Properties

Label 6864.2.a.bf.1.1
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 858)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.70156\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -2.00000 q^{15} -5.40312 q^{17} -7.40312 q^{19} -3.40312 q^{23} -1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} +7.40312 q^{31} -1.00000 q^{33} -5.40312 q^{37} -1.00000 q^{39} +2.00000 q^{41} +2.00000 q^{45} +8.00000 q^{47} -7.00000 q^{49} +5.40312 q^{51} +9.40312 q^{53} +2.00000 q^{55} +7.40312 q^{57} +4.00000 q^{59} +12.8062 q^{61} +2.00000 q^{65} +3.40312 q^{69} +14.8062 q^{71} +13.4031 q^{73} +1.00000 q^{75} +10.8062 q^{79} +1.00000 q^{81} +4.00000 q^{83} -10.8062 q^{85} -2.00000 q^{87} -10.0000 q^{89} -7.40312 q^{93} -14.8062 q^{95} +10.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} - 2 q^{19} + 6 q^{23} - 2 q^{25} - 2 q^{27} + 4 q^{29} + 2 q^{31} - 2 q^{33} + 2 q^{37} - 2 q^{39} + 4 q^{41} + 4 q^{45} + 16 q^{47} - 14 q^{49} - 2 q^{51} + 6 q^{53} + 4 q^{55} + 2 q^{57} + 8 q^{59} + 4 q^{65} - 6 q^{69} + 4 q^{71} + 14 q^{73} + 2 q^{75} - 4 q^{79} + 2 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 20 q^{89} - 2 q^{93} - 4 q^{95} + 20 q^{97} + 2 q^{99}+O(q^{100}) \) 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
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 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
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −5.40312 −1.31045 −0.655225 0.755434i \(-0.727426\pi\)
−0.655225 + 0.755434i \(0.727426\pi\)
\(18\) 0 0
\(19\) −7.40312 −1.69839 −0.849197 0.528077i \(-0.822913\pi\)
−0.849197 + 0.528077i \(0.822913\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.40312 −0.709600 −0.354800 0.934942i \(-0.615451\pi\)
−0.354800 + 0.934942i \(0.615451\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 7.40312 1.32964 0.664820 0.747003i \(-0.268508\pi\)
0.664820 + 0.747003i \(0.268508\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.40312 −0.888268 −0.444134 0.895960i \(-0.646489\pi\)
−0.444134 + 0.895960i \(0.646489\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 5.40312 0.756589
\(52\) 0 0
\(53\) 9.40312 1.29162 0.645809 0.763499i \(-0.276520\pi\)
0.645809 + 0.763499i \(0.276520\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 7.40312 0.980568
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 12.8062 1.63967 0.819836 0.572598i \(-0.194065\pi\)
0.819836 + 0.572598i \(0.194065\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 3.40312 0.409688
\(70\) 0 0
\(71\) 14.8062 1.75718 0.878589 0.477578i \(-0.158485\pi\)
0.878589 + 0.477578i \(0.158485\pi\)
\(72\) 0 0
\(73\) 13.4031 1.56872 0.784359 0.620308i \(-0.212992\pi\)
0.784359 + 0.620308i \(0.212992\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.8062 1.21580 0.607899 0.794014i \(-0.292013\pi\)
0.607899 + 0.794014i \(0.292013\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −10.8062 −1.17210
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −7.40312 −0.767668
\(94\) 0 0
\(95\) −14.8062 −1.51909
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −6.80625 −0.670640 −0.335320 0.942104i \(-0.608844\pi\)
−0.335320 + 0.942104i \(0.608844\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.40312 −0.715687 −0.357844 0.933782i \(-0.616488\pi\)
−0.357844 + 0.933782i \(0.616488\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 5.40312 0.512842
\(112\) 0 0
\(113\) −20.8062 −1.95729 −0.978644 0.205564i \(-0.934097\pi\)
−0.978644 + 0.205564i \(0.934097\pi\)
\(114\) 0 0
\(115\) −6.80625 −0.634686
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 18.8062 1.66878 0.834392 0.551171i \(-0.185819\pi\)
0.834392 + 0.551171i \(0.185819\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 15.4031 1.34578 0.672889 0.739744i \(-0.265053\pi\)
0.672889 + 0.739744i \(0.265053\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.00000 −0.172133
\(136\) 0 0
\(137\) 12.8062 1.09411 0.547056 0.837096i \(-0.315749\pi\)
0.547056 + 0.837096i \(0.315749\pi\)
\(138\) 0 0
\(139\) 6.80625 0.577298 0.288649 0.957435i \(-0.406794\pi\)
0.288649 + 0.957435i \(0.406794\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) 7.00000 0.577350
\(148\) 0 0
\(149\) 12.8062 1.04913 0.524564 0.851371i \(-0.324228\pi\)
0.524564 + 0.851371i \(0.324228\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) −5.40312 −0.436817
\(154\) 0 0
\(155\) 14.8062 1.18927
\(156\) 0 0
\(157\) 4.80625 0.383580 0.191790 0.981436i \(-0.438571\pi\)
0.191790 + 0.981436i \(0.438571\pi\)
\(158\) 0 0
\(159\) −9.40312 −0.745716
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −14.8062 −1.15971 −0.579857 0.814718i \(-0.696892\pi\)
−0.579857 + 0.814718i \(0.696892\pi\)
\(164\) 0 0
\(165\) −2.00000 −0.155700
\(166\) 0 0
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −7.40312 −0.566131
\(172\) 0 0
\(173\) −4.80625 −0.365412 −0.182706 0.983168i \(-0.558486\pi\)
−0.182706 + 0.983168i \(0.558486\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) −16.8062 −1.24920 −0.624599 0.780945i \(-0.714738\pi\)
−0.624599 + 0.780945i \(0.714738\pi\)
\(182\) 0 0
\(183\) −12.8062 −0.946665
\(184\) 0 0
\(185\) −10.8062 −0.794491
\(186\) 0 0
\(187\) −5.40312 −0.395116
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.5969 0.911478 0.455739 0.890113i \(-0.349375\pi\)
0.455739 + 0.890113i \(0.349375\pi\)
\(192\) 0 0
\(193\) −17.4031 −1.25270 −0.626352 0.779540i \(-0.715453\pi\)
−0.626352 + 0.779540i \(0.715453\pi\)
\(194\) 0 0
\(195\) −2.00000 −0.143223
\(196\) 0 0
\(197\) 12.8062 0.912407 0.456204 0.889875i \(-0.349209\pi\)
0.456204 + 0.889875i \(0.349209\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 0 0
\(207\) −3.40312 −0.236533
\(208\) 0 0
\(209\) −7.40312 −0.512085
\(210\) 0 0
\(211\) −6.80625 −0.468561 −0.234281 0.972169i \(-0.575273\pi\)
−0.234281 + 0.972169i \(0.575273\pi\)
\(212\) 0 0
\(213\) −14.8062 −1.01451
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −13.4031 −0.905699
\(220\) 0 0
\(221\) −5.40312 −0.363453
\(222\) 0 0
\(223\) −0.596876 −0.0399698 −0.0199849 0.999800i \(-0.506362\pi\)
−0.0199849 + 0.999800i \(0.506362\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) 2.80625 0.186257 0.0931286 0.995654i \(-0.470313\pi\)
0.0931286 + 0.995654i \(0.470313\pi\)
\(228\) 0 0
\(229\) 2.59688 0.171606 0.0858032 0.996312i \(-0.472654\pi\)
0.0858032 + 0.996312i \(0.472654\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.40312 0.616019 0.308010 0.951383i \(-0.400337\pi\)
0.308010 + 0.951383i \(0.400337\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) 0 0
\(237\) −10.8062 −0.701941
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 21.4031 1.37870 0.689348 0.724430i \(-0.257897\pi\)
0.689348 + 0.724430i \(0.257897\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −14.0000 −0.894427
\(246\) 0 0
\(247\) −7.40312 −0.471050
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −3.40312 −0.213953
\(254\) 0 0
\(255\) 10.8062 0.676714
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 18.8062 1.15526
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 0 0
\(269\) 16.2094 0.988303 0.494151 0.869376i \(-0.335479\pi\)
0.494151 + 0.869376i \(0.335479\pi\)
\(270\) 0 0
\(271\) 22.8062 1.38538 0.692690 0.721235i \(-0.256425\pi\)
0.692690 + 0.721235i \(0.256425\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 0 0
\(279\) 7.40312 0.443213
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) 0 0
\(285\) 14.8062 0.877046
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 12.1938 0.717280
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) −0.806248 −0.0471015 −0.0235508 0.999723i \(-0.507497\pi\)
−0.0235508 + 0.999723i \(0.507497\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −3.40312 −0.196808
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) 25.6125 1.46657
\(306\) 0 0
\(307\) −16.5969 −0.947234 −0.473617 0.880731i \(-0.657052\pi\)
−0.473617 + 0.880731i \(0.657052\pi\)
\(308\) 0 0
\(309\) 6.80625 0.387194
\(310\) 0 0
\(311\) −10.2094 −0.578920 −0.289460 0.957190i \(-0.593476\pi\)
−0.289460 + 0.957190i \(0.593476\pi\)
\(312\) 0 0
\(313\) −28.8062 −1.62823 −0.814113 0.580707i \(-0.802776\pi\)
−0.814113 + 0.580707i \(0.802776\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 26.0000 1.46031 0.730153 0.683284i \(-0.239449\pi\)
0.730153 + 0.683284i \(0.239449\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) 7.40312 0.413202
\(322\) 0 0
\(323\) 40.0000 2.22566
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) −14.0000 −0.774202
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 0 0
\(333\) −5.40312 −0.296089
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.8062 0.915495 0.457747 0.889082i \(-0.348656\pi\)
0.457747 + 0.889082i \(0.348656\pi\)
\(338\) 0 0
\(339\) 20.8062 1.13004
\(340\) 0 0
\(341\) 7.40312 0.400902
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 6.80625 0.366436
\(346\) 0 0
\(347\) 14.2094 0.762799 0.381400 0.924410i \(-0.375442\pi\)
0.381400 + 0.924410i \(0.375442\pi\)
\(348\) 0 0
\(349\) 20.8062 1.11373 0.556866 0.830602i \(-0.312004\pi\)
0.556866 + 0.830602i \(0.312004\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 29.6125 1.57167
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 35.8062 1.88454
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 26.8062 1.40310
\(366\) 0 0
\(367\) 6.80625 0.355283 0.177642 0.984095i \(-0.443153\pi\)
0.177642 + 0.984095i \(0.443153\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −8.80625 −0.455970 −0.227985 0.973665i \(-0.573214\pi\)
−0.227985 + 0.973665i \(0.573214\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) −18.8062 −0.963473
\(382\) 0 0
\(383\) −32.0000 −1.63512 −0.817562 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −28.2094 −1.43027 −0.715136 0.698985i \(-0.753635\pi\)
−0.715136 + 0.698985i \(0.753635\pi\)
\(390\) 0 0
\(391\) 18.3875 0.929896
\(392\) 0 0
\(393\) −15.4031 −0.776985
\(394\) 0 0
\(395\) 21.6125 1.08744
\(396\) 0 0
\(397\) 25.4031 1.27495 0.637473 0.770473i \(-0.279980\pi\)
0.637473 + 0.770473i \(0.279980\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −16.8062 −0.839264 −0.419632 0.907694i \(-0.637841\pi\)
−0.419632 + 0.907694i \(0.637841\pi\)
\(402\) 0 0
\(403\) 7.40312 0.368776
\(404\) 0 0
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) −5.40312 −0.267823
\(408\) 0 0
\(409\) 22.5969 1.11734 0.558672 0.829389i \(-0.311311\pi\)
0.558672 + 0.829389i \(0.311311\pi\)
\(410\) 0 0
\(411\) −12.8062 −0.631686
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) −6.80625 −0.333303
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 32.2094 1.56979 0.784894 0.619630i \(-0.212717\pi\)
0.784894 + 0.619630i \(0.212717\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) 5.40312 0.262090
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.00000 −0.0482805
\(430\) 0 0
\(431\) 21.6125 1.04104 0.520519 0.853850i \(-0.325739\pi\)
0.520519 + 0.853850i \(0.325739\pi\)
\(432\) 0 0
\(433\) 38.4187 1.84629 0.923144 0.384455i \(-0.125611\pi\)
0.923144 + 0.384455i \(0.125611\pi\)
\(434\) 0 0
\(435\) −4.00000 −0.191785
\(436\) 0 0
\(437\) 25.1938 1.20518
\(438\) 0 0
\(439\) −18.8062 −0.897573 −0.448787 0.893639i \(-0.648144\pi\)
−0.448787 + 0.893639i \(0.648144\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) −2.80625 −0.133329 −0.0666644 0.997775i \(-0.521236\pi\)
−0.0666644 + 0.997775i \(0.521236\pi\)
\(444\) 0 0
\(445\) −20.0000 −0.948091
\(446\) 0 0
\(447\) −12.8062 −0.605715
\(448\) 0 0
\(449\) 7.19375 0.339494 0.169747 0.985488i \(-0.445705\pi\)
0.169747 + 0.985488i \(0.445705\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) 0 0
\(453\) 8.00000 0.375873
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −9.40312 −0.439860 −0.219930 0.975516i \(-0.570583\pi\)
−0.219930 + 0.975516i \(0.570583\pi\)
\(458\) 0 0
\(459\) 5.40312 0.252196
\(460\) 0 0
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) 0 0
\(463\) −23.4031 −1.08764 −0.543818 0.839203i \(-0.683022\pi\)
−0.543818 + 0.839203i \(0.683022\pi\)
\(464\) 0 0
\(465\) −14.8062 −0.686623
\(466\) 0 0
\(467\) −5.19375 −0.240338 −0.120169 0.992753i \(-0.538344\pi\)
−0.120169 + 0.992753i \(0.538344\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −4.80625 −0.221460
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 7.40312 0.339679
\(476\) 0 0
\(477\) 9.40312 0.430539
\(478\) 0 0
\(479\) −13.6125 −0.621971 −0.310985 0.950415i \(-0.600659\pi\)
−0.310985 + 0.950415i \(0.600659\pi\)
\(480\) 0 0
\(481\) −5.40312 −0.246361
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 20.0000 0.908153
\(486\) 0 0
\(487\) −23.4031 −1.06050 −0.530248 0.847842i \(-0.677901\pi\)
−0.530248 + 0.847842i \(0.677901\pi\)
\(488\) 0 0
\(489\) 14.8062 0.669562
\(490\) 0 0
\(491\) −22.2094 −1.00229 −0.501147 0.865362i \(-0.667089\pi\)
−0.501147 + 0.865362i \(0.667089\pi\)
\(492\) 0 0
\(493\) −10.8062 −0.486689
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 7.40312 0.326856
\(514\) 0 0
\(515\) −13.6125 −0.599838
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) 4.80625 0.210971
\(520\) 0 0
\(521\) −19.6125 −0.859239 −0.429620 0.903010i \(-0.641352\pi\)
−0.429620 + 0.903010i \(0.641352\pi\)
\(522\) 0 0
\(523\) 24.0000 1.04945 0.524723 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −40.0000 −1.74243
\(528\) 0 0
\(529\) −11.4187 −0.496467
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) −14.8062 −0.640130
\(536\) 0 0
\(537\) −20.0000 −0.863064
\(538\) 0 0
\(539\) −7.00000 −0.301511
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) 16.8062 0.721225
\(544\) 0 0
\(545\) 28.0000 1.19939
\(546\) 0 0
\(547\) 38.8062 1.65924 0.829618 0.558332i \(-0.188558\pi\)
0.829618 + 0.558332i \(0.188558\pi\)
\(548\) 0 0
\(549\) 12.8062 0.546557
\(550\) 0 0
\(551\) −14.8062 −0.630767
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 10.8062 0.458700
\(556\) 0 0
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 5.40312 0.228120
\(562\) 0 0
\(563\) −30.2094 −1.27317 −0.636587 0.771205i \(-0.719654\pi\)
−0.636587 + 0.771205i \(0.719654\pi\)
\(564\) 0 0
\(565\) −41.6125 −1.75065
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −44.2094 −1.85335 −0.926677 0.375860i \(-0.877348\pi\)
−0.926677 + 0.375860i \(0.877348\pi\)
\(570\) 0 0
\(571\) 24.0000 1.00437 0.502184 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) 0 0
\(573\) −12.5969 −0.526242
\(574\) 0 0
\(575\) 3.40312 0.141920
\(576\) 0 0
\(577\) 11.1938 0.466002 0.233001 0.972476i \(-0.425145\pi\)
0.233001 + 0.972476i \(0.425145\pi\)
\(578\) 0 0
\(579\) 17.4031 0.723249
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 9.40312 0.389438
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) −41.6125 −1.71753 −0.858766 0.512368i \(-0.828768\pi\)
−0.858766 + 0.512368i \(0.828768\pi\)
\(588\) 0 0
\(589\) −54.8062 −2.25825
\(590\) 0 0
\(591\) −12.8062 −0.526779
\(592\) 0 0
\(593\) 24.8062 1.01867 0.509335 0.860568i \(-0.329891\pi\)
0.509335 + 0.860568i \(0.329891\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) −4.59688 −0.187823 −0.0939116 0.995581i \(-0.529937\pi\)
−0.0939116 + 0.995581i \(0.529937\pi\)
\(600\) 0 0
\(601\) 0.806248 0.0328876 0.0164438 0.999865i \(-0.494766\pi\)
0.0164438 + 0.999865i \(0.494766\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −40.4187 −1.64055 −0.820273 0.571972i \(-0.806179\pi\)
−0.820273 + 0.571972i \(0.806179\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 35.6125 1.43838 0.719188 0.694816i \(-0.244514\pi\)
0.719188 + 0.694816i \(0.244514\pi\)
\(614\) 0 0
\(615\) −4.00000 −0.161296
\(616\) 0 0
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 0 0
\(619\) −30.8062 −1.23821 −0.619104 0.785309i \(-0.712504\pi\)
−0.619104 + 0.785309i \(0.712504\pi\)
\(620\) 0 0
\(621\) 3.40312 0.136563
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 7.40312 0.295652
\(628\) 0 0
\(629\) 29.1938 1.16403
\(630\) 0 0
\(631\) 37.0156 1.47357 0.736784 0.676128i \(-0.236343\pi\)
0.736784 + 0.676128i \(0.236343\pi\)
\(632\) 0 0
\(633\) 6.80625 0.270524
\(634\) 0 0
\(635\) 37.6125 1.49261
\(636\) 0 0
\(637\) −7.00000 −0.277350
\(638\) 0 0
\(639\) 14.8062 0.585726
\(640\) 0 0
\(641\) −7.19375 −0.284136 −0.142068 0.989857i \(-0.545375\pi\)
−0.142068 + 0.989857i \(0.545375\pi\)
\(642\) 0 0
\(643\) 24.0000 0.946468 0.473234 0.880937i \(-0.343087\pi\)
0.473234 + 0.880937i \(0.343087\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.20937 0.0868594 0.0434297 0.999056i \(-0.486172\pi\)
0.0434297 + 0.999056i \(0.486172\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 39.0156 1.52680 0.763400 0.645926i \(-0.223529\pi\)
0.763400 + 0.645926i \(0.223529\pi\)
\(654\) 0 0
\(655\) 30.8062 1.20370
\(656\) 0 0
\(657\) 13.4031 0.522906
\(658\) 0 0
\(659\) −6.20937 −0.241883 −0.120941 0.992660i \(-0.538591\pi\)
−0.120941 + 0.992660i \(0.538591\pi\)
\(660\) 0 0
\(661\) −5.40312 −0.210157 −0.105079 0.994464i \(-0.533509\pi\)
−0.105079 + 0.994464i \(0.533509\pi\)
\(662\) 0 0
\(663\) 5.40312 0.209840
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −6.80625 −0.263539
\(668\) 0 0
\(669\) 0.596876 0.0230766
\(670\) 0 0
\(671\) 12.8062 0.494380
\(672\) 0 0
\(673\) −27.6125 −1.06438 −0.532192 0.846624i \(-0.678631\pi\)
−0.532192 + 0.846624i \(0.678631\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −4.80625 −0.184719 −0.0923596 0.995726i \(-0.529441\pi\)
−0.0923596 + 0.995726i \(0.529441\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −2.80625 −0.107536
\(682\) 0 0
\(683\) −25.6125 −0.980035 −0.490017 0.871713i \(-0.663010\pi\)
−0.490017 + 0.871713i \(0.663010\pi\)
\(684\) 0 0
\(685\) 25.6125 0.978603
\(686\) 0 0
\(687\) −2.59688 −0.0990770
\(688\) 0 0
\(689\) 9.40312 0.358231
\(690\) 0 0
\(691\) 46.8062 1.78059 0.890297 0.455381i \(-0.150497\pi\)
0.890297 + 0.455381i \(0.150497\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13.6125 0.516351
\(696\) 0 0
\(697\) −10.8062 −0.409316
\(698\) 0 0
\(699\) −9.40312 −0.355659
\(700\) 0 0
\(701\) 24.8062 0.936919 0.468460 0.883485i \(-0.344809\pi\)
0.468460 + 0.883485i \(0.344809\pi\)
\(702\) 0 0
\(703\) 40.0000 1.50863
\(704\) 0 0
\(705\) −16.0000 −0.602595
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.209373 0.00786316 0.00393158 0.999992i \(-0.498749\pi\)
0.00393158 + 0.999992i \(0.498749\pi\)
\(710\) 0 0
\(711\) 10.8062 0.405266
\(712\) 0 0
\(713\) −25.1938 −0.943513
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 33.0156 1.23127 0.615637 0.788030i \(-0.288899\pi\)
0.615637 + 0.788030i \(0.288899\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −21.4031 −0.795991
\(724\) 0 0
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 6.80625 0.252430 0.126215 0.992003i \(-0.459717\pi\)
0.126215 + 0.992003i \(0.459717\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −15.6125 −0.576661 −0.288330 0.957531i \(-0.593100\pi\)
−0.288330 + 0.957531i \(0.593100\pi\)
\(734\) 0 0
\(735\) 14.0000 0.516398
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −22.2094 −0.816985 −0.408492 0.912762i \(-0.633945\pi\)
−0.408492 + 0.912762i \(0.633945\pi\)
\(740\) 0 0
\(741\) 7.40312 0.271961
\(742\) 0 0
\(743\) 13.6125 0.499394 0.249697 0.968324i \(-0.419669\pi\)
0.249697 + 0.968324i \(0.419669\pi\)
\(744\) 0 0
\(745\) 25.6125 0.938369
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −3.19375 −0.116079 −0.0580394 0.998314i \(-0.518485\pi\)
−0.0580394 + 0.998314i \(0.518485\pi\)
\(758\) 0 0
\(759\) 3.40312 0.123526
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −10.8062 −0.390701
\(766\) 0 0
\(767\) 4.00000 0.144432
\(768\) 0 0
\(769\) −1.40312 −0.0505980 −0.0252990 0.999680i \(-0.508054\pi\)
−0.0252990 + 0.999680i \(0.508054\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) −36.8062 −1.32383 −0.661914 0.749579i \(-0.730256\pi\)
−0.661914 + 0.749579i \(0.730256\pi\)
\(774\) 0 0
\(775\) −7.40312 −0.265928
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −14.8062 −0.530489
\(780\) 0 0
\(781\) 14.8062 0.529809
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 9.61250 0.343085
\(786\) 0 0
\(787\) 16.5969 0.591615 0.295807 0.955248i \(-0.404411\pi\)
0.295807 + 0.955248i \(0.404411\pi\)
\(788\) 0 0
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 12.8062 0.454763
\(794\) 0 0
\(795\) −18.8062 −0.666989
\(796\) 0 0
\(797\) −20.2094 −0.715853 −0.357926 0.933750i \(-0.616516\pi\)
−0.357926 + 0.933750i \(0.616516\pi\)
\(798\) 0 0
\(799\) −43.2250 −1.52919
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) 0 0
\(803\) 13.4031 0.472986
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −16.2094 −0.570597
\(808\) 0 0
\(809\) −14.5969 −0.513199 −0.256599 0.966518i \(-0.582602\pi\)
−0.256599 + 0.966518i \(0.582602\pi\)
\(810\) 0 0
\(811\) 14.2094 0.498959 0.249479 0.968380i \(-0.419741\pi\)
0.249479 + 0.968380i \(0.419741\pi\)
\(812\) 0 0
\(813\) −22.8062 −0.799850
\(814\) 0 0
\(815\) −29.6125 −1.03728
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −47.6125 −1.66169 −0.830844 0.556506i \(-0.812142\pi\)
−0.830844 + 0.556506i \(0.812142\pi\)
\(822\) 0 0
\(823\) 45.6125 1.58995 0.794976 0.606641i \(-0.207483\pi\)
0.794976 + 0.606641i \(0.207483\pi\)
\(824\) 0 0
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) −18.8062 −0.653957 −0.326979 0.945032i \(-0.606031\pi\)
−0.326979 + 0.945032i \(0.606031\pi\)
\(828\) 0 0
\(829\) 43.6125 1.51472 0.757362 0.652995i \(-0.226488\pi\)
0.757362 + 0.652995i \(0.226488\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 0 0
\(833\) 37.8219 1.31045
\(834\) 0 0
\(835\) 32.0000 1.10741
\(836\) 0 0
\(837\) −7.40312 −0.255889
\(838\) 0 0
\(839\) 29.6125 1.02234 0.511168 0.859481i \(-0.329213\pi\)
0.511168 + 0.859481i \(0.329213\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 14.0000 0.482186
\(844\) 0 0
\(845\) 2.00000 0.0688021
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) 18.3875 0.630315
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) −14.8062 −0.506363
\(856\) 0 0
\(857\) 17.4031 0.594479 0.297240 0.954803i \(-0.403934\pi\)
0.297240 + 0.954803i \(0.403934\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −52.4187 −1.78435 −0.892177 0.451685i \(-0.850823\pi\)
−0.892177 + 0.451685i \(0.850823\pi\)
\(864\) 0 0
\(865\) −9.61250 −0.326835
\(866\) 0 0
\(867\) −12.1938 −0.414122
\(868\) 0 0
\(869\) 10.8062 0.366577
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −38.4187 −1.29731 −0.648654 0.761083i \(-0.724668\pi\)
−0.648654 + 0.761083i \(0.724668\pi\)
\(878\) 0 0
\(879\) 0.806248 0.0271941
\(880\) 0 0
\(881\) −52.8062 −1.77909 −0.889544 0.456850i \(-0.848978\pi\)
−0.889544 + 0.456850i \(0.848978\pi\)
\(882\) 0 0
\(883\) 49.6125 1.66959 0.834797 0.550558i \(-0.185585\pi\)
0.834797 + 0.550558i \(0.185585\pi\)
\(884\) 0 0
\(885\) −8.00000 −0.268917
\(886\) 0 0
\(887\) 36.4187 1.22282 0.611411 0.791313i \(-0.290602\pi\)
0.611411 + 0.791313i \(0.290602\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) −59.2250 −1.98189
\(894\) 0 0
\(895\) 40.0000 1.33705
\(896\) 0 0
\(897\) 3.40312 0.113627
\(898\) 0 0
\(899\) 14.8062 0.493816
\(900\) 0 0
\(901\) −50.8062 −1.69260
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −33.6125 −1.11732
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 10.2094 0.338252 0.169126 0.985594i \(-0.445906\pi\)
0.169126 + 0.985594i \(0.445906\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) 0 0
\(915\) −25.6125 −0.846723
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 10.8062 0.356465 0.178233 0.983988i \(-0.442962\pi\)
0.178233 + 0.983988i \(0.442962\pi\)
\(920\) 0 0
\(921\) 16.5969 0.546886
\(922\) 0 0
\(923\) 14.8062 0.487354
\(924\) 0 0
\(925\) 5.40312 0.177654
\(926\) 0 0
\(927\) −6.80625 −0.223547
\(928\) 0 0
\(929\) 4.80625 0.157688 0.0788439 0.996887i \(-0.474877\pi\)
0.0788439 + 0.996887i \(0.474877\pi\)
\(930\) 0 0
\(931\) 51.8219 1.69839
\(932\) 0 0
\(933\) 10.2094 0.334240
\(934\) 0 0
\(935\) −10.8062 −0.353402
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 0 0
\(939\) 28.8062 0.940056
\(940\) 0 0
\(941\) 19.6125 0.639349 0.319675 0.947527i \(-0.396426\pi\)
0.319675 + 0.947527i \(0.396426\pi\)
\(942\) 0 0
\(943\) −6.80625 −0.221642
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 33.6125 1.09226 0.546130 0.837701i \(-0.316101\pi\)
0.546130 + 0.837701i \(0.316101\pi\)
\(948\) 0 0
\(949\) 13.4031 0.435084
\(950\) 0 0
\(951\) −26.0000 −0.843108
\(952\) 0 0
\(953\) −44.2094 −1.43208 −0.716041 0.698058i \(-0.754048\pi\)
−0.716041 + 0.698058i \(0.754048\pi\)
\(954\) 0 0
\(955\) 25.1938 0.815251
\(956\) 0 0
\(957\) −2.00000 −0.0646508
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 23.8062 0.767943
\(962\) 0 0
\(963\) −7.40312 −0.238562
\(964\) 0 0
\(965\) −34.8062 −1.12045
\(966\) 0 0
\(967\) −29.6125 −0.952274 −0.476137 0.879371i \(-0.657963\pi\)
−0.476137 + 0.879371i \(0.657963\pi\)
\(968\) 0 0
\(969\) −40.0000 −1.28499
\(970\) 0 0
\(971\) −10.8062 −0.346789 −0.173395 0.984852i \(-0.555474\pi\)
−0.173395 + 0.984852i \(0.555474\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) 3.61250 0.115574 0.0577870 0.998329i \(-0.481596\pi\)
0.0577870 + 0.998329i \(0.481596\pi\)
\(978\) 0 0
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) 36.4187 1.16158 0.580789 0.814054i \(-0.302744\pi\)
0.580789 + 0.814054i \(0.302744\pi\)
\(984\) 0 0
\(985\) 25.6125 0.816082
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) 16.0000 0.507745
\(994\) 0 0
\(995\) −32.0000 −1.01447
\(996\) 0 0
\(997\) 50.4187 1.59678 0.798389 0.602142i \(-0.205686\pi\)
0.798389 + 0.602142i \(0.205686\pi\)
\(998\) 0 0
\(999\) 5.40312 0.170947
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.bf.1.1 2
4.3 odd 2 858.2.a.q.1.1 2
12.11 even 2 2574.2.a.z.1.2 2
44.43 even 2 9438.2.a.bs.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
858.2.a.q.1.1 2 4.3 odd 2
2574.2.a.z.1.2 2 12.11 even 2
6864.2.a.bf.1.1 2 1.1 even 1 trivial
9438.2.a.bs.1.2 2 44.43 even 2