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Properties

Label 54.2.a.a.1.1
Level $54$
Weight $2$
Character 54.1
Self dual yes
Analytic conductor $0.431$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [54,2,Mod(1,54)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("54.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(54, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 54.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.431192170915\)
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 \(\chi\) \(=\) 54.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{10} -3.00000 q^{11} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{19} +3.00000 q^{20} +3.00000 q^{22} -6.00000 q^{23} +4.00000 q^{25} +4.00000 q^{26} -1.00000 q^{28} +6.00000 q^{29} +5.00000 q^{31} -1.00000 q^{32} -3.00000 q^{35} +2.00000 q^{37} -2.00000 q^{38} -3.00000 q^{40} -6.00000 q^{41} -10.0000 q^{43} -3.00000 q^{44} +6.00000 q^{46} +6.00000 q^{47} -6.00000 q^{49} -4.00000 q^{50} -4.00000 q^{52} +9.00000 q^{53} -9.00000 q^{55} +1.00000 q^{56} -6.00000 q^{58} +12.0000 q^{59} +8.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} +14.0000 q^{67} +3.00000 q^{70} -7.00000 q^{73} -2.00000 q^{74} +2.00000 q^{76} +3.00000 q^{77} +8.00000 q^{79} +3.00000 q^{80} +6.00000 q^{82} -3.00000 q^{83} +10.0000 q^{86} +3.00000 q^{88} -18.0000 q^{89} +4.00000 q^{91} -6.00000 q^{92} -6.00000 q^{94} +6.00000 q^{95} -1.00000 q^{97} +6.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −3.00000 −0.948683
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 3.00000 0.670820
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 − 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) −9.00000 −1.21356
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −5.00000 −0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 − 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 3.00000 0.358569
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 − 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 3.00000 0.335410
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) −3.00000 −0.329293 −0.164646 − 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) −18.0000 −1.90800 −0.953998 − 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 − 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 6.00000 0.606092
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) −3.00000 −0.298511 −0.149256 − 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 9.00000 0.870063 0.435031 − 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 9.00000 0.858116
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −18.0000 −1.67851
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −8.00000 −0.724286
\(123\) 0 0
\(124\) 5.00000 0.449013
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 − 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 12.0000 1.05247
\(131\) −15.0000 −1.31056 −0.655278 − 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) −14.0000 −1.20942
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −3.00000 −0.253546
\(141\) 0 0
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 18.0000 1.49482
\(146\) 7.00000 0.579324
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 3.00000 0.245770 0.122885 − 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) 17.0000 1.38344 0.691720 − 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) −3.00000 −0.241747
\(155\) 15.0000 1.20483
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 − 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) −3.00000 −0.237171
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) 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
\(166\) 3.00000 0.232845
\(167\) −6.00000 −0.464294 −0.232147 − 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) −10.0000 −0.762493
\(173\) 15.0000 1.14043 0.570214 − 0.821496i \(-0.306860\pi\)
0.570214 + 0.821496i \(0.306860\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 18.0000 1.34916
\(179\) −9.00000 −0.672692 −0.336346 − 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 − 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) 6.00000 0.442326
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 5.00000 0.359908 0.179954 − 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) 1.00000 0.0717958
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 9.00000 0.641223 0.320612 − 0.947211i \(-0.396112\pi\)
0.320612 + 0.947211i \(0.396112\pi\)
\(198\) 0 0
\(199\) −7.00000 −0.496217 −0.248108 − 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) 3.00000 0.211079
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) −18.0000 −1.25717
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 − 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 9.00000 0.618123
\(213\) 0 0
\(214\) −9.00000 −0.615227
\(215\) −30.0000 −2.04598
\(216\) 0 0
\(217\) −5.00000 −0.339422
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) −9.00000 −0.606780
\(221\) 0 0
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 − 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 18.0000 1.18688
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 18.0000 1.17922 0.589610 − 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 18.0000 1.17419
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) 30.0000 1.94054 0.970269 − 0.242028i \(-0.0778125\pi\)
0.970269 + 0.242028i \(0.0778125\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 2.00000 0.128565
\(243\) 0 0
\(244\) 8.00000 0.512148
\(245\) −18.0000 −1.14998
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) −5.00000 −0.317500
\(249\) 0 0
\(250\) 3.00000 0.189737
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 7.00000 0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 − 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) −12.0000 −0.744208
\(261\) 0 0
\(262\) 15.0000 0.926703
\(263\) −30.0000 −1.84988 −0.924940 − 0.380114i \(-0.875885\pi\)
−0.924940 + 0.380114i \(0.875885\pi\)
\(264\) 0 0
\(265\) 27.0000 1.65860
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) 14.0000 0.855186
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −25.0000 −1.51864 −0.759321 − 0.650716i \(-0.774469\pi\)
−0.759321 + 0.650716i \(0.774469\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) −12.0000 −0.723627
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 − 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 3.00000 0.179284
\(281\) 24.0000 1.43172 0.715860 − 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 − 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) −18.0000 −1.05700
\(291\) 0 0
\(292\) −7.00000 −0.409644
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 36.0000 2.09600
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −3.00000 −0.173785
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 10.0000 0.576390
\(302\) −17.0000 −0.978240
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 24.0000 1.37424
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 − 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 3.00000 0.170941
\(309\) 0 0
\(310\) −15.0000 −0.851943
\(311\) −6.00000 −0.340229 −0.170114 − 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) −19.0000 −1.07394 −0.536972 − 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −3.00000 −0.168497 −0.0842484 − 0.996445i \(-0.526849\pi\)
−0.0842484 + 0.996445i \(0.526849\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) 3.00000 0.167705
\(321\) 0 0
\(322\) −6.00000 −0.334367
\(323\) 0 0
\(324\) 0 0
\(325\) −16.0000 −0.887520
\(326\) −20.0000 −1.10770
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 − 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) −3.00000 −0.164646
\(333\) 0 0
\(334\) 6.00000 0.328305
\(335\) 42.0000 2.29471
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) 0 0
\(341\) −15.0000 −0.812296
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) −15.0000 −0.806405
\(347\) 3.00000 0.161048 0.0805242 − 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) 9.00000 0.475665
\(359\) 18.0000 0.950004 0.475002 − 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 16.0000 0.840941
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −21.0000 −1.09919
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 − 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) −6.00000 −0.311925
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) 32.0000 1.65690 0.828449 − 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 9.00000 0.458682
\(386\) −5.00000 −0.254493
\(387\) 0 0
\(388\) −1.00000 −0.0507673
\(389\) −21.0000 −1.06474 −0.532371 − 0.846511i \(-0.678699\pi\)
−0.532371 + 0.846511i \(0.678699\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 6.00000 0.303046
\(393\) 0 0
\(394\) −9.00000 −0.453413
\(395\) 24.0000 1.20757
\(396\) 0 0
\(397\) 20.0000 1.00377 0.501886 − 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) 7.00000 0.350878
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 12.0000 0.599251 0.299626 − 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) −3.00000 −0.149256
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) 23.0000 1.13728 0.568638 − 0.822588i \(-0.307470\pi\)
0.568638 + 0.822588i \(0.307470\pi\)
\(410\) 18.0000 0.888957
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) −9.00000 −0.441793
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 − 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 22.0000 1.07094
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) 9.00000 0.435031
\(429\) 0 0
\(430\) 30.0000 1.44673
\(431\) 18.0000 0.867029 0.433515 − 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 29.0000 1.39365 0.696826 − 0.717241i \(-0.254595\pi\)
0.696826 + 0.717241i \(0.254595\pi\)
\(434\) 5.00000 0.240008
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) −19.0000 −0.906821 −0.453410 − 0.891302i \(-0.649793\pi\)
−0.453410 + 0.891302i \(0.649793\pi\)
\(440\) 9.00000 0.429058
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −54.0000 −2.55985
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −18.0000 −0.849473 −0.424736 − 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) −1.00000 −0.0467780 −0.0233890 − 0.999726i \(-0.507446\pi\)
−0.0233890 + 0.999726i \(0.507446\pi\)
\(458\) −14.0000 −0.654177
\(459\) 0 0
\(460\) −18.0000 −0.839254
\(461\) −21.0000 −0.978068 −0.489034 − 0.872265i \(-0.662651\pi\)
−0.489034 + 0.872265i \(0.662651\pi\)
\(462\) 0 0
\(463\) −13.0000 −0.604161 −0.302081 − 0.953282i \(-0.597681\pi\)
−0.302081 + 0.953282i \(0.597681\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) −27.0000 −1.24941 −0.624705 − 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) 0 0
\(469\) −14.0000 −0.646460
\(470\) −18.0000 −0.830278
\(471\) 0 0
\(472\) −12.0000 −0.552345
\(473\) 30.0000 1.37940
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) 0 0
\(478\) −30.0000 −1.37217
\(479\) 6.00000 0.274147 0.137073 − 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −3.00000 −0.136223
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 − 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −8.00000 −0.362143
\(489\) 0 0
\(490\) 18.0000 0.813157
\(491\) 39.0000 1.76005 0.880023 − 0.474932i \(-0.157527\pi\)
0.880023 + 0.474932i \(0.157527\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 5.00000 0.224507
\(497\) 0 0
\(498\) 0 0
\(499\) 14.0000 0.626726 0.313363 − 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) −3.00000 −0.134164
\(501\) 0 0
\(502\) 0 0
\(503\) −18.0000 −0.802580 −0.401290 − 0.915951i \(-0.631438\pi\)
−0.401290 + 0.915951i \(0.631438\pi\)
\(504\) 0 0
\(505\) −9.00000 −0.400495
\(506\) −18.0000 −0.800198
\(507\) 0 0
\(508\) −7.00000 −0.310575
\(509\) −15.0000 −0.664863 −0.332432 − 0.943127i \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) 0 0
\(511\) 7.00000 0.309662
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) −18.0000 −0.791639
\(518\) 2.00000 0.0878750
\(519\) 0 0
\(520\) 12.0000 0.526235
\(521\) 36.0000 1.57719 0.788594 − 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 − 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −15.0000 −0.655278
\(525\) 0 0
\(526\) 30.0000 1.30806
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −27.0000 −1.17281
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) 27.0000 1.16731
\(536\) −14.0000 −0.604708
\(537\) 0 0
\(538\) 18.0000 0.776035
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 − 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 25.0000 1.07384
\(543\) 0 0
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 − 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) 12.0000 0.511682
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 27.0000 1.14403 0.572013 − 0.820244i \(-0.306163\pi\)
0.572013 + 0.820244i \(0.306163\pi\)
\(558\) 0 0
\(559\) 40.0000 1.69182
\(560\) −3.00000 −0.126773
\(561\) 0 0
\(562\) −24.0000 −1.01238
\(563\) 3.00000 0.126435 0.0632175 − 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) 0 0
\(569\) −12.0000 −0.503066 −0.251533 − 0.967849i \(-0.580935\pi\)
−0.251533 + 0.967849i \(0.580935\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 − 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) −24.0000 −1.00087
\(576\) 0 0
\(577\) 38.0000 1.58196 0.790980 − 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 17.0000 0.707107
\(579\) 0 0
\(580\) 18.0000 0.747409
\(581\) 3.00000 0.124461
\(582\) 0 0
\(583\) −27.0000 −1.11823
\(584\) 7.00000 0.289662
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −3.00000 −0.123823 −0.0619116 − 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) 0 0
\(589\) 10.0000 0.412043
\(590\) −36.0000 −1.48210
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) 18.0000 0.739171 0.369586 − 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.00000 0.122885
\(597\) 0 0
\(598\) −24.0000 −0.981433
\(599\) −42.0000 −1.71607 −0.858037 − 0.513588i \(-0.828316\pi\)
−0.858037 + 0.513588i \(0.828316\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 − 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) −10.0000 −0.407570
\(603\) 0 0
\(604\) 17.0000 0.691720
\(605\) −6.00000 −0.243935
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 − 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) −24.0000 −0.971732
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) −34.0000 −1.37325 −0.686624 − 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) −42.0000 −1.69086 −0.845428 − 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 15.0000 0.602414
\(621\) 0 0
\(622\) 6.00000 0.240578
\(623\) 18.0000 0.721155
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 19.0000 0.759393
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) 0 0
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 − 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) 3.00000 0.119145
\(635\) −21.0000 −0.833360
\(636\) 0 0
\(637\) 24.0000 0.950915
\(638\) 18.0000 0.712627
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) 42.0000 1.65890 0.829450 − 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 − 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 16.0000 0.627572
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) 39.0000 1.52619 0.763094 − 0.646288i \(-0.223679\pi\)
0.763094 + 0.646288i \(0.223679\pi\)
\(654\) 0 0
\(655\) −45.0000 −1.75830
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 6.00000 0.233904
\(659\) −21.0000 −0.818044 −0.409022 − 0.912525i \(-0.634130\pi\)
−0.409022 + 0.912525i \(0.634130\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 − 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 10.0000 0.388661
\(663\) 0 0
\(664\) 3.00000 0.116423
\(665\) −6.00000 −0.232670
\(666\) 0 0
\(667\) −36.0000 −1.39393
\(668\) −6.00000 −0.232147
\(669\) 0 0
\(670\) −42.0000 −1.62260
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 − 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 42.0000 1.61419 0.807096 − 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 0 0
\(679\) 1.00000 0.0383765
\(680\) 0 0
\(681\) 0 0
\(682\) 15.0000 0.574380
\(683\) −36.0000 −1.37750 −0.688751 − 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) −13.0000 −0.496342
\(687\) 0 0
\(688\) −10.0000 −0.381246
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 − 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 15.0000 0.570214
\(693\) 0 0
\(694\) −3.00000 −0.113878
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) 0 0
\(698\) 10.0000 0.378506
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) 9.00000 0.339925 0.169963 − 0.985451i \(-0.445635\pi\)
0.169963 + 0.985451i \(0.445635\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 3.00000 0.112827
\(708\) 0 0
\(709\) 44.0000 1.65245 0.826227 − 0.563337i \(-0.190483\pi\)
0.826227 + 0.563337i \(0.190483\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 18.0000 0.674579
\(713\) −30.0000 −1.12351
\(714\) 0 0
\(715\) 36.0000 1.34632
\(716\) −9.00000 −0.336346
\(717\) 0 0
\(718\) −18.0000 −0.671754
\(719\) 36.0000 1.34257 0.671287 − 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 15.0000 0.558242
\(723\) 0 0
\(724\) −16.0000 −0.594635
\(725\) 24.0000 0.891338
\(726\) 0 0
\(727\) −1.00000 −0.0370879 −0.0185440 − 0.999828i \(-0.505903\pi\)
−0.0185440 + 0.999828i \(0.505903\pi\)
\(728\) −4.00000 −0.148250
\(729\) 0 0
\(730\) 21.0000 0.777245
\(731\) 0 0
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 − 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −17.0000 −0.627481
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) −42.0000 −1.54709
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 − 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) 9.00000 0.330400
\(743\) 12.0000 0.440237 0.220119 − 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) 9.00000 0.329734
\(746\) −32.0000 −1.17160
\(747\) 0 0
\(748\) 0 0
\(749\) −9.00000 −0.328853
\(750\) 0 0
\(751\) 41.0000 1.49611 0.748056 − 0.663636i \(-0.230988\pi\)
0.748056 + 0.663636i \(0.230988\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 24.0000 0.874028
\(755\) 51.0000 1.85608
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 − 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) −6.00000 −0.217643
\(761\) 48.0000 1.74000 0.869999 − 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) −48.0000 −1.73318
\(768\) 0 0
\(769\) −31.0000 −1.11789 −0.558944 − 0.829205i \(-0.688793\pi\)
−0.558944 + 0.829205i \(0.688793\pi\)
\(770\) −9.00000 −0.324337
\(771\) 0 0
\(772\) 5.00000 0.179954
\(773\) −18.0000 −0.647415 −0.323708 − 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) 20.0000 0.718421
\(776\) 1.00000 0.0358979
\(777\) 0 0
\(778\) 21.0000 0.752886
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) −12.0000 −0.428298
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 − 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 9.00000 0.320612
\(789\) 0 0
\(790\) −24.0000 −0.853882
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) −32.0000 −1.13635
\(794\) −20.0000 −0.709773
\(795\) 0 0
\(796\) −7.00000 −0.248108
\(797\) 39.0000 1.38145 0.690725 − 0.723117i \(-0.257291\pi\)
0.690725 + 0.723117i \(0.257291\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −4.00000 −0.141421
\(801\) 0 0
\(802\) −12.0000 −0.423735
\(803\) 21.0000 0.741074
\(804\) 0 0
\(805\) 18.0000 0.634417
\(806\) 20.0000 0.704470
\(807\) 0 0
\(808\) 3.00000 0.105540
\(809\) 18.0000 0.632846 0.316423 − 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 − 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) 6.00000 0.210300
\(815\) 60.0000 2.10171
\(816\) 0 0
\(817\) −20.0000 −0.699711
\(818\) −23.0000 −0.804176
\(819\) 0 0
\(820\) −18.0000 −0.628587
\(821\) 6.00000 0.209401 0.104701 − 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) −31.0000 −1.08059 −0.540296 − 0.841475i \(-0.681688\pi\)
−0.540296 + 0.841475i \(0.681688\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 9.00000 0.312395
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) 0 0
\(834\) 0 0
\(835\) −18.0000 −0.622916
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) 24.0000 0.828572 0.414286 − 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −8.00000 −0.275698
\(843\) 0 0
\(844\) −22.0000 −0.757271
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 9.00000 0.309061
\(849\) 0 0
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 − 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) −48.0000 −1.63965 −0.819824 − 0.572615i \(-0.805929\pi\)
−0.819824 + 0.572615i \(0.805929\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 − 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) −30.0000 −1.02299
\(861\) 0 0
\(862\) −18.0000 −0.613082
\(863\) −36.0000 −1.22545 −0.612727 − 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) 45.0000 1.53005
\(866\) −29.0000 −0.985460
\(867\) 0 0
\(868\) −5.00000 −0.169711
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) −56.0000 −1.89749
\(872\) −2.00000 −0.0677285
\(873\) 0 0
\(874\) 12.0000 0.405906
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 − 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 19.0000 0.641219
\(879\) 0 0
\(880\) −9.00000 −0.303390
\(881\) −36.0000 −1.21287 −0.606435 − 0.795133i \(-0.707401\pi\)
−0.606435 + 0.795133i \(0.707401\pi\)
\(882\) 0 0
\(883\) 2.00000 0.0673054 0.0336527 − 0.999434i \(-0.489286\pi\)
0.0336527 + 0.999434i \(0.489286\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 12.0000 0.402921 0.201460 − 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) 7.00000 0.234772
\(890\) 54.0000 1.81008
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) −27.0000 −0.902510
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) 30.0000 1.00056
\(900\) 0 0
\(901\) 0 0
\(902\) −18.0000 −0.599334
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −48.0000 −1.59557
\(906\) 0 0
\(907\) 26.0000 0.863316 0.431658 − 0.902037i \(-0.357929\pi\)
0.431658 + 0.902037i \(0.357929\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) −12.0000 −0.397796
\(911\) 6.00000 0.198789 0.0993944 − 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) 0 0
\(913\) 9.00000 0.297857
\(914\) 1.00000 0.0330771
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 15.0000 0.495344
\(918\) 0 0
\(919\) 29.0000 0.956622 0.478311 − 0.878191i \(-0.341249\pi\)
0.478311 + 0.878191i \(0.341249\pi\)
\(920\) 18.0000 0.593442
\(921\) 0 0
\(922\) 21.0000 0.691598
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 13.0000 0.427207
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) −12.0000 −0.393707 −0.196854 − 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 18.0000 0.589610
\(933\) 0 0
\(934\) 27.0000 0.883467
\(935\) 0 0
\(936\) 0 0
\(937\) 11.0000 0.359354 0.179677 − 0.983726i \(-0.442495\pi\)
0.179677 + 0.983726i \(0.442495\pi\)
\(938\) 14.0000 0.457116
\(939\) 0 0
\(940\) 18.0000 0.587095
\(941\) −33.0000 −1.07577 −0.537885 − 0.843018i \(-0.680776\pi\)
−0.537885 + 0.843018i \(0.680776\pi\)
\(942\) 0 0
\(943\) 36.0000 1.17232
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −30.0000 −0.975384
\(947\) 51.0000 1.65728 0.828639 − 0.559784i \(-0.189116\pi\)
0.828639 + 0.559784i \(0.189116\pi\)
\(948\) 0 0
\(949\) 28.0000 0.908918
\(950\) −8.00000 −0.259554
\(951\) 0 0
\(952\) 0 0
\(953\) −36.0000 −1.16615 −0.583077 − 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) −36.0000 −1.16493
\(956\) 30.0000 0.970269
\(957\) 0 0
\(958\) −6.00000 −0.193851
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 8.00000 0.257930
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) 15.0000 0.482867
\(966\) 0 0
\(967\) 23.0000 0.739630 0.369815 − 0.929105i \(-0.379421\pi\)
0.369815 + 0.929105i \(0.379421\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) 3.00000 0.0963242
\(971\) −27.0000 −0.866471 −0.433236 − 0.901281i \(-0.642628\pi\)
−0.433236 + 0.901281i \(0.642628\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) −42.0000 −1.34370 −0.671850 − 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) 54.0000 1.72585
\(980\) −18.0000 −0.574989
\(981\) 0 0
\(982\) −39.0000 −1.24454
\(983\) 6.00000 0.191370 0.0956851 − 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) 0 0
\(985\) 27.0000 0.860292
\(986\) 0 0
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 60.0000 1.90789
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 − 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) −5.00000 −0.158750
\(993\) 0 0
\(994\) 0 0
\(995\) −21.0000 −0.665745
\(996\) 0 0
\(997\) −28.0000 −0.886769 −0.443384 − 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) −14.0000 −0.443162
\(999\) 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 54.2.a.a.1.1 ✓ 1
3.2 odd 2 54.2.a.b.1.1 yes 1
4.3 odd 2 432.2.a.g.1.1 1
5.2 odd 4 1350.2.c.b.649.1 2
5.3 odd 4 1350.2.c.b.649.2 2
5.4 even 2 1350.2.a.r.1.1 1
7.6 odd 2 2646.2.a.a.1.1 1
8.3 odd 2 1728.2.a.d.1.1 1
8.5 even 2 1728.2.a.c.1.1 1
9.2 odd 6 162.2.c.b.109.1 2
9.4 even 3 162.2.c.c.55.1 2
9.5 odd 6 162.2.c.b.55.1 2
9.7 even 3 162.2.c.c.109.1 2
11.10 odd 2 6534.2.a.bc.1.1 1
12.11 even 2 432.2.a.b.1.1 1
13.12 even 2 9126.2.a.u.1.1 1
15.2 even 4 1350.2.c.k.649.2 2
15.8 even 4 1350.2.c.k.649.1 2
15.14 odd 2 1350.2.a.h.1.1 1
21.20 even 2 2646.2.a.bd.1.1 1
24.5 odd 2 1728.2.a.y.1.1 1
24.11 even 2 1728.2.a.z.1.1 1
33.32 even 2 6534.2.a.b.1.1 1
36.7 odd 6 1296.2.i.c.433.1 2
36.11 even 6 1296.2.i.o.433.1 2
36.23 even 6 1296.2.i.o.865.1 2
36.31 odd 6 1296.2.i.c.865.1 2
39.38 odd 2 9126.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
54.2.a.b.1.1 yes 1 3.2 odd 2
162.2.c.b.55.1 2 9.5 odd 6
162.2.c.b.109.1 2 9.2 odd 6
162.2.c.c.55.1 2 9.4 even 3
162.2.c.c.109.1 2 9.7 even 3
432.2.a.b.1.1 1 12.11 even 2
432.2.a.g.1.1 1 4.3 odd 2
1296.2.i.c.433.1 2 36.7 odd 6
1296.2.i.c.865.1 2 36.31 odd 6
1296.2.i.o.433.1 2 36.11 even 6
1296.2.i.o.865.1 2 36.23 even 6
1350.2.a.h.1.1 1 15.14 odd 2
1350.2.a.r.1.1 1 5.4 even 2
1350.2.c.b.649.1 2 5.2 odd 4
1350.2.c.b.649.2 2 5.3 odd 4
1350.2.c.k.649.1 2 15.8 even 4
1350.2.c.k.649.2 2 15.2 even 4
1728.2.a.c.1.1 1 8.5 even 2
1728.2.a.d.1.1 1 8.3 odd 2
1728.2.a.y.1.1 1 24.5 odd 2
1728.2.a.z.1.1 1 24.11 even 2
2646.2.a.a.1.1 1 7.6 odd 2
2646.2.a.bd.1.1 1 21.20 even 2
6534.2.a.b.1.1 1 33.32 even 2
6534.2.a.bc.1.1 1 11.10 odd 2
9126.2.a.r.1.1 1 39.38 odd 2
9126.2.a.u.1.1 1 13.12 even 2