Properties

Label 106.2.a.a.1.1
Level $106$
Weight $2$
Character 106.1
Self dual yes
Analytic conductor $0.846$
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] = [106,2,Mod(1,106)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(106, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("106.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 106 = 2 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 106.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: \(0.846414261426\)
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 \(\chi\) \(=\) 106.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} +4.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +4.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} +2.00000 q^{18} -7.00000 q^{19} -4.00000 q^{20} +4.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} -1.00000 q^{26} +5.00000 q^{27} +5.00000 q^{29} -4.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} -5.00000 q^{34} -2.00000 q^{36} +1.00000 q^{37} +7.00000 q^{38} -1.00000 q^{39} +4.00000 q^{40} -10.0000 q^{41} -10.0000 q^{43} -4.00000 q^{44} +8.00000 q^{45} -1.00000 q^{46} -6.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} -11.0000 q^{50} -5.00000 q^{51} +1.00000 q^{52} -1.00000 q^{53} -5.00000 q^{54} +16.0000 q^{55} +7.00000 q^{57} -5.00000 q^{58} -6.00000 q^{59} +4.00000 q^{60} +4.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{66} +4.00000 q^{67} +5.00000 q^{68} -1.00000 q^{69} +15.0000 q^{71} +2.00000 q^{72} -8.00000 q^{73} -1.00000 q^{74} -11.0000 q^{75} -7.00000 q^{76} +1.00000 q^{78} +1.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -3.00000 q^{83} -20.0000 q^{85} +10.0000 q^{86} -5.00000 q^{87} +4.00000 q^{88} +2.00000 q^{89} -8.00000 q^{90} +1.00000 q^{92} +4.00000 q^{93} +6.00000 q^{94} +28.0000 q^{95} +1.00000 q^{96} +17.0000 q^{97} +7.00000 q^{98} +8.00000 q^{99} +O(q^{100})\)

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 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) 4.00000 1.26491
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 2.00000 0.471405
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −4.00000 −0.894427
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 1.00000 0.204124
\(25\) 11.0000 2.20000
\(26\) −1.00000 −0.196116
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) −4.00000 −0.730297
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) −5.00000 −0.857493
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 7.00000 1.13555
\(39\) −1.00000 −0.160128
\(40\) 4.00000 0.632456
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) −4.00000 −0.603023
\(45\) 8.00000 1.19257
\(46\) −1.00000 −0.147442
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) −11.0000 −1.55563
\(51\) −5.00000 −0.700140
\(52\) 1.00000 0.138675
\(53\) −1.00000 −0.137361
\(54\) −5.00000 −0.680414
\(55\) 16.0000 2.15744
\(56\) 0 0
\(57\) 7.00000 0.927173
\(58\) −5.00000 −0.656532
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 4.00000 0.516398
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) −4.00000 −0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 5.00000 0.606339
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 15.0000 1.78017 0.890086 0.455792i \(-0.150644\pi\)
0.890086 + 0.455792i \(0.150644\pi\)
\(72\) 2.00000 0.235702
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) −1.00000 −0.116248
\(75\) −11.0000 −1.27017
\(76\) −7.00000 −0.802955
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 1.00000 0.112509 0.0562544 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 0 0
\(85\) −20.0000 −2.16930
\(86\) 10.0000 1.07833
\(87\) −5.00000 −0.536056
\(88\) 4.00000 0.426401
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) −8.00000 −0.843274
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 4.00000 0.414781
\(94\) 6.00000 0.618853
\(95\) 28.0000 2.87274
\(96\) 1.00000 0.102062
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 7.00000 0.707107
\(99\) 8.00000 0.804030
\(100\) 11.0000 1.10000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 5.00000 0.495074
\(103\) −15.0000 −1.47799 −0.738997 0.673709i \(-0.764700\pi\)
−0.738997 + 0.673709i \(0.764700\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 1.00000 0.0971286
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 5.00000 0.481125
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) −16.0000 −1.52554
\(111\) −1.00000 −0.0949158
\(112\) 0 0
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) −7.00000 −0.655610
\(115\) −4.00000 −0.373002
\(116\) 5.00000 0.464238
\(117\) −2.00000 −0.184900
\(118\) 6.00000 0.552345
\(119\) 0 0
\(120\) −4.00000 −0.365148
\(121\) 5.00000 0.454545
\(122\) −4.00000 −0.362143
\(123\) 10.0000 0.901670
\(124\) −4.00000 −0.359211
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) −5.00000 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 10.0000 0.880451
\(130\) 4.00000 0.350823
\(131\) −10.0000 −0.873704 −0.436852 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) −20.0000 −1.72133
\(136\) −5.00000 −0.428746
\(137\) 8.00000 0.683486 0.341743 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 1.00000 0.0851257
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) −15.0000 −1.25877
\(143\) −4.00000 −0.334497
\(144\) −2.00000 −0.166667
\(145\) −20.0000 −1.66091
\(146\) 8.00000 0.662085
\(147\) 7.00000 0.577350
\(148\) 1.00000 0.0821995
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 11.0000 0.898146
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 7.00000 0.567775
\(153\) −10.0000 −0.808452
\(154\) 0 0
\(155\) 16.0000 1.28515
\(156\) −1.00000 −0.0800641
\(157\) 8.00000 0.638470 0.319235 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) −1.00000 −0.0795557
\(159\) 1.00000 0.0793052
\(160\) 4.00000 0.316228
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −10.0000 −0.780869
\(165\) −16.0000 −1.24560
\(166\) 3.00000 0.232845
\(167\) 19.0000 1.47026 0.735132 0.677924i \(-0.237120\pi\)
0.735132 + 0.677924i \(0.237120\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 20.0000 1.53393
\(171\) 14.0000 1.07061
\(172\) −10.0000 −0.762493
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 5.00000 0.379049
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 6.00000 0.450988
\(178\) −2.00000 −0.149906
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 8.00000 0.596285
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) −1.00000 −0.0737210
\(185\) −4.00000 −0.294086
\(186\) −4.00000 −0.293294
\(187\) −20.0000 −1.46254
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) −28.0000 −2.03133
\(191\) 13.0000 0.940647 0.470323 0.882494i \(-0.344137\pi\)
0.470323 + 0.882494i \(0.344137\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −12.0000 −0.863779 −0.431889 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) −17.0000 −1.22053
\(195\) 4.00000 0.286446
\(196\) −7.00000 −0.500000
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −8.00000 −0.568535
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −11.0000 −0.777817
\(201\) −4.00000 −0.282138
\(202\) 14.0000 0.985037
\(203\) 0 0
\(204\) −5.00000 −0.350070
\(205\) 40.0000 2.79372
\(206\) 15.0000 1.04510
\(207\) −2.00000 −0.139010
\(208\) 1.00000 0.0693375
\(209\) 28.0000 1.93680
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) −1.00000 −0.0686803
\(213\) −15.0000 −1.02778
\(214\) −6.00000 −0.410152
\(215\) 40.0000 2.72798
\(216\) −5.00000 −0.340207
\(217\) 0 0
\(218\) −16.0000 −1.08366
\(219\) 8.00000 0.540590
\(220\) 16.0000 1.07872
\(221\) 5.00000 0.336336
\(222\) 1.00000 0.0671156
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 0 0
\(225\) −22.0000 −1.46667
\(226\) 9.00000 0.598671
\(227\) 22.0000 1.46019 0.730096 0.683345i \(-0.239475\pi\)
0.730096 + 0.683345i \(0.239475\pi\)
\(228\) 7.00000 0.463586
\(229\) −15.0000 −0.991228 −0.495614 0.868543i \(-0.665057\pi\)
−0.495614 + 0.868543i \(0.665057\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) −5.00000 −0.328266
\(233\) 8.00000 0.524097 0.262049 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 2.00000 0.130744
\(235\) 24.0000 1.56559
\(236\) −6.00000 −0.390567
\(237\) −1.00000 −0.0649570
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 4.00000 0.258199
\(241\) −3.00000 −0.193247 −0.0966235 0.995321i \(-0.530804\pi\)
−0.0966235 + 0.995321i \(0.530804\pi\)
\(242\) −5.00000 −0.321412
\(243\) −16.0000 −1.02640
\(244\) 4.00000 0.256074
\(245\) 28.0000 1.78885
\(246\) −10.0000 −0.637577
\(247\) −7.00000 −0.445399
\(248\) 4.00000 0.254000
\(249\) 3.00000 0.190117
\(250\) 24.0000 1.51789
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 5.00000 0.313728
\(255\) 20.0000 1.25245
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) −10.0000 −0.622573
\(259\) 0 0
\(260\) −4.00000 −0.248069
\(261\) −10.0000 −0.618984
\(262\) 10.0000 0.617802
\(263\) 28.0000 1.72655 0.863277 0.504730i \(-0.168408\pi\)
0.863277 + 0.504730i \(0.168408\pi\)
\(264\) −4.00000 −0.246183
\(265\) 4.00000 0.245718
\(266\) 0 0
\(267\) −2.00000 −0.122398
\(268\) 4.00000 0.244339
\(269\) −19.0000 −1.15845 −0.579225 0.815168i \(-0.696645\pi\)
−0.579225 + 0.815168i \(0.696645\pi\)
\(270\) 20.0000 1.21716
\(271\) 30.0000 1.82237 0.911185 0.411997i \(-0.135169\pi\)
0.911185 + 0.411997i \(0.135169\pi\)
\(272\) 5.00000 0.303170
\(273\) 0 0
\(274\) −8.00000 −0.483298
\(275\) −44.0000 −2.65330
\(276\) −1.00000 −0.0601929
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) 12.0000 0.719712
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −31.0000 −1.84930 −0.924652 0.380812i \(-0.875644\pi\)
−0.924652 + 0.380812i \(0.875644\pi\)
\(282\) −6.00000 −0.357295
\(283\) 13.0000 0.772770 0.386385 0.922338i \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) 15.0000 0.890086
\(285\) −28.0000 −1.65858
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) 2.00000 0.117851
\(289\) 8.00000 0.470588
\(290\) 20.0000 1.17444
\(291\) −17.0000 −0.996558
\(292\) −8.00000 −0.468165
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) −7.00000 −0.408248
\(295\) 24.0000 1.39733
\(296\) −1.00000 −0.0581238
\(297\) −20.0000 −1.16052
\(298\) −15.0000 −0.868927
\(299\) 1.00000 0.0578315
\(300\) −11.0000 −0.635085
\(301\) 0 0
\(302\) 5.00000 0.287718
\(303\) 14.0000 0.804279
\(304\) −7.00000 −0.401478
\(305\) −16.0000 −0.916157
\(306\) 10.0000 0.571662
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 0 0
\(309\) 15.0000 0.853320
\(310\) −16.0000 −0.908739
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 1.00000 0.0566139
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) −8.00000 −0.451466
\(315\) 0 0
\(316\) 1.00000 0.0562544
\(317\) −17.0000 −0.954815 −0.477408 0.878682i \(-0.658423\pi\)
−0.477408 + 0.878682i \(0.658423\pi\)
\(318\) −1.00000 −0.0560772
\(319\) −20.0000 −1.11979
\(320\) −4.00000 −0.223607
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) −35.0000 −1.94745
\(324\) 1.00000 0.0555556
\(325\) 11.0000 0.610170
\(326\) −2.00000 −0.110770
\(327\) −16.0000 −0.884802
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) 16.0000 0.880771
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −3.00000 −0.164646
\(333\) −2.00000 −0.109599
\(334\) −19.0000 −1.03963
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 12.0000 0.652714
\(339\) 9.00000 0.488813
\(340\) −20.0000 −1.08465
\(341\) 16.0000 0.866449
\(342\) −14.0000 −0.757033
\(343\) 0 0
\(344\) 10.0000 0.539164
\(345\) 4.00000 0.215353
\(346\) −2.00000 −0.107521
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) −5.00000 −0.268028
\(349\) 20.0000 1.07058 0.535288 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 4.00000 0.213201
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) −6.00000 −0.318896
\(355\) −60.0000 −3.18447
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 15.0000 0.792775
\(359\) 7.00000 0.369446 0.184723 0.982791i \(-0.440861\pi\)
0.184723 + 0.982791i \(0.440861\pi\)
\(360\) −8.00000 −0.421637
\(361\) 30.0000 1.57895
\(362\) 2.00000 0.105118
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 32.0000 1.67496
\(366\) 4.00000 0.209083
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) 1.00000 0.0521286
\(369\) 20.0000 1.04116
\(370\) 4.00000 0.207950
\(371\) 0 0
\(372\) 4.00000 0.207390
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 20.0000 1.03418
\(375\) 24.0000 1.23935
\(376\) 6.00000 0.309426
\(377\) 5.00000 0.257513
\(378\) 0 0
\(379\) 17.0000 0.873231 0.436616 0.899648i \(-0.356177\pi\)
0.436616 + 0.899648i \(0.356177\pi\)
\(380\) 28.0000 1.43637
\(381\) 5.00000 0.256158
\(382\) −13.0000 −0.665138
\(383\) −32.0000 −1.63512 −0.817562 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 12.0000 0.610784
\(387\) 20.0000 1.01666
\(388\) 17.0000 0.863044
\(389\) 20.0000 1.01404 0.507020 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) −4.00000 −0.202548
\(391\) 5.00000 0.252861
\(392\) 7.00000 0.353553
\(393\) 10.0000 0.504433
\(394\) 2.00000 0.100759
\(395\) −4.00000 −0.201262
\(396\) 8.00000 0.402015
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 4.00000 0.199502
\(403\) −4.00000 −0.199254
\(404\) −14.0000 −0.696526
\(405\) −4.00000 −0.198762
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 5.00000 0.247537
\(409\) −21.0000 −1.03838 −0.519192 0.854658i \(-0.673767\pi\)
−0.519192 + 0.854658i \(0.673767\pi\)
\(410\) −40.0000 −1.97546
\(411\) −8.00000 −0.394611
\(412\) −15.0000 −0.738997
\(413\) 0 0
\(414\) 2.00000 0.0982946
\(415\) 12.0000 0.589057
\(416\) −1.00000 −0.0490290
\(417\) 12.0000 0.587643
\(418\) −28.0000 −1.36952
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) −10.0000 −0.486792
\(423\) 12.0000 0.583460
\(424\) 1.00000 0.0485643
\(425\) 55.0000 2.66789
\(426\) 15.0000 0.726752
\(427\) 0 0
\(428\) 6.00000 0.290021
\(429\) 4.00000 0.193122
\(430\) −40.0000 −1.92897
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 5.00000 0.240563
\(433\) 17.0000 0.816968 0.408484 0.912766i \(-0.366058\pi\)
0.408484 + 0.912766i \(0.366058\pi\)
\(434\) 0 0
\(435\) 20.0000 0.958927
\(436\) 16.0000 0.766261
\(437\) −7.00000 −0.334855
\(438\) −8.00000 −0.382255
\(439\) 2.00000 0.0954548 0.0477274 0.998860i \(-0.484802\pi\)
0.0477274 + 0.998860i \(0.484802\pi\)
\(440\) −16.0000 −0.762770
\(441\) 14.0000 0.666667
\(442\) −5.00000 −0.237826
\(443\) −19.0000 −0.902717 −0.451359 0.892343i \(-0.649060\pi\)
−0.451359 + 0.892343i \(0.649060\pi\)
\(444\) −1.00000 −0.0474579
\(445\) −8.00000 −0.379236
\(446\) 14.0000 0.662919
\(447\) −15.0000 −0.709476
\(448\) 0 0
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 22.0000 1.03709
\(451\) 40.0000 1.88353
\(452\) −9.00000 −0.423324
\(453\) 5.00000 0.234920
\(454\) −22.0000 −1.03251
\(455\) 0 0
\(456\) −7.00000 −0.327805
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) 15.0000 0.700904
\(459\) 25.0000 1.16690
\(460\) −4.00000 −0.186501
\(461\) 3.00000 0.139724 0.0698620 0.997557i \(-0.477744\pi\)
0.0698620 + 0.997557i \(0.477744\pi\)
\(462\) 0 0
\(463\) 39.0000 1.81248 0.906242 0.422760i \(-0.138939\pi\)
0.906242 + 0.422760i \(0.138939\pi\)
\(464\) 5.00000 0.232119
\(465\) −16.0000 −0.741982
\(466\) −8.00000 −0.370593
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) −24.0000 −1.10704
\(471\) −8.00000 −0.368621
\(472\) 6.00000 0.276172
\(473\) 40.0000 1.83920
\(474\) 1.00000 0.0459315
\(475\) −77.0000 −3.53300
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 24.0000 1.09773
\(479\) −3.00000 −0.137073 −0.0685367 0.997649i \(-0.521833\pi\)
−0.0685367 + 0.997649i \(0.521833\pi\)
\(480\) −4.00000 −0.182574
\(481\) 1.00000 0.0455961
\(482\) 3.00000 0.136646
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −68.0000 −3.08772
\(486\) 16.0000 0.725775
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) −4.00000 −0.181071
\(489\) −2.00000 −0.0904431
\(490\) −28.0000 −1.26491
\(491\) 33.0000 1.48927 0.744635 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 10.0000 0.450835
\(493\) 25.0000 1.12594
\(494\) 7.00000 0.314945
\(495\) −32.0000 −1.43829
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) −3.00000 −0.134433
\(499\) 11.0000 0.492428 0.246214 0.969216i \(-0.420813\pi\)
0.246214 + 0.969216i \(0.420813\pi\)
\(500\) −24.0000 −1.07331
\(501\) −19.0000 −0.848857
\(502\) 4.00000 0.178529
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 56.0000 2.49197
\(506\) 4.00000 0.177822
\(507\) 12.0000 0.532939
\(508\) −5.00000 −0.221839
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) −20.0000 −0.885615
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −35.0000 −1.54529
\(514\) 12.0000 0.529297
\(515\) 60.0000 2.64392
\(516\) 10.0000 0.440225
\(517\) 24.0000 1.05552
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 4.00000 0.175412
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 10.0000 0.437688
\(523\) −14.0000 −0.612177 −0.306089 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) −20.0000 −0.871214
\(528\) 4.00000 0.174078
\(529\) −22.0000 −0.956522
\(530\) −4.00000 −0.173749
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) −10.0000 −0.433148
\(534\) 2.00000 0.0865485
\(535\) −24.0000 −1.03761
\(536\) −4.00000 −0.172774
\(537\) 15.0000 0.647298
\(538\) 19.0000 0.819148
\(539\) 28.0000 1.20605
\(540\) −20.0000 −0.860663
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −30.0000 −1.28861
\(543\) 2.00000 0.0858282
\(544\) −5.00000 −0.214373
\(545\) −64.0000 −2.74146
\(546\) 0 0
\(547\) −14.0000 −0.598597 −0.299298 0.954160i \(-0.596753\pi\)
−0.299298 + 0.954160i \(0.596753\pi\)
\(548\) 8.00000 0.341743
\(549\) −8.00000 −0.341432
\(550\) 44.0000 1.87617
\(551\) −35.0000 −1.49105
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 16.0000 0.679775
\(555\) 4.00000 0.169791
\(556\) −12.0000 −0.508913
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) −8.00000 −0.338667
\(559\) −10.0000 −0.422955
\(560\) 0 0
\(561\) 20.0000 0.844401
\(562\) 31.0000 1.30766
\(563\) 45.0000 1.89652 0.948262 0.317489i \(-0.102840\pi\)
0.948262 + 0.317489i \(0.102840\pi\)
\(564\) 6.00000 0.252646
\(565\) 36.0000 1.51453
\(566\) −13.0000 −0.546431
\(567\) 0 0
\(568\) −15.0000 −0.629386
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 28.0000 1.17279
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) −4.00000 −0.167248
\(573\) −13.0000 −0.543083
\(574\) 0 0
\(575\) 11.0000 0.458732
\(576\) −2.00000 −0.0833333
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) −8.00000 −0.332756
\(579\) 12.0000 0.498703
\(580\) −20.0000 −0.830455
\(581\) 0 0
\(582\) 17.0000 0.704673
\(583\) 4.00000 0.165663
\(584\) 8.00000 0.331042
\(585\) 8.00000 0.330759
\(586\) −10.0000 −0.413096
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 7.00000 0.288675
\(589\) 28.0000 1.15372
\(590\) −24.0000 −0.988064
\(591\) 2.00000 0.0822690
\(592\) 1.00000 0.0410997
\(593\) 9.00000 0.369586 0.184793 0.982777i \(-0.440839\pi\)
0.184793 + 0.982777i \(0.440839\pi\)
\(594\) 20.0000 0.820610
\(595\) 0 0
\(596\) 15.0000 0.614424
\(597\) 0 0
\(598\) −1.00000 −0.0408930
\(599\) −10.0000 −0.408589 −0.204294 0.978909i \(-0.565490\pi\)
−0.204294 + 0.978909i \(0.565490\pi\)
\(600\) 11.0000 0.449073
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) −5.00000 −0.203447
\(605\) −20.0000 −0.813116
\(606\) −14.0000 −0.568711
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 7.00000 0.283887
\(609\) 0 0
\(610\) 16.0000 0.647821
\(611\) −6.00000 −0.242734
\(612\) −10.0000 −0.404226
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) −8.00000 −0.322854
\(615\) −40.0000 −1.61296
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) −15.0000 −0.603388
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 16.0000 0.642575
\(621\) 5.00000 0.200643
\(622\) 24.0000 0.962312
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 41.0000 1.64000
\(626\) −22.0000 −0.879297
\(627\) −28.0000 −1.11821
\(628\) 8.00000 0.319235
\(629\) 5.00000 0.199363
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) −1.00000 −0.0397779
\(633\) −10.0000 −0.397464
\(634\) 17.0000 0.675156
\(635\) 20.0000 0.793676
\(636\) 1.00000 0.0396526
\(637\) −7.00000 −0.277350
\(638\) 20.0000 0.791808
\(639\) −30.0000 −1.18678
\(640\) 4.00000 0.158114
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 6.00000 0.236801
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 0 0
\(645\) −40.0000 −1.57500
\(646\) 35.0000 1.37706
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 24.0000 0.942082
\(650\) −11.0000 −0.431455
\(651\) 0 0
\(652\) 2.00000 0.0783260
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 16.0000 0.625650
\(655\) 40.0000 1.56293
\(656\) −10.0000 −0.390434
\(657\) 16.0000 0.624219
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) −16.0000 −0.622799
\(661\) 3.00000 0.116686 0.0583432 0.998297i \(-0.481418\pi\)
0.0583432 + 0.998297i \(0.481418\pi\)
\(662\) 12.0000 0.466393
\(663\) −5.00000 −0.194184
\(664\) 3.00000 0.116423
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 5.00000 0.193601
\(668\) 19.0000 0.735132
\(669\) 14.0000 0.541271
\(670\) 16.0000 0.618134
\(671\) −16.0000 −0.617673
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 10.0000 0.385186
\(675\) 55.0000 2.11695
\(676\) −12.0000 −0.461538
\(677\) −16.0000 −0.614930 −0.307465 0.951559i \(-0.599481\pi\)
−0.307465 + 0.951559i \(0.599481\pi\)
\(678\) −9.00000 −0.345643
\(679\) 0 0
\(680\) 20.0000 0.766965
\(681\) −22.0000 −0.843042
\(682\) −16.0000 −0.612672
\(683\) −2.00000 −0.0765279 −0.0382639 0.999268i \(-0.512183\pi\)
−0.0382639 + 0.999268i \(0.512183\pi\)
\(684\) 14.0000 0.535303
\(685\) −32.0000 −1.22266
\(686\) 0 0
\(687\) 15.0000 0.572286
\(688\) −10.0000 −0.381246
\(689\) −1.00000 −0.0380970
\(690\) −4.00000 −0.152277
\(691\) 43.0000 1.63580 0.817899 0.575362i \(-0.195139\pi\)
0.817899 + 0.575362i \(0.195139\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) 18.0000 0.683271
\(695\) 48.0000 1.82074
\(696\) 5.00000 0.189525
\(697\) −50.0000 −1.89389
\(698\) −20.0000 −0.757011
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) −5.00000 −0.188713
\(703\) −7.00000 −0.264010
\(704\) −4.00000 −0.150756
\(705\) −24.0000 −0.903892
\(706\) −10.0000 −0.376355
\(707\) 0 0
\(708\) 6.00000 0.225494
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 60.0000 2.25176
\(711\) −2.00000 −0.0750059
\(712\) −2.00000 −0.0749532
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 16.0000 0.598366
\(716\) −15.0000 −0.560576
\(717\) 24.0000 0.896296
\(718\) −7.00000 −0.261238
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 8.00000 0.298142
\(721\) 0 0
\(722\) −30.0000 −1.11648
\(723\) 3.00000 0.111571
\(724\) −2.00000 −0.0743294
\(725\) 55.0000 2.04265
\(726\) 5.00000 0.185567
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) −32.0000 −1.18437
\(731\) −50.0000 −1.84932
\(732\) −4.00000 −0.147844
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 18.0000 0.664392
\(735\) −28.0000 −1.03280
\(736\) −1.00000 −0.0368605
\(737\) −16.0000 −0.589368
\(738\) −20.0000 −0.736210
\(739\) −3.00000 −0.110357 −0.0551784 0.998477i \(-0.517573\pi\)
−0.0551784 + 0.998477i \(0.517573\pi\)
\(740\) −4.00000 −0.147043
\(741\) 7.00000 0.257151
\(742\) 0 0
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) −4.00000 −0.146647
\(745\) −60.0000 −2.19823
\(746\) −2.00000 −0.0732252
\(747\) 6.00000 0.219529
\(748\) −20.0000 −0.731272
\(749\) 0 0
\(750\) −24.0000 −0.876356
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) −6.00000 −0.218797
\(753\) 4.00000 0.145768
\(754\) −5.00000 −0.182089
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) 33.0000 1.19941 0.599703 0.800223i \(-0.295286\pi\)
0.599703 + 0.800223i \(0.295286\pi\)
\(758\) −17.0000 −0.617468
\(759\) 4.00000 0.145191
\(760\) −28.0000 −1.01567
\(761\) 40.0000 1.45000 0.724999 0.688749i \(-0.241840\pi\)
0.724999 + 0.688749i \(0.241840\pi\)
\(762\) −5.00000 −0.181131
\(763\) 0 0
\(764\) 13.0000 0.470323
\(765\) 40.0000 1.44620
\(766\) 32.0000 1.15621
\(767\) −6.00000 −0.216647
\(768\) −1.00000 −0.0360844
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) −12.0000 −0.431889
\(773\) 12.0000 0.431610 0.215805 0.976436i \(-0.430762\pi\)
0.215805 + 0.976436i \(0.430762\pi\)
\(774\) −20.0000 −0.718885
\(775\) −44.0000 −1.58053
\(776\) −17.0000 −0.610264
\(777\) 0 0
\(778\) −20.0000 −0.717035
\(779\) 70.0000 2.50801
\(780\) 4.00000 0.143223
\(781\) −60.0000 −2.14697
\(782\) −5.00000 −0.178800
\(783\) 25.0000 0.893427
\(784\) −7.00000 −0.250000
\(785\) −32.0000 −1.14213
\(786\) −10.0000 −0.356688
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −28.0000 −0.996826
\(790\) 4.00000 0.142314
\(791\) 0 0
\(792\) −8.00000 −0.284268
\(793\) 4.00000 0.142044
\(794\) −18.0000 −0.638796
\(795\) −4.00000 −0.141865
\(796\) 0 0
\(797\) −44.0000 −1.55856 −0.779280 0.626676i \(-0.784415\pi\)
−0.779280 + 0.626676i \(0.784415\pi\)
\(798\) 0 0
\(799\) −30.0000 −1.06132
\(800\) −11.0000 −0.388909
\(801\) −4.00000 −0.141333
\(802\) 24.0000 0.847469
\(803\) 32.0000 1.12926
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 19.0000 0.668832
\(808\) 14.0000 0.492518
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 4.00000 0.140546
\(811\) 34.0000 1.19390 0.596951 0.802278i \(-0.296379\pi\)
0.596951 + 0.802278i \(0.296379\pi\)
\(812\) 0 0
\(813\) −30.0000 −1.05215
\(814\) 4.00000 0.140200
\(815\) −8.00000 −0.280228
\(816\) −5.00000 −0.175035
\(817\) 70.0000 2.44899
\(818\) 21.0000 0.734248
\(819\) 0 0
\(820\) 40.0000 1.39686
\(821\) −48.0000 −1.67521 −0.837606 0.546275i \(-0.816045\pi\)
−0.837606 + 0.546275i \(0.816045\pi\)
\(822\) 8.00000 0.279032
\(823\) −18.0000 −0.627441 −0.313720 0.949515i \(-0.601575\pi\)
−0.313720 + 0.949515i \(0.601575\pi\)
\(824\) 15.0000 0.522550
\(825\) 44.0000 1.53188
\(826\) 0 0
\(827\) 45.0000 1.56480 0.782402 0.622774i \(-0.213994\pi\)
0.782402 + 0.622774i \(0.213994\pi\)
\(828\) −2.00000 −0.0695048
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) −12.0000 −0.416526
\(831\) 16.0000 0.555034
\(832\) 1.00000 0.0346688
\(833\) −35.0000 −1.21268
\(834\) −12.0000 −0.415526
\(835\) −76.0000 −2.63009
\(836\) 28.0000 0.968400
\(837\) −20.0000 −0.691301
\(838\) −12.0000 −0.414533
\(839\) −2.00000 −0.0690477 −0.0345238 0.999404i \(-0.510991\pi\)
−0.0345238 + 0.999404i \(0.510991\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 18.0000 0.620321
\(843\) 31.0000 1.06770
\(844\) 10.0000 0.344214
\(845\) 48.0000 1.65125
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) −1.00000 −0.0343401
\(849\) −13.0000 −0.446159
\(850\) −55.0000 −1.88648
\(851\) 1.00000 0.0342796
\(852\) −15.0000 −0.513892
\(853\) 8.00000 0.273915 0.136957 0.990577i \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(854\) 0 0
\(855\) −56.0000 −1.91516
\(856\) −6.00000 −0.205076
\(857\) −14.0000 −0.478231 −0.239115 0.970991i \(-0.576857\pi\)
−0.239115 + 0.970991i \(0.576857\pi\)
\(858\) −4.00000 −0.136558
\(859\) 18.0000 0.614152 0.307076 0.951685i \(-0.400649\pi\)
0.307076 + 0.951685i \(0.400649\pi\)
\(860\) 40.0000 1.36399
\(861\) 0 0
\(862\) 0 0
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) −5.00000 −0.170103
\(865\) −8.00000 −0.272008
\(866\) −17.0000 −0.577684
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) −20.0000 −0.678064
\(871\) 4.00000 0.135535
\(872\) −16.0000 −0.541828
\(873\) −34.0000 −1.15073
\(874\) 7.00000 0.236779
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) 6.00000 0.202606 0.101303 0.994856i \(-0.467699\pi\)
0.101303 + 0.994856i \(0.467699\pi\)
\(878\) −2.00000 −0.0674967
\(879\) −10.0000 −0.337292
\(880\) 16.0000 0.539360
\(881\) −4.00000 −0.134763 −0.0673817 0.997727i \(-0.521465\pi\)
−0.0673817 + 0.997727i \(0.521465\pi\)
\(882\) −14.0000 −0.471405
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 5.00000 0.168168
\(885\) −24.0000 −0.806751
\(886\) 19.0000 0.638317
\(887\) −9.00000 −0.302190 −0.151095 0.988519i \(-0.548280\pi\)
−0.151095 + 0.988519i \(0.548280\pi\)
\(888\) 1.00000 0.0335578
\(889\) 0 0
\(890\) 8.00000 0.268161
\(891\) −4.00000 −0.134005
\(892\) −14.0000 −0.468755
\(893\) 42.0000 1.40548
\(894\) 15.0000 0.501675
\(895\) 60.0000 2.00558
\(896\) 0 0
\(897\) −1.00000 −0.0333890
\(898\) 27.0000 0.901002
\(899\) −20.0000 −0.667037
\(900\) −22.0000 −0.733333
\(901\) −5.00000 −0.166574
\(902\) −40.0000 −1.33185
\(903\) 0 0
\(904\) 9.00000 0.299336
\(905\) 8.00000 0.265929
\(906\) −5.00000 −0.166114
\(907\) −34.0000 −1.12895 −0.564476 0.825450i \(-0.690922\pi\)
−0.564476 + 0.825450i \(0.690922\pi\)
\(908\) 22.0000 0.730096
\(909\) 28.0000 0.928701
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 7.00000 0.231793
\(913\) 12.0000 0.397142
\(914\) 2.00000 0.0661541
\(915\) 16.0000 0.528944
\(916\) −15.0000 −0.495614
\(917\) 0 0
\(918\) −25.0000 −0.825123
\(919\) 9.00000 0.296883 0.148441 0.988921i \(-0.452574\pi\)
0.148441 + 0.988921i \(0.452574\pi\)
\(920\) 4.00000 0.131876
\(921\) −8.00000 −0.263609
\(922\) −3.00000 −0.0987997
\(923\) 15.0000 0.493731
\(924\) 0 0
\(925\) 11.0000 0.361678
\(926\) −39.0000 −1.28162
\(927\) 30.0000 0.985329
\(928\) −5.00000 −0.164133
\(929\) −2.00000 −0.0656179 −0.0328089 0.999462i \(-0.510445\pi\)
−0.0328089 + 0.999462i \(0.510445\pi\)
\(930\) 16.0000 0.524661
\(931\) 49.0000 1.60591
\(932\) 8.00000 0.262049
\(933\) 24.0000 0.785725
\(934\) 28.0000 0.916188
\(935\) 80.0000 2.61628
\(936\) 2.00000 0.0653720
\(937\) 9.00000 0.294017 0.147009 0.989135i \(-0.453036\pi\)
0.147009 + 0.989135i \(0.453036\pi\)
\(938\) 0 0
\(939\) −22.0000 −0.717943
\(940\) 24.0000 0.782794
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 8.00000 0.260654
\(943\) −10.0000 −0.325645
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) −40.0000 −1.30051
\(947\) 32.0000 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) −1.00000 −0.0324785
\(949\) −8.00000 −0.259691
\(950\) 77.0000 2.49821
\(951\) 17.0000 0.551263
\(952\) 0 0
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) −2.00000 −0.0647524
\(955\) −52.0000 −1.68268
\(956\) −24.0000 −0.776215
\(957\) 20.0000 0.646508
\(958\) 3.00000 0.0969256
\(959\) 0 0
\(960\) 4.00000 0.129099
\(961\) −15.0000 −0.483871
\(962\) −1.00000 −0.0322413
\(963\) −12.0000 −0.386695
\(964\) −3.00000 −0.0966235
\(965\) 48.0000 1.54517
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −5.00000 −0.160706
\(969\) 35.0000 1.12436
\(970\) 68.0000 2.18335
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) −16.0000 −0.513200
\(973\) 0 0
\(974\) 12.0000 0.384505
\(975\) −11.0000 −0.352282
\(976\) 4.00000 0.128037
\(977\) 28.0000 0.895799 0.447900 0.894084i \(-0.352172\pi\)
0.447900 + 0.894084i \(0.352172\pi\)
\(978\) 2.00000 0.0639529
\(979\) −8.00000 −0.255681
\(980\) 28.0000 0.894427
\(981\) −32.0000 −1.02168
\(982\) −33.0000 −1.05307
\(983\) −54.0000 −1.72233 −0.861166 0.508323i \(-0.830265\pi\)
−0.861166 + 0.508323i \(0.830265\pi\)
\(984\) −10.0000 −0.318788
\(985\) 8.00000 0.254901
\(986\) −25.0000 −0.796162
\(987\) 0 0
\(988\) −7.00000 −0.222700
\(989\) −10.0000 −0.317982
\(990\) 32.0000 1.01703
\(991\) 42.0000 1.33417 0.667087 0.744980i \(-0.267541\pi\)
0.667087 + 0.744980i \(0.267541\pi\)
\(992\) 4.00000 0.127000
\(993\) 12.0000 0.380808
\(994\) 0 0
\(995\) 0 0
\(996\) 3.00000 0.0950586
\(997\) 23.0000 0.728417 0.364209 0.931317i \(-0.381339\pi\)
0.364209 + 0.931317i \(0.381339\pi\)
\(998\) −11.0000 −0.348199
\(999\) 5.00000 0.158193
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 106.2.a.a.1.1 1
3.2 odd 2 954.2.a.m.1.1 1
4.3 odd 2 848.2.a.d.1.1 1
5.2 odd 4 2650.2.b.f.849.1 2
5.3 odd 4 2650.2.b.f.849.2 2
5.4 even 2 2650.2.a.j.1.1 1
7.6 odd 2 5194.2.a.j.1.1 1
8.3 odd 2 3392.2.a.i.1.1 1
8.5 even 2 3392.2.a.n.1.1 1
12.11 even 2 7632.2.a.r.1.1 1
53.52 even 2 5618.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
106.2.a.a.1.1 1 1.1 even 1 trivial
848.2.a.d.1.1 1 4.3 odd 2
954.2.a.m.1.1 1 3.2 odd 2
2650.2.a.j.1.1 1 5.4 even 2
2650.2.b.f.849.1 2 5.2 odd 4
2650.2.b.f.849.2 2 5.3 odd 4
3392.2.a.i.1.1 1 8.3 odd 2
3392.2.a.n.1.1 1 8.5 even 2
5194.2.a.j.1.1 1 7.6 odd 2
5618.2.a.j.1.1 1 53.52 even 2
7632.2.a.r.1.1 1 12.11 even 2