Properties

Label 6864.2.a.bz.1.2
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $1$
Dimension $4$
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] = [6864,2,Mod(1,6864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6864, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6864.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.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: \(54.8093159474\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.8468.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 3x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 429)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.89122\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.776183 q^{5} -2.69175 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.776183 q^{5} -2.69175 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.776183 q^{15} +2.62136 q^{17} -6.69175 q^{19} -2.69175 q^{21} +8.02655 q^{23} -4.39754 q^{25} +1.00000 q^{27} -8.09695 q^{29} +6.34106 q^{31} +1.00000 q^{33} +2.08929 q^{35} -0.398941 q^{37} +1.00000 q^{39} +6.87313 q^{41} -7.23786 q^{43} -0.776183 q^{45} +4.53692 q^{47} +0.245516 q^{49} +2.62136 q^{51} -2.75448 q^{53} -0.776183 q^{55} -6.69175 q^{57} -10.8104 q^{59} -5.02796 q^{61} -2.69175 q^{63} -0.776183 q^{65} +4.95756 q^{67} +8.02655 q^{69} +5.96382 q^{71} +3.79648 q^{73} -4.39754 q^{75} -2.69175 q^{77} -2.14564 q^{79} +1.00000 q^{81} -3.73375 q^{83} -2.03465 q^{85} -8.09695 q^{87} +2.37724 q^{89} -2.69175 q^{91} +6.34106 q^{93} +5.19402 q^{95} -17.8355 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 2 q^{7} + 4 q^{9} + 4 q^{11} + 4 q^{13} - 8 q^{17} - 18 q^{19} - 2 q^{21} + 4 q^{25} + 4 q^{27} - 10 q^{29} - 12 q^{31} + 4 q^{33} - 22 q^{35} - 2 q^{37} + 4 q^{39} + 2 q^{41} - 28 q^{43} - 6 q^{47} + 8 q^{49} - 8 q^{51} - 4 q^{53} - 18 q^{57} - 16 q^{59} - 10 q^{61} - 2 q^{63} - 10 q^{71} - 6 q^{73} + 4 q^{75} - 2 q^{77} + 8 q^{79} + 4 q^{81} + 8 q^{83} - 18 q^{85} - 10 q^{87} + 6 q^{89} - 2 q^{91} - 12 q^{93} - 22 q^{95} + 10 q^{97} + 4 q^{99}+O(q^{100}) \) 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
\(5\) −0.776183 −0.347120 −0.173560 0.984823i \(-0.555527\pi\)
−0.173560 + 0.984823i \(0.555527\pi\)
\(6\) 0 0
\(7\) −2.69175 −1.01739 −0.508693 0.860948i \(-0.669871\pi\)
−0.508693 + 0.860948i \(0.669871\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −0.776183 −0.200410
\(16\) 0 0
\(17\) 2.62136 0.635773 0.317886 0.948129i \(-0.397027\pi\)
0.317886 + 0.948129i \(0.397027\pi\)
\(18\) 0 0
\(19\) −6.69175 −1.53519 −0.767596 0.640934i \(-0.778547\pi\)
−0.767596 + 0.640934i \(0.778547\pi\)
\(20\) 0 0
\(21\) −2.69175 −0.587388
\(22\) 0 0
\(23\) 8.02655 1.67365 0.836826 0.547469i \(-0.184408\pi\)
0.836826 + 0.547469i \(0.184408\pi\)
\(24\) 0 0
\(25\) −4.39754 −0.879508
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.09695 −1.50357 −0.751783 0.659411i \(-0.770806\pi\)
−0.751783 + 0.659411i \(0.770806\pi\)
\(30\) 0 0
\(31\) 6.34106 1.13889 0.569444 0.822030i \(-0.307158\pi\)
0.569444 + 0.822030i \(0.307158\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 2.08929 0.353154
\(36\) 0 0
\(37\) −0.398941 −0.0655854 −0.0327927 0.999462i \(-0.510440\pi\)
−0.0327927 + 0.999462i \(0.510440\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 6.87313 1.07340 0.536701 0.843772i \(-0.319670\pi\)
0.536701 + 0.843772i \(0.319670\pi\)
\(42\) 0 0
\(43\) −7.23786 −1.10376 −0.551882 0.833923i \(-0.686090\pi\)
−0.551882 + 0.833923i \(0.686090\pi\)
\(44\) 0 0
\(45\) −0.776183 −0.115707
\(46\) 0 0
\(47\) 4.53692 0.661778 0.330889 0.943670i \(-0.392651\pi\)
0.330889 + 0.943670i \(0.392651\pi\)
\(48\) 0 0
\(49\) 0.245516 0.0350737
\(50\) 0 0
\(51\) 2.62136 0.367063
\(52\) 0 0
\(53\) −2.75448 −0.378358 −0.189179 0.981943i \(-0.560583\pi\)
−0.189179 + 0.981943i \(0.560583\pi\)
\(54\) 0 0
\(55\) −0.776183 −0.104660
\(56\) 0 0
\(57\) −6.69175 −0.886344
\(58\) 0 0
\(59\) −10.8104 −1.40739 −0.703697 0.710500i \(-0.748469\pi\)
−0.703697 + 0.710500i \(0.748469\pi\)
\(60\) 0 0
\(61\) −5.02796 −0.643764 −0.321882 0.946780i \(-0.604315\pi\)
−0.321882 + 0.946780i \(0.604315\pi\)
\(62\) 0 0
\(63\) −2.69175 −0.339129
\(64\) 0 0
\(65\) −0.776183 −0.0962736
\(66\) 0 0
\(67\) 4.95756 0.605663 0.302831 0.953044i \(-0.402068\pi\)
0.302831 + 0.953044i \(0.402068\pi\)
\(68\) 0 0
\(69\) 8.02655 0.966284
\(70\) 0 0
\(71\) 5.96382 0.707775 0.353888 0.935288i \(-0.384860\pi\)
0.353888 + 0.935288i \(0.384860\pi\)
\(72\) 0 0
\(73\) 3.79648 0.444344 0.222172 0.975007i \(-0.428685\pi\)
0.222172 + 0.975007i \(0.428685\pi\)
\(74\) 0 0
\(75\) −4.39754 −0.507784
\(76\) 0 0
\(77\) −2.69175 −0.306753
\(78\) 0 0
\(79\) −2.14564 −0.241403 −0.120702 0.992689i \(-0.538514\pi\)
−0.120702 + 0.992689i \(0.538514\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.73375 −0.409832 −0.204916 0.978780i \(-0.565692\pi\)
−0.204916 + 0.978780i \(0.565692\pi\)
\(84\) 0 0
\(85\) −2.03465 −0.220689
\(86\) 0 0
\(87\) −8.09695 −0.868084
\(88\) 0 0
\(89\) 2.37724 0.251987 0.125994 0.992031i \(-0.459788\pi\)
0.125994 + 0.992031i \(0.459788\pi\)
\(90\) 0 0
\(91\) −2.69175 −0.282172
\(92\) 0 0
\(93\) 6.34106 0.657538
\(94\) 0 0
\(95\) 5.19402 0.532895
\(96\) 0 0
\(97\) −17.8355 −1.81093 −0.905463 0.424426i \(-0.860476\pi\)
−0.905463 + 0.424426i \(0.860476\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −11.1096 −1.10545 −0.552723 0.833365i \(-0.686411\pi\)
−0.552723 + 0.833365i \(0.686411\pi\)
\(102\) 0 0
\(103\) −10.1814 −1.00320 −0.501601 0.865099i \(-0.667255\pi\)
−0.501601 + 0.865099i \(0.667255\pi\)
\(104\) 0 0
\(105\) 2.08929 0.203894
\(106\) 0 0
\(107\) −3.06133 −0.295950 −0.147975 0.988991i \(-0.547276\pi\)
−0.147975 + 0.988991i \(0.547276\pi\)
\(108\) 0 0
\(109\) 13.1800 1.26241 0.631207 0.775615i \(-0.282560\pi\)
0.631207 + 0.775615i \(0.282560\pi\)
\(110\) 0 0
\(111\) −0.398941 −0.0378658
\(112\) 0 0
\(113\) 4.50074 0.423394 0.211697 0.977335i \(-0.432101\pi\)
0.211697 + 0.977335i \(0.432101\pi\)
\(114\) 0 0
\(115\) −6.23007 −0.580957
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −7.05604 −0.646826
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.87313 0.619729
\(124\) 0 0
\(125\) 7.29421 0.652414
\(126\) 0 0
\(127\) −19.9687 −1.77193 −0.885967 0.463749i \(-0.846504\pi\)
−0.885967 + 0.463749i \(0.846504\pi\)
\(128\) 0 0
\(129\) −7.23786 −0.637258
\(130\) 0 0
\(131\) −4.92335 −0.430155 −0.215078 0.976597i \(-0.569000\pi\)
−0.215078 + 0.976597i \(0.569000\pi\)
\(132\) 0 0
\(133\) 18.0125 1.56188
\(134\) 0 0
\(135\) −0.776183 −0.0668032
\(136\) 0 0
\(137\) −13.1876 −1.12670 −0.563348 0.826220i \(-0.690487\pi\)
−0.563348 + 0.826220i \(0.690487\pi\)
\(138\) 0 0
\(139\) −11.0203 −0.934729 −0.467365 0.884065i \(-0.654797\pi\)
−0.467365 + 0.884065i \(0.654797\pi\)
\(140\) 0 0
\(141\) 4.53692 0.382078
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) 6.28471 0.521917
\(146\) 0 0
\(147\) 0.245516 0.0202498
\(148\) 0 0
\(149\) 8.25663 0.676409 0.338205 0.941073i \(-0.390180\pi\)
0.338205 + 0.941073i \(0.390180\pi\)
\(150\) 0 0
\(151\) 3.61510 0.294193 0.147096 0.989122i \(-0.453007\pi\)
0.147096 + 0.989122i \(0.453007\pi\)
\(152\) 0 0
\(153\) 2.62136 0.211924
\(154\) 0 0
\(155\) −4.92182 −0.395330
\(156\) 0 0
\(157\) 20.0672 1.60153 0.800766 0.598977i \(-0.204426\pi\)
0.800766 + 0.598977i \(0.204426\pi\)
\(158\) 0 0
\(159\) −2.75448 −0.218445
\(160\) 0 0
\(161\) −21.6055 −1.70275
\(162\) 0 0
\(163\) −17.9827 −1.40852 −0.704258 0.709945i \(-0.748720\pi\)
−0.704258 + 0.709945i \(0.748720\pi\)
\(164\) 0 0
\(165\) −0.776183 −0.0604258
\(166\) 0 0
\(167\) −9.74626 −0.754188 −0.377094 0.926175i \(-0.623077\pi\)
−0.377094 + 0.926175i \(0.623077\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −6.69175 −0.511731
\(172\) 0 0
\(173\) −14.9841 −1.13922 −0.569611 0.821915i \(-0.692906\pi\)
−0.569611 + 0.821915i \(0.692906\pi\)
\(174\) 0 0
\(175\) 11.8371 0.894799
\(176\) 0 0
\(177\) −10.8104 −0.812559
\(178\) 0 0
\(179\) 14.5273 1.08582 0.542911 0.839790i \(-0.317322\pi\)
0.542911 + 0.839790i \(0.317322\pi\)
\(180\) 0 0
\(181\) 0.909310 0.0675885 0.0337942 0.999429i \(-0.489241\pi\)
0.0337942 + 0.999429i \(0.489241\pi\)
\(182\) 0 0
\(183\) −5.02796 −0.371677
\(184\) 0 0
\(185\) 0.309651 0.0227660
\(186\) 0 0
\(187\) 2.62136 0.191693
\(188\) 0 0
\(189\) −2.69175 −0.195796
\(190\) 0 0
\(191\) −12.4448 −0.900477 −0.450238 0.892908i \(-0.648661\pi\)
−0.450238 + 0.892908i \(0.648661\pi\)
\(192\) 0 0
\(193\) 18.3406 1.32019 0.660093 0.751184i \(-0.270517\pi\)
0.660093 + 0.751184i \(0.270517\pi\)
\(194\) 0 0
\(195\) −0.776183 −0.0555836
\(196\) 0 0
\(197\) −7.96535 −0.567508 −0.283754 0.958897i \(-0.591580\pi\)
−0.283754 + 0.958897i \(0.591580\pi\)
\(198\) 0 0
\(199\) 14.6415 1.03791 0.518955 0.854801i \(-0.326321\pi\)
0.518955 + 0.854801i \(0.326321\pi\)
\(200\) 0 0
\(201\) 4.95756 0.349680
\(202\) 0 0
\(203\) 21.7950 1.52971
\(204\) 0 0
\(205\) −5.33481 −0.372599
\(206\) 0 0
\(207\) 8.02655 0.557884
\(208\) 0 0
\(209\) −6.69175 −0.462878
\(210\) 0 0
\(211\) −14.9716 −1.03069 −0.515344 0.856983i \(-0.672336\pi\)
−0.515344 + 0.856983i \(0.672336\pi\)
\(212\) 0 0
\(213\) 5.96382 0.408634
\(214\) 0 0
\(215\) 5.61790 0.383138
\(216\) 0 0
\(217\) −17.0686 −1.15869
\(218\) 0 0
\(219\) 3.79648 0.256542
\(220\) 0 0
\(221\) 2.62136 0.176332
\(222\) 0 0
\(223\) 26.2003 1.75450 0.877250 0.480033i \(-0.159375\pi\)
0.877250 + 0.480033i \(0.159375\pi\)
\(224\) 0 0
\(225\) −4.39754 −0.293169
\(226\) 0 0
\(227\) −24.2441 −1.60914 −0.804569 0.593859i \(-0.797604\pi\)
−0.804569 + 0.593859i \(0.797604\pi\)
\(228\) 0 0
\(229\) −6.43954 −0.425537 −0.212768 0.977103i \(-0.568248\pi\)
−0.212768 + 0.977103i \(0.568248\pi\)
\(230\) 0 0
\(231\) −2.69175 −0.177104
\(232\) 0 0
\(233\) −24.0536 −1.57580 −0.787900 0.615803i \(-0.788832\pi\)
−0.787900 + 0.615803i \(0.788832\pi\)
\(234\) 0 0
\(235\) −3.52148 −0.229716
\(236\) 0 0
\(237\) −2.14564 −0.139374
\(238\) 0 0
\(239\) −5.01391 −0.324323 −0.162162 0.986764i \(-0.551847\pi\)
−0.162162 + 0.986764i \(0.551847\pi\)
\(240\) 0 0
\(241\) −24.9890 −1.60968 −0.804841 0.593491i \(-0.797749\pi\)
−0.804841 + 0.593491i \(0.797749\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.190565 −0.0121748
\(246\) 0 0
\(247\) −6.69175 −0.425786
\(248\) 0 0
\(249\) −3.73375 −0.236617
\(250\) 0 0
\(251\) −18.0406 −1.13871 −0.569356 0.822091i \(-0.692807\pi\)
−0.569356 + 0.822091i \(0.692807\pi\)
\(252\) 0 0
\(253\) 8.02655 0.504625
\(254\) 0 0
\(255\) −2.03465 −0.127415
\(256\) 0 0
\(257\) −21.1298 −1.31804 −0.659019 0.752126i \(-0.729028\pi\)
−0.659019 + 0.752126i \(0.729028\pi\)
\(258\) 0 0
\(259\) 1.07385 0.0667257
\(260\) 0 0
\(261\) −8.09695 −0.501188
\(262\) 0 0
\(263\) −11.8592 −0.731271 −0.365635 0.930758i \(-0.619148\pi\)
−0.365635 + 0.930758i \(0.619148\pi\)
\(264\) 0 0
\(265\) 2.13798 0.131335
\(266\) 0 0
\(267\) 2.37724 0.145485
\(268\) 0 0
\(269\) −0.322165 −0.0196427 −0.00982136 0.999952i \(-0.503126\pi\)
−0.00982136 + 0.999952i \(0.503126\pi\)
\(270\) 0 0
\(271\) −26.2944 −1.59727 −0.798636 0.601814i \(-0.794445\pi\)
−0.798636 + 0.601814i \(0.794445\pi\)
\(272\) 0 0
\(273\) −2.69175 −0.162912
\(274\) 0 0
\(275\) −4.39754 −0.265182
\(276\) 0 0
\(277\) −8.89247 −0.534297 −0.267148 0.963655i \(-0.586081\pi\)
−0.267148 + 0.963655i \(0.586081\pi\)
\(278\) 0 0
\(279\) 6.34106 0.379629
\(280\) 0 0
\(281\) 23.5833 1.40686 0.703432 0.710763i \(-0.251650\pi\)
0.703432 + 0.710763i \(0.251650\pi\)
\(282\) 0 0
\(283\) −8.13754 −0.483727 −0.241863 0.970310i \(-0.577759\pi\)
−0.241863 + 0.970310i \(0.577759\pi\)
\(284\) 0 0
\(285\) 5.19402 0.307667
\(286\) 0 0
\(287\) −18.5007 −1.09206
\(288\) 0 0
\(289\) −10.1285 −0.595793
\(290\) 0 0
\(291\) −17.8355 −1.04554
\(292\) 0 0
\(293\) 19.8649 1.16052 0.580260 0.814431i \(-0.302951\pi\)
0.580260 + 0.814431i \(0.302951\pi\)
\(294\) 0 0
\(295\) 8.39084 0.488534
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 8.02655 0.464188
\(300\) 0 0
\(301\) 19.4825 1.12295
\(302\) 0 0
\(303\) −11.1096 −0.638229
\(304\) 0 0
\(305\) 3.90261 0.223463
\(306\) 0 0
\(307\) 23.3305 1.33154 0.665770 0.746157i \(-0.268103\pi\)
0.665770 + 0.746157i \(0.268103\pi\)
\(308\) 0 0
\(309\) −10.1814 −0.579198
\(310\) 0 0
\(311\) 4.43359 0.251406 0.125703 0.992068i \(-0.459881\pi\)
0.125703 + 0.992068i \(0.459881\pi\)
\(312\) 0 0
\(313\) −17.4129 −0.984233 −0.492116 0.870529i \(-0.663777\pi\)
−0.492116 + 0.870529i \(0.663777\pi\)
\(314\) 0 0
\(315\) 2.08929 0.117718
\(316\) 0 0
\(317\) −30.8752 −1.73412 −0.867062 0.498201i \(-0.833994\pi\)
−0.867062 + 0.498201i \(0.833994\pi\)
\(318\) 0 0
\(319\) −8.09695 −0.453342
\(320\) 0 0
\(321\) −3.06133 −0.170867
\(322\) 0 0
\(323\) −17.5415 −0.976033
\(324\) 0 0
\(325\) −4.39754 −0.243932
\(326\) 0 0
\(327\) 13.1800 0.728855
\(328\) 0 0
\(329\) −12.2123 −0.673284
\(330\) 0 0
\(331\) −6.72749 −0.369776 −0.184888 0.982760i \(-0.559192\pi\)
−0.184888 + 0.982760i \(0.559192\pi\)
\(332\) 0 0
\(333\) −0.398941 −0.0218618
\(334\) 0 0
\(335\) −3.84798 −0.210237
\(336\) 0 0
\(337\) −5.06133 −0.275708 −0.137854 0.990453i \(-0.544021\pi\)
−0.137854 + 0.990453i \(0.544021\pi\)
\(338\) 0 0
\(339\) 4.50074 0.244447
\(340\) 0 0
\(341\) 6.34106 0.343388
\(342\) 0 0
\(343\) 18.1814 0.981702
\(344\) 0 0
\(345\) −6.23007 −0.335416
\(346\) 0 0
\(347\) 4.34303 0.233146 0.116573 0.993182i \(-0.462809\pi\)
0.116573 + 0.993182i \(0.462809\pi\)
\(348\) 0 0
\(349\) 20.2779 1.08545 0.542725 0.839910i \(-0.317393\pi\)
0.542725 + 0.839910i \(0.317393\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −36.5630 −1.94605 −0.973027 0.230691i \(-0.925901\pi\)
−0.973027 + 0.230691i \(0.925901\pi\)
\(354\) 0 0
\(355\) −4.62901 −0.245683
\(356\) 0 0
\(357\) −7.05604 −0.373445
\(358\) 0 0
\(359\) 27.8621 1.47051 0.735253 0.677793i \(-0.237063\pi\)
0.735253 + 0.677793i \(0.237063\pi\)
\(360\) 0 0
\(361\) 25.7795 1.35682
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −2.94676 −0.154241
\(366\) 0 0
\(367\) −10.4476 −0.545362 −0.272681 0.962104i \(-0.587910\pi\)
−0.272681 + 0.962104i \(0.587910\pi\)
\(368\) 0 0
\(369\) 6.87313 0.357801
\(370\) 0 0
\(371\) 7.41438 0.384936
\(372\) 0 0
\(373\) −17.8842 −0.926011 −0.463006 0.886355i \(-0.653229\pi\)
−0.463006 + 0.886355i \(0.653229\pi\)
\(374\) 0 0
\(375\) 7.29421 0.376671
\(376\) 0 0
\(377\) −8.09695 −0.417014
\(378\) 0 0
\(379\) 13.6200 0.699610 0.349805 0.936822i \(-0.386248\pi\)
0.349805 + 0.936822i \(0.386248\pi\)
\(380\) 0 0
\(381\) −19.9687 −1.02303
\(382\) 0 0
\(383\) −4.62901 −0.236532 −0.118266 0.992982i \(-0.537733\pi\)
−0.118266 + 0.992982i \(0.537733\pi\)
\(384\) 0 0
\(385\) 2.08929 0.106480
\(386\) 0 0
\(387\) −7.23786 −0.367921
\(388\) 0 0
\(389\) 5.34290 0.270896 0.135448 0.990784i \(-0.456753\pi\)
0.135448 + 0.990784i \(0.456753\pi\)
\(390\) 0 0
\(391\) 21.0405 1.06406
\(392\) 0 0
\(393\) −4.92335 −0.248350
\(394\) 0 0
\(395\) 1.66541 0.0837958
\(396\) 0 0
\(397\) 28.0700 1.40879 0.704397 0.709806i \(-0.251217\pi\)
0.704397 + 0.709806i \(0.251217\pi\)
\(398\) 0 0
\(399\) 18.0125 0.901754
\(400\) 0 0
\(401\) 32.3942 1.61769 0.808844 0.588023i \(-0.200094\pi\)
0.808844 + 0.588023i \(0.200094\pi\)
\(402\) 0 0
\(403\) 6.34106 0.315871
\(404\) 0 0
\(405\) −0.776183 −0.0385688
\(406\) 0 0
\(407\) −0.398941 −0.0197748
\(408\) 0 0
\(409\) −22.4380 −1.10949 −0.554744 0.832021i \(-0.687184\pi\)
−0.554744 + 0.832021i \(0.687184\pi\)
\(410\) 0 0
\(411\) −13.1876 −0.650498
\(412\) 0 0
\(413\) 29.0989 1.43186
\(414\) 0 0
\(415\) 2.89807 0.142261
\(416\) 0 0
\(417\) −11.0203 −0.539666
\(418\) 0 0
\(419\) −40.6113 −1.98399 −0.991996 0.126272i \(-0.959699\pi\)
−0.991996 + 0.126272i \(0.959699\pi\)
\(420\) 0 0
\(421\) −2.67241 −0.130245 −0.0651227 0.997877i \(-0.520744\pi\)
−0.0651227 + 0.997877i \(0.520744\pi\)
\(422\) 0 0
\(423\) 4.53692 0.220593
\(424\) 0 0
\(425\) −11.5275 −0.559167
\(426\) 0 0
\(427\) 13.5340 0.654956
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 25.9972 1.25224 0.626121 0.779726i \(-0.284642\pi\)
0.626121 + 0.779726i \(0.284642\pi\)
\(432\) 0 0
\(433\) 18.9765 0.911951 0.455975 0.889992i \(-0.349291\pi\)
0.455975 + 0.889992i \(0.349291\pi\)
\(434\) 0 0
\(435\) 6.28471 0.301329
\(436\) 0 0
\(437\) −53.7117 −2.56938
\(438\) 0 0
\(439\) −16.8586 −0.804619 −0.402310 0.915504i \(-0.631792\pi\)
−0.402310 + 0.915504i \(0.631792\pi\)
\(440\) 0 0
\(441\) 0.245516 0.0116912
\(442\) 0 0
\(443\) −34.5105 −1.63964 −0.819821 0.572620i \(-0.805927\pi\)
−0.819821 + 0.572620i \(0.805927\pi\)
\(444\) 0 0
\(445\) −1.84517 −0.0874697
\(446\) 0 0
\(447\) 8.25663 0.390525
\(448\) 0 0
\(449\) 5.12363 0.241799 0.120899 0.992665i \(-0.461422\pi\)
0.120899 + 0.992665i \(0.461422\pi\)
\(450\) 0 0
\(451\) 6.87313 0.323643
\(452\) 0 0
\(453\) 3.61510 0.169852
\(454\) 0 0
\(455\) 2.08929 0.0979474
\(456\) 0 0
\(457\) −19.9248 −0.932045 −0.466022 0.884773i \(-0.654313\pi\)
−0.466022 + 0.884773i \(0.654313\pi\)
\(458\) 0 0
\(459\) 2.62136 0.122354
\(460\) 0 0
\(461\) −10.4661 −0.487454 −0.243727 0.969844i \(-0.578370\pi\)
−0.243727 + 0.969844i \(0.578370\pi\)
\(462\) 0 0
\(463\) 36.5558 1.69889 0.849447 0.527675i \(-0.176936\pi\)
0.849447 + 0.527675i \(0.176936\pi\)
\(464\) 0 0
\(465\) −4.92182 −0.228244
\(466\) 0 0
\(467\) 2.43079 0.112484 0.0562418 0.998417i \(-0.482088\pi\)
0.0562418 + 0.998417i \(0.482088\pi\)
\(468\) 0 0
\(469\) −13.3445 −0.616193
\(470\) 0 0
\(471\) 20.0672 0.924646
\(472\) 0 0
\(473\) −7.23786 −0.332797
\(474\) 0 0
\(475\) 29.4272 1.35021
\(476\) 0 0
\(477\) −2.75448 −0.126119
\(478\) 0 0
\(479\) 1.96535 0.0897990 0.0448995 0.998992i \(-0.485703\pi\)
0.0448995 + 0.998992i \(0.485703\pi\)
\(480\) 0 0
\(481\) −0.398941 −0.0181901
\(482\) 0 0
\(483\) −21.6055 −0.983083
\(484\) 0 0
\(485\) 13.8436 0.628608
\(486\) 0 0
\(487\) −22.4863 −1.01895 −0.509475 0.860486i \(-0.670160\pi\)
−0.509475 + 0.860486i \(0.670160\pi\)
\(488\) 0 0
\(489\) −17.9827 −0.813207
\(490\) 0 0
\(491\) −17.2914 −0.780350 −0.390175 0.920741i \(-0.627586\pi\)
−0.390175 + 0.920741i \(0.627586\pi\)
\(492\) 0 0
\(493\) −21.2250 −0.955926
\(494\) 0 0
\(495\) −0.776183 −0.0348868
\(496\) 0 0
\(497\) −16.0531 −0.720080
\(498\) 0 0
\(499\) −3.71485 −0.166299 −0.0831497 0.996537i \(-0.526498\pi\)
−0.0831497 + 0.996537i \(0.526498\pi\)
\(500\) 0 0
\(501\) −9.74626 −0.435431
\(502\) 0 0
\(503\) 7.68064 0.342463 0.171231 0.985231i \(-0.445225\pi\)
0.171231 + 0.985231i \(0.445225\pi\)
\(504\) 0 0
\(505\) 8.62307 0.383722
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 29.0035 1.28556 0.642778 0.766053i \(-0.277782\pi\)
0.642778 + 0.766053i \(0.277782\pi\)
\(510\) 0 0
\(511\) −10.2192 −0.452070
\(512\) 0 0
\(513\) −6.69175 −0.295448
\(514\) 0 0
\(515\) 7.90261 0.348231
\(516\) 0 0
\(517\) 4.53692 0.199534
\(518\) 0 0
\(519\) −14.9841 −0.657730
\(520\) 0 0
\(521\) 32.5973 1.42811 0.714056 0.700089i \(-0.246856\pi\)
0.714056 + 0.700089i \(0.246856\pi\)
\(522\) 0 0
\(523\) 10.7304 0.469207 0.234603 0.972091i \(-0.424621\pi\)
0.234603 + 0.972091i \(0.424621\pi\)
\(524\) 0 0
\(525\) 11.8371 0.516612
\(526\) 0 0
\(527\) 16.6222 0.724074
\(528\) 0 0
\(529\) 41.4256 1.80111
\(530\) 0 0
\(531\) −10.8104 −0.469131
\(532\) 0 0
\(533\) 6.87313 0.297708
\(534\) 0 0
\(535\) 2.37616 0.102730
\(536\) 0 0
\(537\) 14.5273 0.626899
\(538\) 0 0
\(539\) 0.245516 0.0105751
\(540\) 0 0
\(541\) 12.4910 0.537031 0.268516 0.963275i \(-0.413467\pi\)
0.268516 + 0.963275i \(0.413467\pi\)
\(542\) 0 0
\(543\) 0.909310 0.0390222
\(544\) 0 0
\(545\) −10.2301 −0.438208
\(546\) 0 0
\(547\) −14.2377 −0.608761 −0.304381 0.952550i \(-0.598450\pi\)
−0.304381 + 0.952550i \(0.598450\pi\)
\(548\) 0 0
\(549\) −5.02796 −0.214588
\(550\) 0 0
\(551\) 54.1827 2.30826
\(552\) 0 0
\(553\) 5.77553 0.245600
\(554\) 0 0
\(555\) 0.309651 0.0131439
\(556\) 0 0
\(557\) −28.4633 −1.20603 −0.603014 0.797730i \(-0.706034\pi\)
−0.603014 + 0.797730i \(0.706034\pi\)
\(558\) 0 0
\(559\) −7.23786 −0.306129
\(560\) 0 0
\(561\) 2.62136 0.110674
\(562\) 0 0
\(563\) 29.8998 1.26013 0.630063 0.776544i \(-0.283029\pi\)
0.630063 + 0.776544i \(0.283029\pi\)
\(564\) 0 0
\(565\) −3.49340 −0.146968
\(566\) 0 0
\(567\) −2.69175 −0.113043
\(568\) 0 0
\(569\) 21.5572 0.903726 0.451863 0.892087i \(-0.350760\pi\)
0.451863 + 0.892087i \(0.350760\pi\)
\(570\) 0 0
\(571\) 38.0830 1.59372 0.796862 0.604162i \(-0.206492\pi\)
0.796862 + 0.604162i \(0.206492\pi\)
\(572\) 0 0
\(573\) −12.4448 −0.519890
\(574\) 0 0
\(575\) −35.2971 −1.47199
\(576\) 0 0
\(577\) 19.7463 0.822048 0.411024 0.911625i \(-0.365171\pi\)
0.411024 + 0.911625i \(0.365171\pi\)
\(578\) 0 0
\(579\) 18.3406 0.762210
\(580\) 0 0
\(581\) 10.0503 0.416957
\(582\) 0 0
\(583\) −2.75448 −0.114079
\(584\) 0 0
\(585\) −0.776183 −0.0320912
\(586\) 0 0
\(587\) 5.48001 0.226184 0.113092 0.993585i \(-0.463925\pi\)
0.113092 + 0.993585i \(0.463925\pi\)
\(588\) 0 0
\(589\) −42.4328 −1.74841
\(590\) 0 0
\(591\) −7.96535 −0.327651
\(592\) 0 0
\(593\) 35.0670 1.44003 0.720015 0.693958i \(-0.244135\pi\)
0.720015 + 0.693958i \(0.244135\pi\)
\(594\) 0 0
\(595\) 5.47678 0.224526
\(596\) 0 0
\(597\) 14.6415 0.599238
\(598\) 0 0
\(599\) −17.3401 −0.708497 −0.354249 0.935151i \(-0.615263\pi\)
−0.354249 + 0.935151i \(0.615263\pi\)
\(600\) 0 0
\(601\) 16.0253 0.653685 0.326842 0.945079i \(-0.394015\pi\)
0.326842 + 0.945079i \(0.394015\pi\)
\(602\) 0 0
\(603\) 4.95756 0.201888
\(604\) 0 0
\(605\) −0.776183 −0.0315563
\(606\) 0 0
\(607\) 27.2673 1.10675 0.553373 0.832934i \(-0.313340\pi\)
0.553373 + 0.832934i \(0.313340\pi\)
\(608\) 0 0
\(609\) 21.7950 0.883176
\(610\) 0 0
\(611\) 4.53692 0.183544
\(612\) 0 0
\(613\) −7.92195 −0.319965 −0.159982 0.987120i \(-0.551144\pi\)
−0.159982 + 0.987120i \(0.551144\pi\)
\(614\) 0 0
\(615\) −5.33481 −0.215120
\(616\) 0 0
\(617\) −28.8293 −1.16062 −0.580312 0.814394i \(-0.697069\pi\)
−0.580312 + 0.814394i \(0.697069\pi\)
\(618\) 0 0
\(619\) −3.75794 −0.151044 −0.0755222 0.997144i \(-0.524062\pi\)
−0.0755222 + 0.997144i \(0.524062\pi\)
\(620\) 0 0
\(621\) 8.02655 0.322095
\(622\) 0 0
\(623\) −6.39894 −0.256368
\(624\) 0 0
\(625\) 16.3261 0.653042
\(626\) 0 0
\(627\) −6.69175 −0.267243
\(628\) 0 0
\(629\) −1.04577 −0.0416974
\(630\) 0 0
\(631\) −14.7525 −0.587288 −0.293644 0.955915i \(-0.594868\pi\)
−0.293644 + 0.955915i \(0.594868\pi\)
\(632\) 0 0
\(633\) −14.9716 −0.595068
\(634\) 0 0
\(635\) 15.4993 0.615073
\(636\) 0 0
\(637\) 0.245516 0.00972768
\(638\) 0 0
\(639\) 5.96382 0.235925
\(640\) 0 0
\(641\) −26.5133 −1.04721 −0.523605 0.851961i \(-0.675413\pi\)
−0.523605 + 0.851961i \(0.675413\pi\)
\(642\) 0 0
\(643\) 17.7371 0.699482 0.349741 0.936846i \(-0.386270\pi\)
0.349741 + 0.936846i \(0.386270\pi\)
\(644\) 0 0
\(645\) 5.61790 0.221205
\(646\) 0 0
\(647\) 32.2751 1.26886 0.634432 0.772978i \(-0.281234\pi\)
0.634432 + 0.772978i \(0.281234\pi\)
\(648\) 0 0
\(649\) −10.8104 −0.424345
\(650\) 0 0
\(651\) −17.0686 −0.668969
\(652\) 0 0
\(653\) −3.25803 −0.127497 −0.0637483 0.997966i \(-0.520305\pi\)
−0.0637483 + 0.997966i \(0.520305\pi\)
\(654\) 0 0
\(655\) 3.82142 0.149315
\(656\) 0 0
\(657\) 3.79648 0.148115
\(658\) 0 0
\(659\) −39.0368 −1.52066 −0.760329 0.649539i \(-0.774962\pi\)
−0.760329 + 0.649539i \(0.774962\pi\)
\(660\) 0 0
\(661\) −38.9194 −1.51379 −0.756895 0.653537i \(-0.773285\pi\)
−0.756895 + 0.653537i \(0.773285\pi\)
\(662\) 0 0
\(663\) 2.62136 0.101805
\(664\) 0 0
\(665\) −13.9810 −0.542160
\(666\) 0 0
\(667\) −64.9906 −2.51645
\(668\) 0 0
\(669\) 26.2003 1.01296
\(670\) 0 0
\(671\) −5.02796 −0.194102
\(672\) 0 0
\(673\) 34.5466 1.33167 0.665837 0.746097i \(-0.268074\pi\)
0.665837 + 0.746097i \(0.268074\pi\)
\(674\) 0 0
\(675\) −4.39754 −0.169261
\(676\) 0 0
\(677\) 25.4362 0.977591 0.488796 0.872398i \(-0.337436\pi\)
0.488796 + 0.872398i \(0.337436\pi\)
\(678\) 0 0
\(679\) 48.0088 1.84241
\(680\) 0 0
\(681\) −24.2441 −0.929037
\(682\) 0 0
\(683\) −9.81027 −0.375379 −0.187690 0.982228i \(-0.560100\pi\)
−0.187690 + 0.982228i \(0.560100\pi\)
\(684\) 0 0
\(685\) 10.2360 0.391098
\(686\) 0 0
\(687\) −6.43954 −0.245684
\(688\) 0 0
\(689\) −2.75448 −0.104937
\(690\) 0 0
\(691\) 6.32133 0.240475 0.120237 0.992745i \(-0.461634\pi\)
0.120237 + 0.992745i \(0.461634\pi\)
\(692\) 0 0
\(693\) −2.69175 −0.102251
\(694\) 0 0
\(695\) 8.55377 0.324463
\(696\) 0 0
\(697\) 18.0169 0.682440
\(698\) 0 0
\(699\) −24.0536 −0.909789
\(700\) 0 0
\(701\) 30.4082 1.14850 0.574251 0.818679i \(-0.305293\pi\)
0.574251 + 0.818679i \(0.305293\pi\)
\(702\) 0 0
\(703\) 2.66961 0.100686
\(704\) 0 0
\(705\) −3.52148 −0.132627
\(706\) 0 0
\(707\) 29.9042 1.12466
\(708\) 0 0
\(709\) −38.0014 −1.42717 −0.713586 0.700568i \(-0.752930\pi\)
−0.713586 + 0.700568i \(0.752930\pi\)
\(710\) 0 0
\(711\) −2.14564 −0.0804678
\(712\) 0 0
\(713\) 50.8969 1.90610
\(714\) 0 0
\(715\) −0.776183 −0.0290276
\(716\) 0 0
\(717\) −5.01391 −0.187248
\(718\) 0 0
\(719\) −34.7670 −1.29659 −0.648295 0.761389i \(-0.724518\pi\)
−0.648295 + 0.761389i \(0.724518\pi\)
\(720\) 0 0
\(721\) 27.4057 1.02064
\(722\) 0 0
\(723\) −24.9890 −0.929350
\(724\) 0 0
\(725\) 35.6067 1.32240
\(726\) 0 0
\(727\) −27.5746 −1.02269 −0.511343 0.859377i \(-0.670852\pi\)
−0.511343 + 0.859377i \(0.670852\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −18.9730 −0.701742
\(732\) 0 0
\(733\) 20.5076 0.757464 0.378732 0.925506i \(-0.376360\pi\)
0.378732 + 0.925506i \(0.376360\pi\)
\(734\) 0 0
\(735\) −0.190565 −0.00702910
\(736\) 0 0
\(737\) 4.95756 0.182614
\(738\) 0 0
\(739\) 29.7241 1.09342 0.546710 0.837322i \(-0.315880\pi\)
0.546710 + 0.837322i \(0.315880\pi\)
\(740\) 0 0
\(741\) −6.69175 −0.245828
\(742\) 0 0
\(743\) −5.01986 −0.184161 −0.0920804 0.995752i \(-0.529352\pi\)
−0.0920804 + 0.995752i \(0.529352\pi\)
\(744\) 0 0
\(745\) −6.40865 −0.234795
\(746\) 0 0
\(747\) −3.73375 −0.136611
\(748\) 0 0
\(749\) 8.24035 0.301096
\(750\) 0 0
\(751\) 17.7309 0.647011 0.323506 0.946226i \(-0.395139\pi\)
0.323506 + 0.946226i \(0.395139\pi\)
\(752\) 0 0
\(753\) −18.0406 −0.657436
\(754\) 0 0
\(755\) −2.80598 −0.102120
\(756\) 0 0
\(757\) −32.8369 −1.19348 −0.596740 0.802435i \(-0.703537\pi\)
−0.596740 + 0.802435i \(0.703537\pi\)
\(758\) 0 0
\(759\) 8.02655 0.291345
\(760\) 0 0
\(761\) 40.7015 1.47543 0.737713 0.675114i \(-0.235906\pi\)
0.737713 + 0.675114i \(0.235906\pi\)
\(762\) 0 0
\(763\) −35.4772 −1.28436
\(764\) 0 0
\(765\) −2.03465 −0.0735630
\(766\) 0 0
\(767\) −10.8104 −0.390341
\(768\) 0 0
\(769\) −31.1712 −1.12406 −0.562032 0.827116i \(-0.689980\pi\)
−0.562032 + 0.827116i \(0.689980\pi\)
\(770\) 0 0
\(771\) −21.1298 −0.760970
\(772\) 0 0
\(773\) 9.63553 0.346566 0.173283 0.984872i \(-0.444563\pi\)
0.173283 + 0.984872i \(0.444563\pi\)
\(774\) 0 0
\(775\) −27.8851 −1.00166
\(776\) 0 0
\(777\) 1.07385 0.0385241
\(778\) 0 0
\(779\) −45.9933 −1.64788
\(780\) 0 0
\(781\) 5.96382 0.213402
\(782\) 0 0
\(783\) −8.09695 −0.289361
\(784\) 0 0
\(785\) −15.5758 −0.555923
\(786\) 0 0
\(787\) 29.7241 1.05955 0.529775 0.848138i \(-0.322276\pi\)
0.529775 + 0.848138i \(0.322276\pi\)
\(788\) 0 0
\(789\) −11.8592 −0.422199
\(790\) 0 0
\(791\) −12.1149 −0.430755
\(792\) 0 0
\(793\) −5.02796 −0.178548
\(794\) 0 0
\(795\) 2.13798 0.0758265
\(796\) 0 0
\(797\) 4.82571 0.170935 0.0854677 0.996341i \(-0.472762\pi\)
0.0854677 + 0.996341i \(0.472762\pi\)
\(798\) 0 0
\(799\) 11.8929 0.420741
\(800\) 0 0
\(801\) 2.37724 0.0839957
\(802\) 0 0
\(803\) 3.79648 0.133975
\(804\) 0 0
\(805\) 16.7698 0.591058
\(806\) 0 0
\(807\) −0.322165 −0.0113407
\(808\) 0 0
\(809\) −31.6284 −1.11200 −0.555998 0.831183i \(-0.687664\pi\)
−0.555998 + 0.831183i \(0.687664\pi\)
\(810\) 0 0
\(811\) 30.2655 1.06276 0.531382 0.847132i \(-0.321673\pi\)
0.531382 + 0.847132i \(0.321673\pi\)
\(812\) 0 0
\(813\) −26.2944 −0.922186
\(814\) 0 0
\(815\) 13.9579 0.488923
\(816\) 0 0
\(817\) 48.4339 1.69449
\(818\) 0 0
\(819\) −2.69175 −0.0940573
\(820\) 0 0
\(821\) −18.9833 −0.662521 −0.331261 0.943539i \(-0.607474\pi\)
−0.331261 + 0.943539i \(0.607474\pi\)
\(822\) 0 0
\(823\) 24.6234 0.858318 0.429159 0.903229i \(-0.358810\pi\)
0.429159 + 0.903229i \(0.358810\pi\)
\(824\) 0 0
\(825\) −4.39754 −0.153103
\(826\) 0 0
\(827\) −23.7784 −0.826855 −0.413427 0.910537i \(-0.635668\pi\)
−0.413427 + 0.910537i \(0.635668\pi\)
\(828\) 0 0
\(829\) 44.5973 1.54893 0.774463 0.632619i \(-0.218020\pi\)
0.774463 + 0.632619i \(0.218020\pi\)
\(830\) 0 0
\(831\) −8.89247 −0.308476
\(832\) 0 0
\(833\) 0.643584 0.0222989
\(834\) 0 0
\(835\) 7.56488 0.261793
\(836\) 0 0
\(837\) 6.34106 0.219179
\(838\) 0 0
\(839\) 22.1090 0.763288 0.381644 0.924309i \(-0.375358\pi\)
0.381644 + 0.924309i \(0.375358\pi\)
\(840\) 0 0
\(841\) 36.5606 1.26071
\(842\) 0 0
\(843\) 23.5833 0.812253
\(844\) 0 0
\(845\) −0.776183 −0.0267015
\(846\) 0 0
\(847\) −2.69175 −0.0924896
\(848\) 0 0
\(849\) −8.13754 −0.279280
\(850\) 0 0
\(851\) −3.20212 −0.109767
\(852\) 0 0
\(853\) −17.8689 −0.611820 −0.305910 0.952060i \(-0.598961\pi\)
−0.305910 + 0.952060i \(0.598961\pi\)
\(854\) 0 0
\(855\) 5.19402 0.177632
\(856\) 0 0
\(857\) 11.6521 0.398029 0.199014 0.979997i \(-0.436226\pi\)
0.199014 + 0.979997i \(0.436226\pi\)
\(858\) 0 0
\(859\) −52.8576 −1.80348 −0.901740 0.432279i \(-0.857710\pi\)
−0.901740 + 0.432279i \(0.857710\pi\)
\(860\) 0 0
\(861\) −18.5007 −0.630504
\(862\) 0 0
\(863\) −51.1483 −1.74111 −0.870554 0.492072i \(-0.836240\pi\)
−0.870554 + 0.492072i \(0.836240\pi\)
\(864\) 0 0
\(865\) 11.6304 0.395446
\(866\) 0 0
\(867\) −10.1285 −0.343981
\(868\) 0 0
\(869\) −2.14564 −0.0727859
\(870\) 0 0
\(871\) 4.95756 0.167981
\(872\) 0 0
\(873\) −17.8355 −0.603642
\(874\) 0 0
\(875\) −19.6342 −0.663757
\(876\) 0 0
\(877\) −28.0125 −0.945915 −0.472958 0.881085i \(-0.656814\pi\)
−0.472958 + 0.881085i \(0.656814\pi\)
\(878\) 0 0
\(879\) 19.8649 0.670027
\(880\) 0 0
\(881\) 16.2095 0.546111 0.273055 0.961998i \(-0.411966\pi\)
0.273055 + 0.961998i \(0.411966\pi\)
\(882\) 0 0
\(883\) 50.4100 1.69643 0.848216 0.529650i \(-0.177677\pi\)
0.848216 + 0.529650i \(0.177677\pi\)
\(884\) 0 0
\(885\) 8.39084 0.282055
\(886\) 0 0
\(887\) −49.4717 −1.66110 −0.830548 0.556947i \(-0.811973\pi\)
−0.830548 + 0.556947i \(0.811973\pi\)
\(888\) 0 0
\(889\) 53.7507 1.80274
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) −30.3600 −1.01596
\(894\) 0 0
\(895\) −11.2758 −0.376910
\(896\) 0 0
\(897\) 8.02655 0.267999
\(898\) 0 0
\(899\) −51.3432 −1.71239
\(900\) 0 0
\(901\) −7.22049 −0.240549
\(902\) 0 0
\(903\) 19.4825 0.648337
\(904\) 0 0
\(905\) −0.705791 −0.0234613
\(906\) 0 0
\(907\) 59.3623 1.97109 0.985547 0.169403i \(-0.0541839\pi\)
0.985547 + 0.169403i \(0.0541839\pi\)
\(908\) 0 0
\(909\) −11.1096 −0.368482
\(910\) 0 0
\(911\) 23.7337 0.786334 0.393167 0.919467i \(-0.371379\pi\)
0.393167 + 0.919467i \(0.371379\pi\)
\(912\) 0 0
\(913\) −3.73375 −0.123569
\(914\) 0 0
\(915\) 3.90261 0.129016
\(916\) 0 0
\(917\) 13.2524 0.437634
\(918\) 0 0
\(919\) −37.1293 −1.22478 −0.612392 0.790555i \(-0.709792\pi\)
−0.612392 + 0.790555i \(0.709792\pi\)
\(920\) 0 0
\(921\) 23.3305 0.768765
\(922\) 0 0
\(923\) 5.96382 0.196302
\(924\) 0 0
\(925\) 1.75436 0.0576829
\(926\) 0 0
\(927\) −10.1814 −0.334400
\(928\) 0 0
\(929\) −3.10364 −0.101827 −0.0509136 0.998703i \(-0.516213\pi\)
−0.0509136 + 0.998703i \(0.516213\pi\)
\(930\) 0 0
\(931\) −1.64293 −0.0538448
\(932\) 0 0
\(933\) 4.43359 0.145149
\(934\) 0 0
\(935\) −2.03465 −0.0665403
\(936\) 0 0
\(937\) 23.1072 0.754880 0.377440 0.926034i \(-0.376804\pi\)
0.377440 + 0.926034i \(0.376804\pi\)
\(938\) 0 0
\(939\) −17.4129 −0.568247
\(940\) 0 0
\(941\) 1.36416 0.0444704 0.0222352 0.999753i \(-0.492922\pi\)
0.0222352 + 0.999753i \(0.492922\pi\)
\(942\) 0 0
\(943\) 55.1676 1.79650
\(944\) 0 0
\(945\) 2.08929 0.0679646
\(946\) 0 0
\(947\) 18.6209 0.605098 0.302549 0.953134i \(-0.402162\pi\)
0.302549 + 0.953134i \(0.402162\pi\)
\(948\) 0 0
\(949\) 3.79648 0.123239
\(950\) 0 0
\(951\) −30.8752 −1.00120
\(952\) 0 0
\(953\) 19.4237 0.629194 0.314597 0.949225i \(-0.398131\pi\)
0.314597 + 0.949225i \(0.398131\pi\)
\(954\) 0 0
\(955\) 9.65947 0.312573
\(956\) 0 0
\(957\) −8.09695 −0.261737
\(958\) 0 0
\(959\) 35.4978 1.14628
\(960\) 0 0
\(961\) 9.20907 0.297067
\(962\) 0 0
\(963\) −3.06133 −0.0986501
\(964\) 0 0
\(965\) −14.2357 −0.458263
\(966\) 0 0
\(967\) 10.2413 0.329338 0.164669 0.986349i \(-0.447344\pi\)
0.164669 + 0.986349i \(0.447344\pi\)
\(968\) 0 0
\(969\) −17.5415 −0.563513
\(970\) 0 0
\(971\) −8.71822 −0.279781 −0.139890 0.990167i \(-0.544675\pi\)
−0.139890 + 0.990167i \(0.544675\pi\)
\(972\) 0 0
\(973\) 29.6639 0.950980
\(974\) 0 0
\(975\) −4.39754 −0.140834
\(976\) 0 0
\(977\) −56.9127 −1.82080 −0.910400 0.413730i \(-0.864226\pi\)
−0.910400 + 0.413730i \(0.864226\pi\)
\(978\) 0 0
\(979\) 2.37724 0.0759770
\(980\) 0 0
\(981\) 13.1800 0.420804
\(982\) 0 0
\(983\) −25.1270 −0.801425 −0.400713 0.916204i \(-0.631237\pi\)
−0.400713 + 0.916204i \(0.631237\pi\)
\(984\) 0 0
\(985\) 6.18257 0.196993
\(986\) 0 0
\(987\) −12.2123 −0.388721
\(988\) 0 0
\(989\) −58.0951 −1.84732
\(990\) 0 0
\(991\) 12.8268 0.407458 0.203729 0.979027i \(-0.434694\pi\)
0.203729 + 0.979027i \(0.434694\pi\)
\(992\) 0 0
\(993\) −6.72749 −0.213490
\(994\) 0 0
\(995\) −11.3645 −0.360279
\(996\) 0 0
\(997\) 2.62596 0.0831650 0.0415825 0.999135i \(-0.486760\pi\)
0.0415825 + 0.999135i \(0.486760\pi\)
\(998\) 0 0
\(999\) −0.398941 −0.0126219
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 6864.2.a.bz.1.2 4
4.3 odd 2 429.2.a.h.1.4 4
12.11 even 2 1287.2.a.m.1.1 4
44.43 even 2 4719.2.a.z.1.1 4
52.51 odd 2 5577.2.a.m.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
429.2.a.h.1.4 4 4.3 odd 2
1287.2.a.m.1.1 4 12.11 even 2
4719.2.a.z.1.1 4 44.43 even 2
5577.2.a.m.1.1 4 52.51 odd 2
6864.2.a.bz.1.2 4 1.1 even 1 trivial