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Properties

Label 6018.2.a.f.1.1
Level $6018$
Weight $2$
Character 6018.1
Self dual yes
Analytic conductor $48.054$
Analytic rank $1$
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] = [6018,2,Mod(1,6018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6018, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6018.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: \(48.0539719364\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 6018.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +2.00000 q^{21} -2.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +4.00000 q^{29} +1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} -2.00000 q^{41} +2.00000 q^{42} +8.00000 q^{43} -2.00000 q^{44} +4.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -5.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -2.00000 q^{56} -4.00000 q^{57} +4.00000 q^{58} -1.00000 q^{59} +6.00000 q^{61} -2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -4.00000 q^{67} +1.00000 q^{68} -4.00000 q^{69} +2.00000 q^{71} +1.00000 q^{72} +8.00000 q^{73} -2.00000 q^{74} +5.00000 q^{75} +4.00000 q^{76} +4.00000 q^{77} +2.00000 q^{78} -14.0000 q^{79} +1.00000 q^{81} -2.00000 q^{82} -12.0000 q^{83} +2.00000 q^{84} +8.00000 q^{86} -4.00000 q^{87} -2.00000 q^{88} -10.0000 q^{89} +4.00000 q^{91} +4.00000 q^{92} -8.00000 q^{94} -1.00000 q^{96} +8.00000 q^{97} -3.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) −2.00000 −0.426401
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 2.00000 0.308607
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −5.00000 −0.707107
\(51\) −1.00000 −0.140028
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) −4.00000 −0.529813
\(58\) 4.00000 0.525226
\(59\) −1.00000 −0.130189
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 1.00000 0.121268
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 1.00000 0.117851
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) −2.00000 −0.232495
\(75\) 5.00000 0.577350
\(76\) 4.00000 0.458831
\(77\) 4.00000 0.455842
\(78\) 2.00000 0.226455
\(79\) −14.0000 −1.57512 −0.787562 − 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) −4.00000 −0.428845
\(88\) −2.00000 −0.213201
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −3.00000 −0.303046
\(99\) −2.00000 −0.201008
\(100\) −5.00000 −0.500000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 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
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −2.00000 −0.188982
\(113\) −20.0000 −1.88144 −0.940721 − 0.339182i \(-0.889850\pi\)
−0.940721 + 0.339182i \(0.889850\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) −2.00000 −0.184900
\(118\) −1.00000 −0.0920575
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 6.00000 0.543214
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) −12.0000 −1.06483 −0.532414 − 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −2.00000 −0.174741 −0.0873704 − 0.996176i \(-0.527846\pi\)
−0.0873704 + 0.996176i \(0.527846\pi\)
\(132\) 2.00000 0.174078
\(133\) −8.00000 −0.693688
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) −10.0000 −0.854358 −0.427179 − 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) −4.00000 −0.340503
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 2.00000 0.167836
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 8.00000 0.662085
\(147\) 3.00000 0.247436
\(148\) −2.00000 −0.164399
\(149\) 22.0000 1.80231 0.901155 − 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 5.00000 0.408248
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 4.00000 0.324443
\(153\) 1.00000 0.0808452
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −14.0000 −1.11378
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 1.00000 0.0785674
\(163\) −20.0000 −1.56652 −0.783260 − 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 10.0000 0.773823 0.386912 − 0.922117i \(-0.373542\pi\)
0.386912 + 0.922117i \(0.373542\pi\)
\(168\) 2.00000 0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 8.00000 0.609994
\(173\) −2.00000 −0.152057 −0.0760286 − 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) −4.00000 −0.303239
\(175\) 10.0000 0.755929
\(176\) −2.00000 −0.150756
\(177\) 1.00000 0.0751646
\(178\) −10.0000 −0.749532
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 4.00000 0.296500
\(183\) −6.00000 −0.443533
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) −8.00000 −0.583460
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 − 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 22.0000 1.58359 0.791797 − 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 8.00000 0.569976 0.284988 − 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) −2.00000 −0.142134
\(199\) −10.0000 −0.708881 −0.354441 − 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) −5.00000 −0.353553
\(201\) 4.00000 0.282138
\(202\) −6.00000 −0.422159
\(203\) −8.00000 −0.561490
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) −2.00000 −0.138675
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 − 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) −6.00000 −0.412082
\(213\) −2.00000 −0.137038
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 2.00000 0.134231
\(223\) −24.0000 −1.60716 −0.803579 − 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) −2.00000 −0.133631
\(225\) −5.00000 −0.333333
\(226\) −20.0000 −1.33038
\(227\) 18.0000 1.19470 0.597351 − 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) −4.00000 −0.264906
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 4.00000 0.262613
\(233\) −28.0000 −1.83434 −0.917170 − 0.398495i \(-0.869533\pi\)
−0.917170 + 0.398495i \(0.869533\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −1.00000 −0.0650945
\(237\) 14.0000 0.909398
\(238\) −2.00000 −0.129641
\(239\) 20.0000 1.29369 0.646846 − 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −7.00000 −0.449977
\(243\) −1.00000 −0.0641500
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −2.00000 −0.125988
\(253\) −8.00000 −0.502956
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 − 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) −8.00000 −0.498058
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) −2.00000 −0.123560
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) −8.00000 −0.490511
\(267\) 10.0000 0.611990
\(268\) −4.00000 −0.244339
\(269\) −2.00000 −0.121942 −0.0609711 − 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 1.00000 0.0606339
\(273\) −4.00000 −0.242091
\(274\) −10.0000 −0.604122
\(275\) 10.0000 0.603023
\(276\) −4.00000 −0.240772
\(277\) −8.00000 −0.480673 −0.240337 − 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 − 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 8.00000 0.476393
\(283\) 18.0000 1.06999 0.534994 − 0.844856i \(-0.320314\pi\)
0.534994 + 0.844856i \(0.320314\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 4.00000 0.236113
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −8.00000 −0.468968
\(292\) 8.00000 0.468165
\(293\) −10.0000 −0.584206 −0.292103 − 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 2.00000 0.116052
\(298\) 22.0000 1.27443
\(299\) −8.00000 −0.462652
\(300\) 5.00000 0.288675
\(301\) −16.0000 −0.922225
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 1.00000 0.0571662
\(307\) −4.00000 −0.228292 −0.114146 − 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 − 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 2.00000 0.113228
\(313\) −16.0000 −0.904373 −0.452187 − 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −14.0000 −0.787562
\(317\) −24.0000 −1.34797 −0.673987 − 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 6.00000 0.336463
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) −8.00000 −0.445823
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 10.0000 0.554700
\(326\) −20.0000 −1.10770
\(327\) 10.0000 0.553001
\(328\) −2.00000 −0.110432
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −12.0000 −0.658586
\(333\) −2.00000 −0.109599
\(334\) 10.0000 0.547176
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 20.0000 1.08947 0.544735 − 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) −9.00000 −0.489535
\(339\) 20.0000 1.08625
\(340\) 0 0
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) 20.0000 1.07990
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) −26.0000 −1.39575 −0.697877 − 0.716218i \(-0.745872\pi\)
−0.697877 + 0.716218i \(0.745872\pi\)
\(348\) −4.00000 −0.214423
\(349\) 30.0000 1.60586 0.802932 − 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 10.0000 0.534522
\(351\) 2.00000 0.106752
\(352\) −2.00000 −0.106600
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 1.00000 0.0531494
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 2.00000 0.105851
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 − 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) 36.0000 1.87918 0.939592 − 0.342296i \(-0.111204\pi\)
0.939592 + 0.342296i \(0.111204\pi\)
\(368\) 4.00000 0.208514
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 − 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) −8.00000 −0.412021
\(378\) 2.00000 0.102869
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) −8.00000 −0.409316
\(383\) −8.00000 −0.408781 −0.204390 − 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) 8.00000 0.406663
\(388\) 8.00000 0.406138
\(389\) −26.0000 −1.31825 −0.659126 − 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) −3.00000 −0.151523
\(393\) 2.00000 0.100887
\(394\) 8.00000 0.403034
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) −22.0000 −1.10415 −0.552074 − 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −10.0000 −0.501255
\(399\) 8.00000 0.400501
\(400\) −5.00000 −0.250000
\(401\) −20.0000 −0.998752 −0.499376 − 0.866385i \(-0.666437\pi\)
−0.499376 + 0.866385i \(0.666437\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) 4.00000 0.198273
\(408\) −1.00000 −0.0495074
\(409\) −30.0000 −1.48340 −0.741702 − 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) 0 0
\(413\) 2.00000 0.0984136
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 4.00000 0.195881
\(418\) −8.00000 −0.391293
\(419\) −10.0000 −0.488532 −0.244266 − 0.969708i \(-0.578547\pi\)
−0.244266 + 0.969708i \(0.578547\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 − 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 14.0000 0.681509
\(423\) −8.00000 −0.388973
\(424\) −6.00000 −0.291386
\(425\) −5.00000 −0.242536
\(426\) −2.00000 −0.0969003
\(427\) −12.0000 −0.580721
\(428\) 4.00000 0.193347
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 − 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −34.0000 −1.63394 −0.816968 − 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 16.0000 0.765384
\(438\) −8.00000 −0.382255
\(439\) 26.0000 1.24091 0.620456 − 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −2.00000 −0.0951303
\(443\) 4.00000 0.190046 0.0950229 − 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) −24.0000 −1.13643
\(447\) −22.0000 −1.04056
\(448\) −2.00000 −0.0944911
\(449\) 34.0000 1.60456 0.802280 − 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) −5.00000 −0.235702
\(451\) 4.00000 0.188353
\(452\) −20.0000 −0.940721
\(453\) 0 0
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −22.0000 −1.02799
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) −38.0000 −1.76984 −0.884918 − 0.465746i \(-0.845786\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) −4.00000 −0.186097
\(463\) −40.0000 −1.85896 −0.929479 − 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) −28.0000 −1.29707
\(467\) 8.00000 0.370196 0.185098 − 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) −1.00000 −0.0460287
\(473\) −16.0000 −0.735681
\(474\) 14.0000 0.643041
\(475\) −20.0000 −0.917663
\(476\) −2.00000 −0.0916698
\(477\) −6.00000 −0.274721
\(478\) 20.0000 0.914779
\(479\) 18.0000 0.822441 0.411220 − 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 10.0000 0.455488
\(483\) 8.00000 0.364013
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −18.0000 −0.815658 −0.407829 − 0.913058i \(-0.633714\pi\)
−0.407829 + 0.913058i \(0.633714\pi\)
\(488\) 6.00000 0.271607
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 − 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 2.00000 0.0901670
\(493\) 4.00000 0.180151
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) −4.00000 −0.179425
\(498\) 12.0000 0.537733
\(499\) 8.00000 0.358129 0.179065 − 0.983837i \(-0.442693\pi\)
0.179065 + 0.983837i \(0.442693\pi\)
\(500\) 0 0
\(501\) −10.0000 −0.446767
\(502\) −12.0000 −0.535586
\(503\) 4.00000 0.178351 0.0891756 − 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) −8.00000 −0.355643
\(507\) 9.00000 0.399704
\(508\) −12.0000 −0.532414
\(509\) 30.0000 1.32973 0.664863 − 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) 16.0000 0.703679
\(518\) 4.00000 0.175750
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 − 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 4.00000 0.175075
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −2.00000 −0.0873704
\(525\) −10.0000 −0.436436
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −1.00000 −0.0433963
\(532\) −8.00000 −0.346844
\(533\) 4.00000 0.173259
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) −2.00000 −0.0862261
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 − 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) −32.0000 −1.36822 −0.684111 − 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) −10.0000 −0.427179
\(549\) 6.00000 0.256074
\(550\) 10.0000 0.426401
\(551\) 16.0000 0.681623
\(552\) −4.00000 −0.170251
\(553\) 28.0000 1.19068
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −22.0000 −0.932170 −0.466085 − 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) −10.0000 −0.421825
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) 18.0000 0.756596
\(567\) −2.00000 −0.0839921
\(568\) 2.00000 0.0839181
\(569\) 14.0000 0.586911 0.293455 − 0.955973i \(-0.405195\pi\)
0.293455 + 0.955973i \(0.405195\pi\)
\(570\) 0 0
\(571\) 6.00000 0.251092 0.125546 − 0.992088i \(-0.459932\pi\)
0.125546 + 0.992088i \(0.459932\pi\)
\(572\) 4.00000 0.167248
\(573\) 8.00000 0.334205
\(574\) 4.00000 0.166957
\(575\) −20.0000 −0.834058
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 1.00000 0.0415945
\(579\) −22.0000 −0.914289
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) −8.00000 −0.331611
\(583\) 12.0000 0.496989
\(584\) 8.00000 0.331042
\(585\) 0 0
\(586\) −10.0000 −0.413096
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 3.00000 0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) −8.00000 −0.329076
\(592\) −2.00000 −0.0821995
\(593\) 26.0000 1.06769 0.533846 − 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) 10.0000 0.409273
\(598\) −8.00000 −0.327144
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 5.00000 0.204124
\(601\) −20.0000 −0.815817 −0.407909 − 0.913023i \(-0.633742\pi\)
−0.407909 + 0.913023i \(0.633742\pi\)
\(602\) −16.0000 −0.652111
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) 38.0000 1.54237 0.771186 − 0.636610i \(-0.219664\pi\)
0.771186 + 0.636610i \(0.219664\pi\)
\(608\) 4.00000 0.162221
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) 1.00000 0.0404226
\(613\) 34.0000 1.37325 0.686624 − 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) −10.0000 −0.402585 −0.201292 − 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 18.0000 0.721734
\(623\) 20.0000 0.801283
\(624\) 2.00000 0.0800641
\(625\) 25.0000 1.00000
\(626\) −16.0000 −0.639489
\(627\) 8.00000 0.319489
\(628\) −14.0000 −0.558661
\(629\) −2.00000 −0.0797452
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 − 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) −14.0000 −0.556890
\(633\) −14.0000 −0.556450
\(634\) −24.0000 −0.953162
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 6.00000 0.237729
\(638\) −8.00000 −0.316723
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 − 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) −4.00000 −0.157867
\(643\) 40.0000 1.57745 0.788723 − 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) 12.0000 0.471769 0.235884 − 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 1.00000 0.0392837
\(649\) 2.00000 0.0785069
\(650\) 10.0000 0.392232
\(651\) 0 0
\(652\) −20.0000 −0.783260
\(653\) 16.0000 0.626128 0.313064 − 0.949732i \(-0.398644\pi\)
0.313064 + 0.949732i \(0.398644\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 8.00000 0.312110
\(658\) 16.0000 0.623745
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 − 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 20.0000 0.777322
\(663\) 2.00000 0.0776736
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 16.0000 0.619522
\(668\) 10.0000 0.386912
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 2.00000 0.0771517
\(673\) −44.0000 −1.69608 −0.848038 − 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) 20.0000 0.770371
\(675\) 5.00000 0.192450
\(676\) −9.00000 −0.346154
\(677\) −32.0000 −1.22986 −0.614930 − 0.788582i \(-0.710816\pi\)
−0.614930 + 0.788582i \(0.710816\pi\)
\(678\) 20.0000 0.768095
\(679\) −16.0000 −0.614024
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 0 0
\(683\) −26.0000 −0.994862 −0.497431 − 0.867503i \(-0.665723\pi\)
−0.497431 + 0.867503i \(0.665723\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 22.0000 0.839352
\(688\) 8.00000 0.304997
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 − 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 4.00000 0.151947
\(694\) −26.0000 −0.986947
\(695\) 0 0
\(696\) −4.00000 −0.151620
\(697\) −2.00000 −0.0757554
\(698\) 30.0000 1.13552
\(699\) 28.0000 1.05906
\(700\) 10.0000 0.377964
\(701\) 6.00000 0.226617 0.113308 − 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 2.00000 0.0754851
\(703\) −8.00000 −0.301726
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 12.0000 0.451306
\(708\) 1.00000 0.0375823
\(709\) −36.0000 −1.35201 −0.676004 − 0.736898i \(-0.736290\pi\)
−0.676004 + 0.736898i \(0.736290\pi\)
\(710\) 0 0
\(711\) −14.0000 −0.525041
\(712\) −10.0000 −0.374766
\(713\) 0 0
\(714\) 2.00000 0.0748481
\(715\) 0 0
\(716\) 0 0
\(717\) −20.0000 −0.746914
\(718\) 24.0000 0.895672
\(719\) −16.0000 −0.596699 −0.298350 − 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) −10.0000 −0.371904
\(724\) 0 0
\(725\) −20.0000 −0.742781
\(726\) 7.00000 0.259794
\(727\) 44.0000 1.63187 0.815935 − 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) −6.00000 −0.221766
\(733\) 34.0000 1.25582 0.627909 − 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 36.0000 1.32878
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 8.00000 0.294684
\(738\) −2.00000 −0.0736210
\(739\) −36.0000 −1.32428 −0.662141 − 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) 12.0000 0.440534
\(743\) −6.00000 −0.220119 −0.110059 − 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) −12.0000 −0.439057
\(748\) −2.00000 −0.0731272
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 − 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) −8.00000 −0.291730
\(753\) 12.0000 0.437304
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 22.0000 0.799604 0.399802 − 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −20.0000 −0.726433
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) 38.0000 1.37750 0.688749 − 0.724999i \(-0.258160\pi\)
0.688749 + 0.724999i \(0.258160\pi\)
\(762\) 12.0000 0.434714
\(763\) 20.0000 0.724049
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) 2.00000 0.0722158
\(768\) −1.00000 −0.0360844
\(769\) −18.0000 −0.649097 −0.324548 − 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 22.0000 0.791797
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) 8.00000 0.287183
\(777\) −4.00000 −0.143499
\(778\) −26.0000 −0.932145
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 4.00000 0.143040
\(783\) −4.00000 −0.142948
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 2.00000 0.0713376
\(787\) 40.0000 1.42585 0.712923 − 0.701242i \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) 8.00000 0.284988
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) 40.0000 1.42224
\(792\) −2.00000 −0.0710669
\(793\) −12.0000 −0.426132
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) −26.0000 −0.920967 −0.460484 − 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 8.00000 0.283197
\(799\) −8.00000 −0.283020
\(800\) −5.00000 −0.176777
\(801\) −10.0000 −0.353333
\(802\) −20.0000 −0.706225
\(803\) −16.0000 −0.564628
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) 2.00000 0.0704033
\(808\) −6.00000 −0.211079
\(809\) 48.0000 1.68759 0.843795 − 0.536666i \(-0.180316\pi\)
0.843795 + 0.536666i \(0.180316\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 − 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) −8.00000 −0.280745
\(813\) 0 0
\(814\) 4.00000 0.140200
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) 32.0000 1.11954
\(818\) −30.0000 −1.04893
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 14.0000 0.488603 0.244302 − 0.969699i \(-0.421441\pi\)
0.244302 + 0.969699i \(0.421441\pi\)
\(822\) 10.0000 0.348790
\(823\) 16.0000 0.557725 0.278862 − 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) −10.0000 −0.348155
\(826\) 2.00000 0.0695889
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 4.00000 0.139010
\(829\) 46.0000 1.59765 0.798823 − 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) −2.00000 −0.0693375
\(833\) −3.00000 −0.103944
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) 0 0
\(838\) −10.0000 −0.345444
\(839\) −32.0000 −1.10476 −0.552381 − 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −6.00000 −0.206774
\(843\) 10.0000 0.344418
\(844\) 14.0000 0.481900
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 14.0000 0.481046
\(848\) −6.00000 −0.206041
\(849\) −18.0000 −0.617758
\(850\) −5.00000 −0.171499
\(851\) −8.00000 −0.274236
\(852\) −2.00000 −0.0685189
\(853\) −52.0000 −1.78045 −0.890223 − 0.455525i \(-0.849452\pi\)
−0.890223 + 0.455525i \(0.849452\pi\)
\(854\) −12.0000 −0.410632
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) −4.00000 −0.136558
\(859\) 40.0000 1.36478 0.682391 − 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) −8.00000 −0.272481
\(863\) 8.00000 0.272323 0.136162 − 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 28.0000 0.949835
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −10.0000 −0.338643
\(873\) 8.00000 0.270759
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) −8.00000 −0.270295
\(877\) −52.0000 −1.75592 −0.877958 − 0.478738i \(-0.841094\pi\)
−0.877958 + 0.478738i \(0.841094\pi\)
\(878\) 26.0000 0.877457
\(879\) 10.0000 0.337292
\(880\) 0 0
\(881\) 52.0000 1.75192 0.875962 − 0.482380i \(-0.160227\pi\)
0.875962 + 0.482380i \(0.160227\pi\)
\(882\) −3.00000 −0.101015
\(883\) 36.0000 1.21150 0.605748 − 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 2.00000 0.0671156
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) −24.0000 −0.803579
\(893\) −32.0000 −1.07084
\(894\) −22.0000 −0.735790
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 8.00000 0.267112
\(898\) 34.0000 1.13459
\(899\) 0 0
\(900\) −5.00000 −0.166667
\(901\) −6.00000 −0.199889
\(902\) 4.00000 0.133185
\(903\) 16.0000 0.532447
\(904\) −20.0000 −0.665190
\(905\) 0 0
\(906\) 0 0
\(907\) −48.0000 −1.59381 −0.796907 − 0.604102i \(-0.793532\pi\)
−0.796907 + 0.604102i \(0.793532\pi\)
\(908\) 18.0000 0.597351
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 6.00000 0.198789 0.0993944 − 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) −4.00000 −0.132453
\(913\) 24.0000 0.794284
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) 4.00000 0.132092
\(918\) −1.00000 −0.0330049
\(919\) −8.00000 −0.263896 −0.131948 − 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) −38.0000 −1.25146
\(923\) −4.00000 −0.131662
\(924\) −4.00000 −0.131590
\(925\) 10.0000 0.328798
\(926\) −40.0000 −1.31448
\(927\) 0 0
\(928\) 4.00000 0.131306
\(929\) −52.0000 −1.70606 −0.853032 − 0.521858i \(-0.825239\pi\)
−0.853032 + 0.521858i \(0.825239\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) −28.0000 −0.917170
\(933\) −18.0000 −0.589294
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −58.0000 −1.89478 −0.947389 − 0.320085i \(-0.896288\pi\)
−0.947389 + 0.320085i \(0.896288\pi\)
\(938\) 8.00000 0.261209
\(939\) 16.0000 0.522140
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 − 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 14.0000 0.456145
\(943\) −8.00000 −0.260516
\(944\) −1.00000 −0.0325472
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) −28.0000 −0.909878 −0.454939 − 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 14.0000 0.454699
\(949\) −16.0000 −0.519382
\(950\) −20.0000 −0.648886
\(951\) 24.0000 0.778253
\(952\) −2.00000 −0.0648204
\(953\) 38.0000 1.23094 0.615470 − 0.788160i \(-0.288966\pi\)
0.615470 + 0.788160i \(0.288966\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) 8.00000 0.258603
\(958\) 18.0000 0.581554
\(959\) 20.0000 0.645834
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 4.00000 0.128965
\(963\) 4.00000 0.128898
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −7.00000 −0.224989
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 − 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 8.00000 0.256468
\(974\) −18.0000 −0.576757
\(975\) −10.0000 −0.320256
\(976\) 6.00000 0.192055
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 20.0000 0.639529
\(979\) 20.0000 0.639203
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 36.0000 1.14881
\(983\) 60.0000 1.91370 0.956851 − 0.290578i \(-0.0938475\pi\)
0.956851 + 0.290578i \(0.0938475\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) 4.00000 0.127386
\(987\) −16.0000 −0.509286
\(988\) −8.00000 −0.254514
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 − 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) −4.00000 −0.126872
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) 8.00000 0.253363 0.126681 − 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) 8.00000 0.253236
\(999\) 2.00000 0.0632772
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 6018.2.a.f.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6018.2.a.f.1.1 ✓ 1 1.1 even 1 trivial