Properties

Label 5824.2.a.bf.1.1
Level $5824$
Weight $2$
Character 5824.1
Self dual yes
Analytic conductor $46.505$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5824,2,Mod(1,5824)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5824, 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("5824.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5824 = 2^{6} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5824.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: \(46.5048741372\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5824.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +4.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +4.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +12.0000 q^{15} -6.00000 q^{19} +3.00000 q^{21} +7.00000 q^{23} +11.0000 q^{25} +9.00000 q^{27} +4.00000 q^{29} -7.00000 q^{31} +3.00000 q^{33} +4.00000 q^{35} -9.00000 q^{37} +3.00000 q^{39} -3.00000 q^{41} +4.00000 q^{43} +24.0000 q^{45} -7.00000 q^{47} +1.00000 q^{49} +4.00000 q^{55} -18.0000 q^{57} -10.0000 q^{59} -1.00000 q^{61} +6.00000 q^{63} +4.00000 q^{65} +1.00000 q^{67} +21.0000 q^{69} -16.0000 q^{71} +5.00000 q^{73} +33.0000 q^{75} +1.00000 q^{77} -11.0000 q^{79} +9.00000 q^{81} +12.0000 q^{87} -6.00000 q^{89} +1.00000 q^{91} -21.0000 q^{93} -24.0000 q^{95} -1.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 12.0000 3.09839
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 7.00000 1.45960 0.729800 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) −9.00000 −1.47959 −0.739795 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) 0 0
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 24.0000 3.57771
\(46\) 0 0
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) −18.0000 −2.38416
\(58\) 0 0
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 1.00000 0.122169 0.0610847 0.998133i \(-0.480544\pi\)
0.0610847 + 0.998133i \(0.480544\pi\)
\(68\) 0 0
\(69\) 21.0000 2.52810
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) 5.00000 0.585206 0.292603 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) 0 0
\(75\) 33.0000 3.81051
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −11.0000 −1.23760 −0.618798 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −21.0000 −2.17760
\(94\) 0 0
\(95\) −24.0000 −2.46235
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 5.00000 0.497519 0.248759 0.968565i \(-0.419977\pi\)
0.248759 + 0.968565i \(0.419977\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) 12.0000 1.17108
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −27.0000 −2.56273
\(112\) 0 0
\(113\) −7.00000 −0.658505 −0.329252 0.944242i \(-0.606797\pi\)
−0.329252 + 0.944242i \(0.606797\pi\)
\(114\) 0 0
\(115\) 28.0000 2.61101
\(116\) 0 0
\(117\) 6.00000 0.554700
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) −9.00000 −0.811503
\(124\) 0 0
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) 11.0000 0.976092 0.488046 0.872818i \(-0.337710\pi\)
0.488046 + 0.872818i \(0.337710\pi\)
\(128\) 0 0
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) 36.0000 3.09839
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −21.0000 −1.76852
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) 16.0000 1.32873
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) −9.00000 −0.737309 −0.368654 0.929567i \(-0.620181\pi\)
−0.368654 + 0.929567i \(0.620181\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −28.0000 −2.24901
\(156\) 0 0
\(157\) −5.00000 −0.399043 −0.199522 0.979893i \(-0.563939\pi\)
−0.199522 + 0.979893i \(0.563939\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.00000 0.551677
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 12.0000 0.934199
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −36.0000 −2.75299
\(172\) 0 0
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 11.0000 0.831522
\(176\) 0 0
\(177\) −30.0000 −2.25494
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −15.0000 −1.11494 −0.557471 0.830197i \(-0.688228\pi\)
−0.557471 + 0.830197i \(0.688228\pi\)
\(182\) 0 0
\(183\) −3.00000 −0.221766
\(184\) 0 0
\(185\) −36.0000 −2.64677
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 9.00000 0.654654
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 20.0000 1.43963 0.719816 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) 0 0
\(195\) 12.0000 0.859338
\(196\) 0 0
\(197\) 27.0000 1.92367 0.961835 0.273629i \(-0.0882242\pi\)
0.961835 + 0.273629i \(0.0882242\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 3.00000 0.211604
\(202\) 0 0
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) 42.0000 2.91920
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 0 0
\(213\) −48.0000 −3.28891
\(214\) 0 0
\(215\) 16.0000 1.09119
\(216\) 0 0
\(217\) −7.00000 −0.475191
\(218\) 0 0
\(219\) 15.0000 1.01361
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −21.0000 −1.40626 −0.703132 0.711059i \(-0.748216\pi\)
−0.703132 + 0.711059i \(0.748216\pi\)
\(224\) 0 0
\(225\) 66.0000 4.40000
\(226\) 0 0
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) 24.0000 1.58596 0.792982 0.609245i \(-0.208527\pi\)
0.792982 + 0.609245i \(0.208527\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) −5.00000 −0.327561 −0.163780 0.986497i \(-0.552369\pi\)
−0.163780 + 0.986497i \(0.552369\pi\)
\(234\) 0 0
\(235\) −28.0000 −1.82652
\(236\) 0 0
\(237\) −33.0000 −2.14358
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.00000 0.255551
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) 0 0
\(253\) 7.00000 0.440086
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.0000 1.49708 0.748539 0.663090i \(-0.230755\pi\)
0.748539 + 0.663090i \(0.230755\pi\)
\(258\) 0 0
\(259\) −9.00000 −0.559233
\(260\) 0 0
\(261\) 24.0000 1.48556
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 0 0
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 0 0
\(271\) 23.0000 1.39715 0.698575 0.715537i \(-0.253818\pi\)
0.698575 + 0.715537i \(0.253818\pi\)
\(272\) 0 0
\(273\) 3.00000 0.181568
\(274\) 0 0
\(275\) 11.0000 0.663325
\(276\) 0 0
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 0 0
\(279\) −42.0000 −2.51447
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) 19.0000 1.12943 0.564716 0.825285i \(-0.308986\pi\)
0.564716 + 0.825285i \(0.308986\pi\)
\(284\) 0 0
\(285\) −72.0000 −4.26491
\(286\) 0 0
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −3.00000 −0.175863
\(292\) 0 0
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 0 0
\(295\) −40.0000 −2.32889
\(296\) 0 0
\(297\) 9.00000 0.522233
\(298\) 0 0
\(299\) 7.00000 0.404820
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) 15.0000 0.861727
\(304\) 0 0
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 42.0000 2.38930
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 24.0000 1.35225
\(316\) 0 0
\(317\) −21.0000 −1.17948 −0.589739 0.807594i \(-0.700769\pi\)
−0.589739 + 0.807594i \(0.700769\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 11.0000 0.610170
\(326\) 0 0
\(327\) 42.0000 2.32261
\(328\) 0 0
\(329\) −7.00000 −0.385922
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 0 0
\(333\) −54.0000 −2.95918
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 17.0000 0.926049 0.463025 0.886345i \(-0.346764\pi\)
0.463025 + 0.886345i \(0.346764\pi\)
\(338\) 0 0
\(339\) −21.0000 −1.14056
\(340\) 0 0
\(341\) −7.00000 −0.379071
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 84.0000 4.52241
\(346\) 0 0
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 9.00000 0.480384
\(352\) 0 0
\(353\) 25.0000 1.33062 0.665308 0.746569i \(-0.268300\pi\)
0.665308 + 0.746569i \(0.268300\pi\)
\(354\) 0 0
\(355\) −64.0000 −3.39677
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) −30.0000 −1.57459
\(364\) 0 0
\(365\) 20.0000 1.04685
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 0 0
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 16.0000 0.828449 0.414224 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) 0 0
\(375\) 72.0000 3.71806
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 33.0000 1.69064
\(382\) 0 0
\(383\) 15.0000 0.766464 0.383232 0.923652i \(-0.374811\pi\)
0.383232 + 0.923652i \(0.374811\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) 24.0000 1.21999
\(388\) 0 0
\(389\) 36.0000 1.82527 0.912636 0.408773i \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 24.0000 1.21064
\(394\) 0 0
\(395\) −44.0000 −2.21388
\(396\) 0 0
\(397\) 28.0000 1.40528 0.702640 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) 0 0
\(399\) −18.0000 −0.901127
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) −7.00000 −0.348695
\(404\) 0 0
\(405\) 36.0000 1.78885
\(406\) 0 0
\(407\) −9.00000 −0.446113
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) −10.0000 −0.492068
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) 15.0000 0.732798 0.366399 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) −42.0000 −2.04211
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.00000 −0.0483934
\(428\) 0 0
\(429\) 3.00000 0.144841
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 0 0
\(435\) 48.0000 2.30142
\(436\) 0 0
\(437\) −42.0000 −2.00913
\(438\) 0 0
\(439\) −2.00000 −0.0954548 −0.0477274 0.998860i \(-0.515198\pi\)
−0.0477274 + 0.998860i \(0.515198\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) −24.0000 −1.13771
\(446\) 0 0
\(447\) −27.0000 −1.27706
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) −3.00000 −0.141264
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) −84.0000 −3.89541
\(466\) 0 0
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 0 0
\(469\) 1.00000 0.0461757
\(470\) 0 0
\(471\) −15.0000 −0.691164
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) −66.0000 −3.02829
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −9.00000 −0.410365
\(482\) 0 0
\(483\) 21.0000 0.955533
\(484\) 0 0
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) −38.0000 −1.72194 −0.860972 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 16.0000 0.722070 0.361035 0.932552i \(-0.382424\pi\)
0.361035 + 0.932552i \(0.382424\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 24.0000 1.07872
\(496\) 0 0
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) −5.00000 −0.223831 −0.111915 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 20.0000 0.891756 0.445878 0.895094i \(-0.352892\pi\)
0.445878 + 0.895094i \(0.352892\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 0 0
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 5.00000 0.221187
\(512\) 0 0
\(513\) −54.0000 −2.38416
\(514\) 0 0
\(515\) 56.0000 2.46765
\(516\) 0 0
\(517\) −7.00000 −0.307860
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 27.0000 1.18063 0.590314 0.807174i \(-0.299004\pi\)
0.590314 + 0.807174i \(0.299004\pi\)
\(524\) 0 0
\(525\) 33.0000 1.44024
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) −60.0000 −2.60378
\(532\) 0 0
\(533\) −3.00000 −0.129944
\(534\) 0 0
\(535\) −16.0000 −0.691740
\(536\) 0 0
\(537\) −18.0000 −0.776757
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) −45.0000 −1.93113
\(544\) 0 0
\(545\) 56.0000 2.39878
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) −11.0000 −0.467768
\(554\) 0 0
\(555\) −108.000 −4.58434
\(556\) 0 0
\(557\) −9.00000 −0.381342 −0.190671 0.981654i \(-0.561066\pi\)
−0.190671 + 0.981654i \(0.561066\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 11.0000 0.463595 0.231797 0.972764i \(-0.425539\pi\)
0.231797 + 0.972764i \(0.425539\pi\)
\(564\) 0 0
\(565\) −28.0000 −1.17797
\(566\) 0 0
\(567\) 9.00000 0.377964
\(568\) 0 0
\(569\) −35.0000 −1.46728 −0.733638 0.679540i \(-0.762179\pi\)
−0.733638 + 0.679540i \(0.762179\pi\)
\(570\) 0 0
\(571\) 6.00000 0.251092 0.125546 0.992088i \(-0.459932\pi\)
0.125546 + 0.992088i \(0.459932\pi\)
\(572\) 0 0
\(573\) 24.0000 1.00261
\(574\) 0 0
\(575\) 77.0000 3.21112
\(576\) 0 0
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) 0 0
\(579\) 60.0000 2.49351
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 24.0000 0.992278
\(586\) 0 0
\(587\) 14.0000 0.577842 0.288921 0.957353i \(-0.406704\pi\)
0.288921 + 0.957353i \(0.406704\pi\)
\(588\) 0 0
\(589\) 42.0000 1.73058
\(590\) 0 0
\(591\) 81.0000 3.33189
\(592\) 0 0
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 12.0000 0.491127
\(598\) 0 0
\(599\) 29.0000 1.18491 0.592454 0.805604i \(-0.298159\pi\)
0.592454 + 0.805604i \(0.298159\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 6.00000 0.244339
\(604\) 0 0
\(605\) −40.0000 −1.62623
\(606\) 0 0
\(607\) −6.00000 −0.243532 −0.121766 0.992559i \(-0.538856\pi\)
−0.121766 + 0.992559i \(0.538856\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) −7.00000 −0.283190
\(612\) 0 0
\(613\) 49.0000 1.97909 0.989546 0.144220i \(-0.0460672\pi\)
0.989546 + 0.144220i \(0.0460672\pi\)
\(614\) 0 0
\(615\) −36.0000 −1.45166
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) 22.0000 0.884255 0.442127 0.896952i \(-0.354224\pi\)
0.442127 + 0.896952i \(0.354224\pi\)
\(620\) 0 0
\(621\) 63.0000 2.52810
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) −18.0000 −0.718851
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 2.00000 0.0796187 0.0398094 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) 0 0
\(633\) −30.0000 −1.19239
\(634\) 0 0
\(635\) 44.0000 1.74609
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −96.0000 −3.79770
\(640\) 0 0
\(641\) 7.00000 0.276483 0.138242 0.990399i \(-0.455855\pi\)
0.138242 + 0.990399i \(0.455855\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 0 0
\(645\) 48.0000 1.89000
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) −10.0000 −0.392534
\(650\) 0 0
\(651\) −21.0000 −0.823055
\(652\) 0 0
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) 32.0000 1.25034
\(656\) 0 0
\(657\) 30.0000 1.17041
\(658\) 0 0
\(659\) −18.0000 −0.701180 −0.350590 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) 0 0
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −24.0000 −0.930680
\(666\) 0 0
\(667\) 28.0000 1.08416
\(668\) 0 0
\(669\) −63.0000 −2.43572
\(670\) 0 0
\(671\) −1.00000 −0.0386046
\(672\) 0 0
\(673\) 7.00000 0.269830 0.134915 0.990857i \(-0.456924\pi\)
0.134915 + 0.990857i \(0.456924\pi\)
\(674\) 0 0
\(675\) 99.0000 3.81051
\(676\) 0 0
\(677\) −39.0000 −1.49889 −0.749446 0.662066i \(-0.769680\pi\)
−0.749446 + 0.662066i \(0.769680\pi\)
\(678\) 0 0
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) 72.0000 2.75905
\(682\) 0 0
\(683\) 1.00000 0.0382639 0.0191320 0.999817i \(-0.493910\pi\)
0.0191320 + 0.999817i \(0.493910\pi\)
\(684\) 0 0
\(685\) −24.0000 −0.916993
\(686\) 0 0
\(687\) 72.0000 2.74697
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 0 0
\(693\) 6.00000 0.227921
\(694\) 0 0
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −15.0000 −0.567352
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 54.0000 2.03665
\(704\) 0 0
\(705\) −84.0000 −3.16362
\(706\) 0 0
\(707\) 5.00000 0.188044
\(708\) 0 0
\(709\) 13.0000 0.488225 0.244113 0.969747i \(-0.421503\pi\)
0.244113 + 0.969747i \(0.421503\pi\)
\(710\) 0 0
\(711\) −66.0000 −2.47519
\(712\) 0 0
\(713\) −49.0000 −1.83506
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 0 0
\(717\) −18.0000 −0.672222
\(718\) 0 0
\(719\) −2.00000 −0.0745874 −0.0372937 0.999304i \(-0.511874\pi\)
−0.0372937 + 0.999304i \(0.511874\pi\)
\(720\) 0 0
\(721\) 14.0000 0.521387
\(722\) 0 0
\(723\) −54.0000 −2.00828
\(724\) 0 0
\(725\) 44.0000 1.63412
\(726\) 0 0
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 0 0
\(735\) 12.0000 0.442627
\(736\) 0 0
\(737\) 1.00000 0.0368355
\(738\) 0 0
\(739\) 24.0000 0.882854 0.441427 0.897297i \(-0.354472\pi\)
0.441427 + 0.897297i \(0.354472\pi\)
\(740\) 0 0
\(741\) −18.0000 −0.661247
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) −36.0000 −1.31894
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 27.0000 0.985244 0.492622 0.870243i \(-0.336039\pi\)
0.492622 + 0.870243i \(0.336039\pi\)
\(752\) 0 0
\(753\) 9.00000 0.327978
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −54.0000 −1.96266 −0.981332 0.192323i \(-0.938398\pi\)
−0.981332 + 0.192323i \(0.938398\pi\)
\(758\) 0 0
\(759\) 21.0000 0.762252
\(760\) 0 0
\(761\) 45.0000 1.63125 0.815624 0.578582i \(-0.196394\pi\)
0.815624 + 0.578582i \(0.196394\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −10.0000 −0.361079
\(768\) 0 0
\(769\) 21.0000 0.757279 0.378640 0.925544i \(-0.376392\pi\)
0.378640 + 0.925544i \(0.376392\pi\)
\(770\) 0 0
\(771\) 72.0000 2.59302
\(772\) 0 0
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 0 0
\(775\) −77.0000 −2.76592
\(776\) 0 0
\(777\) −27.0000 −0.968620
\(778\) 0 0
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 36.0000 1.28654
\(784\) 0 0
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 0 0
\(789\) −72.0000 −2.56327
\(790\) 0 0
\(791\) −7.00000 −0.248891
\(792\) 0 0
\(793\) −1.00000 −0.0355110
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −15.0000 −0.531327 −0.265664 0.964066i \(-0.585591\pi\)
−0.265664 + 0.964066i \(0.585591\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −36.0000 −1.27200
\(802\) 0 0
\(803\) 5.00000 0.176446
\(804\) 0 0
\(805\) 28.0000 0.986870
\(806\) 0 0
\(807\) −27.0000 −0.950445
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 69.0000 2.41994
\(814\) 0 0
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) −24.0000 −0.839654
\(818\) 0 0
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 0 0
\(823\) 3.00000 0.104573 0.0522867 0.998632i \(-0.483349\pi\)
0.0522867 + 0.998632i \(0.483349\pi\)
\(824\) 0 0
\(825\) 33.0000 1.14891
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 54.0000 1.87324
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −63.0000 −2.17760
\(838\) 0 0
\(839\) −17.0000 −0.586905 −0.293453 0.955974i \(-0.594804\pi\)
−0.293453 + 0.955974i \(0.594804\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) −24.0000 −0.826604
\(844\) 0 0
\(845\) 4.00000 0.137604
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 0 0
\(849\) 57.0000 1.95623
\(850\) 0 0
\(851\) −63.0000 −2.15961
\(852\) 0 0
\(853\) −16.0000 −0.547830 −0.273915 0.961754i \(-0.588319\pi\)
−0.273915 + 0.961754i \(0.588319\pi\)
\(854\) 0 0
\(855\) −144.000 −4.92470
\(856\) 0 0
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) 5.00000 0.170598 0.0852989 0.996355i \(-0.472815\pi\)
0.0852989 + 0.996355i \(0.472815\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) 0 0
\(863\) −28.0000 −0.953131 −0.476566 0.879139i \(-0.658119\pi\)
−0.476566 + 0.879139i \(0.658119\pi\)
\(864\) 0 0
\(865\) −8.00000 −0.272008
\(866\) 0 0
\(867\) −51.0000 −1.73205
\(868\) 0 0
\(869\) −11.0000 −0.373149
\(870\) 0 0
\(871\) 1.00000 0.0338837
\(872\) 0 0
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) 45.0000 1.51954 0.759771 0.650191i \(-0.225311\pi\)
0.759771 + 0.650191i \(0.225311\pi\)
\(878\) 0 0
\(879\) 78.0000 2.63087
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) −14.0000 −0.471138 −0.235569 0.971858i \(-0.575695\pi\)
−0.235569 + 0.971858i \(0.575695\pi\)
\(884\) 0 0
\(885\) −120.000 −4.03376
\(886\) 0 0
\(887\) 38.0000 1.27592 0.637958 0.770072i \(-0.279780\pi\)
0.637958 + 0.770072i \(0.279780\pi\)
\(888\) 0 0
\(889\) 11.0000 0.368928
\(890\) 0 0
\(891\) 9.00000 0.301511
\(892\) 0 0
\(893\) 42.0000 1.40548
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) 21.0000 0.701170
\(898\) 0 0
\(899\) −28.0000 −0.933852
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 12.0000 0.399335
\(904\) 0 0
\(905\) −60.0000 −1.99447
\(906\) 0 0
\(907\) −30.0000 −0.996134 −0.498067 0.867139i \(-0.665957\pi\)
−0.498067 + 0.867139i \(0.665957\pi\)
\(908\) 0 0
\(909\) 30.0000 0.995037
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −12.0000 −0.396708
\(916\) 0 0
\(917\) 8.00000 0.264183
\(918\) 0 0
\(919\) −27.0000 −0.890648 −0.445324 0.895370i \(-0.646911\pi\)
−0.445324 + 0.895370i \(0.646911\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 0 0
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) −99.0000 −3.25510
\(926\) 0 0
\(927\) 84.0000 2.75892
\(928\) 0 0
\(929\) −15.0000 −0.492134 −0.246067 0.969253i \(-0.579138\pi\)
−0.246067 + 0.969253i \(0.579138\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) −90.0000 −2.94647
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −36.0000 −1.17607 −0.588034 0.808836i \(-0.700098\pi\)
−0.588034 + 0.808836i \(0.700098\pi\)
\(938\) 0 0
\(939\) 42.0000 1.37062
\(940\) 0 0
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 0 0
\(943\) −21.0000 −0.683854
\(944\) 0 0
\(945\) 36.0000 1.17108
\(946\) 0 0
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 0 0
\(949\) 5.00000 0.162307
\(950\) 0 0
\(951\) −63.0000 −2.04291
\(952\) 0 0
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) 32.0000 1.03550
\(956\) 0 0
\(957\) 12.0000 0.387905
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) −24.0000 −0.773389
\(964\) 0 0
\(965\) 80.0000 2.57529
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) 0 0
\(975\) 33.0000 1.05685
\(976\) 0 0
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 84.0000 2.68191
\(982\) 0 0
\(983\) 40.0000 1.27580 0.637901 0.770118i \(-0.279803\pi\)
0.637901 + 0.770118i \(0.279803\pi\)
\(984\) 0 0
\(985\) 108.000 3.44117
\(986\) 0 0
\(987\) −21.0000 −0.668437
\(988\) 0 0
\(989\) 28.0000 0.890348
\(990\) 0 0
\(991\) 57.0000 1.81066 0.905332 0.424704i \(-0.139622\pi\)
0.905332 + 0.424704i \(0.139622\pi\)
\(992\) 0 0
\(993\) −21.0000 −0.666415
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) −5.00000 −0.158352 −0.0791758 0.996861i \(-0.525229\pi\)
−0.0791758 + 0.996861i \(0.525229\pi\)
\(998\) 0 0
\(999\) −81.0000 −2.56273
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 5824.2.a.bf.1.1 1
4.3 odd 2 5824.2.a.b.1.1 1
8.3 odd 2 182.2.a.e.1.1 1
8.5 even 2 1456.2.a.a.1.1 1
24.11 even 2 1638.2.a.j.1.1 1
40.19 odd 2 4550.2.a.a.1.1 1
56.3 even 6 1274.2.f.k.79.1 2
56.11 odd 6 1274.2.f.b.79.1 2
56.19 even 6 1274.2.f.k.1145.1 2
56.27 even 2 1274.2.a.h.1.1 1
56.51 odd 6 1274.2.f.b.1145.1 2
104.51 odd 2 2366.2.a.h.1.1 1
104.83 even 4 2366.2.d.j.337.1 2
104.99 even 4 2366.2.d.j.337.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.a.e.1.1 1 8.3 odd 2
1274.2.a.h.1.1 1 56.27 even 2
1274.2.f.b.79.1 2 56.11 odd 6
1274.2.f.b.1145.1 2 56.51 odd 6
1274.2.f.k.79.1 2 56.3 even 6
1274.2.f.k.1145.1 2 56.19 even 6
1456.2.a.a.1.1 1 8.5 even 2
1638.2.a.j.1.1 1 24.11 even 2
2366.2.a.h.1.1 1 104.51 odd 2
2366.2.d.j.337.1 2 104.83 even 4
2366.2.d.j.337.2 2 104.99 even 4
4550.2.a.a.1.1 1 40.19 odd 2
5824.2.a.b.1.1 1 4.3 odd 2
5824.2.a.bf.1.1 1 1.1 even 1 trivial