↑

Properties

Label 5082.2.a.ba.1.1
Level $5082$
Weight $2$
Character 5082.1
Self dual yes
Analytic conductor $40.580$
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] = [5082,2,Mod(1,5082)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5082, 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("5082.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5082.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: \(40.5799743072\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5082.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{21} -4.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -6.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} -4.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} -6.00000 q^{38} -6.00000 q^{39} +4.00000 q^{41} +1.00000 q^{42} -8.00000 q^{43} -4.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -5.00000 q^{50} -4.00000 q^{51} -6.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} +1.00000 q^{56} -6.00000 q^{57} -6.00000 q^{58} +2.00000 q^{61} -2.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{67} -4.00000 q^{68} -4.00000 q^{69} +16.0000 q^{71} +1.00000 q^{72} -12.0000 q^{73} +10.0000 q^{74} -5.00000 q^{75} -6.00000 q^{76} -6.00000 q^{78} +16.0000 q^{79} +1.00000 q^{81} +4.00000 q^{82} +2.00000 q^{83} +1.00000 q^{84} -8.00000 q^{86} -6.00000 q^{87} -6.00000 q^{89} -6.00000 q^{91} -4.00000 q^{92} -2.00000 q^{93} -6.00000 q^{94} +1.00000 q^{96} -6.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) −6.00000 −0.973329
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 1.00000 0.154303
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −5.00000 −0.707107
\(51\) −4.00000 −0.560112
\(52\) −6.00000 −0.832050
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −6.00000 −0.794719
\(58\) −6.00000 −0.787839
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −2.00000 −0.254000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −4.00000 −0.485071
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 − 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 1.00000 0.117851
\(73\) −12.0000 −1.40449 −0.702247 − 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 10.0000 1.16248
\(75\) −5.00000 −0.577350
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) −4.00000 −0.417029
\(93\) −2.00000 −0.207390
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −5.00000 −0.500000
\(101\) 14.0000 1.39305 0.696526 − 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −4.00000 −0.396059
\(103\) −14.0000 −1.37946 −0.689730 − 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) 14.0000 1.34096 0.670478 − 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 1.00000 0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) −6.00000 −0.561951
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 0 0
\(122\) 2.00000 0.181071
\(123\) 4.00000 0.360668
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 22.0000 1.92215 0.961074 − 0.276289i \(-0.0891049\pi\)
0.961074 + 0.276289i \(0.0891049\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 22.0000 1.87959 0.939793 − 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) −4.00000 −0.340503
\(139\) 14.0000 1.18746 0.593732 − 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 16.0000 1.34269
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) 1.00000 0.0824786
\(148\) 10.0000 0.821995
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −5.00000 −0.408248
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −6.00000 −0.486664
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) 12.0000 0.957704 0.478852 − 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 16.0000 1.27289
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 4.00000 0.312348
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 1.00000 0.0771517
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) −8.00000 −0.609994
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −6.00000 −0.454859
\(175\) −5.00000 −0.377964
\(176\) 0 0
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 4.00000 0.298974 0.149487 − 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 − 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) −6.00000 −0.444750
\(183\) 2.00000 0.147844
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −2.00000 −0.146647
\(187\) 0 0
\(188\) −6.00000 −0.437595
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) 6.00000 0.431889 0.215945 − 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 − 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) −5.00000 −0.353553
\(201\) −4.00000 −0.282138
\(202\) 14.0000 0.985037
\(203\) −6.00000 −0.421117
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) −4.00000 −0.278019
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 − 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −10.0000 −0.686803
\(213\) 16.0000 1.09630
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −2.00000 −0.135769
\(218\) 14.0000 0.948200
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 10.0000 0.671156
\(223\) −26.0000 −1.74109 −0.870544 − 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 1.00000 0.0668153
\(225\) −5.00000 −0.333333
\(226\) −2.00000 −0.133038
\(227\) −18.0000 −1.19470 −0.597351 − 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) −6.00000 −0.397360
\(229\) 8.00000 0.528655 0.264327 − 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −22.0000 −1.44127 −0.720634 − 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 0 0
\(237\) 16.0000 1.03931
\(238\) −4.00000 −0.259281
\(239\) −8.00000 −0.517477 −0.258738 − 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 − 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 4.00000 0.255031
\(247\) 36.0000 2.29063
\(248\) −2.00000 −0.127000
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.0000 0.623783 0.311891 − 0.950118i \(-0.399037\pi\)
0.311891 + 0.950118i \(0.399037\pi\)
\(258\) −8.00000 −0.498058
\(259\) 10.0000 0.621370
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 22.0000 1.35916
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.00000 −0.367884
\(267\) −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
\(271\) −12.0000 −0.728948 −0.364474 − 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) −4.00000 −0.242536
\(273\) −6.00000 −0.363137
\(274\) 22.0000 1.32907
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) 6.00000 0.360505 0.180253 − 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 14.0000 0.839664
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 − 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) −6.00000 −0.357295
\(283\) −18.0000 −1.06999 −0.534994 − 0.844856i \(-0.679686\pi\)
−0.534994 + 0.844856i \(0.679686\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) −12.0000 −0.702247
\(293\) −14.0000 −0.817889 −0.408944 − 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 24.0000 1.38796
\(300\) −5.00000 −0.288675
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) 30.0000 1.71219 0.856095 − 0.516818i \(-0.172884\pi\)
0.856095 + 0.516818i \(0.172884\pi\)
\(308\) 0 0
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 − 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −6.00000 −0.339683
\(313\) 6.00000 0.339140 0.169570 − 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 12.0000 0.677199
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −10.0000 −0.560772
\(319\) 0 0
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) −4.00000 −0.222911
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) 30.0000 1.66410
\(326\) −4.00000 −0.221540
\(327\) 14.0000 0.774202
\(328\) 4.00000 0.220863
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 2.00000 0.109764
\(333\) 10.0000 0.547997
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 23.0000 1.25104
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 0 0
\(342\) −6.00000 −0.324443
\(343\) 1.00000 0.0539949
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −16.0000 −0.858925 −0.429463 − 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) −6.00000 −0.321634
\(349\) −26.0000 −1.39175 −0.695874 − 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) −5.00000 −0.267261
\(351\) −6.00000 −0.320256
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −4.00000 −0.211702
\(358\) 4.00000 0.211407
\(359\) −8.00000 −0.422224 −0.211112 − 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 8.00000 0.420471
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) −2.00000 −0.104399 −0.0521996 − 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) −4.00000 −0.208514
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) −10.0000 −0.519174
\(372\) −2.00000 −0.103695
\(373\) 10.0000 0.517780 0.258890 − 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) 36.0000 1.85409
\(378\) 1.00000 0.0514344
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) −12.0000 −0.613973
\(383\) −34.0000 −1.73732 −0.868659 − 0.495410i \(-0.835018\pi\)
−0.868659 + 0.495410i \(0.835018\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) −8.00000 −0.406663
\(388\) −6.00000 −0.304604
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 1.00000 0.0505076
\(393\) 22.0000 1.10975
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) −4.00000 −0.200754 −0.100377 − 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) −14.0000 −0.701757
\(399\) −6.00000 −0.300376
\(400\) −5.00000 −0.250000
\(401\) 30.0000 1.49813 0.749064 − 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −4.00000 −0.199502
\(403\) 12.0000 0.597763
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) −4.00000 −0.198030
\(409\) −16.0000 −0.791149 −0.395575 − 0.918434i \(-0.629455\pi\)
−0.395575 + 0.918434i \(0.629455\pi\)
\(410\) 0 0
\(411\) 22.0000 1.08518
\(412\) −14.0000 −0.689730
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) 14.0000 0.685583
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 − 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −8.00000 −0.389434
\(423\) −6.00000 −0.291730
\(424\) −10.0000 −0.485643
\(425\) 20.0000 0.970143
\(426\) 16.0000 0.775203
\(427\) 2.00000 0.0967868
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) −40.0000 −1.92673 −0.963366 − 0.268190i \(-0.913575\pi\)
−0.963366 + 0.268190i \(0.913575\pi\)
\(432\) 1.00000 0.0481125
\(433\) 26.0000 1.24948 0.624740 − 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 24.0000 1.14808
\(438\) −12.0000 −0.573382
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 24.0000 1.14156
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) −26.0000 −1.23114
\(447\) 6.00000 0.283790
\(448\) 1.00000 0.0472456
\(449\) −34.0000 −1.60456 −0.802280 − 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) −5.00000 −0.235702
\(451\) 0 0
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) −6.00000 −0.280976
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 8.00000 0.373815
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 − 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) 28.0000 1.29569 0.647843 − 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) −6.00000 −0.277350
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 12.0000 0.552931
\(472\) 0 0
\(473\) 0 0
\(474\) 16.0000 0.734904
\(475\) 30.0000 1.37649
\(476\) −4.00000 −0.183340
\(477\) −10.0000 −0.457869
\(478\) −8.00000 −0.365911
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −60.0000 −2.73576
\(482\) 20.0000 0.910975
\(483\) −4.00000 −0.182006
\(484\) 0 0
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 20.0000 0.906287 0.453143 − 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 2.00000 0.0905357
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 4.00000 0.180334
\(493\) 24.0000 1.08091
\(494\) 36.0000 1.61972
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 16.0000 0.717698
\(498\) 2.00000 0.0896221
\(499\) 36.0000 1.61158 0.805791 − 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) 23.0000 1.02147
\(508\) 16.0000 0.709885
\(509\) −16.0000 −0.709188 −0.354594 − 0.935020i \(-0.615381\pi\)
−0.354594 + 0.935020i \(0.615381\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) 1.00000 0.0441942
\(513\) −6.00000 −0.264906
\(514\) 10.0000 0.441081
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) 0 0
\(518\) 10.0000 0.439375
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 2.00000 0.0876216 0.0438108 − 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) −6.00000 −0.262613
\(523\) 6.00000 0.262362 0.131181 − 0.991358i \(-0.458123\pi\)
0.131181 + 0.991358i \(0.458123\pi\)
\(524\) 22.0000 0.961074
\(525\) −5.00000 −0.218218
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) −6.00000 −0.260133
\(533\) −24.0000 −1.03956
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 4.00000 0.172613
\(538\) −24.0000 −1.03471
\(539\) 0 0
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 − 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) −12.0000 −0.515444
\(543\) 8.00000 0.343313
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) −6.00000 −0.256776
\(547\) −12.0000 −0.513083 −0.256541 − 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 22.0000 0.939793
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 36.0000 1.53365
\(552\) −4.00000 −0.170251
\(553\) 16.0000 0.680389
\(554\) 6.00000 0.254916
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) 2.00000 0.0847427 0.0423714 − 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 48.0000 2.03018
\(560\) 0 0
\(561\) 0 0
\(562\) −14.0000 −0.590554
\(563\) 22.0000 0.927189 0.463595 − 0.886047i \(-0.346559\pi\)
0.463595 + 0.886047i \(0.346559\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) −18.0000 −0.756596
\(567\) 1.00000 0.0419961
\(568\) 16.0000 0.671345
\(569\) 42.0000 1.76073 0.880366 − 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 − 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 4.00000 0.166957
\(575\) 20.0000 0.834058
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 − 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 6.00000 0.249351
\(580\) 0 0
\(581\) 2.00000 0.0829740
\(582\) −6.00000 −0.248708
\(583\) 0 0
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 24.0000 0.990586 0.495293 − 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 1.00000 0.0412393
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 10.0000 0.410997
\(593\) −20.0000 −0.821302 −0.410651 − 0.911793i \(-0.634698\pi\)
−0.410651 + 0.911793i \(0.634698\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) −14.0000 −0.572982
\(598\) 24.0000 0.981433
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −5.00000 −0.204124
\(601\) 32.0000 1.30531 0.652654 − 0.757656i \(-0.273656\pi\)
0.652654 + 0.757656i \(0.273656\pi\)
\(602\) −8.00000 −0.326056
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 0 0
\(606\) 14.0000 0.568711
\(607\) −24.0000 −0.974130 −0.487065 − 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) −6.00000 −0.243332
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 36.0000 1.45640
\(612\) −4.00000 −0.161690
\(613\) 26.0000 1.05013 0.525065 − 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 30.0000 1.21070
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 − 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) −14.0000 −0.563163
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −18.0000 −0.721734
\(623\) −6.00000 −0.240385
\(624\) −6.00000 −0.240192
\(625\) 25.0000 1.00000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 12.0000 0.478852
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 − 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 16.0000 0.636446
\(633\) −8.00000 −0.317971
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) 16.0000 0.632950
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 − 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) −8.00000 −0.315735
\(643\) −24.0000 −0.946468 −0.473234 − 0.880937i \(-0.656913\pi\)
−0.473234 + 0.880937i \(0.656913\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) −34.0000 −1.33668 −0.668339 − 0.743857i \(-0.732994\pi\)
−0.668339 + 0.743857i \(0.732994\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 30.0000 1.17670
\(651\) −2.00000 −0.0783862
\(652\) −4.00000 −0.156652
\(653\) −6.00000 −0.234798 −0.117399 − 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) 4.00000 0.156174
\(657\) −12.0000 −0.468165
\(658\) −6.00000 −0.233904
\(659\) −16.0000 −0.623272 −0.311636 − 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 − 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) −4.00000 −0.155464
\(663\) 24.0000 0.932083
\(664\) 2.00000 0.0776151
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) 24.0000 0.929284
\(668\) −12.0000 −0.464294
\(669\) −26.0000 −1.00522
\(670\) 0 0
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) 26.0000 1.00223 0.501113 − 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −22.0000 −0.847408
\(675\) −5.00000 −0.192450
\(676\) 23.0000 0.884615
\(677\) −30.0000 −1.15299 −0.576497 − 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) −2.00000 −0.0768095
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 8.00000 0.305219
\(688\) −8.00000 −0.304997
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −16.0000 −0.607352
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) −16.0000 −0.606043
\(698\) −26.0000 −0.984115
\(699\) −22.0000 −0.832116
\(700\) −5.00000 −0.188982
\(701\) 46.0000 1.73740 0.868698 − 0.495342i \(-0.164957\pi\)
0.868698 + 0.495342i \(0.164957\pi\)
\(702\) −6.00000 −0.226455
\(703\) −60.0000 −2.26294
\(704\) 0 0
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 14.0000 0.526524
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 − 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) −6.00000 −0.224860
\(713\) 8.00000 0.299602
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) −8.00000 −0.298765
\(718\) −8.00000 −0.298557
\(719\) −50.0000 −1.86469 −0.932343 − 0.361576i \(-0.882239\pi\)
−0.932343 + 0.361576i \(0.882239\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 17.0000 0.632674
\(723\) 20.0000 0.743808
\(724\) 8.00000 0.297318
\(725\) 30.0000 1.11417
\(726\) 0 0
\(727\) 22.0000 0.815935 0.407967 − 0.912996i \(-0.366238\pi\)
0.407967 + 0.912996i \(0.366238\pi\)
\(728\) −6.00000 −0.222375
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 32.0000 1.18356
\(732\) 2.00000 0.0739221
\(733\) −6.00000 −0.221615 −0.110808 − 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) −2.00000 −0.0738213
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 0 0
\(738\) 4.00000 0.147242
\(739\) 52.0000 1.91285 0.956425 − 0.291977i \(-0.0943129\pi\)
0.956425 + 0.291977i \(0.0943129\pi\)
\(740\) 0 0
\(741\) 36.0000 1.32249
\(742\) −10.0000 −0.367112
\(743\) −16.0000 −0.586983 −0.293492 − 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 0 0
\(746\) 10.0000 0.366126
\(747\) 2.00000 0.0731762
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 − 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) 36.0000 1.31104
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) −14.0000 −0.508839 −0.254419 − 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) −8.00000 −0.290000 −0.145000 − 0.989432i \(-0.546318\pi\)
−0.145000 + 0.989432i \(0.546318\pi\)
\(762\) 16.0000 0.579619
\(763\) 14.0000 0.506834
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −34.0000 −1.22847
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −16.0000 −0.576975 −0.288487 − 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 10.0000 0.360141
\(772\) 6.00000 0.215945
\(773\) 20.0000 0.719350 0.359675 − 0.933078i \(-0.382888\pi\)
0.359675 + 0.933078i \(0.382888\pi\)
\(774\) −8.00000 −0.287554
\(775\) 10.0000 0.359211
\(776\) −6.00000 −0.215387
\(777\) 10.0000 0.358748
\(778\) −30.0000 −1.07555
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 16.0000 0.572159
\(783\) −6.00000 −0.214423
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 22.0000 0.784714
\(787\) −22.0000 −0.784215 −0.392108 − 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) −4.00000 −0.141955
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) −20.0000 −0.708436 −0.354218 − 0.935163i \(-0.615253\pi\)
−0.354218 + 0.935163i \(0.615253\pi\)
\(798\) −6.00000 −0.212398
\(799\) 24.0000 0.849059
\(800\) −5.00000 −0.176777
\(801\) −6.00000 −0.212000
\(802\) 30.0000 1.05934
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 12.0000 0.422682
\(807\) −24.0000 −0.844840
\(808\) 14.0000 0.492518
\(809\) −18.0000 −0.632846 −0.316423 − 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −18.0000 −0.632065 −0.316033 − 0.948748i \(-0.602351\pi\)
−0.316033 + 0.948748i \(0.602351\pi\)
\(812\) −6.00000 −0.210559
\(813\) −12.0000 −0.420858
\(814\) 0 0
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) 48.0000 1.67931
\(818\) −16.0000 −0.559427
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 38.0000 1.32621 0.663105 − 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) 22.0000 0.767338
\(823\) 28.0000 0.976019 0.488009 − 0.872838i \(-0.337723\pi\)
0.488009 + 0.872838i \(0.337723\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −4.00000 −0.139010
\(829\) 24.0000 0.833554 0.416777 − 0.909009i \(-0.363160\pi\)
0.416777 + 0.909009i \(0.363160\pi\)
\(830\) 0 0
\(831\) 6.00000 0.208138
\(832\) −6.00000 −0.208013
\(833\) −4.00000 −0.138592
\(834\) 14.0000 0.484780
\(835\) 0 0
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 4.00000 0.138178
\(839\) −42.0000 −1.45000 −0.725001 − 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) −14.0000 −0.482186
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) 0 0
\(848\) −10.0000 −0.343401
\(849\) −18.0000 −0.617758
\(850\) 20.0000 0.685994
\(851\) −40.0000 −1.37118
\(852\) 16.0000 0.548151
\(853\) 42.0000 1.43805 0.719026 − 0.694983i \(-0.244588\pi\)
0.719026 + 0.694983i \(0.244588\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) −24.0000 −0.819824 −0.409912 − 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 0 0
\(859\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 4.00000 0.136320
\(862\) −40.0000 −1.36241
\(863\) 12.0000 0.408485 0.204242 − 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) −1.00000 −0.0339618
\(868\) −2.00000 −0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 14.0000 0.474100
\(873\) −6.00000 −0.203069
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) −12.0000 −0.405442
\(877\) −18.0000 −0.607817 −0.303908 − 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 20.0000 0.674967
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) −50.0000 −1.68454 −0.842271 − 0.539054i \(-0.818782\pi\)
−0.842271 + 0.539054i \(0.818782\pi\)
\(882\) 1.00000 0.0336718
\(883\) −28.0000 −0.942275 −0.471138 − 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) −24.0000 −0.805841 −0.402921 − 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 10.0000 0.335578
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) −26.0000 −0.870544
\(893\) 36.0000 1.20469
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 24.0000 0.801337
\(898\) −34.0000 −1.13459
\(899\) 12.0000 0.400222
\(900\) −5.00000 −0.166667
\(901\) 40.0000 1.33259
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 − 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) −18.0000 −0.597351
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 − 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) −6.00000 −0.198680
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 8.00000 0.264327
\(917\) 22.0000 0.726504
\(918\) −4.00000 −0.132020
\(919\) 8.00000 0.263896 0.131948 − 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 30.0000 0.988534
\(922\) −2.00000 −0.0658665
\(923\) −96.0000 −3.15988
\(924\) 0 0
\(925\) −50.0000 −1.64399
\(926\) 16.0000 0.525793
\(927\) −14.0000 −0.459820
\(928\) −6.00000 −0.196960
\(929\) 10.0000 0.328089 0.164045 − 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −22.0000 −0.720634
\(933\) −18.0000 −0.589294
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 12.0000 0.392023 0.196011 − 0.980602i \(-0.437201\pi\)
0.196011 + 0.980602i \(0.437201\pi\)
\(938\) −4.00000 −0.130605
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 − 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 12.0000 0.390981
\(943\) −16.0000 −0.521032
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 16.0000 0.519656
\(949\) 72.0000 2.33722
\(950\) 30.0000 0.973329
\(951\) −18.0000 −0.583690
\(952\) −4.00000 −0.129641
\(953\) −18.0000 −0.583077 −0.291539 − 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 22.0000 0.710417
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −60.0000 −1.93448
\(963\) −8.00000 −0.257796
\(964\) 20.0000 0.644157
\(965\) 0 0
\(966\) −4.00000 −0.128698
\(967\) −24.0000 −0.771788 −0.385894 − 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) 0 0
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 − 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 1.00000 0.0320750
\(973\) 14.0000 0.448819
\(974\) 20.0000 0.640841
\(975\) 30.0000 0.960769
\(976\) 2.00000 0.0640184
\(977\) −46.0000 −1.47167 −0.735835 − 0.677161i \(-0.763210\pi\)
−0.735835 + 0.677161i \(0.763210\pi\)
\(978\) −4.00000 −0.127906
\(979\) 0 0
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) −12.0000 −0.382935
\(983\) 10.0000 0.318950 0.159475 − 0.987202i \(-0.449020\pi\)
0.159475 + 0.987202i \(0.449020\pi\)
\(984\) 4.00000 0.127515
\(985\) 0 0
\(986\) 24.0000 0.764316
\(987\) −6.00000 −0.190982
\(988\) 36.0000 1.14531
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) −2.00000 −0.0635001
\(993\) −4.00000 −0.126936
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) 2.00000 0.0633724
\(997\) 46.0000 1.45683 0.728417 − 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) 36.0000 1.13956
\(999\) 10.0000 0.316386
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 5082.2.a.ba.1.1 1
11.10 odd 2 462.2.a.d.1.1 ✓ 1
33.32 even 2 1386.2.a.j.1.1 1
44.43 even 2 3696.2.a.j.1.1 1
77.76 even 2 3234.2.a.b.1.1 1
231.230 odd 2 9702.2.a.bp.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.d.1.1 ✓ 1 11.10 odd 2
1386.2.a.j.1.1 1 33.32 even 2
3234.2.a.b.1.1 1 77.76 even 2
3696.2.a.j.1.1 1 44.43 even 2
5082.2.a.ba.1.1 1 1.1 even 1 trivial
9702.2.a.bp.1.1 1 231.230 odd 2