Properties

Label 8512.2.a.t.1.1
Level $8512$
Weight $2$
Character 8512.1
Self dual yes
Analytic conductor $67.969$
Analytic rank $0$
Dimension $2$
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] = [8512,2,Mod(1,8512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8512, 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("8512.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8512.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: \(67.9686622005\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 532)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.79129\) of defining polynomial
Character \(\chi\) \(=\) 8512.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.79129 q^{3} -3.00000 q^{5} +1.00000 q^{7} +0.208712 q^{9} +O(q^{10})\) \(q-1.79129 q^{3} -3.00000 q^{5} +1.00000 q^{7} +0.208712 q^{9} -0.791288 q^{11} +1.00000 q^{13} +5.37386 q^{15} -0.791288 q^{17} -1.00000 q^{19} -1.79129 q^{21} -4.58258 q^{23} +4.00000 q^{25} +5.00000 q^{27} +0.791288 q^{29} -6.37386 q^{31} +1.41742 q^{33} -3.00000 q^{35} -5.00000 q^{37} -1.79129 q^{39} +0.791288 q^{41} -2.00000 q^{43} -0.626136 q^{45} +1.41742 q^{47} +1.00000 q^{49} +1.41742 q^{51} -5.37386 q^{53} +2.37386 q^{55} +1.79129 q^{57} +6.16515 q^{59} +1.00000 q^{61} +0.208712 q^{63} -3.00000 q^{65} -4.37386 q^{67} +8.20871 q^{69} +6.16515 q^{71} +2.62614 q^{73} -7.16515 q^{75} -0.791288 q^{77} -10.0000 q^{79} -9.58258 q^{81} -0.626136 q^{83} +2.37386 q^{85} -1.41742 q^{87} -1.58258 q^{89} +1.00000 q^{91} +11.4174 q^{93} +3.00000 q^{95} -7.00000 q^{97} -0.165151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 6 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 6 q^{5} + 2 q^{7} + 5 q^{9} + 3 q^{11} + 2 q^{13} - 3 q^{15} + 3 q^{17} - 2 q^{19} + q^{21} + 8 q^{25} + 10 q^{27} - 3 q^{29} + q^{31} + 12 q^{33} - 6 q^{35} - 10 q^{37} + q^{39} - 3 q^{41} - 4 q^{43} - 15 q^{45} + 12 q^{47} + 2 q^{49} + 12 q^{51} + 3 q^{53} - 9 q^{55} - q^{57} - 6 q^{59} + 2 q^{61} + 5 q^{63} - 6 q^{65} + 5 q^{67} + 21 q^{69} - 6 q^{71} + 19 q^{73} + 4 q^{75} + 3 q^{77} - 20 q^{79} - 10 q^{81} - 15 q^{83} - 9 q^{85} - 12 q^{87} + 6 q^{89} + 2 q^{91} + 32 q^{93} + 6 q^{95} - 14 q^{97} + 18 q^{99}+O(q^{100}) \) 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
\(3\) −1.79129 −1.03420 −0.517100 0.855925i \(-0.672989\pi\)
−0.517100 + 0.855925i \(0.672989\pi\)
\(4\) 0 0
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0.208712 0.0695707
\(10\) 0 0
\(11\) −0.791288 −0.238582 −0.119291 0.992859i \(-0.538062\pi\)
−0.119291 + 0.992859i \(0.538062\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 5.37386 1.38753
\(16\) 0 0
\(17\) −0.791288 −0.191915 −0.0959577 0.995385i \(-0.530591\pi\)
−0.0959577 + 0.995385i \(0.530591\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.79129 −0.390891
\(22\) 0 0
\(23\) −4.58258 −0.955533 −0.477767 0.878487i \(-0.658554\pi\)
−0.477767 + 0.878487i \(0.658554\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 0.791288 0.146938 0.0734692 0.997297i \(-0.476593\pi\)
0.0734692 + 0.997297i \(0.476593\pi\)
\(30\) 0 0
\(31\) −6.37386 −1.14478 −0.572390 0.819982i \(-0.693984\pi\)
−0.572390 + 0.819982i \(0.693984\pi\)
\(32\) 0 0
\(33\) 1.41742 0.246742
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) −1.79129 −0.286836
\(40\) 0 0
\(41\) 0.791288 0.123578 0.0617892 0.998089i \(-0.480319\pi\)
0.0617892 + 0.998089i \(0.480319\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) −0.626136 −0.0933389
\(46\) 0 0
\(47\) 1.41742 0.206753 0.103376 0.994642i \(-0.467035\pi\)
0.103376 + 0.994642i \(0.467035\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.41742 0.198479
\(52\) 0 0
\(53\) −5.37386 −0.738157 −0.369078 0.929398i \(-0.620327\pi\)
−0.369078 + 0.929398i \(0.620327\pi\)
\(54\) 0 0
\(55\) 2.37386 0.320092
\(56\) 0 0
\(57\) 1.79129 0.237262
\(58\) 0 0
\(59\) 6.16515 0.802634 0.401317 0.915939i \(-0.368553\pi\)
0.401317 + 0.915939i \(0.368553\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 0.208712 0.0262953
\(64\) 0 0
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) −4.37386 −0.534352 −0.267176 0.963648i \(-0.586091\pi\)
−0.267176 + 0.963648i \(0.586091\pi\)
\(68\) 0 0
\(69\) 8.20871 0.988213
\(70\) 0 0
\(71\) 6.16515 0.731669 0.365834 0.930680i \(-0.380784\pi\)
0.365834 + 0.930680i \(0.380784\pi\)
\(72\) 0 0
\(73\) 2.62614 0.307366 0.153683 0.988120i \(-0.450887\pi\)
0.153683 + 0.988120i \(0.450887\pi\)
\(74\) 0 0
\(75\) −7.16515 −0.827360
\(76\) 0 0
\(77\) −0.791288 −0.0901756
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) −9.58258 −1.06473
\(82\) 0 0
\(83\) −0.626136 −0.0687274 −0.0343637 0.999409i \(-0.510940\pi\)
−0.0343637 + 0.999409i \(0.510940\pi\)
\(84\) 0 0
\(85\) 2.37386 0.257482
\(86\) 0 0
\(87\) −1.41742 −0.151964
\(88\) 0 0
\(89\) −1.58258 −0.167753 −0.0838763 0.996476i \(-0.526730\pi\)
−0.0838763 + 0.996476i \(0.526730\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 11.4174 1.18393
\(94\) 0 0
\(95\) 3.00000 0.307794
\(96\) 0 0
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 0 0
\(99\) −0.165151 −0.0165983
\(100\) 0 0
\(101\) −19.7477 −1.96497 −0.982486 0.186336i \(-0.940339\pi\)
−0.982486 + 0.186336i \(0.940339\pi\)
\(102\) 0 0
\(103\) −13.0000 −1.28093 −0.640464 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 0 0
\(105\) 5.37386 0.524435
\(106\) 0 0
\(107\) 18.1652 1.75609 0.878046 0.478577i \(-0.158847\pi\)
0.878046 + 0.478577i \(0.158847\pi\)
\(108\) 0 0
\(109\) −15.7477 −1.50836 −0.754179 0.656668i \(-0.771965\pi\)
−0.754179 + 0.656668i \(0.771965\pi\)
\(110\) 0 0
\(111\) 8.95644 0.850108
\(112\) 0 0
\(113\) −5.37386 −0.505531 −0.252765 0.967528i \(-0.581340\pi\)
−0.252765 + 0.967528i \(0.581340\pi\)
\(114\) 0 0
\(115\) 13.7477 1.28198
\(116\) 0 0
\(117\) 0.208712 0.0192954
\(118\) 0 0
\(119\) −0.791288 −0.0725372
\(120\) 0 0
\(121\) −10.3739 −0.943079
\(122\) 0 0
\(123\) −1.41742 −0.127805
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −20.7477 −1.84106 −0.920532 0.390668i \(-0.872244\pi\)
−0.920532 + 0.390668i \(0.872244\pi\)
\(128\) 0 0
\(129\) 3.58258 0.315428
\(130\) 0 0
\(131\) −19.1216 −1.67066 −0.835331 0.549748i \(-0.814724\pi\)
−0.835331 + 0.549748i \(0.814724\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) −15.0000 −1.29099
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 17.7477 1.50534 0.752671 0.658396i \(-0.228765\pi\)
0.752671 + 0.658396i \(0.228765\pi\)
\(140\) 0 0
\(141\) −2.53901 −0.213824
\(142\) 0 0
\(143\) −0.791288 −0.0661708
\(144\) 0 0
\(145\) −2.37386 −0.197139
\(146\) 0 0
\(147\) −1.79129 −0.147743
\(148\) 0 0
\(149\) −3.00000 −0.245770 −0.122885 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(150\) 0 0
\(151\) −0.373864 −0.0304246 −0.0152123 0.999884i \(-0.504842\pi\)
−0.0152123 + 0.999884i \(0.504842\pi\)
\(152\) 0 0
\(153\) −0.165151 −0.0133517
\(154\) 0 0
\(155\) 19.1216 1.53588
\(156\) 0 0
\(157\) −21.1216 −1.68569 −0.842843 0.538159i \(-0.819120\pi\)
−0.842843 + 0.538159i \(0.819120\pi\)
\(158\) 0 0
\(159\) 9.62614 0.763402
\(160\) 0 0
\(161\) −4.58258 −0.361158
\(162\) 0 0
\(163\) 9.37386 0.734218 0.367109 0.930178i \(-0.380348\pi\)
0.367109 + 0.930178i \(0.380348\pi\)
\(164\) 0 0
\(165\) −4.25227 −0.331039
\(166\) 0 0
\(167\) 22.5826 1.74749 0.873746 0.486382i \(-0.161684\pi\)
0.873746 + 0.486382i \(0.161684\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −0.208712 −0.0159606
\(172\) 0 0
\(173\) 4.74773 0.360963 0.180482 0.983578i \(-0.442234\pi\)
0.180482 + 0.983578i \(0.442234\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) −11.0436 −0.830085
\(178\) 0 0
\(179\) 11.2087 0.837778 0.418889 0.908037i \(-0.362420\pi\)
0.418889 + 0.908037i \(0.362420\pi\)
\(180\) 0 0
\(181\) −4.37386 −0.325107 −0.162553 0.986700i \(-0.551973\pi\)
−0.162553 + 0.986700i \(0.551973\pi\)
\(182\) 0 0
\(183\) −1.79129 −0.132416
\(184\) 0 0
\(185\) 15.0000 1.10282
\(186\) 0 0
\(187\) 0.626136 0.0457876
\(188\) 0 0
\(189\) 5.00000 0.363696
\(190\) 0 0
\(191\) −8.53901 −0.617861 −0.308931 0.951085i \(-0.599971\pi\)
−0.308931 + 0.951085i \(0.599971\pi\)
\(192\) 0 0
\(193\) −6.37386 −0.458801 −0.229400 0.973332i \(-0.573677\pi\)
−0.229400 + 0.973332i \(0.573677\pi\)
\(194\) 0 0
\(195\) 5.37386 0.384830
\(196\) 0 0
\(197\) −11.3739 −0.810354 −0.405177 0.914238i \(-0.632790\pi\)
−0.405177 + 0.914238i \(0.632790\pi\)
\(198\) 0 0
\(199\) 3.74773 0.265669 0.132835 0.991138i \(-0.457592\pi\)
0.132835 + 0.991138i \(0.457592\pi\)
\(200\) 0 0
\(201\) 7.83485 0.552628
\(202\) 0 0
\(203\) 0.791288 0.0555375
\(204\) 0 0
\(205\) −2.37386 −0.165798
\(206\) 0 0
\(207\) −0.956439 −0.0664771
\(208\) 0 0
\(209\) 0.791288 0.0547345
\(210\) 0 0
\(211\) 10.6261 0.731533 0.365767 0.930707i \(-0.380807\pi\)
0.365767 + 0.930707i \(0.380807\pi\)
\(212\) 0 0
\(213\) −11.0436 −0.756692
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) −6.37386 −0.432686
\(218\) 0 0
\(219\) −4.70417 −0.317878
\(220\) 0 0
\(221\) −0.791288 −0.0532278
\(222\) 0 0
\(223\) 3.74773 0.250966 0.125483 0.992096i \(-0.459952\pi\)
0.125483 + 0.992096i \(0.459952\pi\)
\(224\) 0 0
\(225\) 0.834849 0.0556566
\(226\) 0 0
\(227\) −0.791288 −0.0525196 −0.0262598 0.999655i \(-0.508360\pi\)
−0.0262598 + 0.999655i \(0.508360\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 1.41742 0.0932597
\(232\) 0 0
\(233\) 13.1216 0.859624 0.429812 0.902918i \(-0.358580\pi\)
0.429812 + 0.902918i \(0.358580\pi\)
\(234\) 0 0
\(235\) −4.25227 −0.277388
\(236\) 0 0
\(237\) 17.9129 1.16357
\(238\) 0 0
\(239\) 19.7477 1.27737 0.638687 0.769467i \(-0.279478\pi\)
0.638687 + 0.769467i \(0.279478\pi\)
\(240\) 0 0
\(241\) 18.7477 1.20765 0.603824 0.797118i \(-0.293643\pi\)
0.603824 + 0.797118i \(0.293643\pi\)
\(242\) 0 0
\(243\) 2.16515 0.138895
\(244\) 0 0
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 0 0
\(249\) 1.12159 0.0710779
\(250\) 0 0
\(251\) −5.20871 −0.328771 −0.164385 0.986396i \(-0.552564\pi\)
−0.164385 + 0.986396i \(0.552564\pi\)
\(252\) 0 0
\(253\) 3.62614 0.227973
\(254\) 0 0
\(255\) −4.25227 −0.266288
\(256\) 0 0
\(257\) 15.7913 0.985033 0.492517 0.870303i \(-0.336077\pi\)
0.492517 + 0.870303i \(0.336077\pi\)
\(258\) 0 0
\(259\) −5.00000 −0.310685
\(260\) 0 0
\(261\) 0.165151 0.0102226
\(262\) 0 0
\(263\) 2.20871 0.136195 0.0680975 0.997679i \(-0.478307\pi\)
0.0680975 + 0.997679i \(0.478307\pi\)
\(264\) 0 0
\(265\) 16.1216 0.990341
\(266\) 0 0
\(267\) 2.83485 0.173490
\(268\) 0 0
\(269\) −12.9564 −0.789968 −0.394984 0.918688i \(-0.629250\pi\)
−0.394984 + 0.918688i \(0.629250\pi\)
\(270\) 0 0
\(271\) 27.1216 1.64752 0.823760 0.566939i \(-0.191873\pi\)
0.823760 + 0.566939i \(0.191873\pi\)
\(272\) 0 0
\(273\) −1.79129 −0.108414
\(274\) 0 0
\(275\) −3.16515 −0.190866
\(276\) 0 0
\(277\) −30.7477 −1.84745 −0.923726 0.383054i \(-0.874872\pi\)
−0.923726 + 0.383054i \(0.874872\pi\)
\(278\) 0 0
\(279\) −1.33030 −0.0796431
\(280\) 0 0
\(281\) 28.5826 1.70509 0.852547 0.522651i \(-0.175057\pi\)
0.852547 + 0.522651i \(0.175057\pi\)
\(282\) 0 0
\(283\) −7.37386 −0.438331 −0.219165 0.975688i \(-0.570333\pi\)
−0.219165 + 0.975688i \(0.570333\pi\)
\(284\) 0 0
\(285\) −5.37386 −0.318320
\(286\) 0 0
\(287\) 0.791288 0.0467082
\(288\) 0 0
\(289\) −16.3739 −0.963168
\(290\) 0 0
\(291\) 12.5390 0.735050
\(292\) 0 0
\(293\) 7.74773 0.452627 0.226314 0.974055i \(-0.427333\pi\)
0.226314 + 0.974055i \(0.427333\pi\)
\(294\) 0 0
\(295\) −18.4955 −1.07685
\(296\) 0 0
\(297\) −3.95644 −0.229576
\(298\) 0 0
\(299\) −4.58258 −0.265017
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 0 0
\(303\) 35.3739 2.03218
\(304\) 0 0
\(305\) −3.00000 −0.171780
\(306\) 0 0
\(307\) 13.6261 0.777685 0.388842 0.921304i \(-0.372875\pi\)
0.388842 + 0.921304i \(0.372875\pi\)
\(308\) 0 0
\(309\) 23.2867 1.32474
\(310\) 0 0
\(311\) 30.7913 1.74601 0.873007 0.487708i \(-0.162167\pi\)
0.873007 + 0.487708i \(0.162167\pi\)
\(312\) 0 0
\(313\) −14.7477 −0.833591 −0.416795 0.909000i \(-0.636847\pi\)
−0.416795 + 0.909000i \(0.636847\pi\)
\(314\) 0 0
\(315\) −0.626136 −0.0352788
\(316\) 0 0
\(317\) −33.1652 −1.86274 −0.931370 0.364073i \(-0.881386\pi\)
−0.931370 + 0.364073i \(0.881386\pi\)
\(318\) 0 0
\(319\) −0.626136 −0.0350569
\(320\) 0 0
\(321\) −32.5390 −1.81615
\(322\) 0 0
\(323\) 0.791288 0.0440284
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) 28.2087 1.55995
\(328\) 0 0
\(329\) 1.41742 0.0781451
\(330\) 0 0
\(331\) −14.6261 −0.803925 −0.401963 0.915656i \(-0.631672\pi\)
−0.401963 + 0.915656i \(0.631672\pi\)
\(332\) 0 0
\(333\) −1.04356 −0.0571868
\(334\) 0 0
\(335\) 13.1216 0.716909
\(336\) 0 0
\(337\) −12.3739 −0.674047 −0.337024 0.941496i \(-0.609420\pi\)
−0.337024 + 0.941496i \(0.609420\pi\)
\(338\) 0 0
\(339\) 9.62614 0.522820
\(340\) 0 0
\(341\) 5.04356 0.273124
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −24.6261 −1.32583
\(346\) 0 0
\(347\) 3.79129 0.203527 0.101763 0.994809i \(-0.467552\pi\)
0.101763 + 0.994809i \(0.467552\pi\)
\(348\) 0 0
\(349\) 15.3739 0.822944 0.411472 0.911422i \(-0.365015\pi\)
0.411472 + 0.911422i \(0.365015\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) 23.5390 1.25286 0.626428 0.779480i \(-0.284516\pi\)
0.626428 + 0.779480i \(0.284516\pi\)
\(354\) 0 0
\(355\) −18.4955 −0.981637
\(356\) 0 0
\(357\) 1.41742 0.0750180
\(358\) 0 0
\(359\) −14.3739 −0.758624 −0.379312 0.925269i \(-0.623839\pi\)
−0.379312 + 0.925269i \(0.623839\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 18.5826 0.975332
\(364\) 0 0
\(365\) −7.87841 −0.412375
\(366\) 0 0
\(367\) 32.4955 1.69625 0.848124 0.529797i \(-0.177732\pi\)
0.848124 + 0.529797i \(0.177732\pi\)
\(368\) 0 0
\(369\) 0.165151 0.00859744
\(370\) 0 0
\(371\) −5.37386 −0.278997
\(372\) 0 0
\(373\) −2.62614 −0.135976 −0.0679881 0.997686i \(-0.521658\pi\)
−0.0679881 + 0.997686i \(0.521658\pi\)
\(374\) 0 0
\(375\) −5.37386 −0.277505
\(376\) 0 0
\(377\) 0.791288 0.0407534
\(378\) 0 0
\(379\) 5.74773 0.295241 0.147620 0.989044i \(-0.452839\pi\)
0.147620 + 0.989044i \(0.452839\pi\)
\(380\) 0 0
\(381\) 37.1652 1.90403
\(382\) 0 0
\(383\) 21.0000 1.07305 0.536525 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 0 0
\(385\) 2.37386 0.120983
\(386\) 0 0
\(387\) −0.417424 −0.0212189
\(388\) 0 0
\(389\) 32.3739 1.64142 0.820710 0.571345i \(-0.193578\pi\)
0.820710 + 0.571345i \(0.193578\pi\)
\(390\) 0 0
\(391\) 3.62614 0.183382
\(392\) 0 0
\(393\) 34.2523 1.72780
\(394\) 0 0
\(395\) 30.0000 1.50946
\(396\) 0 0
\(397\) 32.7477 1.64356 0.821781 0.569804i \(-0.192981\pi\)
0.821781 + 0.569804i \(0.192981\pi\)
\(398\) 0 0
\(399\) 1.79129 0.0896766
\(400\) 0 0
\(401\) −26.3739 −1.31705 −0.658524 0.752560i \(-0.728819\pi\)
−0.658524 + 0.752560i \(0.728819\pi\)
\(402\) 0 0
\(403\) −6.37386 −0.317505
\(404\) 0 0
\(405\) 28.7477 1.42849
\(406\) 0 0
\(407\) 3.95644 0.196113
\(408\) 0 0
\(409\) −14.1216 −0.698268 −0.349134 0.937073i \(-0.613524\pi\)
−0.349134 + 0.937073i \(0.613524\pi\)
\(410\) 0 0
\(411\) −10.7477 −0.530146
\(412\) 0 0
\(413\) 6.16515 0.303367
\(414\) 0 0
\(415\) 1.87841 0.0922075
\(416\) 0 0
\(417\) −31.7913 −1.55683
\(418\) 0 0
\(419\) −1.74773 −0.0853821 −0.0426910 0.999088i \(-0.513593\pi\)
−0.0426910 + 0.999088i \(0.513593\pi\)
\(420\) 0 0
\(421\) 19.0000 0.926003 0.463002 0.886357i \(-0.346772\pi\)
0.463002 + 0.886357i \(0.346772\pi\)
\(422\) 0 0
\(423\) 0.295834 0.0143839
\(424\) 0 0
\(425\) −3.16515 −0.153532
\(426\) 0 0
\(427\) 1.00000 0.0483934
\(428\) 0 0
\(429\) 1.41742 0.0684339
\(430\) 0 0
\(431\) −6.00000 −0.289010 −0.144505 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) 0 0
\(433\) −17.7477 −0.852901 −0.426451 0.904511i \(-0.640236\pi\)
−0.426451 + 0.904511i \(0.640236\pi\)
\(434\) 0 0
\(435\) 4.25227 0.203881
\(436\) 0 0
\(437\) 4.58258 0.219214
\(438\) 0 0
\(439\) −13.4955 −0.644103 −0.322051 0.946722i \(-0.604372\pi\)
−0.322051 + 0.946722i \(0.604372\pi\)
\(440\) 0 0
\(441\) 0.208712 0.00993867
\(442\) 0 0
\(443\) 36.9564 1.75585 0.877927 0.478795i \(-0.158926\pi\)
0.877927 + 0.478795i \(0.158926\pi\)
\(444\) 0 0
\(445\) 4.74773 0.225064
\(446\) 0 0
\(447\) 5.37386 0.254175
\(448\) 0 0
\(449\) −16.1216 −0.760825 −0.380412 0.924817i \(-0.624218\pi\)
−0.380412 + 0.924817i \(0.624218\pi\)
\(450\) 0 0
\(451\) −0.626136 −0.0294836
\(452\) 0 0
\(453\) 0.669697 0.0314651
\(454\) 0 0
\(455\) −3.00000 −0.140642
\(456\) 0 0
\(457\) 28.8693 1.35045 0.675225 0.737612i \(-0.264047\pi\)
0.675225 + 0.737612i \(0.264047\pi\)
\(458\) 0 0
\(459\) −3.95644 −0.184671
\(460\) 0 0
\(461\) −20.5390 −0.956597 −0.478299 0.878197i \(-0.658746\pi\)
−0.478299 + 0.878197i \(0.658746\pi\)
\(462\) 0 0
\(463\) 20.4955 0.952505 0.476252 0.879309i \(-0.341995\pi\)
0.476252 + 0.879309i \(0.341995\pi\)
\(464\) 0 0
\(465\) −34.2523 −1.58841
\(466\) 0 0
\(467\) 25.2867 1.17013 0.585065 0.810986i \(-0.301069\pi\)
0.585065 + 0.810986i \(0.301069\pi\)
\(468\) 0 0
\(469\) −4.37386 −0.201966
\(470\) 0 0
\(471\) 37.8348 1.74334
\(472\) 0 0
\(473\) 1.58258 0.0727669
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −1.12159 −0.0513541
\(478\) 0 0
\(479\) 26.3739 1.20505 0.602526 0.798099i \(-0.294161\pi\)
0.602526 + 0.798099i \(0.294161\pi\)
\(480\) 0 0
\(481\) −5.00000 −0.227980
\(482\) 0 0
\(483\) 8.20871 0.373509
\(484\) 0 0
\(485\) 21.0000 0.953561
\(486\) 0 0
\(487\) 11.0000 0.498458 0.249229 0.968445i \(-0.419823\pi\)
0.249229 + 0.968445i \(0.419823\pi\)
\(488\) 0 0
\(489\) −16.7913 −0.759328
\(490\) 0 0
\(491\) −3.16515 −0.142841 −0.0714206 0.997446i \(-0.522753\pi\)
−0.0714206 + 0.997446i \(0.522753\pi\)
\(492\) 0 0
\(493\) −0.626136 −0.0281998
\(494\) 0 0
\(495\) 0.495454 0.0222690
\(496\) 0 0
\(497\) 6.16515 0.276545
\(498\) 0 0
\(499\) −31.3739 −1.40449 −0.702244 0.711937i \(-0.747818\pi\)
−0.702244 + 0.711937i \(0.747818\pi\)
\(500\) 0 0
\(501\) −40.4519 −1.80726
\(502\) 0 0
\(503\) 30.3303 1.35236 0.676181 0.736736i \(-0.263634\pi\)
0.676181 + 0.736736i \(0.263634\pi\)
\(504\) 0 0
\(505\) 59.2432 2.63629
\(506\) 0 0
\(507\) 21.4955 0.954647
\(508\) 0 0
\(509\) 36.3303 1.61031 0.805156 0.593063i \(-0.202081\pi\)
0.805156 + 0.593063i \(0.202081\pi\)
\(510\) 0 0
\(511\) 2.62614 0.116173
\(512\) 0 0
\(513\) −5.00000 −0.220755
\(514\) 0 0
\(515\) 39.0000 1.71855
\(516\) 0 0
\(517\) −1.12159 −0.0493275
\(518\) 0 0
\(519\) −8.50455 −0.373308
\(520\) 0 0
\(521\) −22.5826 −0.989361 −0.494680 0.869075i \(-0.664715\pi\)
−0.494680 + 0.869075i \(0.664715\pi\)
\(522\) 0 0
\(523\) −27.7477 −1.21332 −0.606662 0.794960i \(-0.707492\pi\)
−0.606662 + 0.794960i \(0.707492\pi\)
\(524\) 0 0
\(525\) −7.16515 −0.312713
\(526\) 0 0
\(527\) 5.04356 0.219701
\(528\) 0 0
\(529\) −2.00000 −0.0869565
\(530\) 0 0
\(531\) 1.28674 0.0558398
\(532\) 0 0
\(533\) 0.791288 0.0342745
\(534\) 0 0
\(535\) −54.4955 −2.35604
\(536\) 0 0
\(537\) −20.0780 −0.866431
\(538\) 0 0
\(539\) −0.791288 −0.0340832
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 7.83485 0.336226
\(544\) 0 0
\(545\) 47.2432 2.02368
\(546\) 0 0
\(547\) −1.37386 −0.0587422 −0.0293711 0.999569i \(-0.509350\pi\)
−0.0293711 + 0.999569i \(0.509350\pi\)
\(548\) 0 0
\(549\) 0.208712 0.00890762
\(550\) 0 0
\(551\) −0.791288 −0.0337100
\(552\) 0 0
\(553\) −10.0000 −0.425243
\(554\) 0 0
\(555\) −26.8693 −1.14054
\(556\) 0 0
\(557\) −14.3739 −0.609040 −0.304520 0.952506i \(-0.598496\pi\)
−0.304520 + 0.952506i \(0.598496\pi\)
\(558\) 0 0
\(559\) −2.00000 −0.0845910
\(560\) 0 0
\(561\) −1.12159 −0.0473536
\(562\) 0 0
\(563\) −6.16515 −0.259830 −0.129915 0.991525i \(-0.541470\pi\)
−0.129915 + 0.991525i \(0.541470\pi\)
\(564\) 0 0
\(565\) 16.1216 0.678240
\(566\) 0 0
\(567\) −9.58258 −0.402430
\(568\) 0 0
\(569\) 11.8348 0.496143 0.248071 0.968742i \(-0.420203\pi\)
0.248071 + 0.968742i \(0.420203\pi\)
\(570\) 0 0
\(571\) 22.4955 0.941405 0.470703 0.882292i \(-0.344000\pi\)
0.470703 + 0.882292i \(0.344000\pi\)
\(572\) 0 0
\(573\) 15.2958 0.638993
\(574\) 0 0
\(575\) −18.3303 −0.764426
\(576\) 0 0
\(577\) 18.1216 0.754412 0.377206 0.926129i \(-0.376885\pi\)
0.377206 + 0.926129i \(0.376885\pi\)
\(578\) 0 0
\(579\) 11.4174 0.474492
\(580\) 0 0
\(581\) −0.626136 −0.0259765
\(582\) 0 0
\(583\) 4.25227 0.176111
\(584\) 0 0
\(585\) −0.626136 −0.0258876
\(586\) 0 0
\(587\) −9.16515 −0.378286 −0.189143 0.981950i \(-0.560571\pi\)
−0.189143 + 0.981950i \(0.560571\pi\)
\(588\) 0 0
\(589\) 6.37386 0.262630
\(590\) 0 0
\(591\) 20.3739 0.838069
\(592\) 0 0
\(593\) 16.9129 0.694529 0.347264 0.937767i \(-0.387111\pi\)
0.347264 + 0.937767i \(0.387111\pi\)
\(594\) 0 0
\(595\) 2.37386 0.0973189
\(596\) 0 0
\(597\) −6.71326 −0.274755
\(598\) 0 0
\(599\) −6.62614 −0.270737 −0.135368 0.990795i \(-0.543222\pi\)
−0.135368 + 0.990795i \(0.543222\pi\)
\(600\) 0 0
\(601\) 25.3739 1.03502 0.517511 0.855677i \(-0.326859\pi\)
0.517511 + 0.855677i \(0.326859\pi\)
\(602\) 0 0
\(603\) −0.912878 −0.0371753
\(604\) 0 0
\(605\) 31.1216 1.26527
\(606\) 0 0
\(607\) −28.4955 −1.15659 −0.578297 0.815826i \(-0.696283\pi\)
−0.578297 + 0.815826i \(0.696283\pi\)
\(608\) 0 0
\(609\) −1.41742 −0.0574369
\(610\) 0 0
\(611\) 1.41742 0.0573428
\(612\) 0 0
\(613\) −20.6261 −0.833082 −0.416541 0.909117i \(-0.636758\pi\)
−0.416541 + 0.909117i \(0.636758\pi\)
\(614\) 0 0
\(615\) 4.25227 0.171468
\(616\) 0 0
\(617\) 15.9564 0.642382 0.321191 0.947014i \(-0.395917\pi\)
0.321191 + 0.947014i \(0.395917\pi\)
\(618\) 0 0
\(619\) 9.37386 0.376767 0.188384 0.982096i \(-0.439675\pi\)
0.188384 + 0.982096i \(0.439675\pi\)
\(620\) 0 0
\(621\) −22.9129 −0.919462
\(622\) 0 0
\(623\) −1.58258 −0.0634046
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) −1.41742 −0.0566065
\(628\) 0 0
\(629\) 3.95644 0.157754
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 0 0
\(633\) −19.0345 −0.756552
\(634\) 0 0
\(635\) 62.2432 2.47005
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 1.28674 0.0509027
\(640\) 0 0
\(641\) −24.7913 −0.979197 −0.489598 0.871948i \(-0.662857\pi\)
−0.489598 + 0.871948i \(0.662857\pi\)
\(642\) 0 0
\(643\) −23.0000 −0.907031 −0.453516 0.891248i \(-0.649830\pi\)
−0.453516 + 0.891248i \(0.649830\pi\)
\(644\) 0 0
\(645\) −10.7477 −0.423191
\(646\) 0 0
\(647\) −22.5826 −0.887813 −0.443906 0.896073i \(-0.646408\pi\)
−0.443906 + 0.896073i \(0.646408\pi\)
\(648\) 0 0
\(649\) −4.87841 −0.191494
\(650\) 0 0
\(651\) 11.4174 0.447484
\(652\) 0 0
\(653\) −28.5826 −1.11852 −0.559261 0.828991i \(-0.688915\pi\)
−0.559261 + 0.828991i \(0.688915\pi\)
\(654\) 0 0
\(655\) 57.3648 2.24143
\(656\) 0 0
\(657\) 0.548107 0.0213837
\(658\) 0 0
\(659\) 14.0436 0.547059 0.273530 0.961864i \(-0.411809\pi\)
0.273530 + 0.961864i \(0.411809\pi\)
\(660\) 0 0
\(661\) 34.0000 1.32245 0.661223 0.750189i \(-0.270038\pi\)
0.661223 + 0.750189i \(0.270038\pi\)
\(662\) 0 0
\(663\) 1.41742 0.0550482
\(664\) 0 0
\(665\) 3.00000 0.116335
\(666\) 0 0
\(667\) −3.62614 −0.140405
\(668\) 0 0
\(669\) −6.71326 −0.259550
\(670\) 0 0
\(671\) −0.791288 −0.0305473
\(672\) 0 0
\(673\) 24.1216 0.929819 0.464909 0.885358i \(-0.346087\pi\)
0.464909 + 0.885358i \(0.346087\pi\)
\(674\) 0 0
\(675\) 20.0000 0.769800
\(676\) 0 0
\(677\) −0.956439 −0.0367589 −0.0183795 0.999831i \(-0.505851\pi\)
−0.0183795 + 0.999831i \(0.505851\pi\)
\(678\) 0 0
\(679\) −7.00000 −0.268635
\(680\) 0 0
\(681\) 1.41742 0.0543158
\(682\) 0 0
\(683\) 17.8348 0.682432 0.341216 0.939985i \(-0.389161\pi\)
0.341216 + 0.939985i \(0.389161\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) −39.4083 −1.50352
\(688\) 0 0
\(689\) −5.37386 −0.204728
\(690\) 0 0
\(691\) −21.2523 −0.808475 −0.404237 0.914654i \(-0.632463\pi\)
−0.404237 + 0.914654i \(0.632463\pi\)
\(692\) 0 0
\(693\) −0.165151 −0.00627358
\(694\) 0 0
\(695\) −53.2432 −2.01963
\(696\) 0 0
\(697\) −0.626136 −0.0237166
\(698\) 0 0
\(699\) −23.5045 −0.889024
\(700\) 0 0
\(701\) −5.66970 −0.214142 −0.107071 0.994251i \(-0.534147\pi\)
−0.107071 + 0.994251i \(0.534147\pi\)
\(702\) 0 0
\(703\) 5.00000 0.188579
\(704\) 0 0
\(705\) 7.61704 0.286875
\(706\) 0 0
\(707\) −19.7477 −0.742690
\(708\) 0 0
\(709\) 51.2432 1.92448 0.962239 0.272206i \(-0.0877532\pi\)
0.962239 + 0.272206i \(0.0877532\pi\)
\(710\) 0 0
\(711\) −2.08712 −0.0782732
\(712\) 0 0
\(713\) 29.2087 1.09387
\(714\) 0 0
\(715\) 2.37386 0.0887775
\(716\) 0 0
\(717\) −35.3739 −1.32106
\(718\) 0 0
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) −13.0000 −0.484145
\(722\) 0 0
\(723\) −33.5826 −1.24895
\(724\) 0 0
\(725\) 3.16515 0.117551
\(726\) 0 0
\(727\) 11.0000 0.407967 0.203984 0.978974i \(-0.434611\pi\)
0.203984 + 0.978974i \(0.434611\pi\)
\(728\) 0 0
\(729\) 24.8693 0.921086
\(730\) 0 0
\(731\) 1.58258 0.0585337
\(732\) 0 0
\(733\) 13.4955 0.498466 0.249233 0.968444i \(-0.419822\pi\)
0.249233 + 0.968444i \(0.419822\pi\)
\(734\) 0 0
\(735\) 5.37386 0.198218
\(736\) 0 0
\(737\) 3.46099 0.127487
\(738\) 0 0
\(739\) 8.25227 0.303565 0.151782 0.988414i \(-0.451499\pi\)
0.151782 + 0.988414i \(0.451499\pi\)
\(740\) 0 0
\(741\) 1.79129 0.0658046
\(742\) 0 0
\(743\) −15.3303 −0.562414 −0.281207 0.959647i \(-0.590735\pi\)
−0.281207 + 0.959647i \(0.590735\pi\)
\(744\) 0 0
\(745\) 9.00000 0.329734
\(746\) 0 0
\(747\) −0.130682 −0.00478141
\(748\) 0 0
\(749\) 18.1652 0.663740
\(750\) 0 0
\(751\) 25.3739 0.925905 0.462953 0.886383i \(-0.346790\pi\)
0.462953 + 0.886383i \(0.346790\pi\)
\(752\) 0 0
\(753\) 9.33030 0.340015
\(754\) 0 0
\(755\) 1.12159 0.0408189
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) −6.49545 −0.235770
\(760\) 0 0
\(761\) −2.66970 −0.0967764 −0.0483882 0.998829i \(-0.515408\pi\)
−0.0483882 + 0.998829i \(0.515408\pi\)
\(762\) 0 0
\(763\) −15.7477 −0.570106
\(764\) 0 0
\(765\) 0.495454 0.0179132
\(766\) 0 0
\(767\) 6.16515 0.222611
\(768\) 0 0
\(769\) 41.4955 1.49636 0.748182 0.663493i \(-0.230927\pi\)
0.748182 + 0.663493i \(0.230927\pi\)
\(770\) 0 0
\(771\) −28.2867 −1.01872
\(772\) 0 0
\(773\) 10.7477 0.386569 0.193284 0.981143i \(-0.438086\pi\)
0.193284 + 0.981143i \(0.438086\pi\)
\(774\) 0 0
\(775\) −25.4955 −0.915824
\(776\) 0 0
\(777\) 8.95644 0.321310
\(778\) 0 0
\(779\) −0.791288 −0.0283508
\(780\) 0 0
\(781\) −4.87841 −0.174563
\(782\) 0 0
\(783\) 3.95644 0.141392
\(784\) 0 0
\(785\) 63.3648 2.26159
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 0 0
\(789\) −3.95644 −0.140853
\(790\) 0 0
\(791\) −5.37386 −0.191073
\(792\) 0 0
\(793\) 1.00000 0.0355110
\(794\) 0 0
\(795\) −28.8784 −1.02421
\(796\) 0 0
\(797\) −35.8693 −1.27056 −0.635278 0.772283i \(-0.719115\pi\)
−0.635278 + 0.772283i \(0.719115\pi\)
\(798\) 0 0
\(799\) −1.12159 −0.0396790
\(800\) 0 0
\(801\) −0.330303 −0.0116707
\(802\) 0 0
\(803\) −2.07803 −0.0733321
\(804\) 0 0
\(805\) 13.7477 0.484544
\(806\) 0 0
\(807\) 23.2087 0.816985
\(808\) 0 0
\(809\) 22.9129 0.805574 0.402787 0.915294i \(-0.368042\pi\)
0.402787 + 0.915294i \(0.368042\pi\)
\(810\) 0 0
\(811\) −23.4955 −0.825037 −0.412518 0.910949i \(-0.635351\pi\)
−0.412518 + 0.910949i \(0.635351\pi\)
\(812\) 0 0
\(813\) −48.5826 −1.70387
\(814\) 0 0
\(815\) −28.1216 −0.985056
\(816\) 0 0
\(817\) 2.00000 0.0699711
\(818\) 0 0
\(819\) 0.208712 0.00729299
\(820\) 0 0
\(821\) −5.66970 −0.197874 −0.0989369 0.995094i \(-0.531544\pi\)
−0.0989369 + 0.995094i \(0.531544\pi\)
\(822\) 0 0
\(823\) 28.2432 0.984495 0.492248 0.870455i \(-0.336175\pi\)
0.492248 + 0.870455i \(0.336175\pi\)
\(824\) 0 0
\(825\) 5.66970 0.197394
\(826\) 0 0
\(827\) 33.6606 1.17049 0.585247 0.810855i \(-0.300998\pi\)
0.585247 + 0.810855i \(0.300998\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) 55.0780 1.91064
\(832\) 0 0
\(833\) −0.791288 −0.0274165
\(834\) 0 0
\(835\) −67.7477 −2.34451
\(836\) 0 0
\(837\) −31.8693 −1.10156
\(838\) 0 0
\(839\) 27.1652 0.937845 0.468923 0.883239i \(-0.344642\pi\)
0.468923 + 0.883239i \(0.344642\pi\)
\(840\) 0 0
\(841\) −28.3739 −0.978409
\(842\) 0 0
\(843\) −51.1996 −1.76341
\(844\) 0 0
\(845\) 36.0000 1.23844
\(846\) 0 0
\(847\) −10.3739 −0.356450
\(848\) 0 0
\(849\) 13.2087 0.453322
\(850\) 0 0
\(851\) 22.9129 0.785443
\(852\) 0 0
\(853\) 32.1216 1.09982 0.549911 0.835223i \(-0.314662\pi\)
0.549911 + 0.835223i \(0.314662\pi\)
\(854\) 0 0
\(855\) 0.626136 0.0214134
\(856\) 0 0
\(857\) 38.7042 1.32211 0.661055 0.750338i \(-0.270109\pi\)
0.661055 + 0.750338i \(0.270109\pi\)
\(858\) 0 0
\(859\) 6.37386 0.217473 0.108737 0.994071i \(-0.465319\pi\)
0.108737 + 0.994071i \(0.465319\pi\)
\(860\) 0 0
\(861\) −1.41742 −0.0483057
\(862\) 0 0
\(863\) 11.7042 0.398414 0.199207 0.979957i \(-0.436163\pi\)
0.199207 + 0.979957i \(0.436163\pi\)
\(864\) 0 0
\(865\) −14.2432 −0.484283
\(866\) 0 0
\(867\) 29.3303 0.996109
\(868\) 0 0
\(869\) 7.91288 0.268426
\(870\) 0 0
\(871\) −4.37386 −0.148203
\(872\) 0 0
\(873\) −1.46099 −0.0494469
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) 11.2523 0.379962 0.189981 0.981788i \(-0.439157\pi\)
0.189981 + 0.981788i \(0.439157\pi\)
\(878\) 0 0
\(879\) −13.8784 −0.468107
\(880\) 0 0
\(881\) −11.3739 −0.383195 −0.191598 0.981474i \(-0.561367\pi\)
−0.191598 + 0.981474i \(0.561367\pi\)
\(882\) 0 0
\(883\) −51.7477 −1.74145 −0.870725 0.491771i \(-0.836350\pi\)
−0.870725 + 0.491771i \(0.836350\pi\)
\(884\) 0 0
\(885\) 33.1307 1.11368
\(886\) 0 0
\(887\) −4.58258 −0.153868 −0.0769339 0.997036i \(-0.524513\pi\)
−0.0769339 + 0.997036i \(0.524513\pi\)
\(888\) 0 0
\(889\) −20.7477 −0.695856
\(890\) 0 0
\(891\) 7.58258 0.254026
\(892\) 0 0
\(893\) −1.41742 −0.0474323
\(894\) 0 0
\(895\) −33.6261 −1.12400
\(896\) 0 0
\(897\) 8.20871 0.274081
\(898\) 0 0
\(899\) −5.04356 −0.168212
\(900\) 0 0
\(901\) 4.25227 0.141664
\(902\) 0 0
\(903\) 3.58258 0.119221
\(904\) 0 0
\(905\) 13.1216 0.436176
\(906\) 0 0
\(907\) 51.2432 1.70150 0.850751 0.525569i \(-0.176148\pi\)
0.850751 + 0.525569i \(0.176148\pi\)
\(908\) 0 0
\(909\) −4.12159 −0.136705
\(910\) 0 0
\(911\) −51.4955 −1.70612 −0.853060 0.521812i \(-0.825256\pi\)
−0.853060 + 0.521812i \(0.825256\pi\)
\(912\) 0 0
\(913\) 0.495454 0.0163971
\(914\) 0 0
\(915\) 5.37386 0.177654
\(916\) 0 0
\(917\) −19.1216 −0.631451
\(918\) 0 0
\(919\) 20.4955 0.676083 0.338041 0.941131i \(-0.390236\pi\)
0.338041 + 0.941131i \(0.390236\pi\)
\(920\) 0 0
\(921\) −24.4083 −0.804282
\(922\) 0 0
\(923\) 6.16515 0.202928
\(924\) 0 0
\(925\) −20.0000 −0.657596
\(926\) 0 0
\(927\) −2.71326 −0.0891151
\(928\) 0 0
\(929\) −20.8693 −0.684700 −0.342350 0.939572i \(-0.611223\pi\)
−0.342350 + 0.939572i \(0.611223\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) −55.1561 −1.80573
\(934\) 0 0
\(935\) −1.87841 −0.0614306
\(936\) 0 0
\(937\) −1.62614 −0.0531236 −0.0265618 0.999647i \(-0.508456\pi\)
−0.0265618 + 0.999647i \(0.508456\pi\)
\(938\) 0 0
\(939\) 26.4174 0.862100
\(940\) 0 0
\(941\) 27.4955 0.896326 0.448163 0.893952i \(-0.352078\pi\)
0.448163 + 0.893952i \(0.352078\pi\)
\(942\) 0 0
\(943\) −3.62614 −0.118083
\(944\) 0 0
\(945\) −15.0000 −0.487950
\(946\) 0 0
\(947\) −43.6170 −1.41736 −0.708682 0.705528i \(-0.750710\pi\)
−0.708682 + 0.705528i \(0.750710\pi\)
\(948\) 0 0
\(949\) 2.62614 0.0852480
\(950\) 0 0
\(951\) 59.4083 1.92645
\(952\) 0 0
\(953\) 53.5390 1.73430 0.867149 0.498048i \(-0.165950\pi\)
0.867149 + 0.498048i \(0.165950\pi\)
\(954\) 0 0
\(955\) 25.6170 0.828948
\(956\) 0 0
\(957\) 1.12159 0.0362559
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 9.62614 0.310521
\(962\) 0 0
\(963\) 3.79129 0.122173
\(964\) 0 0
\(965\) 19.1216 0.615546
\(966\) 0 0
\(967\) −29.1216 −0.936487 −0.468244 0.883599i \(-0.655113\pi\)
−0.468244 + 0.883599i \(0.655113\pi\)
\(968\) 0 0
\(969\) −1.41742 −0.0455342
\(970\) 0 0
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) 17.7477 0.568966
\(974\) 0 0
\(975\) −7.16515 −0.229468
\(976\) 0 0
\(977\) −43.5826 −1.39433 −0.697165 0.716911i \(-0.745556\pi\)
−0.697165 + 0.716911i \(0.745556\pi\)
\(978\) 0 0
\(979\) 1.25227 0.0400228
\(980\) 0 0
\(981\) −3.28674 −0.104938
\(982\) 0 0
\(983\) 45.3303 1.44581 0.722906 0.690946i \(-0.242806\pi\)
0.722906 + 0.690946i \(0.242806\pi\)
\(984\) 0 0
\(985\) 34.1216 1.08720
\(986\) 0 0
\(987\) −2.53901 −0.0808177
\(988\) 0 0
\(989\) 9.16515 0.291435
\(990\) 0 0
\(991\) 18.7477 0.595541 0.297771 0.954637i \(-0.403757\pi\)
0.297771 + 0.954637i \(0.403757\pi\)
\(992\) 0 0
\(993\) 26.1996 0.831420
\(994\) 0 0
\(995\) −11.2432 −0.356433
\(996\) 0 0
\(997\) 44.1216 1.39734 0.698672 0.715442i \(-0.253775\pi\)
0.698672 + 0.715442i \(0.253775\pi\)
\(998\) 0 0
\(999\) −25.0000 −0.790965
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 8512.2.a.t.1.1 2
4.3 odd 2 8512.2.a.m.1.2 2
8.3 odd 2 2128.2.a.k.1.1 2
8.5 even 2 532.2.a.c.1.2 2
24.5 odd 2 4788.2.a.g.1.1 2
56.13 odd 2 3724.2.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
532.2.a.c.1.2 2 8.5 even 2
2128.2.a.k.1.1 2 8.3 odd 2
3724.2.a.e.1.1 2 56.13 odd 2
4788.2.a.g.1.1 2 24.5 odd 2
8512.2.a.m.1.2 2 4.3 odd 2
8512.2.a.t.1.1 2 1.1 even 1 trivial