Properties

Label 1150.2.a.e.1.1
Level $1150$
Weight $2$
Character 1150.1
Self dual yes
Analytic conductor $9.183$
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] = [1150,2,Mod(1,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1150.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.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: \(9.18279623245\)
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 \(\chi\) \(=\) 1150.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +3.00000 q^{11} -3.00000 q^{12} +6.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -5.00000 q^{17} +6.00000 q^{18} -1.00000 q^{19} +12.0000 q^{21} +3.00000 q^{22} -1.00000 q^{23} -3.00000 q^{24} +6.00000 q^{26} -9.00000 q^{27} -4.00000 q^{28} -8.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} -9.00000 q^{33} -5.00000 q^{34} +6.00000 q^{36} -2.00000 q^{37} -1.00000 q^{38} -18.0000 q^{39} -7.00000 q^{41} +12.0000 q^{42} -4.00000 q^{43} +3.00000 q^{44} -1.00000 q^{46} -10.0000 q^{47} -3.00000 q^{48} +9.00000 q^{49} +15.0000 q^{51} +6.00000 q^{52} +12.0000 q^{53} -9.00000 q^{54} -4.00000 q^{56} +3.00000 q^{57} -8.00000 q^{58} +4.00000 q^{59} -8.00000 q^{61} -8.00000 q^{62} -24.0000 q^{63} +1.00000 q^{64} -9.00000 q^{66} -3.00000 q^{67} -5.00000 q^{68} +3.00000 q^{69} +4.00000 q^{71} +6.00000 q^{72} +7.00000 q^{73} -2.00000 q^{74} -1.00000 q^{76} -12.0000 q^{77} -18.0000 q^{78} -6.00000 q^{79} +9.00000 q^{81} -7.00000 q^{82} -11.0000 q^{83} +12.0000 q^{84} -4.00000 q^{86} +24.0000 q^{87} +3.00000 q^{88} -3.00000 q^{89} -24.0000 q^{91} -1.00000 q^{92} +24.0000 q^{93} -10.0000 q^{94} -3.00000 q^{96} +14.0000 q^{97} +9.00000 q^{98} +18.0000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −3.00000 −0.866025
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 6.00000 1.41421
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 12.0000 2.61861
\(22\) 3.00000 0.639602
\(23\) −1.00000 −0.208514
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) −9.00000 −1.73205
\(28\) −4.00000 −0.755929
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) −9.00000 −1.56670
\(34\) −5.00000 −0.857493
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −1.00000 −0.162221
\(39\) −18.0000 −2.88231
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 12.0000 1.85164
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −10.0000 −1.45865 −0.729325 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) −3.00000 −0.433013
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 15.0000 2.10042
\(52\) 6.00000 0.832050
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) −9.00000 −1.22474
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 3.00000 0.397360
\(58\) −8.00000 −1.05045
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −8.00000 −1.01600
\(63\) −24.0000 −3.02372
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −9.00000 −1.10782
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) −5.00000 −0.606339
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 6.00000 0.707107
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −12.0000 −1.36753
\(78\) −18.0000 −2.03810
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −7.00000 −0.773021
\(83\) −11.0000 −1.20741 −0.603703 0.797209i \(-0.706309\pi\)
−0.603703 + 0.797209i \(0.706309\pi\)
\(84\) 12.0000 1.30931
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 24.0000 2.57307
\(88\) 3.00000 0.319801
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) −24.0000 −2.51588
\(92\) −1.00000 −0.104257
\(93\) 24.0000 2.48868
\(94\) −10.0000 −1.03142
\(95\) 0 0
\(96\) −3.00000 −0.306186
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 9.00000 0.909137
\(99\) 18.0000 1.80907
\(100\) 0 0
\(101\) 4.00000 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 15.0000 1.48522
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) −5.00000 −0.483368 −0.241684 0.970355i \(-0.577700\pi\)
−0.241684 + 0.970355i \(0.577700\pi\)
\(108\) −9.00000 −0.866025
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) −4.00000 −0.377964
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 3.00000 0.280976
\(115\) 0 0
\(116\) −8.00000 −0.742781
\(117\) 36.0000 3.32820
\(118\) 4.00000 0.368230
\(119\) 20.0000 1.83340
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −8.00000 −0.724286
\(123\) 21.0000 1.89351
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) −24.0000 −2.13809
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −9.00000 −0.783349
\(133\) 4.00000 0.346844
\(134\) −3.00000 −0.259161
\(135\) 0 0
\(136\) −5.00000 −0.428746
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 3.00000 0.255377
\(139\) −19.0000 −1.61156 −0.805779 0.592216i \(-0.798253\pi\)
−0.805779 + 0.592216i \(0.798253\pi\)
\(140\) 0 0
\(141\) 30.0000 2.52646
\(142\) 4.00000 0.335673
\(143\) 18.0000 1.50524
\(144\) 6.00000 0.500000
\(145\) 0 0
\(146\) 7.00000 0.579324
\(147\) −27.0000 −2.22692
\(148\) −2.00000 −0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −30.0000 −2.42536
\(154\) −12.0000 −0.966988
\(155\) 0 0
\(156\) −18.0000 −1.44115
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −6.00000 −0.477334
\(159\) −36.0000 −2.85499
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 9.00000 0.707107
\(163\) −5.00000 −0.391630 −0.195815 0.980641i \(-0.562735\pi\)
−0.195815 + 0.980641i \(0.562735\pi\)
\(164\) −7.00000 −0.546608
\(165\) 0 0
\(166\) −11.0000 −0.853766
\(167\) −22.0000 −1.70241 −0.851206 0.524832i \(-0.824128\pi\)
−0.851206 + 0.524832i \(0.824128\pi\)
\(168\) 12.0000 0.925820
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) −4.00000 −0.304997
\(173\) −4.00000 −0.304114 −0.152057 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 24.0000 1.81944
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) −12.0000 −0.901975
\(178\) −3.00000 −0.224860
\(179\) −3.00000 −0.224231 −0.112115 0.993695i \(-0.535763\pi\)
−0.112115 + 0.993695i \(0.535763\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −24.0000 −1.77900
\(183\) 24.0000 1.77413
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 24.0000 1.75977
\(187\) −15.0000 −1.09691
\(188\) −10.0000 −0.729325
\(189\) 36.0000 2.61861
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) −3.00000 −0.216506
\(193\) 9.00000 0.647834 0.323917 0.946085i \(-0.395000\pi\)
0.323917 + 0.946085i \(0.395000\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 18.0000 1.27920
\(199\) 18.0000 1.27599 0.637993 0.770042i \(-0.279765\pi\)
0.637993 + 0.770042i \(0.279765\pi\)
\(200\) 0 0
\(201\) 9.00000 0.634811
\(202\) 4.00000 0.281439
\(203\) 32.0000 2.24596
\(204\) 15.0000 1.05021
\(205\) 0 0
\(206\) −10.0000 −0.696733
\(207\) −6.00000 −0.417029
\(208\) 6.00000 0.416025
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 12.0000 0.824163
\(213\) −12.0000 −0.822226
\(214\) −5.00000 −0.341793
\(215\) 0 0
\(216\) −9.00000 −0.612372
\(217\) 32.0000 2.17230
\(218\) −10.0000 −0.677285
\(219\) −21.0000 −1.41905
\(220\) 0 0
\(221\) −30.0000 −2.01802
\(222\) 6.00000 0.402694
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −15.0000 −0.997785
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 3.00000 0.198680
\(229\) −30.0000 −1.98246 −0.991228 0.132164i \(-0.957808\pi\)
−0.991228 + 0.132164i \(0.957808\pi\)
\(230\) 0 0
\(231\) 36.0000 2.36863
\(232\) −8.00000 −0.525226
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 36.0000 2.35339
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 18.0000 1.16923
\(238\) 20.0000 1.29641
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 23.0000 1.48156 0.740780 0.671748i \(-0.234456\pi\)
0.740780 + 0.671748i \(0.234456\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) 21.0000 1.33891
\(247\) −6.00000 −0.381771
\(248\) −8.00000 −0.508001
\(249\) 33.0000 2.09129
\(250\) 0 0
\(251\) −7.00000 −0.441836 −0.220918 0.975292i \(-0.570905\pi\)
−0.220918 + 0.975292i \(0.570905\pi\)
\(252\) −24.0000 −1.51186
\(253\) −3.00000 −0.188608
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 12.0000 0.747087
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −48.0000 −2.97113
\(262\) 12.0000 0.741362
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) −9.00000 −0.553912
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) 9.00000 0.550791
\(268\) −3.00000 −0.183254
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 10.0000 0.607457 0.303728 0.952759i \(-0.401768\pi\)
0.303728 + 0.952759i \(0.401768\pi\)
\(272\) −5.00000 −0.303170
\(273\) 72.0000 4.35764
\(274\) 3.00000 0.181237
\(275\) 0 0
\(276\) 3.00000 0.180579
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) −19.0000 −1.13954
\(279\) −48.0000 −2.87368
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 30.0000 1.78647
\(283\) 33.0000 1.96165 0.980823 0.194900i \(-0.0624381\pi\)
0.980823 + 0.194900i \(0.0624381\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) 18.0000 1.06436
\(287\) 28.0000 1.65279
\(288\) 6.00000 0.353553
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −42.0000 −2.46208
\(292\) 7.00000 0.409644
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) −27.0000 −1.57467
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) −27.0000 −1.56670
\(298\) 6.00000 0.347571
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 20.0000 1.15087
\(303\) −12.0000 −0.689382
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −30.0000 −1.71499
\(307\) 9.00000 0.513657 0.256829 0.966457i \(-0.417322\pi\)
0.256829 + 0.966457i \(0.417322\pi\)
\(308\) −12.0000 −0.683763
\(309\) 30.0000 1.70664
\(310\) 0 0
\(311\) 2.00000 0.113410 0.0567048 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) −18.0000 −1.01905
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) −36.0000 −2.01878
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) 15.0000 0.837218
\(322\) 4.00000 0.222911
\(323\) 5.00000 0.278207
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −5.00000 −0.276924
\(327\) 30.0000 1.65900
\(328\) −7.00000 −0.386510
\(329\) 40.0000 2.20527
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) −11.0000 −0.603703
\(333\) −12.0000 −0.657596
\(334\) −22.0000 −1.20379
\(335\) 0 0
\(336\) 12.0000 0.654654
\(337\) −3.00000 −0.163420 −0.0817102 0.996656i \(-0.526038\pi\)
−0.0817102 + 0.996656i \(0.526038\pi\)
\(338\) 23.0000 1.25104
\(339\) 45.0000 2.44406
\(340\) 0 0
\(341\) −24.0000 −1.29967
\(342\) −6.00000 −0.324443
\(343\) −8.00000 −0.431959
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −4.00000 −0.215041
\(347\) −7.00000 −0.375780 −0.187890 0.982190i \(-0.560165\pi\)
−0.187890 + 0.982190i \(0.560165\pi\)
\(348\) 24.0000 1.28654
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −54.0000 −2.88231
\(352\) 3.00000 0.159901
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −3.00000 −0.159000
\(357\) −60.0000 −3.17554
\(358\) −3.00000 −0.158555
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −22.0000 −1.15629
\(363\) 6.00000 0.314918
\(364\) −24.0000 −1.25794
\(365\) 0 0
\(366\) 24.0000 1.25450
\(367\) −2.00000 −0.104399 −0.0521996 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −42.0000 −2.18643
\(370\) 0 0
\(371\) −48.0000 −2.49204
\(372\) 24.0000 1.24434
\(373\) 12.0000 0.621336 0.310668 0.950518i \(-0.399447\pi\)
0.310668 + 0.950518i \(0.399447\pi\)
\(374\) −15.0000 −0.775632
\(375\) 0 0
\(376\) −10.0000 −0.515711
\(377\) −48.0000 −2.47213
\(378\) 36.0000 1.85164
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) 0 0
\(381\) −24.0000 −1.22956
\(382\) 12.0000 0.613973
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 9.00000 0.458088
\(387\) −24.0000 −1.21999
\(388\) 14.0000 0.710742
\(389\) −36.0000 −1.82527 −0.912636 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 0 0
\(391\) 5.00000 0.252861
\(392\) 9.00000 0.454569
\(393\) −36.0000 −1.81596
\(394\) 8.00000 0.403034
\(395\) 0 0
\(396\) 18.0000 0.904534
\(397\) 12.0000 0.602263 0.301131 0.953583i \(-0.402636\pi\)
0.301131 + 0.953583i \(0.402636\pi\)
\(398\) 18.0000 0.902258
\(399\) −12.0000 −0.600751
\(400\) 0 0
\(401\) −5.00000 −0.249688 −0.124844 0.992176i \(-0.539843\pi\)
−0.124844 + 0.992176i \(0.539843\pi\)
\(402\) 9.00000 0.448879
\(403\) −48.0000 −2.39105
\(404\) 4.00000 0.199007
\(405\) 0 0
\(406\) 32.0000 1.58813
\(407\) −6.00000 −0.297409
\(408\) 15.0000 0.742611
\(409\) 21.0000 1.03838 0.519192 0.854658i \(-0.326233\pi\)
0.519192 + 0.854658i \(0.326233\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) −10.0000 −0.492665
\(413\) −16.0000 −0.787309
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 57.0000 2.79130
\(418\) −3.00000 −0.146735
\(419\) 3.00000 0.146560 0.0732798 0.997311i \(-0.476653\pi\)
0.0732798 + 0.997311i \(0.476653\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) −1.00000 −0.0486792
\(423\) −60.0000 −2.91730
\(424\) 12.0000 0.582772
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) 32.0000 1.54859
\(428\) −5.00000 −0.241684
\(429\) −54.0000 −2.60714
\(430\) 0 0
\(431\) −4.00000 −0.192673 −0.0963366 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) −9.00000 −0.433013
\(433\) −29.0000 −1.39365 −0.696826 0.717241i \(-0.745405\pi\)
−0.696826 + 0.717241i \(0.745405\pi\)
\(434\) 32.0000 1.53605
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 1.00000 0.0478365
\(438\) −21.0000 −1.00342
\(439\) −6.00000 −0.286364 −0.143182 0.989696i \(-0.545733\pi\)
−0.143182 + 0.989696i \(0.545733\pi\)
\(440\) 0 0
\(441\) 54.0000 2.57143
\(442\) −30.0000 −1.42695
\(443\) 39.0000 1.85295 0.926473 0.376361i \(-0.122825\pi\)
0.926473 + 0.376361i \(0.122825\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) 0 0
\(447\) −18.0000 −0.851371
\(448\) −4.00000 −0.188982
\(449\) 5.00000 0.235965 0.117982 0.993016i \(-0.462357\pi\)
0.117982 + 0.993016i \(0.462357\pi\)
\(450\) 0 0
\(451\) −21.0000 −0.988851
\(452\) −15.0000 −0.705541
\(453\) −60.0000 −2.81905
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) 7.00000 0.327446 0.163723 0.986506i \(-0.447650\pi\)
0.163723 + 0.986506i \(0.447650\pi\)
\(458\) −30.0000 −1.40181
\(459\) 45.0000 2.10042
\(460\) 0 0
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 36.0000 1.67487
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 36.0000 1.66410
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 42.0000 1.93526
\(472\) 4.00000 0.184115
\(473\) −12.0000 −0.551761
\(474\) 18.0000 0.826767
\(475\) 0 0
\(476\) 20.0000 0.916698
\(477\) 72.0000 3.29665
\(478\) −6.00000 −0.274434
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 23.0000 1.04762
\(483\) −12.0000 −0.546019
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 0 0
\(487\) 30.0000 1.35943 0.679715 0.733476i \(-0.262104\pi\)
0.679715 + 0.733476i \(0.262104\pi\)
\(488\) −8.00000 −0.362143
\(489\) 15.0000 0.678323
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 21.0000 0.946753
\(493\) 40.0000 1.80151
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −16.0000 −0.717698
\(498\) 33.0000 1.47877
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) 66.0000 2.94866
\(502\) −7.00000 −0.312425
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −24.0000 −1.06904
\(505\) 0 0
\(506\) −3.00000 −0.133366
\(507\) −69.0000 −3.06440
\(508\) 8.00000 0.354943
\(509\) 36.0000 1.59567 0.797836 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) 0 0
\(511\) −28.0000 −1.23865
\(512\) 1.00000 0.0441942
\(513\) 9.00000 0.397360
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) −30.0000 −1.31940
\(518\) 8.00000 0.351500
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) 31.0000 1.35813 0.679067 0.734076i \(-0.262384\pi\)
0.679067 + 0.734076i \(0.262384\pi\)
\(522\) −48.0000 −2.10090
\(523\) −37.0000 −1.61790 −0.808949 0.587879i \(-0.799963\pi\)
−0.808949 + 0.587879i \(0.799963\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) 40.0000 1.74243
\(528\) −9.00000 −0.391675
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 4.00000 0.173422
\(533\) −42.0000 −1.81922
\(534\) 9.00000 0.389468
\(535\) 0 0
\(536\) −3.00000 −0.129580
\(537\) 9.00000 0.388379
\(538\) −6.00000 −0.258678
\(539\) 27.0000 1.16297
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 10.0000 0.429537
\(543\) 66.0000 2.83233
\(544\) −5.00000 −0.214373
\(545\) 0 0
\(546\) 72.0000 3.08132
\(547\) 17.0000 0.726868 0.363434 0.931620i \(-0.381604\pi\)
0.363434 + 0.931620i \(0.381604\pi\)
\(548\) 3.00000 0.128154
\(549\) −48.0000 −2.04859
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 3.00000 0.127688
\(553\) 24.0000 1.02058
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) −19.0000 −0.805779
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) −48.0000 −2.03200
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 45.0000 1.89990
\(562\) −30.0000 −1.26547
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 30.0000 1.26323
\(565\) 0 0
\(566\) 33.0000 1.38709
\(567\) −36.0000 −1.51186
\(568\) 4.00000 0.167836
\(569\) −9.00000 −0.377300 −0.188650 0.982044i \(-0.560411\pi\)
−0.188650 + 0.982044i \(0.560411\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 18.0000 0.752618
\(573\) −36.0000 −1.50392
\(574\) 28.0000 1.16870
\(575\) 0 0
\(576\) 6.00000 0.250000
\(577\) −17.0000 −0.707719 −0.353860 0.935299i \(-0.615131\pi\)
−0.353860 + 0.935299i \(0.615131\pi\)
\(578\) 8.00000 0.332756
\(579\) −27.0000 −1.12208
\(580\) 0 0
\(581\) 44.0000 1.82543
\(582\) −42.0000 −1.74096
\(583\) 36.0000 1.49097
\(584\) 7.00000 0.289662
\(585\) 0 0
\(586\) −24.0000 −0.991431
\(587\) −17.0000 −0.701665 −0.350833 0.936438i \(-0.614101\pi\)
−0.350833 + 0.936438i \(0.614101\pi\)
\(588\) −27.0000 −1.11346
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) −2.00000 −0.0821995
\(593\) 35.0000 1.43728 0.718639 0.695383i \(-0.244765\pi\)
0.718639 + 0.695383i \(0.244765\pi\)
\(594\) −27.0000 −1.10782
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) −54.0000 −2.21007
\(598\) −6.00000 −0.245358
\(599\) −20.0000 −0.817178 −0.408589 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 0 0
\(601\) 31.0000 1.26452 0.632258 0.774758i \(-0.282128\pi\)
0.632258 + 0.774758i \(0.282128\pi\)
\(602\) 16.0000 0.652111
\(603\) −18.0000 −0.733017
\(604\) 20.0000 0.813788
\(605\) 0 0
\(606\) −12.0000 −0.487467
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −96.0000 −3.89012
\(610\) 0 0
\(611\) −60.0000 −2.42734
\(612\) −30.0000 −1.21268
\(613\) 8.00000 0.323117 0.161558 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) 9.00000 0.363210
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 30.0000 1.20678
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 9.00000 0.361158
\(622\) 2.00000 0.0801927
\(623\) 12.0000 0.480770
\(624\) −18.0000 −0.720577
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 9.00000 0.359425
\(628\) −14.0000 −0.558661
\(629\) 10.0000 0.398726
\(630\) 0 0
\(631\) 30.0000 1.19428 0.597141 0.802137i \(-0.296303\pi\)
0.597141 + 0.802137i \(0.296303\pi\)
\(632\) −6.00000 −0.238667
\(633\) 3.00000 0.119239
\(634\) −12.0000 −0.476581
\(635\) 0 0
\(636\) −36.0000 −1.42749
\(637\) 54.0000 2.13956
\(638\) −24.0000 −0.950169
\(639\) 24.0000 0.949425
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 15.0000 0.592003
\(643\) −32.0000 −1.26196 −0.630978 0.775800i \(-0.717346\pi\)
−0.630978 + 0.775800i \(0.717346\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 5.00000 0.196722
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 9.00000 0.353553
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) −96.0000 −3.76254
\(652\) −5.00000 −0.195815
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 30.0000 1.17309
\(655\) 0 0
\(656\) −7.00000 −0.273304
\(657\) 42.0000 1.63858
\(658\) 40.0000 1.55936
\(659\) 5.00000 0.194772 0.0973862 0.995247i \(-0.468952\pi\)
0.0973862 + 0.995247i \(0.468952\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 25.0000 0.971653
\(663\) 90.0000 3.49531
\(664\) −11.0000 −0.426883
\(665\) 0 0
\(666\) −12.0000 −0.464991
\(667\) 8.00000 0.309761
\(668\) −22.0000 −0.851206
\(669\) 0 0
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 12.0000 0.462910
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −3.00000 −0.115556
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 45.0000 1.72821
\(679\) −56.0000 −2.14908
\(680\) 0 0
\(681\) −24.0000 −0.919682
\(682\) −24.0000 −0.919007
\(683\) −35.0000 −1.33924 −0.669619 0.742705i \(-0.733543\pi\)
−0.669619 + 0.742705i \(0.733543\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 90.0000 3.43371
\(688\) −4.00000 −0.152499
\(689\) 72.0000 2.74298
\(690\) 0 0
\(691\) 29.0000 1.10321 0.551606 0.834105i \(-0.314015\pi\)
0.551606 + 0.834105i \(0.314015\pi\)
\(692\) −4.00000 −0.152057
\(693\) −72.0000 −2.73505
\(694\) −7.00000 −0.265716
\(695\) 0 0
\(696\) 24.0000 0.909718
\(697\) 35.0000 1.32572
\(698\) 26.0000 0.984115
\(699\) −42.0000 −1.58859
\(700\) 0 0
\(701\) −8.00000 −0.302156 −0.151078 0.988522i \(-0.548274\pi\)
−0.151078 + 0.988522i \(0.548274\pi\)
\(702\) −54.0000 −2.03810
\(703\) 2.00000 0.0754314
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) −16.0000 −0.601742
\(708\) −12.0000 −0.450988
\(709\) 16.0000 0.600893 0.300446 0.953799i \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) 0 0
\(711\) −36.0000 −1.35011
\(712\) −3.00000 −0.112430
\(713\) 8.00000 0.299602
\(714\) −60.0000 −2.24544
\(715\) 0 0
\(716\) −3.00000 −0.112115
\(717\) 18.0000 0.672222
\(718\) −20.0000 −0.746393
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 40.0000 1.48968
\(722\) −18.0000 −0.669891
\(723\) −69.0000 −2.56614
\(724\) −22.0000 −0.817624
\(725\) 0 0
\(726\) 6.00000 0.222681
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) −24.0000 −0.889499
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 20.0000 0.739727
\(732\) 24.0000 0.887066
\(733\) −38.0000 −1.40356 −0.701781 0.712393i \(-0.747612\pi\)
−0.701781 + 0.712393i \(0.747612\pi\)
\(734\) −2.00000 −0.0738213
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −9.00000 −0.331519
\(738\) −42.0000 −1.54604
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 18.0000 0.661247
\(742\) −48.0000 −1.76214
\(743\) −26.0000 −0.953847 −0.476924 0.878945i \(-0.658248\pi\)
−0.476924 + 0.878945i \(0.658248\pi\)
\(744\) 24.0000 0.879883
\(745\) 0 0
\(746\) 12.0000 0.439351
\(747\) −66.0000 −2.41481
\(748\) −15.0000 −0.548454
\(749\) 20.0000 0.730784
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) −10.0000 −0.364662
\(753\) 21.0000 0.765283
\(754\) −48.0000 −1.74806
\(755\) 0 0
\(756\) 36.0000 1.30931
\(757\) −16.0000 −0.581530 −0.290765 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) 5.00000 0.181608
\(759\) 9.00000 0.326679
\(760\) 0 0
\(761\) −15.0000 −0.543750 −0.271875 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) −24.0000 −0.869428
\(763\) 40.0000 1.44810
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) 24.0000 0.866590
\(768\) −3.00000 −0.108253
\(769\) −21.0000 −0.757279 −0.378640 0.925544i \(-0.623608\pi\)
−0.378640 + 0.925544i \(0.623608\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 9.00000 0.323917
\(773\) −4.00000 −0.143870 −0.0719350 0.997409i \(-0.522917\pi\)
−0.0719350 + 0.997409i \(0.522917\pi\)
\(774\) −24.0000 −0.862662
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) −24.0000 −0.860995
\(778\) −36.0000 −1.29066
\(779\) 7.00000 0.250801
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 5.00000 0.178800
\(783\) 72.0000 2.57307
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) −36.0000 −1.28408
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) 8.00000 0.284988
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 60.0000 2.13335
\(792\) 18.0000 0.639602
\(793\) −48.0000 −1.70453
\(794\) 12.0000 0.425864
\(795\) 0 0
\(796\) 18.0000 0.637993
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) −12.0000 −0.424795
\(799\) 50.0000 1.76887
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) −5.00000 −0.176556
\(803\) 21.0000 0.741074
\(804\) 9.00000 0.317406
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) 18.0000 0.633630
\(808\) 4.00000 0.140720
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 32.0000 1.12298
\(813\) −30.0000 −1.05215
\(814\) −6.00000 −0.210300
\(815\) 0 0
\(816\) 15.0000 0.525105
\(817\) 4.00000 0.139942
\(818\) 21.0000 0.734248
\(819\) −144.000 −5.03177
\(820\) 0 0
\(821\) 16.0000 0.558404 0.279202 0.960232i \(-0.409930\pi\)
0.279202 + 0.960232i \(0.409930\pi\)
\(822\) −9.00000 −0.313911
\(823\) 26.0000 0.906303 0.453152 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) −10.0000 −0.348367
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) 17.0000 0.591148 0.295574 0.955320i \(-0.404489\pi\)
0.295574 + 0.955320i \(0.404489\pi\)
\(828\) −6.00000 −0.208514
\(829\) 24.0000 0.833554 0.416777 0.909009i \(-0.363160\pi\)
0.416777 + 0.909009i \(0.363160\pi\)
\(830\) 0 0
\(831\) −84.0000 −2.91393
\(832\) 6.00000 0.208013
\(833\) −45.0000 −1.55916
\(834\) 57.0000 1.97375
\(835\) 0 0
\(836\) −3.00000 −0.103757
\(837\) 72.0000 2.48868
\(838\) 3.00000 0.103633
\(839\) 26.0000 0.897620 0.448810 0.893627i \(-0.351848\pi\)
0.448810 + 0.893627i \(0.351848\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) −14.0000 −0.482472
\(843\) 90.0000 3.09976
\(844\) −1.00000 −0.0344214
\(845\) 0 0
\(846\) −60.0000 −2.06284
\(847\) 8.00000 0.274883
\(848\) 12.0000 0.412082
\(849\) −99.0000 −3.39767
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) −12.0000 −0.411113
\(853\) −12.0000 −0.410872 −0.205436 0.978671i \(-0.565861\pi\)
−0.205436 + 0.978671i \(0.565861\pi\)
\(854\) 32.0000 1.09502
\(855\) 0 0
\(856\) −5.00000 −0.170896
\(857\) 21.0000 0.717346 0.358673 0.933463i \(-0.383229\pi\)
0.358673 + 0.933463i \(0.383229\pi\)
\(858\) −54.0000 −1.84353
\(859\) −11.0000 −0.375315 −0.187658 0.982235i \(-0.560090\pi\)
−0.187658 + 0.982235i \(0.560090\pi\)
\(860\) 0 0
\(861\) −84.0000 −2.86271
\(862\) −4.00000 −0.136241
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −9.00000 −0.306186
\(865\) 0 0
\(866\) −29.0000 −0.985460
\(867\) −24.0000 −0.815083
\(868\) 32.0000 1.08615
\(869\) −18.0000 −0.610608
\(870\) 0 0
\(871\) −18.0000 −0.609907
\(872\) −10.0000 −0.338643
\(873\) 84.0000 2.84297
\(874\) 1.00000 0.0338255
\(875\) 0 0
\(876\) −21.0000 −0.709524
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) −6.00000 −0.202490
\(879\) 72.0000 2.42850
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 54.0000 1.81827
\(883\) 37.0000 1.24515 0.622575 0.782560i \(-0.286087\pi\)
0.622575 + 0.782560i \(0.286087\pi\)
\(884\) −30.0000 −1.00901
\(885\) 0 0
\(886\) 39.0000 1.31023
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 6.00000 0.201347
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) 27.0000 0.904534
\(892\) 0 0
\(893\) 10.0000 0.334637
\(894\) −18.0000 −0.602010
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 18.0000 0.601003
\(898\) 5.00000 0.166852
\(899\) 64.0000 2.13452
\(900\) 0 0
\(901\) −60.0000 −1.99889
\(902\) −21.0000 −0.699224
\(903\) −48.0000 −1.59734
\(904\) −15.0000 −0.498893
\(905\) 0 0
\(906\) −60.0000 −1.99337
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 8.00000 0.265489
\(909\) 24.0000 0.796030
\(910\) 0 0
\(911\) −10.0000 −0.331315 −0.165657 0.986183i \(-0.552975\pi\)
−0.165657 + 0.986183i \(0.552975\pi\)
\(912\) 3.00000 0.0993399
\(913\) −33.0000 −1.09214
\(914\) 7.00000 0.231539
\(915\) 0 0
\(916\) −30.0000 −0.991228
\(917\) −48.0000 −1.58510
\(918\) 45.0000 1.48522
\(919\) 52.0000 1.71532 0.857661 0.514216i \(-0.171917\pi\)
0.857661 + 0.514216i \(0.171917\pi\)
\(920\) 0 0
\(921\) −27.0000 −0.889680
\(922\) −24.0000 −0.790398
\(923\) 24.0000 0.789970
\(924\) 36.0000 1.18431
\(925\) 0 0
\(926\) −26.0000 −0.854413
\(927\) −60.0000 −1.97066
\(928\) −8.00000 −0.262613
\(929\) −22.0000 −0.721797 −0.360898 0.932605i \(-0.617530\pi\)
−0.360898 + 0.932605i \(0.617530\pi\)
\(930\) 0 0
\(931\) −9.00000 −0.294963
\(932\) 14.0000 0.458585
\(933\) −6.00000 −0.196431
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) 36.0000 1.17670
\(937\) 9.00000 0.294017 0.147009 0.989135i \(-0.453036\pi\)
0.147009 + 0.989135i \(0.453036\pi\)
\(938\) 12.0000 0.391814
\(939\) −30.0000 −0.979013
\(940\) 0 0
\(941\) 4.00000 0.130396 0.0651981 0.997872i \(-0.479232\pi\)
0.0651981 + 0.997872i \(0.479232\pi\)
\(942\) 42.0000 1.36843
\(943\) 7.00000 0.227951
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 18.0000 0.584613
\(949\) 42.0000 1.36338
\(950\) 0 0
\(951\) 36.0000 1.16738
\(952\) 20.0000 0.648204
\(953\) 27.0000 0.874616 0.437308 0.899312i \(-0.355932\pi\)
0.437308 + 0.899312i \(0.355932\pi\)
\(954\) 72.0000 2.33109
\(955\) 0 0
\(956\) −6.00000 −0.194054
\(957\) 72.0000 2.32743
\(958\) −4.00000 −0.129234
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −12.0000 −0.386896
\(963\) −30.0000 −0.966736
\(964\) 23.0000 0.740780
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) 28.0000 0.900419 0.450210 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(968\) −2.00000 −0.0642824
\(969\) −15.0000 −0.481869
\(970\) 0 0
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 0 0
\(973\) 76.0000 2.43645
\(974\) 30.0000 0.961262
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) −17.0000 −0.543878 −0.271939 0.962314i \(-0.587665\pi\)
−0.271939 + 0.962314i \(0.587665\pi\)
\(978\) 15.0000 0.479647
\(979\) −9.00000 −0.287641
\(980\) 0 0
\(981\) −60.0000 −1.91565
\(982\) 0 0
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 21.0000 0.669456
\(985\) 0 0
\(986\) 40.0000 1.27386
\(987\) −120.000 −3.81964
\(988\) −6.00000 −0.190885
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) −8.00000 −0.254000
\(993\) −75.0000 −2.38005
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) 33.0000 1.04565
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) −12.0000 −0.379853
\(999\) 18.0000 0.569495
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 1150.2.a.e.1.1 yes 1
4.3 odd 2 9200.2.a.bl.1.1 1
5.2 odd 4 1150.2.b.a.599.2 2
5.3 odd 4 1150.2.b.a.599.1 2
5.4 even 2 1150.2.a.d.1.1 1
20.19 odd 2 9200.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.2.a.d.1.1 1 5.4 even 2
1150.2.a.e.1.1 yes 1 1.1 even 1 trivial
1150.2.b.a.599.1 2 5.3 odd 4
1150.2.b.a.599.2 2 5.2 odd 4
9200.2.a.a.1.1 1 20.19 odd 2
9200.2.a.bl.1.1 1 4.3 odd 2