Properties

Label 4650.2.a.x.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} +1.00000 q^{18} +4.00000 q^{21} +2.00000 q^{22} +6.00000 q^{23} -1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} +1.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} +1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{39} -10.0000 q^{41} +4.00000 q^{42} +4.00000 q^{43} +2.00000 q^{44} +6.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{56} -4.00000 q^{59} +1.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} -4.00000 q^{67} -6.00000 q^{69} -16.0000 q^{71} +1.00000 q^{72} -4.00000 q^{73} +2.00000 q^{74} -8.00000 q^{77} +2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{81} -10.0000 q^{82} -8.00000 q^{83} +4.00000 q^{84} +4.00000 q^{86} +2.00000 q^{88} +6.00000 q^{89} +8.00000 q^{91} +6.00000 q^{92} -1.00000 q^{93} -4.00000 q^{94} -1.00000 q^{96} -14.0000 q^{97} +9.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −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
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 2.00000 0.426401
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −4.00000 −0.755929
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 4.00000 0.617213
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 1.00000 0.127000
\(63\) −4.00000 −0.503953
\(64\) 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 \(0.578571\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 0 0
\(77\) −8.00000 −0.911685
\(78\) 2.00000 0.226455
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 6.00000 0.625543
\(93\) −1.00000 −0.103695
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 9.00000 0.909137
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −4.00000 −0.377964
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 10.0000 0.901670
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) −6.00000 −0.510754
\(139\) −6.00000 −0.508913 −0.254457 0.967084i \(-0.581897\pi\)
−0.254457 + 0.967084i \(0.581897\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) −16.0000 −1.34269
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) −9.00000 −0.742307
\(148\) 2.00000 0.164399
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −8.00000 −0.644658
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 4.00000 0.318223
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 1.00000 0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) −14.0000 −1.08335 −0.541676 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) 4.00000 0.308607
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 4.00000 0.300658
\(178\) 6.00000 0.449719
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 8.00000 0.592999
\(183\) 0 0
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) 0 0
\(188\) −4.00000 −0.291730
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 2.00000 0.142134
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) −10.0000 −0.703598
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 12.0000 0.836080
\(207\) 6.00000 0.417029
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 16.0000 1.09630
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −4.00000 −0.271538
\(218\) −2.00000 −0.135457
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) 10.0000 0.669650 0.334825 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 0 0
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) −7.00000 −0.449977
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) 0 0
\(248\) 1.00000 0.0635001
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) −4.00000 −0.251976
\(253\) 12.0000 0.754434
\(254\) −18.0000 −1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −4.00000 −0.249029
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) −10.0000 −0.616626 −0.308313 0.951285i \(-0.599764\pi\)
−0.308313 + 0.951285i \(0.599764\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) −4.00000 −0.244339
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) −8.00000 −0.484182
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −6.00000 −0.359856
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 4.00000 0.238197
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 40.0000 2.36113
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) −4.00000 −0.234082
\(293\) 2.00000 0.116841 0.0584206 0.998292i \(-0.481394\pi\)
0.0584206 + 0.998292i \(0.481394\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) −2.00000 −0.116052
\(298\) −2.00000 −0.115857
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 4.00000 0.230174
\(303\) 10.0000 0.574485
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) −8.00000 −0.455842
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 2.00000 0.113228
\(313\) −28.0000 −1.58265 −0.791327 0.611393i \(-0.790609\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) −24.0000 −1.33747
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) 2.00000 0.110600
\(328\) −10.0000 −0.552158
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 18.0000 0.989369 0.494685 0.869072i \(-0.335284\pi\)
0.494685 + 0.869072i \(0.335284\pi\)
\(332\) −8.00000 −0.439057
\(333\) 2.00000 0.109599
\(334\) −14.0000 −0.766046
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) −9.00000 −0.489535
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 2.00000 0.106600
\(353\) −36.0000 −1.91609 −0.958043 0.286623i \(-0.907467\pi\)
−0.958043 + 0.286623i \(0.907467\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −2.00000 −0.105703
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −20.0000 −1.05118
\(363\) 7.00000 0.367405
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) 0 0
\(367\) −26.0000 −1.35719 −0.678594 0.734513i \(-0.737411\pi\)
−0.678594 + 0.734513i \(0.737411\pi\)
\(368\) 6.00000 0.312772
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) −1.00000 −0.0518476
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) 0 0
\(378\) 4.00000 0.205738
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) 0 0
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 4.00000 0.203331
\(388\) −14.0000 −0.710742
\(389\) 20.0000 1.01404 0.507020 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.00000 0.454569
\(393\) −4.00000 −0.201773
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 4.00000 0.199502
\(403\) −2.00000 −0.0996271
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 12.0000 0.591198
\(413\) 16.0000 0.787309
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 6.00000 0.293821
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) −4.00000 −0.194717
\(423\) −4.00000 −0.194487
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 40.0000 1.92673 0.963366 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 20.0000 0.961139 0.480569 0.876957i \(-0.340430\pi\)
0.480569 + 0.876957i \(0.340430\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 10.0000 0.473514
\(447\) 2.00000 0.0945968
\(448\) −4.00000 −0.188982
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −20.0000 −0.941763
\(452\) −2.00000 −0.0940721
\(453\) −4.00000 −0.187936
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −20.0000 −0.934539
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 8.00000 0.372194
\(463\) 18.0000 0.836531 0.418265 0.908325i \(-0.362638\pi\)
0.418265 + 0.908325i \(0.362638\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 22.0000 1.01913
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) −4.00000 −0.184115
\(473\) 8.00000 0.367840
\(474\) −4.00000 −0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −8.00000 −0.365911
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −22.0000 −1.00207
\(483\) 24.0000 1.09204
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −10.0000 −0.453143 −0.226572 0.973995i \(-0.572752\pi\)
−0.226572 + 0.973995i \(0.572752\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 10.0000 0.450835
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 64.0000 2.87079
\(498\) 8.00000 0.358489
\(499\) 22.0000 0.984855 0.492428 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) 0 0
\(501\) 14.0000 0.625474
\(502\) 2.00000 0.0892644
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 12.0000 0.533465
\(507\) 9.00000 0.399704
\(508\) −18.0000 −0.798621
\(509\) −4.00000 −0.177297 −0.0886484 0.996063i \(-0.528255\pi\)
−0.0886484 + 0.996063i \(0.528255\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) −8.00000 −0.351840
\(518\) −8.00000 −0.351500
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −10.0000 −0.436021
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 20.0000 0.866296
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 2.00000 0.0863064
\(538\) −24.0000 −1.03471
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) 0 0
\(543\) 20.0000 0.858282
\(544\) 0 0
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) 12.0000 0.512615
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) −6.00000 −0.255377
\(553\) −16.0000 −0.680389
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −6.00000 −0.254457
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 1.00000 0.0423334
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 4.00000 0.168430
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) −4.00000 −0.167984
\(568\) −16.0000 −0.671345
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −26.0000 −1.08807 −0.544033 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) 40.0000 1.66957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 6.00000 0.249783 0.124892 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) −17.0000 −0.707107
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) 32.0000 1.32758
\(582\) 14.0000 0.580319
\(583\) −12.0000 −0.496989
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 2.00000 0.0826192
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) −9.00000 −0.371154
\(589\) 0 0
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 2.00000 0.0821995
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −2.00000 −0.0819232
\(597\) −20.0000 −0.818546
\(598\) −12.0000 −0.490716
\(599\) −48.0000 −1.96123 −0.980613 0.195952i \(-0.937220\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −16.0000 −0.652111
\(603\) −4.00000 −0.162893
\(604\) 4.00000 0.162758
\(605\) 0 0
\(606\) 10.0000 0.406222
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 42.0000 1.69636 0.848182 0.529705i \(-0.177697\pi\)
0.848182 + 0.529705i \(0.177697\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) −8.00000 −0.322329
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) −12.0000 −0.482711
\(619\) 34.0000 1.36658 0.683288 0.730149i \(-0.260549\pi\)
0.683288 + 0.730149i \(0.260549\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) −24.0000 −0.961540
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −28.0000 −1.11911
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) 0 0
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 4.00000 0.159111
\(633\) 4.00000 0.158986
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) −18.0000 −0.713186
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 12.0000 0.473602
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) −24.0000 −0.945732
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 1.00000 0.0392837
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 12.0000 0.469956
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) −4.00000 −0.156055
\(658\) 16.0000 0.623745
\(659\) −28.0000 −1.09073 −0.545363 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 18.0000 0.699590
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 0 0
\(668\) −14.0000 −0.541676
\(669\) −10.0000 −0.386622
\(670\) 0 0
\(671\) 0 0
\(672\) 4.00000 0.154303
\(673\) 40.0000 1.54189 0.770943 0.636904i \(-0.219785\pi\)
0.770943 + 0.636904i \(0.219785\pi\)
\(674\) 32.0000 1.23259
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 2.00000 0.0768095
\(679\) 56.0000 2.14908
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 2.00000 0.0765840
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 20.0000 0.763048
\(688\) 4.00000 0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 14.0000 0.532200
\(693\) −8.00000 −0.303895
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −14.0000 −0.529908
\(699\) −22.0000 −0.832116
\(700\) 0 0
\(701\) −38.0000 −1.43524 −0.717620 0.696435i \(-0.754769\pi\)
−0.717620 + 0.696435i \(0.754769\pi\)
\(702\) 2.00000 0.0754851
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −36.0000 −1.35488
\(707\) 40.0000 1.50435
\(708\) 4.00000 0.150329
\(709\) −32.0000 −1.20179 −0.600893 0.799330i \(-0.705188\pi\)
−0.600893 + 0.799330i \(0.705188\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 6.00000 0.224860
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) −2.00000 −0.0747435
\(717\) 8.00000 0.298765
\(718\) 24.0000 0.895672
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) −19.0000 −0.707107
\(723\) 22.0000 0.818189
\(724\) −20.0000 −0.743294
\(725\) 0 0
\(726\) 7.00000 0.259794
\(727\) −48.0000 −1.78022 −0.890111 0.455744i \(-0.849373\pi\)
−0.890111 + 0.455744i \(0.849373\pi\)
\(728\) 8.00000 0.296500
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −26.0000 −0.959678
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) −8.00000 −0.294684
\(738\) −10.0000 −0.368105
\(739\) 2.00000 0.0735712 0.0367856 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) −26.0000 −0.953847 −0.476924 0.878945i \(-0.658248\pi\)
−0.476924 + 0.878945i \(0.658248\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) −8.00000 −0.292705
\(748\) 0 0
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) −4.00000 −0.145865
\(753\) −2.00000 −0.0728841
\(754\) 0 0
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) −16.0000 −0.581146
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) −26.0000 −0.942499 −0.471250 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) 18.0000 0.652071
\(763\) 8.00000 0.289619
\(764\) 0 0
\(765\) 0 0
\(766\) −6.00000 −0.216789
\(767\) 8.00000 0.288863
\(768\) −1.00000 −0.0360844
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 10.0000 0.359908
\(773\) 2.00000 0.0719350 0.0359675 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) 8.00000 0.286998
\(778\) 20.0000 0.717035
\(779\) 0 0
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) 40.0000 1.42585 0.712923 0.701242i \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) 6.00000 0.213741
\(789\) 10.0000 0.356009
\(790\) 0 0
\(791\) 8.00000 0.284447
\(792\) 2.00000 0.0710669
\(793\) 0 0
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −38.0000 −1.34603 −0.673015 0.739629i \(-0.735001\pi\)
−0.673015 + 0.739629i \(0.735001\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 30.0000 1.05934
\(803\) −8.00000 −0.282314
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −2.00000 −0.0704470
\(807\) 24.0000 0.844840
\(808\) −10.0000 −0.351799
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 32.0000 1.12367 0.561836 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 4.00000 0.140200
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −18.0000 −0.629355
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) −12.0000 −0.418548
\(823\) −18.0000 −0.627441 −0.313720 0.949515i \(-0.601575\pi\)
−0.313720 + 0.949515i \(0.601575\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) 16.0000 0.556711
\(827\) 56.0000 1.94731 0.973655 0.228024i \(-0.0732266\pi\)
0.973655 + 0.228024i \(0.0732266\pi\)
\(828\) 6.00000 0.208514
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 6.00000 0.207763
\(835\) 0 0
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 36.0000 1.24360
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −30.0000 −1.03387
\(843\) −6.00000 −0.206651
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) −4.00000 −0.137523
\(847\) 28.0000 0.962091
\(848\) −6.00000 −0.206041
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 16.0000 0.548151
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 46.0000 1.57133 0.785665 0.618652i \(-0.212321\pi\)
0.785665 + 0.618652i \(0.212321\pi\)
\(858\) 4.00000 0.136558
\(859\) −2.00000 −0.0682391 −0.0341196 0.999418i \(-0.510863\pi\)
−0.0341196 + 0.999418i \(0.510863\pi\)
\(860\) 0 0
\(861\) −40.0000 −1.36320
\(862\) 40.0000 1.36241
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 20.0000 0.679628
\(867\) 17.0000 0.577350
\(868\) −4.00000 −0.135769
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −2.00000 −0.0677285
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 24.0000 0.809961
\(879\) −2.00000 −0.0674583
\(880\) 0 0
\(881\) 22.0000 0.741199 0.370599 0.928793i \(-0.379152\pi\)
0.370599 + 0.928793i \(0.379152\pi\)
\(882\) 9.00000 0.303046
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 28.0000 0.940148 0.470074 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 72.0000 2.41480
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 10.0000 0.334825
\(893\) 0 0
\(894\) 2.00000 0.0668900
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 12.0000 0.400668
\(898\) 6.00000 0.200223
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −20.0000 −0.665927
\(903\) 16.0000 0.532447
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) −4.00000 −0.132891
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 20.0000 0.663723
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) −20.0000 −0.660819
\(917\) −16.0000 −0.528367
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) 0 0
\(923\) 32.0000 1.05329
\(924\) 8.00000 0.263181
\(925\) 0 0
\(926\) 18.0000 0.591517
\(927\) 12.0000 0.394132
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 22.0000 0.720634
\(933\) 0 0
\(934\) 20.0000 0.654420
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 16.0000 0.522419
\(939\) 28.0000 0.913745
\(940\) 0 0
\(941\) 28.0000 0.912774 0.456387 0.889781i \(-0.349143\pi\)
0.456387 + 0.889781i \(0.349143\pi\)
\(942\) 18.0000 0.586472
\(943\) −60.0000 −1.95387
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) −4.00000 −0.129914
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) 12.0000 0.388718 0.194359 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) 16.0000 0.516937
\(959\) −48.0000 −1.55000
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −4.00000 −0.128965
\(963\) −12.0000 −0.386695
\(964\) −22.0000 −0.708572
\(965\) 0 0
\(966\) 24.0000 0.772187
\(967\) −26.0000 −0.836104 −0.418052 0.908423i \(-0.637287\pi\)
−0.418052 + 0.908423i \(0.637287\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 0 0
\(971\) 48.0000 1.54039 0.770197 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 24.0000 0.769405
\(974\) −10.0000 −0.320421
\(975\) 0 0
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −12.0000 −0.383718
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 6.00000 0.191468
\(983\) −10.0000 −0.318950 −0.159475 0.987202i \(-0.550980\pi\)
−0.159475 + 0.987202i \(0.550980\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) 0 0
\(987\) −16.0000 −0.509286
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 28.0000 0.889449 0.444725 0.895667i \(-0.353302\pi\)
0.444725 + 0.895667i \(0.353302\pi\)
\(992\) 1.00000 0.0317500
\(993\) −18.0000 −0.571213
\(994\) 64.0000 2.02996
\(995\) 0 0
\(996\) 8.00000 0.253490
\(997\) −6.00000 −0.190022 −0.0950110 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) 22.0000 0.696398
\(999\) −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 4650.2.a.x.1.1 1
5.2 odd 4 4650.2.d.h.3349.2 2
5.3 odd 4 4650.2.d.h.3349.1 2
5.4 even 2 930.2.a.j.1.1 1
15.14 odd 2 2790.2.a.v.1.1 1
20.19 odd 2 7440.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.j.1.1 1 5.4 even 2
2790.2.a.v.1.1 1 15.14 odd 2
4650.2.a.x.1.1 1 1.1 even 1 trivial
4650.2.d.h.3349.1 2 5.3 odd 4
4650.2.d.h.3349.2 2 5.2 odd 4
7440.2.a.g.1.1 1 20.19 odd 2