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Properties

Label 4650.2.a.w.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
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] = [4650,2,Mod(1,4650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4650, 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("4650.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.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: \(37.1304369399\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4650.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +4.00000 q^{21} -4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -2.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +10.0000 q^{41} +4.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} -4.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +2.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} +8.00000 q^{59} +10.0000 q^{61} -1.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} +12.0000 q^{67} -2.00000 q^{68} +4.00000 q^{69} +1.00000 q^{72} -14.0000 q^{73} +6.00000 q^{74} +4.00000 q^{76} +16.0000 q^{77} +2.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +10.0000 q^{82} -4.00000 q^{83} +4.00000 q^{84} +4.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} -6.00000 q^{89} +8.00000 q^{91} -4.00000 q^{92} +1.00000 q^{93} -8.00000 q^{94} -1.00000 q^{96} +6.00000 q^{97} +9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 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
\(21\) 4.00000 0.872872
\(22\) −4.00000 −0.852803
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −4.00000 −0.755929
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 4.00000 0.648886
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 4.00000 0.617213
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) 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
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) −4.00000 −0.529813
\(58\) −2.00000 −0.262613
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −1.00000 −0.127000
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −2.00000 −0.242536
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 16.0000 1.82337
\(78\) 2.00000 0.226455
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 2.00000 0.214423
\(88\) −4.00000 −0.426401
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) −4.00000 −0.417029
\(93\) 1.00000 0.103695
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 9.00000 0.909137
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 − 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 2.00000 0.198030
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −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
\(109\) 6.00000 0.574696 0.287348 − 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) −4.00000 −0.377964
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) 8.00000 0.736460
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) −10.0000 −0.901670
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 − 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 4.00000 0.348155
\(133\) −16.0000 −1.38738
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 4.00000 0.340503
\(139\) 16.0000 1.35710 0.678551 − 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) −9.00000 −0.742307
\(148\) 6.00000 0.493197
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 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
\(153\) −2.00000 −0.161690
\(154\) 16.0000 1.28932
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 22.0000 1.75579 0.877896 − 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) −8.00000 −0.636446
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 1.00000 0.0785674
\(163\) −12.0000 −0.939913 −0.469956 − 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 20.0000 1.54765 0.773823 − 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 4.00000 0.308607
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) −8.00000 −0.601317
\(178\) −6.00000 −0.449719
\(179\) −20.0000 −1.49487 −0.747435 − 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 − 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 8.00000 0.592999
\(183\) −10.0000 −0.739221
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 8.00000 0.585018
\(188\) −8.00000 −0.583460
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 14.0000 0.997459 0.498729 − 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) −4.00000 −0.284268
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 18.0000 1.26648
\(203\) 8.00000 0.561490
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) −4.00000 −0.278019
\(208\) −2.00000 −0.138675
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 4.00000 0.271538
\(218\) 6.00000 0.406371
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) −6.00000 −0.402694
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −4.00000 −0.265489 −0.132745 − 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) −4.00000 −0.264906
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) −2.00000 −0.131306
\(233\) 10.0000 0.655122 0.327561 − 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) 8.00000 0.519656
\(238\) 8.00000 0.518563
\(239\) 8.00000 0.517477 0.258738 − 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 − 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) −8.00000 −0.509028
\(248\) −1.00000 −0.0635001
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 − 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) −4.00000 −0.251976
\(253\) 16.0000 1.00591
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −4.00000 −0.249029
\(259\) −24.0000 −1.49129
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) −8.00000 −0.494242
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) −16.0000 −0.981023
\(267\) 6.00000 0.367194
\(268\) 12.0000 0.733017
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 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
\(273\) −8.00000 −0.484182
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 22.0000 1.32185 0.660926 − 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 16.0000 0.959616
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 − 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 8.00000 0.476393
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) −40.0000 −2.36113
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) −14.0000 −0.819288
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 4.00000 0.232104
\(298\) −6.00000 −0.347571
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 8.00000 0.460348
\(303\) −18.0000 −1.03407
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 16.0000 0.911685
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 − 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 2.00000 0.113228
\(313\) 34.0000 1.92179 0.960897 − 0.276907i \(-0.0893093\pi\)
0.960897 + 0.276907i \(0.0893093\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −2.00000 −0.112331 −0.0561656 − 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −6.00000 −0.336463
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 16.0000 0.891645
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) −6.00000 −0.331801
\(328\) 10.0000 0.552158
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) 24.0000 1.31916 0.659580 − 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) −4.00000 −0.219529
\(333\) 6.00000 0.328798
\(334\) 20.0000 1.09435
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) 2.00000 0.108947 0.0544735 − 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −9.00000 −0.489535
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 4.00000 0.216295
\(343\) −8.00000 −0.431959
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) −20.0000 −1.07366 −0.536828 − 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 2.00000 0.107211
\(349\) 30.0000 1.60586 0.802932 − 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −4.00000 −0.213201
\(353\) −10.0000 −0.532246 −0.266123 − 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) −8.00000 −0.425195
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −8.00000 −0.423405
\(358\) −20.0000 −1.05703
\(359\) 16.0000 0.844448 0.422224 − 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −6.00000 −0.315353
\(363\) −5.00000 −0.262432
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 8.00000 0.417597 0.208798 − 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −4.00000 −0.208514
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) −24.0000 −1.24602
\(372\) 1.00000 0.0518476
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 4.00000 0.206010
\(378\) 4.00000 0.205738
\(379\) 4.00000 0.205466 0.102733 − 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 24.0000 1.22795
\(383\) 4.00000 0.204390 0.102195 − 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 4.00000 0.203331
\(388\) 6.00000 0.304604
\(389\) −10.0000 −0.507020 −0.253510 − 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 9.00000 0.454569
\(393\) 8.00000 0.403547
\(394\) 14.0000 0.705310
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) −2.00000 −0.100377 −0.0501886 − 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −8.00000 −0.401004
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) 34.0000 1.69788 0.848939 − 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) −12.0000 −0.598506
\(403\) 2.00000 0.0996271
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) −24.0000 −1.18964
\(408\) 2.00000 0.0990148
\(409\) 18.0000 0.890043 0.445021 − 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) −4.00000 −0.197066
\(413\) −32.0000 −1.57462
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −16.0000 −0.783523
\(418\) −16.0000 −0.782586
\(419\) 8.00000 0.390826 0.195413 − 0.980721i \(-0.437395\pi\)
0.195413 + 0.980721i \(0.437395\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 − 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) −12.0000 −0.584151
\(423\) −8.00000 −0.388973
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) −40.0000 −1.93574
\(428\) 4.00000 0.193347
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 26.0000 1.24948 0.624740 − 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) −16.0000 −0.765384
\(438\) 14.0000 0.668946
\(439\) 32.0000 1.52728 0.763638 − 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 4.00000 0.190261
\(443\) −28.0000 −1.33032 −0.665160 − 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 6.00000 0.283790
\(448\) −4.00000 −0.188982
\(449\) −14.0000 −0.660701 −0.330350 − 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) −6.00000 −0.282216
\(453\) −8.00000 −0.375873
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 18.0000 0.842004 0.421002 − 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −14.0000 −0.654177
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 38.0000 1.76984 0.884918 − 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(462\) −16.0000 −0.744387
\(463\) 24.0000 1.11537 0.557687 − 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −48.0000 −2.21643
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 8.00000 0.368230
\(473\) −16.0000 −0.735681
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) 6.00000 0.274721
\(478\) 8.00000 0.365911
\(479\) −16.0000 −0.731059 −0.365529 − 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −14.0000 −0.637683
\(483\) −16.0000 −0.728025
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −24.0000 −1.08754 −0.543772 − 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 10.0000 0.452679
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 − 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) −10.0000 −0.450835
\(493\) 4.00000 0.180151
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) −24.0000 −1.07439 −0.537194 − 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 0 0
\(501\) −20.0000 −0.893534
\(502\) −28.0000 −1.24970
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) 9.00000 0.399704
\(508\) 8.00000 0.354943
\(509\) 22.0000 0.975133 0.487566 − 0.873086i \(-0.337885\pi\)
0.487566 + 0.873086i \(0.337885\pi\)
\(510\) 0 0
\(511\) 56.0000 2.47729
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 32.0000 1.40736
\(518\) −24.0000 −1.05450
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 − 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −36.0000 −1.57417 −0.787085 − 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 2.00000 0.0871214
\(528\) 4.00000 0.174078
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) −16.0000 −0.693688
\(533\) −20.0000 −0.866296
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 20.0000 0.863064
\(538\) −18.0000 −0.776035
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 − 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 8.00000 0.343629
\(543\) 6.00000 0.257485
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) 28.0000 1.19719 0.598597 − 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 6.00000 0.256307
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 4.00000 0.170251
\(553\) 32.0000 1.36078
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) −42.0000 −1.77960 −0.889799 − 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) −1.00000 −0.0423334
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) −30.0000 −1.26547
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) −4.00000 −0.167984
\(568\) 0 0
\(569\) 26.0000 1.08998 0.544988 − 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 8.00000 0.334497
\(573\) −24.0000 −1.00261
\(574\) −40.0000 −1.66957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 30.0000 1.24892 0.624458 − 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) −13.0000 −0.540729
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 16.0000 0.663792
\(582\) −6.00000 −0.248708
\(583\) −24.0000 −0.993978
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −28.0000 −1.15568 −0.577842 − 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) −9.00000 −0.371154
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) −14.0000 −0.575883
\(592\) 6.00000 0.246598
\(593\) 18.0000 0.739171 0.369586 − 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 8.00000 0.327418
\(598\) 8.00000 0.327144
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 34.0000 1.38689 0.693444 − 0.720510i \(-0.256092\pi\)
0.693444 + 0.720510i \(0.256092\pi\)
\(602\) −16.0000 −0.652111
\(603\) 12.0000 0.488678
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) −18.0000 −0.731200
\(607\) −28.0000 −1.13648 −0.568242 − 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 4.00000 0.162221
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) −2.00000 −0.0808452
\(613\) −26.0000 −1.05013 −0.525065 − 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 16.0000 0.644658
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 4.00000 0.160904
\(619\) −32.0000 −1.28619 −0.643094 − 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 24.0000 0.962312
\(623\) 24.0000 0.961540
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 34.0000 1.35891
\(627\) 16.0000 0.638978
\(628\) 22.0000 0.877896
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −8.00000 −0.318223
\(633\) 12.0000 0.476957
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) −18.0000 −0.713186
\(638\) 8.00000 0.316723
\(639\) 0 0
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 − 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −4.00000 −0.157867
\(643\) 20.0000 0.788723 0.394362 − 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 28.0000 1.10079 0.550397 − 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 1.00000 0.0392837
\(649\) −32.0000 −1.25611
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) −12.0000 −0.469956
\(653\) −18.0000 −0.704394 −0.352197 − 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −6.00000 −0.234619
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) −14.0000 −0.546192
\(658\) 32.0000 1.24749
\(659\) −8.00000 −0.311636 −0.155818 − 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 0 0
\(661\) 46.0000 1.78919 0.894596 − 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) 24.0000 0.932786
\(663\) −4.00000 −0.155347
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 8.00000 0.309761
\(668\) 20.0000 0.773823
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) 4.00000 0.154303
\(673\) −14.0000 −0.539660 −0.269830 − 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 6.00000 0.230429
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 4.00000 0.153168
\(683\) −28.0000 −1.07139 −0.535695 − 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 14.0000 0.534133
\(688\) 4.00000 0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 14.0000 0.532200
\(693\) 16.0000 0.607790
\(694\) −20.0000 −0.759190
\(695\) 0 0
\(696\) 2.00000 0.0758098
\(697\) −20.0000 −0.757554
\(698\) 30.0000 1.13552
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) −46.0000 −1.73740 −0.868698 − 0.495342i \(-0.835043\pi\)
−0.868698 + 0.495342i \(0.835043\pi\)
\(702\) 2.00000 0.0754851
\(703\) 24.0000 0.905177
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −10.0000 −0.376355
\(707\) −72.0000 −2.70784
\(708\) −8.00000 −0.300658
\(709\) 42.0000 1.57734 0.788672 − 0.614815i \(-0.210769\pi\)
0.788672 + 0.614815i \(0.210769\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −6.00000 −0.224860
\(713\) 4.00000 0.149801
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) −8.00000 −0.298765
\(718\) 16.0000 0.597115
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) −3.00000 −0.111648
\(723\) 14.0000 0.520666
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) −4.00000 −0.148352 −0.0741759 − 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) 8.00000 0.296500
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) −10.0000 −0.369611
\(733\) −18.0000 −0.664845 −0.332423 − 0.943131i \(-0.607866\pi\)
−0.332423 + 0.943131i \(0.607866\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −48.0000 −1.76810
\(738\) 10.0000 0.368105
\(739\) −16.0000 −0.588570 −0.294285 − 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) −24.0000 −0.881068
\(743\) 4.00000 0.146746 0.0733729 − 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) 14.0000 0.512576
\(747\) −4.00000 −0.146352
\(748\) 8.00000 0.292509
\(749\) −16.0000 −0.584627
\(750\) 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
\(753\) 28.0000 1.02038
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 54.0000 1.96266 0.981332 − 0.192323i \(-0.0616021\pi\)
0.981332 + 0.192323i \(0.0616021\pi\)
\(758\) 4.00000 0.145287
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 − 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) −8.00000 −0.289809
\(763\) −24.0000 −0.868858
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) 4.00000 0.144526
\(767\) −16.0000 −0.577727
\(768\) −1.00000 −0.0360844
\(769\) −46.0000 −1.65880 −0.829401 − 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 14.0000 0.503871
\(773\) −26.0000 −0.935155 −0.467578 − 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 24.0000 0.860995
\(778\) −10.0000 −0.358517
\(779\) 40.0000 1.43315
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) 2.00000 0.0714742
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 8.00000 0.285351
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 14.0000 0.498729
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) −4.00000 −0.142134
\(793\) −20.0000 −0.710221
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 16.0000 0.566394
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 34.0000 1.20058
\(803\) 56.0000 1.97620
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) 18.0000 0.633630
\(808\) 18.0000 0.633238
\(809\) −38.0000 −1.33601 −0.668004 − 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 8.00000 0.280745
\(813\) −8.00000 −0.280572
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) 16.0000 0.559769
\(818\) 18.0000 0.629355
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 − 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) −6.00000 −0.209274
\(823\) −40.0000 −1.39431 −0.697156 − 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −32.0000 −1.11342
\(827\) 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 \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) −2.00000 −0.0693375
\(833\) −18.0000 −0.623663
\(834\) −16.0000 −0.554035
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 1.00000 0.0345651
\(838\) 8.00000 0.276355
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 14.0000 0.482472
\(843\) 30.0000 1.03325
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) −20.0000 −0.687208
\(848\) 6.00000 0.206041
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 − 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) −40.0000 −1.36877
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) −22.0000 −0.751506 −0.375753 − 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) −8.00000 −0.273115
\(859\) 40.0000 1.36478 0.682391 − 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 40.0000 1.36320
\(862\) −24.0000 −0.817443
\(863\) −4.00000 −0.136162 −0.0680808 − 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) 13.0000 0.441503
\(868\) 4.00000 0.135769
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 6.00000 0.203186
\(873\) 6.00000 0.203069
\(874\) −16.0000 −0.541208
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) 38.0000 1.28317 0.641584 − 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 32.0000 1.07995
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 − 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 9.00000 0.303046
\(883\) 52.0000 1.74994 0.874970 − 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −28.0000 −0.940678
\(887\) −40.0000 −1.34307 −0.671534 − 0.740973i \(-0.734364\pi\)
−0.671534 + 0.740973i \(0.734364\pi\)
\(888\) −6.00000 −0.201347
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) −16.0000 −0.535720
\(893\) −32.0000 −1.07084
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) −8.00000 −0.267112
\(898\) −14.0000 −0.467186
\(899\) 2.00000 0.0667037
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) −40.0000 −1.33185
\(903\) 16.0000 0.532447
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 28.0000 0.929725 0.464862 − 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −4.00000 −0.132745
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −4.00000 −0.132453
\(913\) 16.0000 0.529523
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 32.0000 1.05673
\(918\) 2.00000 0.0660098
\(919\) −8.00000 −0.263896 −0.131948 − 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 38.0000 1.25146
\(923\) 0 0
\(924\) −16.0000 −0.526361
\(925\) 0 0
\(926\) 24.0000 0.788689
\(927\) −4.00000 −0.131377
\(928\) −2.00000 −0.0656532
\(929\) −46.0000 −1.50921 −0.754606 − 0.656179i \(-0.772172\pi\)
−0.754606 + 0.656179i \(0.772172\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) 10.0000 0.327561
\(933\) −24.0000 −0.785725
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 14.0000 0.457360 0.228680 − 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) −48.0000 −1.56726
\(939\) −34.0000 −1.10955
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 − 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) −22.0000 −0.716799
\(943\) −40.0000 −1.30258
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) −52.0000 −1.68977 −0.844886 − 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 8.00000 0.259828
\(949\) 28.0000 0.908918
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 8.00000 0.259281
\(953\) 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
\(956\) 8.00000 0.258738
\(957\) −8.00000 −0.258603
\(958\) −16.0000 −0.516937
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −12.0000 −0.386896
\(963\) 4.00000 0.128898
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) −48.0000 −1.54358 −0.771788 − 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 5.00000 0.160706
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 40.0000 1.28366 0.641831 − 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −64.0000 −2.05175
\(974\) −24.0000 −0.769010
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 10.0000 0.319928 0.159964 − 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) 12.0000 0.383718
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) −20.0000 −0.638226
\(983\) 36.0000 1.14822 0.574111 − 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) −10.0000 −0.318788
\(985\) 0 0
\(986\) 4.00000 0.127386
\(987\) −32.0000 −1.01857
\(988\) −8.00000 −0.254514
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) −1.00000 −0.0317500
\(993\) −24.0000 −0.761617
\(994\) 0 0
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) −18.0000 −0.570066 −0.285033 − 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) −24.0000 −0.759707
\(999\) −6.00000 −0.189832
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 4650.2.a.w.1.1 1
5.2 odd 4 4650.2.d.c.3349.2 2
5.3 odd 4 4650.2.d.c.3349.1 2
5.4 even 2 930.2.a.g.1.1 ✓ 1
15.14 odd 2 2790.2.a.bc.1.1 1
20.19 odd 2 7440.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.g.1.1 ✓ 1 5.4 even 2
2790.2.a.bc.1.1 1 15.14 odd 2
4650.2.a.w.1.1 1 1.1 even 1 trivial
4650.2.d.c.3349.1 2 5.3 odd 4
4650.2.d.c.3349.2 2 5.2 odd 4
7440.2.a.a.1.1 1 20.19 odd 2