 q^{17} - 2 q^{19} - 4 q^{21} - 6 q^{23} + 17 q^{25} - 3 q^{27} - 8 q^{29} + 4 q^{31} + 3 q^{33} + 12 q^{35} + 2 q^{37} + 8 q^{41} - 10 q^{43} - 18 q^{47} + 27 q^{49} - 8 q^{51} + 4 q^{53} + 2 q^{57} - 8 q^{61} + 4 q^{63} + 32 q^{65} + 4 q^{67} + 6 q^{69} - 30 q^{71} + 10 q^{73} - 17 q^{75} - 4 q^{77} + 16 q^{79} + 3 q^{81} + 24 q^{83} + 20 q^{85} + 8 q^{87} + 6 q^{89} + 12 q^{91} - 4 q^{93} - 20 q^{95} + 6 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −0.508203 −0.227275 −0.113638 0.993522i \(-0.536250\pi\)
−0.113638 + 0.993522i \(0.536250\pi\)
\(6\) 0 0
\(7\) −3.87086 −1.46305 −0.731525 0.681815i \(-0.761191\pi\)
−0.731525 + 0.681815i \(0.761191\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −0.508203 −0.140950 −0.0704751 0.997514i \(-0.522452\pi\)
−0.0704751 + 0.997514i \(0.522452\pi\)
\(14\) 0 0
\(15\) 0.508203 0.131218
\(16\) 0 0
\(17\) 7.36266 1.78571 0.892854 0.450347i \(-0.148700\pi\)
0.892854 + 0.450347i \(0.148700\pi\)
\(18\) 0 0
\(19\) −5.36266 −1.23028 −0.615139 0.788418i \(-0.710900\pi\)
−0.615139 + 0.788418i \(0.710900\pi\)
\(20\) 0 0
\(21\) 3.87086 0.844692
\(22\) 0 0
\(23\) −2.50820 −0.522997 −0.261498 0.965204i \(-0.584217\pi\)
−0.261498 + 0.965204i \(0.584217\pi\)
\(24\) 0 0
\(25\) −4.74173 −0.948346
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.36266 −1.36721 −0.683606 0.729851i \(-0.739589\pi\)
−0.683606 + 0.729851i \(0.739589\pi\)
\(30\) 0 0
\(31\) 10.7253 1.92632 0.963162 0.268920i \(-0.0866669\pi\)
0.963162 + 0.268920i \(0.0866669\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 1.96719 0.332515
\(36\) 0 0
\(37\) −8.72532 −1.43443 −0.717217 0.696850i \(-0.754584\pi\)
−0.717217 + 0.696850i \(0.754584\pi\)
\(38\) 0 0
\(39\) 0.508203 0.0813777
\(40\) 0 0
\(41\) 8.37907 1.30859 0.654295 0.756239i \(-0.272965\pi\)
0.654295 + 0.756239i \(0.272965\pi\)
\(42\) 0 0
\(43\) 2.37907 0.362804 0.181402 0.983409i \(-0.441936\pi\)
0.181402 + 0.983409i \(0.441936\pi\)
\(44\) 0 0
\(45\) −0.508203 −0.0757585
\(46\) 0 0
\(47\) −5.49180 −0.801061 −0.400530 0.916283i \(-0.631174\pi\)
−0.400530 + 0.916283i \(0.631174\pi\)
\(48\) 0 0
\(49\) 7.98359 1.14051
\(50\) 0 0
\(51\) −7.36266 −1.03098
\(52\) 0 0
\(53\) 12.2499 1.68266 0.841329 0.540524i \(-0.181774\pi\)
0.841329 + 0.540524i \(0.181774\pi\)
\(54\) 0 0
\(55\) 0.508203 0.0685261
\(56\) 0 0
\(57\) 5.36266 0.710302
\(58\) 0 0
\(59\) 1.01641 0.132325 0.0661624 0.997809i \(-0.478924\pi\)
0.0661624 + 0.997809i \(0.478924\pi\)
\(60\) 0 0
\(61\) 7.23353 0.926158 0.463079 0.886317i \(-0.346745\pi\)
0.463079 + 0.886317i \(0.346745\pi\)
\(62\) 0 0
\(63\) −3.87086 −0.487683
\(64\) 0 0
\(65\) 0.258271 0.0320345
\(66\) 0 0
\(67\) 12.7581 1.55865 0.779327 0.626617i \(-0.215561\pi\)
0.779327 + 0.626617i \(0.215561\pi\)
\(68\) 0 0
\(69\) 2.50820 0.301952
\(70\) 0 0
\(71\) −10.5082 −1.24709 −0.623547 0.781786i \(-0.714309\pi\)
−0.623547 + 0.781786i \(0.714309\pi\)
\(72\) 0 0
\(73\) 14.7581 1.72731 0.863655 0.504084i \(-0.168170\pi\)
0.863655 + 0.504084i \(0.168170\pi\)
\(74\) 0 0
\(75\) 4.74173 0.547528
\(76\) 0 0
\(77\) 3.87086 0.441126
\(78\) 0 0
\(79\) 1.14554 0.128884 0.0644418 0.997921i \(-0.479473\pi\)
0.0644418 + 0.997921i \(0.479473\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) −3.74173 −0.405848
\(86\) 0 0
\(87\) 7.36266 0.789360
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 1.96719 0.206217
\(92\) 0 0
\(93\) −10.7253 −1.11216
\(94\) 0 0
\(95\) 2.72532 0.279612
\(96\) 0 0
\(97\) 3.01641 0.306270 0.153135 0.988205i \(-0.451063\pi\)
0.153135 + 0.988205i \(0.451063\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 0.637339 0.0634176 0.0317088 0.999497i \(-0.489905\pi\)
0.0317088 + 0.999497i \(0.489905\pi\)
\(102\) 0 0
\(103\) 13.0164 1.28254 0.641272 0.767313i \(-0.278407\pi\)
0.641272 + 0.767313i \(0.278407\pi\)
\(104\) 0 0
\(105\) −1.96719 −0.191978
\(106\) 0 0
\(107\) 6.72532 0.650161 0.325081 0.945686i \(-0.394609\pi\)
0.325081 + 0.945686i \(0.394609\pi\)
\(108\) 0 0
\(109\) −0.508203 −0.0486771 −0.0243385 0.999704i \(-0.507748\pi\)
−0.0243385 + 0.999704i \(0.507748\pi\)
\(110\) 0 0
\(111\) 8.72532 0.828171
\(112\) 0 0
\(113\) −0.725323 −0.0682326 −0.0341163 0.999418i \(-0.510862\pi\)
−0.0341163 + 0.999418i \(0.510862\pi\)
\(114\) 0 0
\(115\) 1.27468 0.118864
\(116\) 0 0
\(117\) −0.508203 −0.0469834
\(118\) 0 0
\(119\) −28.4999 −2.61258
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −8.37907 −0.755515
\(124\) 0 0
\(125\) 4.95078 0.442811
\(126\) 0 0
\(127\) 1.14554 0.101650 0.0508252 0.998708i \(-0.483815\pi\)
0.0508252 + 0.998708i \(0.483815\pi\)
\(128\) 0 0
\(129\) −2.37907 −0.209465
\(130\) 0 0
\(131\) −10.7253 −0.937076 −0.468538 0.883443i \(-0.655219\pi\)
−0.468538 + 0.883443i \(0.655219\pi\)
\(132\) 0 0
\(133\) 20.7581 1.79996
\(134\) 0 0
\(135\) 0.508203 0.0437392
\(136\) 0 0
\(137\) −11.7089 −1.00036 −0.500180 0.865921i \(-0.666733\pi\)
−0.500180 + 0.865921i \(0.666733\pi\)
\(138\) 0 0
\(139\) −7.39547 −0.627276 −0.313638 0.949543i \(-0.601548\pi\)
−0.313638 + 0.949543i \(0.601548\pi\)
\(140\) 0 0
\(141\) 5.49180 0.462493
\(142\) 0 0
\(143\) 0.508203 0.0424981
\(144\) 0 0
\(145\) 3.74173 0.310734
\(146\) 0 0
\(147\) −7.98359 −0.658476
\(148\) 0 0
\(149\) 10.4119 0.852975 0.426487 0.904494i \(-0.359751\pi\)
0.426487 + 0.904494i \(0.359751\pi\)
\(150\) 0 0
\(151\) 16.6290 1.35325 0.676624 0.736328i \(-0.263442\pi\)
0.676624 + 0.736328i \(0.263442\pi\)
\(152\) 0 0
\(153\) 7.36266 0.595236
\(154\) 0 0
\(155\) −5.45065 −0.437806
\(156\) 0 0
\(157\) 7.01641 0.559970 0.279985 0.960004i \(-0.409670\pi\)
0.279985 + 0.960004i \(0.409670\pi\)
\(158\) 0 0
\(159\) −12.2499 −0.971483
\(160\) 0 0
\(161\) 9.70892 0.765170
\(162\) 0 0
\(163\) −8.75814 −0.685990 −0.342995 0.939337i \(-0.611441\pi\)
−0.342995 + 0.939337i \(0.611441\pi\)
\(164\) 0 0
\(165\) −0.508203 −0.0395636
\(166\) 0 0
\(167\) 5.70892 0.441769 0.220885 0.975300i \(-0.429106\pi\)
0.220885 + 0.975300i \(0.429106\pi\)
\(168\) 0 0
\(169\) −12.7417 −0.980133
\(170\) 0 0
\(171\) −5.36266 −0.410093
\(172\) 0 0
\(173\) −7.36266 −0.559773 −0.279886 0.960033i \(-0.590297\pi\)
−0.279886 + 0.960033i \(0.590297\pi\)
\(174\) 0 0
\(175\) 18.3546 1.38748
\(176\) 0 0
\(177\) −1.01641 −0.0763978
\(178\) 0 0
\(179\) 6.98359 0.521978 0.260989 0.965342i \(-0.415951\pi\)
0.260989 + 0.965342i \(0.415951\pi\)
\(180\) 0 0
\(181\) 4.98359 0.370428 0.185214 0.982698i \(-0.440702\pi\)
0.185214 + 0.982698i \(0.440702\pi\)
\(182\) 0 0
\(183\) −7.23353 −0.534718
\(184\) 0 0
\(185\) 4.43424 0.326012
\(186\) 0 0
\(187\) −7.36266 −0.538411
\(188\) 0 0
\(189\) 3.87086 0.281564
\(190\) 0 0
\(191\) 4.54102 0.328576 0.164288 0.986412i \(-0.447467\pi\)
0.164288 + 0.986412i \(0.447467\pi\)
\(192\) 0 0
\(193\) 20.4671 1.47325 0.736625 0.676301i \(-0.236418\pi\)
0.736625 + 0.676301i \(0.236418\pi\)
\(194\) 0 0
\(195\) −0.258271 −0.0184951
\(196\) 0 0
\(197\) 8.63734 0.615385 0.307692 0.951486i \(-0.400443\pi\)
0.307692 + 0.951486i \(0.400443\pi\)
\(198\) 0 0
\(199\) −7.74173 −0.548797 −0.274398 0.961616i \(-0.588479\pi\)
−0.274398 + 0.961616i \(0.588479\pi\)
\(200\) 0 0
\(201\) −12.7581 −0.899890
\(202\) 0 0
\(203\) 28.4999 2.00030
\(204\) 0 0
\(205\) −4.25827 −0.297411
\(206\) 0 0
\(207\) −2.50820 −0.174332
\(208\) 0 0
\(209\) 5.36266 0.370943
\(210\) 0 0
\(211\) 19.0716 1.31294 0.656471 0.754351i \(-0.272049\pi\)
0.656471 + 0.754351i \(0.272049\pi\)
\(212\) 0 0
\(213\) 10.5082 0.720010
\(214\) 0 0
\(215\) −1.20905 −0.0824566
\(216\) 0 0
\(217\) −41.5163 −2.81831
\(218\) 0 0
\(219\) −14.7581 −0.997262
\(220\) 0 0
\(221\) −3.74173 −0.251696
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −4.74173 −0.316115
\(226\) 0 0
\(227\) 22.2088 1.47405 0.737024 0.675866i \(-0.236230\pi\)
0.737024 + 0.675866i \(0.236230\pi\)
\(228\) 0 0
\(229\) 15.7089 1.03807 0.519037 0.854752i \(-0.326291\pi\)
0.519037 + 0.854752i \(0.326291\pi\)
\(230\) 0 0
\(231\) −3.87086 −0.254684
\(232\) 0 0
\(233\) −23.1044 −1.51362 −0.756809 0.653636i \(-0.773243\pi\)
−0.756809 + 0.653636i \(0.773243\pi\)
\(234\) 0 0
\(235\) 2.79095 0.182061
\(236\) 0 0
\(237\) −1.14554 −0.0744110
\(238\) 0 0
\(239\) −15.7417 −1.01825 −0.509124 0.860693i \(-0.670031\pi\)
−0.509124 + 0.860693i \(0.670031\pi\)
\(240\) 0 0
\(241\) −5.74173 −0.369857 −0.184929 0.982752i \(-0.559205\pi\)
−0.184929 + 0.982752i \(0.559205\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −4.05729 −0.259211
\(246\) 0 0
\(247\) 2.72532 0.173408
\(248\) 0 0
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) −30.4671 −1.92306 −0.961532 0.274694i \(-0.911423\pi\)
−0.961532 + 0.274694i \(0.911423\pi\)
\(252\) 0 0
\(253\) 2.50820 0.157689
\(254\) 0 0
\(255\) 3.74173 0.234316
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 33.7745 2.09865
\(260\) 0 0
\(261\) −7.36266 −0.455737
\(262\) 0 0
\(263\) 26.7253 1.64795 0.823977 0.566623i \(-0.191750\pi\)
0.823977 + 0.566623i \(0.191750\pi\)
\(264\) 0 0
\(265\) −6.22546 −0.382427
\(266\) 0 0
\(267\) −2.00000 −0.122398
\(268\) 0 0
\(269\) −27.2335 −1.66046 −0.830229 0.557423i \(-0.811790\pi\)
−0.830229 + 0.557423i \(0.811790\pi\)
\(270\) 0 0
\(271\) 12.8216 0.778859 0.389430 0.921056i \(-0.372672\pi\)
0.389430 + 0.921056i \(0.372672\pi\)
\(272\) 0 0
\(273\) −1.96719 −0.119060
\(274\) 0 0
\(275\) 4.74173 0.285937
\(276\) 0 0
\(277\) 12.5082 0.751545 0.375773 0.926712i \(-0.377377\pi\)
0.375773 + 0.926712i \(0.377377\pi\)
\(278\) 0 0
\(279\) 10.7253 0.642108
\(280\) 0 0
\(281\) −13.3955 −0.799107 −0.399554 0.916710i \(-0.630835\pi\)
−0.399554 + 0.916710i \(0.630835\pi\)
\(282\) 0 0
\(283\) −31.1372 −1.85091 −0.925457 0.378852i \(-0.876319\pi\)
−0.925457 + 0.378852i \(0.876319\pi\)
\(284\) 0 0
\(285\) −2.72532 −0.161434
\(286\) 0 0
\(287\) −32.4342 −1.91453
\(288\) 0 0
\(289\) 37.2088 2.18875
\(290\) 0 0
\(291\) −3.01641 −0.176825
\(292\) 0 0
\(293\) 21.8297 1.27531 0.637653 0.770324i \(-0.279905\pi\)
0.637653 + 0.770324i \(0.279905\pi\)
\(294\) 0 0
\(295\) −0.516541 −0.0300742
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 1.27468 0.0737165
\(300\) 0 0
\(301\) −9.20905 −0.530801
\(302\) 0 0
\(303\) −0.637339 −0.0366142
\(304\) 0 0
\(305\) −3.67610 −0.210493
\(306\) 0 0
\(307\) 14.0552 0.802171 0.401085 0.916041i \(-0.368633\pi\)
0.401085 + 0.916041i \(0.368633\pi\)
\(308\) 0 0
\(309\) −13.0164 −0.740478
\(310\) 0 0
\(311\) −10.5082 −0.595866 −0.297933 0.954587i \(-0.596297\pi\)
−0.297933 + 0.954587i \(0.596297\pi\)
\(312\) 0 0
\(313\) 2.69251 0.152190 0.0760948 0.997101i \(-0.475755\pi\)
0.0760948 + 0.997101i \(0.475755\pi\)
\(314\) 0 0
\(315\) 1.96719 0.110838
\(316\) 0 0
\(317\) 6.54102 0.367380 0.183690 0.982984i \(-0.441196\pi\)
0.183690 + 0.982984i \(0.441196\pi\)
\(318\) 0 0
\(319\) 7.36266 0.412230
\(320\) 0 0
\(321\) −6.72532 −0.375371
\(322\) 0 0
\(323\) −39.4835 −2.19692
\(324\) 0 0
\(325\) 2.40976 0.133670
\(326\) 0 0
\(327\) 0.508203 0.0281037
\(328\) 0 0
\(329\) 21.2580 1.17199
\(330\) 0 0
\(331\) 30.7253 1.68882 0.844408 0.535700i \(-0.179952\pi\)
0.844408 + 0.535700i \(0.179952\pi\)
\(332\) 0 0
\(333\) −8.72532 −0.478145
\(334\) 0 0
\(335\) −6.48373 −0.354244
\(336\) 0 0
\(337\) −30.1760 −1.64379 −0.821895 0.569639i \(-0.807083\pi\)
−0.821895 + 0.569639i \(0.807083\pi\)
\(338\) 0 0
\(339\) 0.725323 0.0393941
\(340\) 0 0
\(341\) −10.7253 −0.580809
\(342\) 0 0
\(343\) −3.80736 −0.205578
\(344\) 0 0
\(345\) −1.27468 −0.0686263
\(346\) 0 0
\(347\) 18.0328 0.968052 0.484026 0.875054i \(-0.339174\pi\)
0.484026 + 0.875054i \(0.339174\pi\)
\(348\) 0 0
\(349\) 2.21712 0.118680 0.0593398 0.998238i \(-0.481100\pi\)
0.0593398 + 0.998238i \(0.481100\pi\)
\(350\) 0 0
\(351\) 0.508203 0.0271259
\(352\) 0 0
\(353\) −13.4835 −0.717652 −0.358826 0.933404i \(-0.616823\pi\)
−0.358826 + 0.933404i \(0.616823\pi\)
\(354\) 0 0
\(355\) 5.34030 0.283434
\(356\) 0 0
\(357\) 28.4999 1.50837
\(358\) 0 0
\(359\) 20.4999 1.08194 0.540971 0.841041i \(-0.318057\pi\)
0.540971 + 0.841041i \(0.318057\pi\)
\(360\) 0 0
\(361\) 9.75814 0.513586
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −7.50013 −0.392575
\(366\) 0 0
\(367\) 18.0328 0.941305 0.470653 0.882319i \(-0.344019\pi\)
0.470653 + 0.882319i \(0.344019\pi\)
\(368\) 0 0
\(369\) 8.37907 0.436197
\(370\) 0 0
\(371\) −47.4178 −2.46181
\(372\) 0 0
\(373\) −11.2335 −0.581650 −0.290825 0.956776i \(-0.593930\pi\)
−0.290825 + 0.956776i \(0.593930\pi\)
\(374\) 0 0
\(375\) −4.95078 −0.255657
\(376\) 0 0
\(377\) 3.74173 0.192709
\(378\) 0 0
\(379\) 7.24186 0.371990 0.185995 0.982551i \(-0.440449\pi\)
0.185995 + 0.982551i \(0.440449\pi\)
\(380\) 0 0
\(381\) −1.14554 −0.0586879
\(382\) 0 0
\(383\) −23.9588 −1.22424 −0.612120 0.790765i \(-0.709683\pi\)
−0.612120 + 0.790765i \(0.709683\pi\)
\(384\) 0 0
\(385\) −1.96719 −0.100257
\(386\) 0 0
\(387\) 2.37907 0.120935
\(388\) 0 0
\(389\) −6.47539 −0.328315 −0.164158 0.986434i \(-0.552491\pi\)
−0.164158 + 0.986434i \(0.552491\pi\)
\(390\) 0 0
\(391\) −18.4671 −0.933919
\(392\) 0 0
\(393\) 10.7253 0.541021
\(394\) 0 0
\(395\) −0.582168 −0.0292921
\(396\) 0 0
\(397\) −19.1924 −0.963238 −0.481619 0.876381i \(-0.659951\pi\)
−0.481619 + 0.876381i \(0.659951\pi\)
\(398\) 0 0
\(399\) −20.7581 −1.03921
\(400\) 0 0
\(401\) 11.7745 0.587993 0.293996 0.955807i \(-0.405015\pi\)
0.293996 + 0.955807i \(0.405015\pi\)
\(402\) 0 0
\(403\) −5.45065 −0.271516
\(404\) 0 0
\(405\) −0.508203 −0.0252528
\(406\) 0 0
\(407\) 8.72532 0.432498
\(408\) 0 0
\(409\) −35.1924 −1.74015 −0.870075 0.492919i \(-0.835930\pi\)
−0.870075 + 0.492919i \(0.835930\pi\)
\(410\) 0 0
\(411\) 11.7089 0.577558
\(412\) 0 0
\(413\) −3.93437 −0.193598
\(414\) 0 0
\(415\) −4.06563 −0.199574
\(416\) 0 0
\(417\) 7.39547 0.362158
\(418\) 0 0
\(419\) 14.9836 0.731996 0.365998 0.930616i \(-0.380728\pi\)
0.365998 + 0.930616i \(0.380728\pi\)
\(420\) 0 0
\(421\) −25.4178 −1.23879 −0.619395 0.785080i \(-0.712622\pi\)
−0.619395 + 0.785080i \(0.712622\pi\)
\(422\) 0 0
\(423\) −5.49180 −0.267020
\(424\) 0 0
\(425\) −34.9117 −1.69347
\(426\) 0 0
\(427\) −28.0000 −1.35501
\(428\) 0 0
\(429\) −0.508203 −0.0245363
\(430\) 0 0
\(431\) −26.2088 −1.26243 −0.631216 0.775607i \(-0.717444\pi\)
−0.631216 + 0.775607i \(0.717444\pi\)
\(432\) 0 0
\(433\) 15.2747 0.734054 0.367027 0.930210i \(-0.380376\pi\)
0.367027 + 0.930210i \(0.380376\pi\)
\(434\) 0 0
\(435\) −3.74173 −0.179402
\(436\) 0 0
\(437\) 13.4506 0.643432
\(438\) 0 0
\(439\) 6.42022 0.306420 0.153210 0.988194i \(-0.451039\pi\)
0.153210 + 0.988194i \(0.451039\pi\)
\(440\) 0 0
\(441\) 7.98359 0.380171
\(442\) 0 0
\(443\) 16.4999 0.783932 0.391966 0.919980i \(-0.371795\pi\)
0.391966 + 0.919980i \(0.371795\pi\)
\(444\) 0 0
\(445\) −1.01641 −0.0481823
\(446\) 0 0
\(447\) −10.4119 −0.492465
\(448\) 0 0
\(449\) −10.4999 −0.495519 −0.247760 0.968822i \(-0.579694\pi\)
−0.247760 + 0.968822i \(0.579694\pi\)
\(450\) 0 0
\(451\) −8.37907 −0.394555
\(452\) 0 0
\(453\) −16.6290 −0.781299
\(454\) 0 0
\(455\) −0.999731 −0.0468681
\(456\) 0 0
\(457\) −15.7745 −0.737902 −0.368951 0.929449i \(-0.620283\pi\)
−0.368951 + 0.929449i \(0.620283\pi\)
\(458\) 0 0
\(459\) −7.36266 −0.343660
\(460\) 0 0
\(461\) −17.6537 −0.822217 −0.411108 0.911586i \(-0.634858\pi\)
−0.411108 + 0.911586i \(0.634858\pi\)
\(462\) 0 0
\(463\) 36.4999 1.69629 0.848146 0.529762i \(-0.177719\pi\)
0.848146 + 0.529762i \(0.177719\pi\)
\(464\) 0 0
\(465\) 5.45065 0.252768
\(466\) 0 0
\(467\) −6.03281 −0.279165 −0.139583 0.990210i \(-0.544576\pi\)
−0.139583 + 0.990210i \(0.544576\pi\)
\(468\) 0 0
\(469\) −49.3850 −2.28039
\(470\) 0 0
\(471\) −7.01641 −0.323299
\(472\) 0 0
\(473\) −2.37907 −0.109390
\(474\) 0 0
\(475\) 25.4283 1.16673
\(476\) 0 0
\(477\) 12.2499 0.560886
\(478\) 0 0
\(479\) 6.65970 0.304289 0.152145 0.988358i \(-0.451382\pi\)
0.152145 + 0.988358i \(0.451382\pi\)
\(480\) 0 0
\(481\) 4.43424 0.202184
\(482\) 0 0
\(483\) −9.70892 −0.441771
\(484\) 0 0
\(485\) −1.53295 −0.0696076
\(486\) 0 0
\(487\) −16.2583 −0.736733 −0.368366 0.929681i \(-0.620083\pi\)
−0.368366 + 0.929681i \(0.620083\pi\)
\(488\) 0 0
\(489\) 8.75814 0.396057
\(490\) 0 0
\(491\) −16.7581 −0.756284 −0.378142 0.925748i \(-0.623437\pi\)
−0.378142 + 0.925748i \(0.623437\pi\)
\(492\) 0 0
\(493\) −54.2088 −2.44144
\(494\) 0 0
\(495\) 0.508203 0.0228420
\(496\) 0 0
\(497\) 40.6758 1.82456
\(498\) 0 0
\(499\) −14.0328 −0.628195 −0.314098 0.949391i \(-0.601702\pi\)
−0.314098 + 0.949391i \(0.601702\pi\)
\(500\) 0 0
\(501\) −5.70892 −0.255056
\(502\) 0 0
\(503\) −7.74173 −0.345187 −0.172593 0.984993i \(-0.555215\pi\)
−0.172593 + 0.984993i \(0.555215\pi\)
\(504\) 0 0
\(505\) −0.323898 −0.0144133
\(506\) 0 0
\(507\) 12.7417 0.565880
\(508\) 0 0
\(509\) 25.7006 1.13916 0.569579 0.821937i \(-0.307106\pi\)
0.569579 + 0.821937i \(0.307106\pi\)
\(510\) 0 0
\(511\) −57.1267 −2.52714
\(512\) 0 0
\(513\) 5.36266 0.236767
\(514\) 0 0
\(515\) −6.61498 −0.291491
\(516\) 0 0
\(517\) 5.49180 0.241529
\(518\) 0 0
\(519\) 7.36266 0.323185
\(520\) 0 0
\(521\) −19.7089 −0.863463 −0.431732 0.902002i \(-0.642097\pi\)
−0.431732 + 0.902002i \(0.642097\pi\)
\(522\) 0 0
\(523\) 24.3463 1.06459 0.532294 0.846560i \(-0.321330\pi\)
0.532294 + 0.846560i \(0.321330\pi\)
\(524\) 0 0
\(525\) −18.3546 −0.801060
\(526\) 0 0
\(527\) 78.9669 3.43985
\(528\) 0 0
\(529\) −16.7089 −0.726475
\(530\) 0 0
\(531\) 1.01641 0.0441083
\(532\) 0 0
\(533\) −4.25827 −0.184446
\(534\) 0 0
\(535\) −3.41783 −0.147766
\(536\) 0 0
\(537\) −6.98359 −0.301364
\(538\) 0 0
\(539\) −7.98359 −0.343878
\(540\) 0 0
\(541\) 14.9753 0.643837 0.321918 0.946767i \(-0.395672\pi\)
0.321918 + 0.946767i \(0.395672\pi\)
\(542\) 0 0
\(543\) −4.98359 −0.213866
\(544\) 0 0
\(545\) 0.258271 0.0110631
\(546\) 0 0
\(547\) 20.8461 0.891316 0.445658 0.895203i \(-0.352970\pi\)
0.445658 + 0.895203i \(0.352970\pi\)
\(548\) 0 0
\(549\) 7.23353 0.308719
\(550\) 0 0
\(551\) 39.4835 1.68205
\(552\) 0 0
\(553\) −4.43424 −0.188563
\(554\) 0 0
\(555\) −4.43424 −0.188223
\(556\) 0 0
\(557\) −1.13720 −0.0481849 −0.0240924 0.999710i \(-0.507670\pi\)
−0.0240924 + 0.999710i \(0.507670\pi\)
\(558\) 0 0
\(559\) −1.20905 −0.0511374
\(560\) 0 0
\(561\) 7.36266 0.310852
\(562\) 0 0
\(563\) −32.9341 −1.38801 −0.694003 0.719972i \(-0.744155\pi\)
−0.694003 + 0.719972i \(0.744155\pi\)
\(564\) 0 0
\(565\) 0.368611 0.0155076
\(566\) 0 0
\(567\) −3.87086 −0.162561
\(568\) 0 0
\(569\) 9.91202 0.415533 0.207767 0.978178i \(-0.433381\pi\)
0.207767 + 0.978178i \(0.433381\pi\)
\(570\) 0 0
\(571\) −32.0880 −1.34284 −0.671420 0.741077i \(-0.734315\pi\)
−0.671420 + 0.741077i \(0.734315\pi\)
\(572\) 0 0
\(573\) −4.54102 −0.189704
\(574\) 0 0
\(575\) 11.8932 0.495982
\(576\) 0 0
\(577\) −31.1924 −1.29856 −0.649278 0.760551i \(-0.724929\pi\)
−0.649278 + 0.760551i \(0.724929\pi\)
\(578\) 0 0
\(579\) −20.4671 −0.850582
\(580\) 0 0
\(581\) −30.9669 −1.28472
\(582\) 0 0
\(583\) −12.2499 −0.507340
\(584\) 0 0
\(585\) 0.258271 0.0106782
\(586\) 0 0
\(587\) −22.4671 −0.927314 −0.463657 0.886015i \(-0.653463\pi\)
−0.463657 + 0.886015i \(0.653463\pi\)
\(588\) 0 0
\(589\) −57.5163 −2.36992
\(590\) 0 0
\(591\) −8.63734 −0.355293
\(592\) 0 0
\(593\) 10.6045 0.435476 0.217738 0.976007i \(-0.430132\pi\)
0.217738 + 0.976007i \(0.430132\pi\)
\(594\) 0 0
\(595\) 14.4837 0.593775
\(596\) 0 0
\(597\) 7.74173 0.316848
\(598\) 0 0
\(599\) −10.5082 −0.429354 −0.214677 0.976685i \(-0.568870\pi\)
−0.214677 + 0.976685i \(0.568870\pi\)
\(600\) 0 0
\(601\) 27.5163 1.12241 0.561206 0.827676i \(-0.310338\pi\)
0.561206 + 0.827676i \(0.310338\pi\)
\(602\) 0 0
\(603\) 12.7581 0.519551
\(604\) 0 0
\(605\) −0.508203 −0.0206614
\(606\) 0 0
\(607\) 46.0796 1.87032 0.935158 0.354232i \(-0.115258\pi\)
0.935158 + 0.354232i \(0.115258\pi\)
\(608\) 0 0
\(609\) −28.4999 −1.15487
\(610\) 0 0
\(611\) 2.79095 0.112910
\(612\) 0 0
\(613\) 31.2335 1.26151 0.630755 0.775982i \(-0.282745\pi\)
0.630755 + 0.775982i \(0.282745\pi\)
\(614\) 0 0
\(615\) 4.25827 0.171710
\(616\) 0 0
\(617\) 49.6594 1.99921 0.999607 0.0280467i \(-0.00892871\pi\)
0.999607 + 0.0280467i \(0.00892871\pi\)
\(618\) 0 0
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) 2.50820 0.100651
\(622\) 0 0
\(623\) −7.74173 −0.310166
\(624\) 0 0
\(625\) 21.1926 0.847706
\(626\) 0 0
\(627\) −5.36266 −0.214164
\(628\) 0 0
\(629\) −64.2416 −2.56148
\(630\) 0 0
\(631\) 32.1760 1.28091 0.640453 0.767998i \(-0.278747\pi\)
0.640453 + 0.767998i \(0.278747\pi\)
\(632\) 0 0
\(633\) −19.0716 −0.758027
\(634\) 0 0
\(635\) −0.582168 −0.0231026
\(636\) 0 0
\(637\) −4.05729 −0.160756
\(638\) 0 0
\(639\) −10.5082 −0.415698
\(640\) 0 0
\(641\) 24.5327 0.968983 0.484491 0.874796i \(-0.339005\pi\)
0.484491 + 0.874796i \(0.339005\pi\)
\(642\) 0 0
\(643\) 19.4178 0.765765 0.382882 0.923797i \(-0.374931\pi\)
0.382882 + 0.923797i \(0.374931\pi\)
\(644\) 0 0
\(645\) 1.20905 0.0476063
\(646\) 0 0
\(647\) 4.97526 0.195597 0.0977987 0.995206i \(-0.468820\pi\)
0.0977987 + 0.995206i \(0.468820\pi\)
\(648\) 0 0
\(649\) −1.01641 −0.0398975
\(650\) 0 0
\(651\) 41.5163 1.62715
\(652\) 0 0
\(653\) −32.2499 −1.26204 −0.631019 0.775768i \(-0.717363\pi\)
−0.631019 + 0.775768i \(0.717363\pi\)
\(654\) 0 0
\(655\) 5.45065 0.212974
\(656\) 0 0
\(657\) 14.7581 0.575770
\(658\) 0 0
\(659\) 11.4835 0.447332 0.223666 0.974666i \(-0.428198\pi\)
0.223666 + 0.974666i \(0.428198\pi\)
\(660\) 0 0
\(661\) 45.9833 1.78854 0.894272 0.447524i \(-0.147694\pi\)
0.894272 + 0.447524i \(0.147694\pi\)
\(662\) 0 0
\(663\) 3.74173 0.145317
\(664\) 0 0
\(665\) −10.5494 −0.409086
\(666\) 0 0
\(667\) 18.4671 0.715047
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −7.23353 −0.279247
\(672\) 0 0
\(673\) 24.4014 0.940606 0.470303 0.882505i \(-0.344145\pi\)
0.470303 + 0.882505i \(0.344145\pi\)
\(674\) 0 0
\(675\) 4.74173 0.182509
\(676\) 0 0
\(677\) 9.15388 0.351812 0.175906 0.984407i \(-0.443714\pi\)
0.175906 + 0.984407i \(0.443714\pi\)
\(678\) 0 0
\(679\) −11.6761 −0.448088
\(680\) 0 0
\(681\) −22.2088 −0.851042
\(682\) 0 0
\(683\) 26.4014 1.01022 0.505111 0.863054i \(-0.331451\pi\)
0.505111 + 0.863054i \(0.331451\pi\)
\(684\) 0 0
\(685\) 5.95051 0.227357
\(686\) 0 0
\(687\) −15.7089 −0.599333
\(688\) 0 0
\(689\) −6.22546 −0.237171
\(690\) 0 0
\(691\) 25.5163 0.970685 0.485342 0.874324i \(-0.338695\pi\)
0.485342 + 0.874324i \(0.338695\pi\)
\(692\) 0 0
\(693\) 3.87086 0.147042
\(694\) 0 0
\(695\) 3.75841 0.142564
\(696\) 0 0
\(697\) 61.6922 2.33676
\(698\) 0 0
\(699\) 23.1044 0.873888
\(700\) 0 0
\(701\) −7.10439 −0.268329 −0.134165 0.990959i \(-0.542835\pi\)
−0.134165 + 0.990959i \(0.542835\pi\)
\(702\) 0 0
\(703\) 46.7909 1.76475
\(704\) 0 0
\(705\) −2.79095 −0.105113
\(706\) 0 0
\(707\) −2.46705 −0.0927830
\(708\) 0 0
\(709\) 3.77454 0.141756 0.0708780 0.997485i \(-0.477420\pi\)
0.0708780 + 0.997485i \(0.477420\pi\)
\(710\) 0 0
\(711\) 1.14554 0.0429612
\(712\) 0 0
\(713\) −26.9013 −1.00746
\(714\) 0 0
\(715\) −0.258271 −0.00965878
\(716\) 0 0
\(717\) 15.7417 0.587886
\(718\) 0 0
\(719\) 18.9424 0.706434 0.353217 0.935541i \(-0.385088\pi\)
0.353217 + 0.935541i \(0.385088\pi\)
\(720\) 0 0
\(721\) −50.3847 −1.87643
\(722\) 0 0
\(723\) 5.74173 0.213537
\(724\) 0 0
\(725\) 34.9117 1.29659
\(726\) 0 0
\(727\) −16.2583 −0.602986 −0.301493 0.953468i \(-0.597485\pi\)
−0.301493 + 0.953468i \(0.597485\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 17.5163 0.647863
\(732\) 0 0
\(733\) 15.4918 0.572203 0.286101 0.958199i \(-0.407641\pi\)
0.286101 + 0.958199i \(0.407641\pi\)
\(734\) 0 0
\(735\) 4.05729 0.149655
\(736\) 0 0
\(737\) −12.7581 −0.469952
\(738\) 0 0
\(739\) 2.89561 0.106517 0.0532584 0.998581i \(-0.483039\pi\)
0.0532584 + 0.998581i \(0.483039\pi\)
\(740\) 0 0
\(741\) −2.72532 −0.100117
\(742\) 0 0
\(743\) 5.53295 0.202984 0.101492 0.994836i \(-0.467638\pi\)
0.101492 + 0.994836i \(0.467638\pi\)
\(744\) 0 0
\(745\) −5.29135 −0.193860
\(746\) 0 0
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) −26.0328 −0.951218
\(750\) 0 0
\(751\) −12.2416 −0.446702 −0.223351 0.974738i \(-0.571700\pi\)
−0.223351 + 0.974738i \(0.571700\pi\)
\(752\) 0 0
\(753\) 30.4671 1.11028
\(754\) 0 0
\(755\) −8.45091 −0.307560
\(756\) 0 0
\(757\) 24.9669 0.907438 0.453719 0.891145i \(-0.350097\pi\)
0.453719 + 0.891145i \(0.350097\pi\)
\(758\) 0 0
\(759\) −2.50820 −0.0910420
\(760\) 0 0
\(761\) −31.6043 −1.14565 −0.572827 0.819677i \(-0.694153\pi\)
−0.572827 + 0.819677i \(0.694153\pi\)
\(762\) 0 0
\(763\) 1.96719 0.0712169
\(764\) 0 0
\(765\) −3.74173 −0.135283
\(766\) 0 0
\(767\) −0.516541 −0.0186512
\(768\) 0 0
\(769\) −5.04922 −0.182080 −0.0910398 0.995847i \(-0.529019\pi\)
−0.0910398 + 0.995847i \(0.529019\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 0 0
\(773\) 38.2827 1.37693 0.688467 0.725267i \(-0.258284\pi\)
0.688467 + 0.725267i \(0.258284\pi\)
\(774\) 0 0
\(775\) −50.8566 −1.82682
\(776\) 0 0
\(777\) −33.7745 −1.21165
\(778\) 0 0
\(779\) −44.9341 −1.60993
\(780\) 0 0
\(781\) 10.5082 0.376013
\(782\) 0 0
\(783\) 7.36266 0.263120
\(784\) 0 0
\(785\) −3.56576 −0.127267
\(786\) 0 0
\(787\) −28.5878 −1.01905 −0.509523 0.860457i \(-0.670178\pi\)
−0.509523 + 0.860457i \(0.670178\pi\)
\(788\) 0 0
\(789\) −26.7253 −0.951447
\(790\) 0 0
\(791\) 2.80763 0.0998277
\(792\) 0 0
\(793\) −3.67610 −0.130542
\(794\) 0 0
\(795\) 6.22546 0.220794
\(796\) 0 0
\(797\) 7.49180 0.265373 0.132687 0.991158i \(-0.457640\pi\)
0.132687 + 0.991158i \(0.457640\pi\)
\(798\) 0 0
\(799\) −40.4342 −1.43046
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) 0 0
\(803\) −14.7581 −0.520803
\(804\) 0 0
\(805\) −4.93410 −0.173904
\(806\) 0 0
\(807\) 27.2335 0.958666
\(808\) 0 0
\(809\) 5.65375 0.198775 0.0993876 0.995049i \(-0.468312\pi\)
0.0993876 + 0.995049i \(0.468312\pi\)
\(810\) 0 0
\(811\) −24.3463 −0.854913 −0.427456 0.904036i \(-0.640590\pi\)
−0.427456 + 0.904036i \(0.640590\pi\)
\(812\) 0 0
\(813\) −12.8216 −0.449675
\(814\) 0 0
\(815\) 4.45091 0.155909
\(816\) 0 0
\(817\) −12.7581 −0.446351
\(818\) 0 0
\(819\) 1.96719 0.0687391
\(820\) 0 0
\(821\) −6.84612 −0.238931 −0.119466 0.992838i \(-0.538118\pi\)
−0.119466 + 0.992838i \(0.538118\pi\)
\(822\) 0 0
\(823\) −46.9669 −1.63716 −0.818582 0.574390i \(-0.805239\pi\)
−0.818582 + 0.574390i \(0.805239\pi\)
\(824\) 0 0
\(825\) −4.74173 −0.165086
\(826\) 0 0
\(827\) 35.5491 1.23616 0.618081 0.786114i \(-0.287910\pi\)
0.618081 + 0.786114i \(0.287910\pi\)
\(828\) 0 0
\(829\) −28.1432 −0.977452 −0.488726 0.872437i \(-0.662538\pi\)
−0.488726 + 0.872437i \(0.662538\pi\)
\(830\) 0 0
\(831\) −12.5082 −0.433905
\(832\) 0 0
\(833\) 58.7805 2.03662
\(834\) 0 0
\(835\) −2.90129 −0.100403
\(836\) 0 0
\(837\) −10.7253 −0.370721
\(838\) 0 0
\(839\) −46.3603 −1.60053 −0.800267 0.599644i \(-0.795309\pi\)
−0.800267 + 0.599644i \(0.795309\pi\)
\(840\) 0 0
\(841\) 25.2088 0.869268
\(842\) 0 0
\(843\) 13.3955 0.461365
\(844\) 0 0
\(845\) 6.47539 0.222760
\(846\) 0 0
\(847\) −3.87086 −0.133004
\(848\) 0 0
\(849\) 31.1372 1.06863
\(850\) 0 0
\(851\) 21.8849 0.750204
\(852\) 0 0
\(853\) −21.7006 −0.743014 −0.371507 0.928430i \(-0.621159\pi\)
−0.371507 + 0.928430i \(0.621159\pi\)
\(854\) 0 0
\(855\) 2.72532 0.0932041
\(856\) 0 0
\(857\) −47.7969 −1.63271 −0.816355 0.577550i \(-0.804009\pi\)
−0.816355 + 0.577550i \(0.804009\pi\)
\(858\) 0 0
\(859\) 0.758136 0.0258673 0.0129336 0.999916i \(-0.495883\pi\)
0.0129336 + 0.999916i \(0.495883\pi\)
\(860\) 0 0
\(861\) 32.4342 1.10536
\(862\) 0 0
\(863\) 15.4423 0.525662 0.262831 0.964842i \(-0.415344\pi\)
0.262831 + 0.964842i \(0.415344\pi\)
\(864\) 0 0
\(865\) 3.74173 0.127223
\(866\) 0 0
\(867\) −37.2088 −1.26368
\(868\) 0 0
\(869\) −1.14554 −0.0388599
\(870\) 0 0
\(871\) −6.48373 −0.219693
\(872\) 0 0
\(873\) 3.01641 0.102090
\(874\) 0 0
\(875\) −19.1638 −0.647855
\(876\) 0 0
\(877\) 28.2499 0.953932 0.476966 0.878922i \(-0.341736\pi\)
0.476966 + 0.878922i \(0.341736\pi\)
\(878\) 0 0
\(879\) −21.8297 −0.736298
\(880\) 0 0
\(881\) 3.20905 0.108116 0.0540578 0.998538i \(-0.482784\pi\)
0.0540578 + 0.998538i \(0.482784\pi\)
\(882\) 0 0
\(883\) 56.1760 1.89047 0.945236 0.326388i \(-0.105832\pi\)
0.945236 + 0.326388i \(0.105832\pi\)
\(884\) 0 0
\(885\) 0.516541 0.0173633
\(886\) 0 0
\(887\) 51.9833 1.74543 0.872715 0.488231i \(-0.162358\pi\)
0.872715 + 0.488231i \(0.162358\pi\)
\(888\) 0 0
\(889\) −4.43424 −0.148720
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 29.4506 0.985528
\(894\) 0 0
\(895\) −3.54909 −0.118633
\(896\) 0 0
\(897\) −1.27468 −0.0425602
\(898\) 0 0
\(899\) −78.9669 −2.63369
\(900\) 0 0
\(901\) 90.1921 3.00473
\(902\) 0 0
\(903\) 9.20905 0.306458
\(904\) 0 0
\(905\) −2.53268 −0.0841891
\(906\) 0 0
\(907\) −48.8685 −1.62265 −0.811326 0.584595i \(-0.801254\pi\)
−0.811326 + 0.584595i \(0.801254\pi\)
\(908\) 0 0
\(909\) 0.637339 0.0211392
\(910\) 0 0
\(911\) −20.0245 −0.663440 −0.331720 0.943378i \(-0.607629\pi\)
−0.331720 + 0.943378i \(0.607629\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) 0 0
\(915\) 3.67610 0.121528
\(916\) 0 0
\(917\) 41.5163 1.37099
\(918\) 0 0
\(919\) 20.3051 0.669804 0.334902 0.942253i \(-0.391297\pi\)
0.334902 + 0.942253i \(0.391297\pi\)
\(920\) 0 0
\(921\) −14.0552 −0.463134
\(922\) 0 0
\(923\) 5.34030 0.175778
\(924\) 0 0
\(925\) 41.3731 1.36034
\(926\) 0 0
\(927\) 13.0164 0.427515
\(928\) 0 0
\(929\) −29.7417 −0.975794 −0.487897 0.872901i \(-0.662236\pi\)
−0.487897 + 0.872901i \(0.662236\pi\)
\(930\) 0 0
\(931\) −42.8133 −1.40315
\(932\) 0 0
\(933\) 10.5082 0.344023
\(934\) 0 0
\(935\) 3.74173 0.122368
\(936\) 0 0
\(937\) 11.5163 0.376220 0.188110 0.982148i \(-0.439764\pi\)
0.188110 + 0.982148i \(0.439764\pi\)
\(938\) 0 0
\(939\) −2.69251 −0.0878667
\(940\) 0 0
\(941\) 30.0880 0.980840 0.490420 0.871486i \(-0.336843\pi\)
0.490420 + 0.871486i \(0.336843\pi\)
\(942\) 0 0
\(943\) −21.0164 −0.684388
\(944\) 0 0
\(945\) −1.96719 −0.0639926
\(946\) 0 0
\(947\) 4.51654 0.146768 0.0733839 0.997304i \(-0.476620\pi\)
0.0733839 + 0.997304i \(0.476620\pi\)
\(948\) 0 0
\(949\) −7.50013 −0.243465
\(950\) 0 0
\(951\) −6.54102 −0.212107
\(952\) 0 0
\(953\) 31.1700 1.00970 0.504848 0.863208i \(-0.331549\pi\)
0.504848 + 0.863208i \(0.331549\pi\)
\(954\) 0 0
\(955\) −2.30776 −0.0746774
\(956\) 0 0
\(957\) −7.36266 −0.238001
\(958\) 0 0
\(959\) 45.3236 1.46358
\(960\) 0 0
\(961\) 84.0325 2.71073
\(962\) 0 0
\(963\) 6.72532 0.216720
\(964\) 0 0
\(965\) −10.4014 −0.334834
\(966\) 0 0
\(967\) 30.0796 0.967296 0.483648 0.875263i \(-0.339311\pi\)
0.483648 + 0.875263i \(0.339311\pi\)
\(968\) 0 0
\(969\) 39.4835 1.26839
\(970\) 0 0
\(971\) −48.4999 −1.55643 −0.778217 0.627995i \(-0.783876\pi\)
−0.778217 + 0.627995i \(0.783876\pi\)
\(972\) 0 0
\(973\) 28.6269 0.917736
\(974\) 0 0
\(975\) −2.40976 −0.0771742
\(976\) 0 0
\(977\) −53.2252 −1.70282 −0.851412 0.524497i \(-0.824253\pi\)
−0.851412 + 0.524497i \(0.824253\pi\)
\(978\) 0 0
\(979\) −2.00000 −0.0639203
\(980\) 0 0
\(981\) −0.508203 −0.0162257
\(982\) 0 0
\(983\) 31.0081 0.989004 0.494502 0.869177i \(-0.335350\pi\)
0.494502 + 0.869177i \(0.335350\pi\)
\(984\) 0 0
\(985\) −4.38952 −0.139862
\(986\) 0 0
\(987\) −21.2580 −0.676650
\(988\) 0 0
\(989\) −5.96719 −0.189745
\(990\) 0 0
\(991\) −36.2416 −1.15125 −0.575626 0.817713i \(-0.695242\pi\)
−0.575626 + 0.817713i \(0.695242\pi\)
\(992\) 0 0
\(993\) −30.7253 −0.975039
\(994\) 0 0
\(995\) 3.93437 0.124728
\(996\) 0 0
\(997\) 4.76647 0.150956 0.0754779 0.997147i \(-0.475952\pi\)
0.0754779 + 0.997147i \(0.475952\pi\)
\(998\) 0 0
\(999\) 8.72532 0.276057
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 2112.2.a.bg.1.2 3
3.2 odd 2 6336.2.a.cz.1.2 3
4.3 odd 2 2112.2.a.bh.1.2 3
8.3 odd 2 1056.2.a.m.1.2 3
8.5 even 2 1056.2.a.n.1.2 yes 3
12.11 even 2 6336.2.a.cy.1.2 3
24.5 odd 2 3168.2.a.bh.1.2 3
24.11 even 2 3168.2.a.bg.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1056.2.a.m.1.2 3 8.3 odd 2
1056.2.a.n.1.2 yes 3 8.5 even 2
2112.2.a.bg.1.2 3 1.1 even 1 trivial
2112.2.a.bh.1.2 3 4.3 odd 2
3168.2.a.bg.1.2 3 24.11 even 2
3168.2.a.bh.1.2 3 24.5 odd 2
6336.2.a.cy.1.2 3 12.11 even 2
6336.2.a.cz.1.2 3 3.2 odd 2