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Properties

Label 254.2.a.d.1.1
Level $254$
Weight $2$
Character 254.1
Self dual yes
Analytic conductor $2.028$
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] = [254,2,Mod(1,254)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(254, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("254.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 254 = 2 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 254.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: \(2.02820021134\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 254.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{8} -3.00000 q^{9} +2.00000 q^{10} +4.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} -3.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} +4.00000 q^{22} -1.00000 q^{25} -2.00000 q^{26} -6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} -3.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} +2.00000 q^{40} -6.00000 q^{41} +4.00000 q^{44} -6.00000 q^{45} -8.00000 q^{47} -7.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -6.00000 q^{53} +8.00000 q^{55} -6.00000 q^{58} +8.00000 q^{59} -2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -8.00000 q^{67} +2.00000 q^{68} -3.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} -4.00000 q^{76} +16.0000 q^{79} +2.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} +4.00000 q^{85} +4.00000 q^{88} -6.00000 q^{89} -6.00000 q^{90} -8.00000 q^{94} -8.00000 q^{95} +10.0000 q^{97} -7.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 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 \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) −3.00000 −1.00000
\(10\) 2.00000 0.632456
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −3.00000 −0.707107
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 4.00000 0.603023
\(45\) −6.00000 −0.894427
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −3.00000 −0.353553
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 2.00000 0.223607
\(81\) 9.00000 1.00000
\(82\) −6.00000 −0.662589
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −6.00000 −0.632456
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −7.00000 −0.707107
\(99\) −12.0000 −1.20605
\(100\) −1.00000 −0.100000
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 − 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 8.00000 0.762770
\(111\) 0 0
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 6.00000 0.554700
\(118\) 8.00000 0.736460
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −14.0000 −1.19610 −0.598050 − 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 − 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) −3.00000 −0.250000
\(145\) −12.0000 −0.996546
\(146\) 10.0000 0.827606
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −2.00000 −0.163846 −0.0819232 − 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −4.00000 −0.324443
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 16.0000 1.28515
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 16.0000 1.27289
\(159\) 0 0
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 9.00000 0.707107
\(163\) 20.0000 1.56652 0.783260 − 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 − 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 4.00000 0.306786
\(171\) 12.0000 0.917663
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −20.0000 −1.49487 −0.747435 − 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) −6.00000 −0.447214
\(181\) 26.0000 1.93256 0.966282 − 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 − 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) −10.0000 −0.712470 −0.356235 − 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) −12.0000 −0.852803
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 8.00000 0.539360
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 2.00000 0.133038
\(227\) −20.0000 −1.32745 −0.663723 − 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 − 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −22.0000 −1.44127 −0.720634 − 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 6.00000 0.392232
\(235\) −16.0000 −1.04372
\(236\) 8.00000 0.520756
\(237\) 0 0
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 − 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) −14.0000 −0.894427
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.00000 −0.0627456
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 − 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −4.00000 −0.248069
\(261\) 18.0000 1.11417
\(262\) 4.00000 0.247121
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 30.0000 1.82913 0.914566 − 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −14.0000 −0.845771
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 − 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 16.0000 0.959616
\(279\) −24.0000 −1.43684
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 − 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) −24.0000 −1.42665 −0.713326 − 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) −3.00000 −0.176777
\(289\) −13.0000 −0.764706
\(290\) −12.0000 −0.704664
\(291\) 0 0
\(292\) 10.0000 0.585206
\(293\) −14.0000 −0.817889 −0.408944 − 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −2.00000 −0.115857
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) −4.00000 −0.229039
\(306\) −6.00000 −0.342997
\(307\) −8.00000 −0.456584 −0.228292 − 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 16.0000 0.908739
\(311\) 16.0000 0.907277 0.453638 − 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 − 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) 26.0000 1.46031 0.730153 − 0.683284i \(-0.239449\pi\)
0.730153 + 0.683284i \(0.239449\pi\)
\(318\) 0 0
\(319\) −24.0000 −1.34374
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 9.00000 0.500000
\(325\) 2.00000 0.110940
\(326\) 20.0000 1.10770
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 16.0000 0.879440 0.439720 − 0.898135i \(-0.355078\pi\)
0.439720 + 0.898135i \(0.355078\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 8.00000 0.437741
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 − 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) 32.0000 1.73290
\(342\) 12.0000 0.648886
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 18.0000 0.963518 0.481759 − 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) −14.0000 −0.745145 −0.372572 − 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −6.00000 −0.316228
\(361\) −3.00000 −0.157895
\(362\) 26.0000 1.36653
\(363\) 0 0
\(364\) 0 0
\(365\) 20.0000 1.04685
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 − 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) −4.00000 −0.207950
\(371\) 0 0
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 − 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) 0 0
\(383\) 32.0000 1.63512 0.817562 − 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) 0 0
\(388\) 10.0000 0.507673
\(389\) 6.00000 0.304212 0.152106 − 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −7.00000 −0.353553
\(393\) 0 0
\(394\) −10.0000 −0.503793
\(395\) 32.0000 1.61009
\(396\) −12.0000 −0.603023
\(397\) 38.0000 1.90717 0.953583 − 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −22.0000 −1.09863 −0.549314 − 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) −16.0000 −0.797017
\(404\) 10.0000 0.497519
\(405\) 18.0000 0.894427
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 34.0000 1.68119 0.840596 − 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) −12.0000 −0.592638
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) −16.0000 −0.782586
\(419\) 4.00000 0.195413 0.0977064 − 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 − 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 4.00000 0.194717
\(423\) 24.0000 1.16692
\(424\) −6.00000 −0.291386
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 18.0000 0.865025 0.432512 − 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 0 0
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 − 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 8.00000 0.381385
\(441\) 21.0000 1.00000
\(442\) −4.00000 −0.190261
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 3.00000 0.141421
\(451\) −24.0000 −1.13012
\(452\) 2.00000 0.0940721
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 10.0000 0.467269
\(459\) 0 0
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 − 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 − 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) −8.00000 −0.370196 −0.185098 − 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) −16.0000 −0.738025
\(471\) 0 0
\(472\) 8.00000 0.368230
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 18.0000 0.824163
\(478\) 8.00000 0.365911
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 20.0000 0.908153
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 − 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) −14.0000 −0.632456
\(491\) 40.0000 1.80517 0.902587 − 0.430507i \(-0.141665\pi\)
0.902587 + 0.430507i \(0.141665\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 8.00000 0.359937
\(495\) −24.0000 −1.07872
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) −24.0000 −1.07439 −0.537194 − 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 0 0
\(507\) 0 0
\(508\) −1.00000 −0.0443678
\(509\) 6.00000 0.265945 0.132973 − 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) 0 0
\(517\) −32.0000 −1.40736
\(518\) 0 0
\(519\) 0 0
\(520\) −4.00000 −0.175412
\(521\) −22.0000 −0.963837 −0.481919 − 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 18.0000 0.787839
\(523\) −28.0000 −1.22435 −0.612177 − 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −12.0000 −0.521247
\(531\) −24.0000 −1.04151
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 30.0000 1.29339
\(539\) −28.0000 −1.20605
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 − 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 − 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −14.0000 −0.598050
\(549\) 6.00000 0.256074
\(550\) −4.00000 −0.170561
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) −10.0000 −0.423714 −0.211857 − 0.977301i \(-0.567951\pi\)
−0.211857 + 0.977301i \(0.567951\pi\)
\(558\) −24.0000 −1.01600
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −22.0000 −0.928014
\(563\) 32.0000 1.34864 0.674320 − 0.738440i \(-0.264437\pi\)
0.674320 + 0.738440i \(0.264437\pi\)
\(564\) 0 0
\(565\) 4.00000 0.168281
\(566\) −24.0000 −1.00880
\(567\) 0 0
\(568\) 0 0
\(569\) 42.0000 1.76073 0.880366 − 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 − 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −3.00000 −0.125000
\(577\) −30.0000 −1.24892 −0.624458 − 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) −12.0000 −0.498273
\(581\) 0 0
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) 10.0000 0.413803
\(585\) 12.0000 0.496139
\(586\) −14.0000 −0.578335
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 16.0000 0.658710
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −2.00000 −0.0819232
\(597\) 0 0
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 − 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 − 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 24.0000 0.977356
\(604\) 8.00000 0.325515
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) −48.0000 −1.94826 −0.974130 − 0.225989i \(-0.927439\pi\)
−0.974130 + 0.225989i \(0.927439\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) 16.0000 0.647291
\(612\) −6.00000 −0.242536
\(613\) −14.0000 −0.565455 −0.282727 − 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 16.0000 0.642575
\(621\) 0 0
\(622\) 16.0000 0.641542
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 − 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 16.0000 0.636446
\(633\) 0 0
\(634\) 26.0000 1.03259
\(635\) −2.00000 −0.0793676
\(636\) 0 0
\(637\) 14.0000 0.554700
\(638\) −24.0000 −0.950169
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) 26.0000 1.02694 0.513469 − 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) −36.0000 −1.41970 −0.709851 − 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 9.00000 0.353553
\(649\) 32.0000 1.25611
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) −26.0000 −1.01746 −0.508729 − 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 0 0
\(655\) 8.00000 0.312586
\(656\) −6.00000 −0.234261
\(657\) −30.0000 −1.17041
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 16.0000 0.621858
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 0 0
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) −16.0000 −0.618134
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 18.0000 0.693849 0.346925 − 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) 18.0000 0.693334
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 30.0000 1.15299 0.576497 − 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) 0 0
\(682\) 32.0000 1.22534
\(683\) 40.0000 1.53056 0.765279 − 0.643699i \(-0.222601\pi\)
0.765279 + 0.643699i \(0.222601\pi\)
\(684\) 12.0000 0.458831
\(685\) −28.0000 −1.06983
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 − 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 0 0
\(695\) 32.0000 1.21383
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 18.0000 0.681310
\(699\) 0 0
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 − 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 0 0
\(708\) 0 0
\(709\) 46.0000 1.72757 0.863783 − 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) −48.0000 −1.80014
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) −20.0000 −0.747435
\(717\) 0 0
\(718\) 0 0
\(719\) −40.0000 −1.49175 −0.745874 − 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) −6.00000 −0.223607
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) 26.0000 0.966282
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 20.0000 0.740233
\(731\) 0 0
\(732\) 0 0
\(733\) 46.0000 1.69905 0.849524 − 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) −32.0000 −1.17874
\(738\) 18.0000 0.662589
\(739\) 12.0000 0.441427 0.220714 − 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 − 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) −4.00000 −0.146549
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 − 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −12.0000 −0.433861
\(766\) 32.0000 1.15621
\(767\) −16.0000 −0.577727
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 − 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −22.0000 −0.791797
\(773\) −42.0000 −1.51064 −0.755318 − 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −7.00000 −0.250000
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −10.0000 −0.356235
\(789\) 0 0
\(790\) 32.0000 1.13851
\(791\) 0 0
\(792\) −12.0000 −0.426401
\(793\) 4.00000 0.142044
\(794\) 38.0000 1.34857
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) −42.0000 −1.48772 −0.743858 − 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) −1.00000 −0.0353553
\(801\) 18.0000 0.635999
\(802\) −22.0000 −0.776847
\(803\) 40.0000 1.41157
\(804\) 0 0
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 0 0
\(808\) 10.0000 0.351799
\(809\) −54.0000 −1.89854 −0.949269 − 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 18.0000 0.632456
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) 40.0000 1.40114
\(816\) 0 0
\(817\) 0 0
\(818\) 34.0000 1.18878
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) −22.0000 −0.767805 −0.383903 − 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 − 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −32.0000 −1.11275 −0.556375 − 0.830932i \(-0.687808\pi\)
−0.556375 + 0.830932i \(0.687808\pi\)
\(828\) 0 0
\(829\) 26.0000 0.903017 0.451509 − 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) 16.0000 0.553703
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) 4.00000 0.138178
\(839\) −24.0000 −0.828572 −0.414286 − 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −6.00000 −0.206774
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) −18.0000 −0.619219
\(846\) 24.0000 0.825137
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 0 0
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) 24.0000 0.820783
\(856\) 12.0000 0.410152
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −8.00000 −0.272956 −0.136478 − 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 36.0000 1.22404
\(866\) 18.0000 0.611665
\(867\) 0 0
\(868\) 0 0
\(869\) 64.0000 2.17105
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 10.0000 0.338643
\(873\) −30.0000 −1.01535
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 6.00000 0.202606 0.101303 − 0.994856i \(-0.467699\pi\)
0.101303 + 0.994856i \(0.467699\pi\)
\(878\) 16.0000 0.539974
\(879\) 0 0
\(880\) 8.00000 0.269680
\(881\) −46.0000 −1.54978 −0.774890 − 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) 21.0000 0.707107
\(883\) −20.0000 −0.673054 −0.336527 − 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 56.0000 1.88030 0.940148 − 0.340766i \(-0.110687\pi\)
0.940148 + 0.340766i \(0.110687\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −12.0000 −0.402241
\(891\) 36.0000 1.20605
\(892\) −16.0000 −0.535720
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) −40.0000 −1.33705
\(896\) 0 0
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) −48.0000 −1.60089
\(900\) 3.00000 0.100000
\(901\) −12.0000 −0.399778
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 52.0000 1.72854
\(906\) 0 0
\(907\) −4.00000 −0.132818 −0.0664089 − 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) −20.0000 −0.663723
\(909\) −30.0000 −0.995037
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 − 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 2.00000 0.0658665
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) 10.0000 0.328089 0.164045 − 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 0 0
\(931\) 28.0000 0.917663
\(932\) −22.0000 −0.720634
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) 16.0000 0.523256
\(936\) 6.00000 0.196116
\(937\) −22.0000 −0.718709 −0.359354 − 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −16.0000 −0.521862
\(941\) −10.0000 −0.325991 −0.162995 − 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) 0 0
\(947\) 24.0000 0.779895 0.389948 − 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 0 0
\(953\) −54.0000 −1.74923 −0.874616 − 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 18.0000 0.582772
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 4.00000 0.128965
\(963\) −36.0000 −1.16008
\(964\) 10.0000 0.322078
\(965\) −44.0000 −1.41641
\(966\) 0 0
\(967\) 56.0000 1.80084 0.900419 − 0.435023i \(-0.143260\pi\)
0.900419 + 0.435023i \(0.143260\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 20.0000 0.642161
\(971\) −36.0000 −1.15529 −0.577647 − 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −40.0000 −1.28168
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 50.0000 1.59964 0.799821 − 0.600239i \(-0.204928\pi\)
0.799821 + 0.600239i \(0.204928\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) −14.0000 −0.447214
\(981\) −30.0000 −0.957826
\(982\) 40.0000 1.27645
\(983\) −8.00000 −0.255160 −0.127580 − 0.991828i \(-0.540721\pi\)
−0.127580 + 0.991828i \(0.540721\pi\)
\(984\) 0 0
\(985\) −20.0000 −0.637253
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) −24.0000 −0.762770
\(991\) −40.0000 −1.27064 −0.635321 − 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) 2.00000 0.0633406 0.0316703 − 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) −24.0000 −0.759707
\(999\) 0 0
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 254.2.a.d.1.1 ✓ 1
3.2 odd 2 2286.2.a.a.1.1 1
4.3 odd 2 2032.2.a.c.1.1 1
5.4 even 2 6350.2.a.b.1.1 1
8.3 odd 2 8128.2.a.f.1.1 1
8.5 even 2 8128.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
254.2.a.d.1.1 ✓ 1 1.1 even 1 trivial
2032.2.a.c.1.1 1 4.3 odd 2
2286.2.a.a.1.1 1 3.2 odd 2
6350.2.a.b.1.1 1 5.4 even 2
8128.2.a.e.1.1 1 8.5 even 2
8128.2.a.f.1.1 1 8.3 odd 2