Properties

Label 8670.2.a.be.1.2
Level $8670$
Weight $2$
Character 8670.1
Self dual yes
Analytic conductor $69.230$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8670,2,Mod(1,8670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8670, 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("8670.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8670.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: \(69.2302985525\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 510)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 8670.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +4.89898 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +4.89898 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} +6.89898 q^{13} -4.89898 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +4.89898 q^{21} +4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -6.89898 q^{26} +1.00000 q^{27} +4.89898 q^{28} -6.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -4.89898 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} +6.89898 q^{39} +1.00000 q^{40} +2.89898 q^{41} -4.89898 q^{42} +8.89898 q^{43} -1.00000 q^{45} -4.00000 q^{46} +9.79796 q^{47} +1.00000 q^{48} +17.0000 q^{49} -1.00000 q^{50} +6.89898 q^{52} -7.79796 q^{53} -1.00000 q^{54} -4.89898 q^{56} +4.00000 q^{57} +6.00000 q^{58} +4.89898 q^{59} -1.00000 q^{60} -11.7980 q^{61} +4.00000 q^{62} +4.89898 q^{63} +1.00000 q^{64} -6.89898 q^{65} +0.898979 q^{67} +4.00000 q^{69} +4.89898 q^{70} +8.89898 q^{71} -1.00000 q^{72} +10.8990 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -6.89898 q^{78} +5.79796 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.89898 q^{82} +13.7980 q^{83} +4.89898 q^{84} -8.89898 q^{86} -6.00000 q^{87} -7.79796 q^{89} +1.00000 q^{90} +33.7980 q^{91} +4.00000 q^{92} -4.00000 q^{93} -9.79796 q^{94} -4.00000 q^{95} -1.00000 q^{96} +12.6969 q^{97} -17.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 4 q^{13} - 2 q^{15} + 2 q^{16} - 2 q^{18} + 8 q^{19} - 2 q^{20} + 8 q^{23} - 2 q^{24} + 2 q^{25} - 4 q^{26} + 2 q^{27} - 12 q^{29} + 2 q^{30} - 8 q^{31} - 2 q^{32} + 2 q^{36} - 12 q^{37} - 8 q^{38} + 4 q^{39} + 2 q^{40} - 4 q^{41} + 8 q^{43} - 2 q^{45} - 8 q^{46} + 2 q^{48} + 34 q^{49} - 2 q^{50} + 4 q^{52} + 4 q^{53} - 2 q^{54} + 8 q^{57} + 12 q^{58} - 2 q^{60} - 4 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} + 8 q^{69} + 8 q^{71} - 2 q^{72} + 12 q^{73} + 12 q^{74} + 2 q^{75} + 8 q^{76} - 4 q^{78} - 8 q^{79} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} - 8 q^{86} - 12 q^{87} + 4 q^{89} + 2 q^{90} + 48 q^{91} + 8 q^{92} - 8 q^{93} - 8 q^{95} - 2 q^{96} - 4 q^{97} - 34 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) 4.89898 1.85164 0.925820 0.377964i \(-0.123376\pi\)
0.925820 + 0.377964i \(0.123376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 6.89898 1.91343 0.956716 0.291022i \(-0.0939953\pi\)
0.956716 + 0.291022i \(0.0939953\pi\)
\(14\) −4.89898 −1.30931
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) 4.89898 1.06904
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −6.89898 −1.35300
\(27\) 1.00000 0.192450
\(28\) 4.89898 0.925820
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) −4.89898 −0.828079
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −4.00000 −0.648886
\(39\) 6.89898 1.10472
\(40\) 1.00000 0.158114
\(41\) 2.89898 0.452745 0.226372 0.974041i \(-0.427313\pi\)
0.226372 + 0.974041i \(0.427313\pi\)
\(42\) −4.89898 −0.755929
\(43\) 8.89898 1.35708 0.678541 0.734563i \(-0.262613\pi\)
0.678541 + 0.734563i \(0.262613\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) −4.00000 −0.589768
\(47\) 9.79796 1.42918 0.714590 0.699544i \(-0.246613\pi\)
0.714590 + 0.699544i \(0.246613\pi\)
\(48\) 1.00000 0.144338
\(49\) 17.0000 2.42857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 6.89898 0.956716
\(53\) −7.79796 −1.07113 −0.535566 0.844493i \(-0.679902\pi\)
−0.535566 + 0.844493i \(0.679902\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −4.89898 −0.654654
\(57\) 4.00000 0.529813
\(58\) 6.00000 0.787839
\(59\) 4.89898 0.637793 0.318896 0.947790i \(-0.396688\pi\)
0.318896 + 0.947790i \(0.396688\pi\)
\(60\) −1.00000 −0.129099
\(61\) −11.7980 −1.51057 −0.755287 0.655394i \(-0.772502\pi\)
−0.755287 + 0.655394i \(0.772502\pi\)
\(62\) 4.00000 0.508001
\(63\) 4.89898 0.617213
\(64\) 1.00000 0.125000
\(65\) −6.89898 −0.855713
\(66\) 0 0
\(67\) 0.898979 0.109828 0.0549139 0.998491i \(-0.482512\pi\)
0.0549139 + 0.998491i \(0.482512\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 4.89898 0.585540
\(71\) 8.89898 1.05611 0.528057 0.849209i \(-0.322921\pi\)
0.528057 + 0.849209i \(0.322921\pi\)
\(72\) −1.00000 −0.117851
\(73\) 10.8990 1.27563 0.637815 0.770190i \(-0.279839\pi\)
0.637815 + 0.770190i \(0.279839\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −6.89898 −0.781156
\(79\) 5.79796 0.652321 0.326161 0.945314i \(-0.394245\pi\)
0.326161 + 0.945314i \(0.394245\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −2.89898 −0.320139
\(83\) 13.7980 1.51452 0.757261 0.653112i \(-0.226537\pi\)
0.757261 + 0.653112i \(0.226537\pi\)
\(84\) 4.89898 0.534522
\(85\) 0 0
\(86\) −8.89898 −0.959602
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −7.79796 −0.826582 −0.413291 0.910599i \(-0.635621\pi\)
−0.413291 + 0.910599i \(0.635621\pi\)
\(90\) 1.00000 0.105409
\(91\) 33.7980 3.54299
\(92\) 4.00000 0.417029
\(93\) −4.00000 −0.414781
\(94\) −9.79796 −1.01058
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) 12.6969 1.28918 0.644589 0.764529i \(-0.277028\pi\)
0.644589 + 0.764529i \(0.277028\pi\)
\(98\) −17.0000 −1.71726
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −18.8990 −1.88052 −0.940259 0.340459i \(-0.889418\pi\)
−0.940259 + 0.340459i \(0.889418\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) −6.89898 −0.676501
\(105\) −4.89898 −0.478091
\(106\) 7.79796 0.757405
\(107\) −5.79796 −0.560510 −0.280255 0.959926i \(-0.590419\pi\)
−0.280255 + 0.959926i \(0.590419\pi\)
\(108\) 1.00000 0.0962250
\(109\) −11.7980 −1.13004 −0.565020 0.825077i \(-0.691131\pi\)
−0.565020 + 0.825077i \(0.691131\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 4.89898 0.462910
\(113\) −7.79796 −0.733570 −0.366785 0.930306i \(-0.619542\pi\)
−0.366785 + 0.930306i \(0.619542\pi\)
\(114\) −4.00000 −0.374634
\(115\) −4.00000 −0.373002
\(116\) −6.00000 −0.557086
\(117\) 6.89898 0.637811
\(118\) −4.89898 −0.450988
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −11.0000 −1.00000
\(122\) 11.7980 1.06814
\(123\) 2.89898 0.261392
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) −4.89898 −0.436436
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.89898 0.783511
\(130\) 6.89898 0.605081
\(131\) −9.79796 −0.856052 −0.428026 0.903767i \(-0.640791\pi\)
−0.428026 + 0.903767i \(0.640791\pi\)
\(132\) 0 0
\(133\) 19.5959 1.69918
\(134\) −0.898979 −0.0776600
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 14.0000 1.19610 0.598050 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) −4.00000 −0.340503
\(139\) −13.7980 −1.17033 −0.585164 0.810915i \(-0.698970\pi\)
−0.585164 + 0.810915i \(0.698970\pi\)
\(140\) −4.89898 −0.414039
\(141\) 9.79796 0.825137
\(142\) −8.89898 −0.746786
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) −10.8990 −0.902006
\(147\) 17.0000 1.40214
\(148\) −6.00000 −0.493197
\(149\) −18.8990 −1.54826 −0.774132 0.633024i \(-0.781814\pi\)
−0.774132 + 0.633024i \(0.781814\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −9.79796 −0.797347 −0.398673 0.917093i \(-0.630529\pi\)
−0.398673 + 0.917093i \(0.630529\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 6.89898 0.552360
\(157\) −1.10102 −0.0878710 −0.0439355 0.999034i \(-0.513990\pi\)
−0.0439355 + 0.999034i \(0.513990\pi\)
\(158\) −5.79796 −0.461261
\(159\) −7.79796 −0.618418
\(160\) 1.00000 0.0790569
\(161\) 19.5959 1.54437
\(162\) −1.00000 −0.0785674
\(163\) 2.20204 0.172477 0.0862386 0.996275i \(-0.472515\pi\)
0.0862386 + 0.996275i \(0.472515\pi\)
\(164\) 2.89898 0.226372
\(165\) 0 0
\(166\) −13.7980 −1.07093
\(167\) −2.20204 −0.170399 −0.0851995 0.996364i \(-0.527153\pi\)
−0.0851995 + 0.996364i \(0.527153\pi\)
\(168\) −4.89898 −0.377964
\(169\) 34.5959 2.66122
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 8.89898 0.678541
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 6.00000 0.454859
\(175\) 4.89898 0.370328
\(176\) 0 0
\(177\) 4.89898 0.368230
\(178\) 7.79796 0.584482
\(179\) −4.89898 −0.366167 −0.183083 0.983097i \(-0.558608\pi\)
−0.183083 + 0.983097i \(0.558608\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 4.20204 0.312335 0.156168 0.987731i \(-0.450086\pi\)
0.156168 + 0.987731i \(0.450086\pi\)
\(182\) −33.7980 −2.50527
\(183\) −11.7980 −0.872130
\(184\) −4.00000 −0.294884
\(185\) 6.00000 0.441129
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) 9.79796 0.714590
\(189\) 4.89898 0.356348
\(190\) 4.00000 0.290191
\(191\) −9.79796 −0.708955 −0.354478 0.935064i \(-0.615341\pi\)
−0.354478 + 0.935064i \(0.615341\pi\)
\(192\) 1.00000 0.0721688
\(193\) 1.10102 0.0792532 0.0396266 0.999215i \(-0.487383\pi\)
0.0396266 + 0.999215i \(0.487383\pi\)
\(194\) −12.6969 −0.911587
\(195\) −6.89898 −0.494046
\(196\) 17.0000 1.21429
\(197\) 13.5959 0.968669 0.484335 0.874883i \(-0.339062\pi\)
0.484335 + 0.874883i \(0.339062\pi\)
\(198\) 0 0
\(199\) −15.5959 −1.10557 −0.552783 0.833325i \(-0.686434\pi\)
−0.552783 + 0.833325i \(0.686434\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0.898979 0.0634091
\(202\) 18.8990 1.32973
\(203\) −29.3939 −2.06305
\(204\) 0 0
\(205\) −2.89898 −0.202474
\(206\) −4.00000 −0.278693
\(207\) 4.00000 0.278019
\(208\) 6.89898 0.478358
\(209\) 0 0
\(210\) 4.89898 0.338062
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −7.79796 −0.535566
\(213\) 8.89898 0.609748
\(214\) 5.79796 0.396340
\(215\) −8.89898 −0.606905
\(216\) −1.00000 −0.0680414
\(217\) −19.5959 −1.33026
\(218\) 11.7980 0.799059
\(219\) 10.8990 0.736485
\(220\) 0 0
\(221\) 0 0
\(222\) 6.00000 0.402694
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −4.89898 −0.327327
\(225\) 1.00000 0.0666667
\(226\) 7.79796 0.518713
\(227\) 21.7980 1.44678 0.723391 0.690439i \(-0.242583\pi\)
0.723391 + 0.690439i \(0.242583\pi\)
\(228\) 4.00000 0.264906
\(229\) −13.5959 −0.898444 −0.449222 0.893420i \(-0.648299\pi\)
−0.449222 + 0.893420i \(0.648299\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) −6.89898 −0.451000
\(235\) −9.79796 −0.639148
\(236\) 4.89898 0.318896
\(237\) 5.79796 0.376618
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −13.5959 −0.875790 −0.437895 0.899026i \(-0.644276\pi\)
−0.437895 + 0.899026i \(0.644276\pi\)
\(242\) 11.0000 0.707107
\(243\) 1.00000 0.0641500
\(244\) −11.7980 −0.755287
\(245\) −17.0000 −1.08609
\(246\) −2.89898 −0.184832
\(247\) 27.5959 1.75589
\(248\) 4.00000 0.254000
\(249\) 13.7980 0.874410
\(250\) 1.00000 0.0632456
\(251\) 4.89898 0.309221 0.154610 0.987976i \(-0.450588\pi\)
0.154610 + 0.987976i \(0.450588\pi\)
\(252\) 4.89898 0.308607
\(253\) 0 0
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) −8.89898 −0.554026
\(259\) −29.3939 −1.82645
\(260\) −6.89898 −0.427857
\(261\) −6.00000 −0.371391
\(262\) 9.79796 0.605320
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) 7.79796 0.479025
\(266\) −19.5959 −1.20150
\(267\) −7.79796 −0.477227
\(268\) 0.898979 0.0549139
\(269\) −9.59592 −0.585073 −0.292537 0.956254i \(-0.594499\pi\)
−0.292537 + 0.956254i \(0.594499\pi\)
\(270\) 1.00000 0.0608581
\(271\) −1.79796 −0.109218 −0.0546091 0.998508i \(-0.517391\pi\)
−0.0546091 + 0.998508i \(0.517391\pi\)
\(272\) 0 0
\(273\) 33.7980 2.04555
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) −7.79796 −0.468534 −0.234267 0.972172i \(-0.575269\pi\)
−0.234267 + 0.972172i \(0.575269\pi\)
\(278\) 13.7980 0.827547
\(279\) −4.00000 −0.239474
\(280\) 4.89898 0.292770
\(281\) 27.7980 1.65829 0.829144 0.559036i \(-0.188829\pi\)
0.829144 + 0.559036i \(0.188829\pi\)
\(282\) −9.79796 −0.583460
\(283\) −23.5959 −1.40263 −0.701316 0.712851i \(-0.747404\pi\)
−0.701316 + 0.712851i \(0.747404\pi\)
\(284\) 8.89898 0.528057
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) 14.2020 0.838320
\(288\) −1.00000 −0.0589256
\(289\) 0 0
\(290\) −6.00000 −0.352332
\(291\) 12.6969 0.744308
\(292\) 10.8990 0.637815
\(293\) 13.5959 0.794282 0.397141 0.917758i \(-0.370002\pi\)
0.397141 + 0.917758i \(0.370002\pi\)
\(294\) −17.0000 −0.991460
\(295\) −4.89898 −0.285230
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 18.8990 1.09479
\(299\) 27.5959 1.59591
\(300\) 1.00000 0.0577350
\(301\) 43.5959 2.51283
\(302\) 9.79796 0.563809
\(303\) −18.8990 −1.08572
\(304\) 4.00000 0.229416
\(305\) 11.7980 0.675549
\(306\) 0 0
\(307\) 0.898979 0.0513075 0.0256537 0.999671i \(-0.491833\pi\)
0.0256537 + 0.999671i \(0.491833\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) −4.00000 −0.227185
\(311\) 7.10102 0.402662 0.201331 0.979523i \(-0.435473\pi\)
0.201331 + 0.979523i \(0.435473\pi\)
\(312\) −6.89898 −0.390578
\(313\) −6.89898 −0.389953 −0.194977 0.980808i \(-0.562463\pi\)
−0.194977 + 0.980808i \(0.562463\pi\)
\(314\) 1.10102 0.0621342
\(315\) −4.89898 −0.276026
\(316\) 5.79796 0.326161
\(317\) −14.0000 −0.786318 −0.393159 0.919470i \(-0.628618\pi\)
−0.393159 + 0.919470i \(0.628618\pi\)
\(318\) 7.79796 0.437288
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −5.79796 −0.323611
\(322\) −19.5959 −1.09204
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 6.89898 0.382687
\(326\) −2.20204 −0.121960
\(327\) −11.7980 −0.652429
\(328\) −2.89898 −0.160069
\(329\) 48.0000 2.64633
\(330\) 0 0
\(331\) −5.79796 −0.318685 −0.159342 0.987223i \(-0.550937\pi\)
−0.159342 + 0.987223i \(0.550937\pi\)
\(332\) 13.7980 0.757261
\(333\) −6.00000 −0.328798
\(334\) 2.20204 0.120490
\(335\) −0.898979 −0.0491165
\(336\) 4.89898 0.267261
\(337\) −32.6969 −1.78112 −0.890558 0.454870i \(-0.849686\pi\)
−0.890558 + 0.454870i \(0.849686\pi\)
\(338\) −34.5959 −1.88177
\(339\) −7.79796 −0.423527
\(340\) 0 0
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 48.9898 2.64520
\(344\) −8.89898 −0.479801
\(345\) −4.00000 −0.215353
\(346\) 6.00000 0.322562
\(347\) −21.7980 −1.17018 −0.585088 0.810970i \(-0.698940\pi\)
−0.585088 + 0.810970i \(0.698940\pi\)
\(348\) −6.00000 −0.321634
\(349\) 4.20204 0.224930 0.112465 0.993656i \(-0.464125\pi\)
0.112465 + 0.993656i \(0.464125\pi\)
\(350\) −4.89898 −0.261861
\(351\) 6.89898 0.368240
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) −4.89898 −0.260378
\(355\) −8.89898 −0.472309
\(356\) −7.79796 −0.413291
\(357\) 0 0
\(358\) 4.89898 0.258919
\(359\) 37.3939 1.97357 0.986787 0.162025i \(-0.0518025\pi\)
0.986787 + 0.162025i \(0.0518025\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) −4.20204 −0.220854
\(363\) −11.0000 −0.577350
\(364\) 33.7980 1.77149
\(365\) −10.8990 −0.570479
\(366\) 11.7980 0.616689
\(367\) 27.1010 1.41466 0.707331 0.706883i \(-0.249899\pi\)
0.707331 + 0.706883i \(0.249899\pi\)
\(368\) 4.00000 0.208514
\(369\) 2.89898 0.150915
\(370\) −6.00000 −0.311925
\(371\) −38.2020 −1.98335
\(372\) −4.00000 −0.207390
\(373\) 24.6969 1.27876 0.639380 0.768891i \(-0.279191\pi\)
0.639380 + 0.768891i \(0.279191\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) −9.79796 −0.505291
\(377\) −41.3939 −2.13189
\(378\) −4.89898 −0.251976
\(379\) 29.7980 1.53062 0.765309 0.643663i \(-0.222586\pi\)
0.765309 + 0.643663i \(0.222586\pi\)
\(380\) −4.00000 −0.205196
\(381\) −12.0000 −0.614779
\(382\) 9.79796 0.501307
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −1.10102 −0.0560405
\(387\) 8.89898 0.452361
\(388\) 12.6969 0.644589
\(389\) −38.4949 −1.95177 −0.975884 0.218288i \(-0.929953\pi\)
−0.975884 + 0.218288i \(0.929953\pi\)
\(390\) 6.89898 0.349343
\(391\) 0 0
\(392\) −17.0000 −0.858630
\(393\) −9.79796 −0.494242
\(394\) −13.5959 −0.684952
\(395\) −5.79796 −0.291727
\(396\) 0 0
\(397\) −1.59592 −0.0800968 −0.0400484 0.999198i \(-0.512751\pi\)
−0.0400484 + 0.999198i \(0.512751\pi\)
\(398\) 15.5959 0.781753
\(399\) 19.5959 0.981023
\(400\) 1.00000 0.0500000
\(401\) −22.8990 −1.14352 −0.571760 0.820421i \(-0.693739\pi\)
−0.571760 + 0.820421i \(0.693739\pi\)
\(402\) −0.898979 −0.0448370
\(403\) −27.5959 −1.37465
\(404\) −18.8990 −0.940259
\(405\) −1.00000 −0.0496904
\(406\) 29.3939 1.45879
\(407\) 0 0
\(408\) 0 0
\(409\) −17.5959 −0.870062 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(410\) 2.89898 0.143170
\(411\) 14.0000 0.690569
\(412\) 4.00000 0.197066
\(413\) 24.0000 1.18096
\(414\) −4.00000 −0.196589
\(415\) −13.7980 −0.677315
\(416\) −6.89898 −0.338250
\(417\) −13.7980 −0.675689
\(418\) 0 0
\(419\) 17.7980 0.869487 0.434744 0.900554i \(-0.356839\pi\)
0.434744 + 0.900554i \(0.356839\pi\)
\(420\) −4.89898 −0.239046
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) −12.0000 −0.584151
\(423\) 9.79796 0.476393
\(424\) 7.79796 0.378702
\(425\) 0 0
\(426\) −8.89898 −0.431157
\(427\) −57.7980 −2.79704
\(428\) −5.79796 −0.280255
\(429\) 0 0
\(430\) 8.89898 0.429147
\(431\) 23.1010 1.11274 0.556369 0.830936i \(-0.312194\pi\)
0.556369 + 0.830936i \(0.312194\pi\)
\(432\) 1.00000 0.0481125
\(433\) −35.3939 −1.70092 −0.850461 0.526039i \(-0.823677\pi\)
−0.850461 + 0.526039i \(0.823677\pi\)
\(434\) 19.5959 0.940634
\(435\) 6.00000 0.287678
\(436\) −11.7980 −0.565020
\(437\) 16.0000 0.765384
\(438\) −10.8990 −0.520773
\(439\) −5.79796 −0.276721 −0.138361 0.990382i \(-0.544183\pi\)
−0.138361 + 0.990382i \(0.544183\pi\)
\(440\) 0 0
\(441\) 17.0000 0.809524
\(442\) 0 0
\(443\) 37.7980 1.79584 0.897918 0.440164i \(-0.145080\pi\)
0.897918 + 0.440164i \(0.145080\pi\)
\(444\) −6.00000 −0.284747
\(445\) 7.79796 0.369659
\(446\) 4.00000 0.189405
\(447\) −18.8990 −0.893891
\(448\) 4.89898 0.231455
\(449\) −6.89898 −0.325583 −0.162791 0.986660i \(-0.552050\pi\)
−0.162791 + 0.986660i \(0.552050\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) −7.79796 −0.366785
\(453\) −9.79796 −0.460348
\(454\) −21.7980 −1.02303
\(455\) −33.7980 −1.58447
\(456\) −4.00000 −0.187317
\(457\) 16.2020 0.757900 0.378950 0.925417i \(-0.376285\pi\)
0.378950 + 0.925417i \(0.376285\pi\)
\(458\) 13.5959 0.635296
\(459\) 0 0
\(460\) −4.00000 −0.186501
\(461\) −28.6969 −1.33655 −0.668275 0.743914i \(-0.732967\pi\)
−0.668275 + 0.743914i \(0.732967\pi\)
\(462\) 0 0
\(463\) 7.59592 0.353012 0.176506 0.984300i \(-0.443520\pi\)
0.176506 + 0.984300i \(0.443520\pi\)
\(464\) −6.00000 −0.278543
\(465\) 4.00000 0.185496
\(466\) 14.0000 0.648537
\(467\) −7.59592 −0.351497 −0.175749 0.984435i \(-0.556235\pi\)
−0.175749 + 0.984435i \(0.556235\pi\)
\(468\) 6.89898 0.318905
\(469\) 4.40408 0.203362
\(470\) 9.79796 0.451946
\(471\) −1.10102 −0.0507323
\(472\) −4.89898 −0.225494
\(473\) 0 0
\(474\) −5.79796 −0.266309
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −7.79796 −0.357044
\(478\) 0 0
\(479\) 32.8990 1.50319 0.751596 0.659623i \(-0.229284\pi\)
0.751596 + 0.659623i \(0.229284\pi\)
\(480\) 1.00000 0.0456435
\(481\) −41.3939 −1.88740
\(482\) 13.5959 0.619277
\(483\) 19.5959 0.891645
\(484\) −11.0000 −0.500000
\(485\) −12.6969 −0.576538
\(486\) −1.00000 −0.0453609
\(487\) 20.8990 0.947023 0.473512 0.880788i \(-0.342986\pi\)
0.473512 + 0.880788i \(0.342986\pi\)
\(488\) 11.7980 0.534069
\(489\) 2.20204 0.0995797
\(490\) 17.0000 0.767982
\(491\) −1.30306 −0.0588063 −0.0294032 0.999568i \(-0.509361\pi\)
−0.0294032 + 0.999568i \(0.509361\pi\)
\(492\) 2.89898 0.130696
\(493\) 0 0
\(494\) −27.5959 −1.24160
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 43.5959 1.95554
\(498\) −13.7980 −0.618301
\(499\) 39.5959 1.77256 0.886278 0.463153i \(-0.153282\pi\)
0.886278 + 0.463153i \(0.153282\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −2.20204 −0.0983799
\(502\) −4.89898 −0.218652
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) −4.89898 −0.218218
\(505\) 18.8990 0.840994
\(506\) 0 0
\(507\) 34.5959 1.53646
\(508\) −12.0000 −0.532414
\(509\) −15.3031 −0.678296 −0.339148 0.940733i \(-0.610139\pi\)
−0.339148 + 0.940733i \(0.610139\pi\)
\(510\) 0 0
\(511\) 53.3939 2.36201
\(512\) −1.00000 −0.0441942
\(513\) 4.00000 0.176604
\(514\) 2.00000 0.0882162
\(515\) −4.00000 −0.176261
\(516\) 8.89898 0.391756
\(517\) 0 0
\(518\) 29.3939 1.29149
\(519\) −6.00000 −0.263371
\(520\) 6.89898 0.302540
\(521\) 40.2929 1.76526 0.882631 0.470066i \(-0.155770\pi\)
0.882631 + 0.470066i \(0.155770\pi\)
\(522\) 6.00000 0.262613
\(523\) 18.6969 0.817560 0.408780 0.912633i \(-0.365954\pi\)
0.408780 + 0.912633i \(0.365954\pi\)
\(524\) −9.79796 −0.428026
\(525\) 4.89898 0.213809
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −7.79796 −0.338722
\(531\) 4.89898 0.212598
\(532\) 19.5959 0.849591
\(533\) 20.0000 0.866296
\(534\) 7.79796 0.337451
\(535\) 5.79796 0.250668
\(536\) −0.898979 −0.0388300
\(537\) −4.89898 −0.211407
\(538\) 9.59592 0.413709
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) 7.79796 0.335260 0.167630 0.985850i \(-0.446389\pi\)
0.167630 + 0.985850i \(0.446389\pi\)
\(542\) 1.79796 0.0772290
\(543\) 4.20204 0.180327
\(544\) 0 0
\(545\) 11.7980 0.505369
\(546\) −33.7980 −1.44642
\(547\) −0.404082 −0.0172773 −0.00863865 0.999963i \(-0.502750\pi\)
−0.00863865 + 0.999963i \(0.502750\pi\)
\(548\) 14.0000 0.598050
\(549\) −11.7980 −0.503525
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) −4.00000 −0.170251
\(553\) 28.4041 1.20786
\(554\) 7.79796 0.331304
\(555\) 6.00000 0.254686
\(556\) −13.7980 −0.585164
\(557\) 19.7980 0.838866 0.419433 0.907786i \(-0.362229\pi\)
0.419433 + 0.907786i \(0.362229\pi\)
\(558\) 4.00000 0.169334
\(559\) 61.3939 2.59668
\(560\) −4.89898 −0.207020
\(561\) 0 0
\(562\) −27.7980 −1.17259
\(563\) 21.7980 0.918674 0.459337 0.888262i \(-0.348087\pi\)
0.459337 + 0.888262i \(0.348087\pi\)
\(564\) 9.79796 0.412568
\(565\) 7.79796 0.328063
\(566\) 23.5959 0.991810
\(567\) 4.89898 0.205738
\(568\) −8.89898 −0.373393
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 4.00000 0.167542
\(571\) −29.7980 −1.24701 −0.623503 0.781821i \(-0.714291\pi\)
−0.623503 + 0.781821i \(0.714291\pi\)
\(572\) 0 0
\(573\) −9.79796 −0.409316
\(574\) −14.2020 −0.592782
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) −27.3939 −1.14042 −0.570211 0.821498i \(-0.693139\pi\)
−0.570211 + 0.821498i \(0.693139\pi\)
\(578\) 0 0
\(579\) 1.10102 0.0457569
\(580\) 6.00000 0.249136
\(581\) 67.5959 2.80435
\(582\) −12.6969 −0.526305
\(583\) 0 0
\(584\) −10.8990 −0.451003
\(585\) −6.89898 −0.285238
\(586\) −13.5959 −0.561642
\(587\) −41.3939 −1.70851 −0.854254 0.519856i \(-0.825986\pi\)
−0.854254 + 0.519856i \(0.825986\pi\)
\(588\) 17.0000 0.701068
\(589\) −16.0000 −0.659269
\(590\) 4.89898 0.201688
\(591\) 13.5959 0.559261
\(592\) −6.00000 −0.246598
\(593\) 1.59592 0.0655365 0.0327682 0.999463i \(-0.489568\pi\)
0.0327682 + 0.999463i \(0.489568\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18.8990 −0.774132
\(597\) −15.5959 −0.638298
\(598\) −27.5959 −1.12848
\(599\) 21.3939 0.874130 0.437065 0.899430i \(-0.356018\pi\)
0.437065 + 0.899430i \(0.356018\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −16.2020 −0.660895 −0.330448 0.943824i \(-0.607200\pi\)
−0.330448 + 0.943824i \(0.607200\pi\)
\(602\) −43.5959 −1.77684
\(603\) 0.898979 0.0366093
\(604\) −9.79796 −0.398673
\(605\) 11.0000 0.447214
\(606\) 18.8990 0.767719
\(607\) 32.4949 1.31893 0.659464 0.751736i \(-0.270783\pi\)
0.659464 + 0.751736i \(0.270783\pi\)
\(608\) −4.00000 −0.162221
\(609\) −29.3939 −1.19110
\(610\) −11.7980 −0.477685
\(611\) 67.5959 2.73464
\(612\) 0 0
\(613\) −14.4949 −0.585443 −0.292722 0.956198i \(-0.594561\pi\)
−0.292722 + 0.956198i \(0.594561\pi\)
\(614\) −0.898979 −0.0362799
\(615\) −2.89898 −0.116898
\(616\) 0 0
\(617\) 37.5959 1.51355 0.756777 0.653673i \(-0.226773\pi\)
0.756777 + 0.653673i \(0.226773\pi\)
\(618\) −4.00000 −0.160904
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 4.00000 0.160644
\(621\) 4.00000 0.160514
\(622\) −7.10102 −0.284725
\(623\) −38.2020 −1.53053
\(624\) 6.89898 0.276180
\(625\) 1.00000 0.0400000
\(626\) 6.89898 0.275739
\(627\) 0 0
\(628\) −1.10102 −0.0439355
\(629\) 0 0
\(630\) 4.89898 0.195180
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −5.79796 −0.230630
\(633\) 12.0000 0.476957
\(634\) 14.0000 0.556011
\(635\) 12.0000 0.476205
\(636\) −7.79796 −0.309209
\(637\) 117.283 4.64691
\(638\) 0 0
\(639\) 8.89898 0.352038
\(640\) 1.00000 0.0395285
\(641\) 20.6969 0.817480 0.408740 0.912651i \(-0.365968\pi\)
0.408740 + 0.912651i \(0.365968\pi\)
\(642\) 5.79796 0.228827
\(643\) −29.7980 −1.17512 −0.587558 0.809182i \(-0.699911\pi\)
−0.587558 + 0.809182i \(0.699911\pi\)
\(644\) 19.5959 0.772187
\(645\) −8.89898 −0.350397
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −6.89898 −0.270600
\(651\) −19.5959 −0.768025
\(652\) 2.20204 0.0862386
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 11.7980 0.461337
\(655\) 9.79796 0.382838
\(656\) 2.89898 0.113186
\(657\) 10.8990 0.425210
\(658\) −48.0000 −1.87123
\(659\) 30.6969 1.19578 0.597891 0.801577i \(-0.296005\pi\)
0.597891 + 0.801577i \(0.296005\pi\)
\(660\) 0 0
\(661\) −19.7980 −0.770051 −0.385026 0.922906i \(-0.625807\pi\)
−0.385026 + 0.922906i \(0.625807\pi\)
\(662\) 5.79796 0.225344
\(663\) 0 0
\(664\) −13.7980 −0.535465
\(665\) −19.5959 −0.759897
\(666\) 6.00000 0.232495
\(667\) −24.0000 −0.929284
\(668\) −2.20204 −0.0851995
\(669\) −4.00000 −0.154649
\(670\) 0.898979 0.0347306
\(671\) 0 0
\(672\) −4.89898 −0.188982
\(673\) 33.1010 1.27595 0.637975 0.770057i \(-0.279772\pi\)
0.637975 + 0.770057i \(0.279772\pi\)
\(674\) 32.6969 1.25944
\(675\) 1.00000 0.0384900
\(676\) 34.5959 1.33061
\(677\) 13.5959 0.522534 0.261267 0.965267i \(-0.415860\pi\)
0.261267 + 0.965267i \(0.415860\pi\)
\(678\) 7.79796 0.299479
\(679\) 62.2020 2.38710
\(680\) 0 0
\(681\) 21.7980 0.835300
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 4.00000 0.152944
\(685\) −14.0000 −0.534913
\(686\) −48.9898 −1.87044
\(687\) −13.5959 −0.518717
\(688\) 8.89898 0.339270
\(689\) −53.7980 −2.04954
\(690\) 4.00000 0.152277
\(691\) 27.1918 1.03443 0.517213 0.855857i \(-0.326969\pi\)
0.517213 + 0.855857i \(0.326969\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 21.7980 0.827439
\(695\) 13.7980 0.523386
\(696\) 6.00000 0.227429
\(697\) 0 0
\(698\) −4.20204 −0.159050
\(699\) −14.0000 −0.529529
\(700\) 4.89898 0.185164
\(701\) 14.8990 0.562727 0.281363 0.959601i \(-0.409213\pi\)
0.281363 + 0.959601i \(0.409213\pi\)
\(702\) −6.89898 −0.260385
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) −9.79796 −0.369012
\(706\) 26.0000 0.978523
\(707\) −92.5857 −3.48204
\(708\) 4.89898 0.184115
\(709\) 31.7980 1.19420 0.597099 0.802168i \(-0.296320\pi\)
0.597099 + 0.802168i \(0.296320\pi\)
\(710\) 8.89898 0.333973
\(711\) 5.79796 0.217440
\(712\) 7.79796 0.292241
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) −4.89898 −0.183083
\(717\) 0 0
\(718\) −37.3939 −1.39553
\(719\) 12.4949 0.465981 0.232991 0.972479i \(-0.425149\pi\)
0.232991 + 0.972479i \(0.425149\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 19.5959 0.729790
\(722\) 3.00000 0.111648
\(723\) −13.5959 −0.505638
\(724\) 4.20204 0.156168
\(725\) −6.00000 −0.222834
\(726\) 11.0000 0.408248
\(727\) 0.404082 0.0149866 0.00749329 0.999972i \(-0.497615\pi\)
0.00749329 + 0.999972i \(0.497615\pi\)
\(728\) −33.7980 −1.25264
\(729\) 1.00000 0.0370370
\(730\) 10.8990 0.403389
\(731\) 0 0
\(732\) −11.7980 −0.436065
\(733\) −46.4949 −1.71733 −0.858664 0.512539i \(-0.828705\pi\)
−0.858664 + 0.512539i \(0.828705\pi\)
\(734\) −27.1010 −1.00032
\(735\) −17.0000 −0.627054
\(736\) −4.00000 −0.147442
\(737\) 0 0
\(738\) −2.89898 −0.106713
\(739\) −9.39388 −0.345559 −0.172780 0.984960i \(-0.555275\pi\)
−0.172780 + 0.984960i \(0.555275\pi\)
\(740\) 6.00000 0.220564
\(741\) 27.5959 1.01376
\(742\) 38.2020 1.40244
\(743\) 9.39388 0.344628 0.172314 0.985042i \(-0.444876\pi\)
0.172314 + 0.985042i \(0.444876\pi\)
\(744\) 4.00000 0.146647
\(745\) 18.8990 0.692405
\(746\) −24.6969 −0.904219
\(747\) 13.7980 0.504841
\(748\) 0 0
\(749\) −28.4041 −1.03786
\(750\) 1.00000 0.0365148
\(751\) −21.7980 −0.795419 −0.397709 0.917511i \(-0.630195\pi\)
−0.397709 + 0.917511i \(0.630195\pi\)
\(752\) 9.79796 0.357295
\(753\) 4.89898 0.178529
\(754\) 41.3939 1.50748
\(755\) 9.79796 0.356584
\(756\) 4.89898 0.178174
\(757\) −4.69694 −0.170713 −0.0853566 0.996350i \(-0.527203\pi\)
−0.0853566 + 0.996350i \(0.527203\pi\)
\(758\) −29.7980 −1.08231
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) −1.59592 −0.0578520 −0.0289260 0.999582i \(-0.509209\pi\)
−0.0289260 + 0.999582i \(0.509209\pi\)
\(762\) 12.0000 0.434714
\(763\) −57.7980 −2.09243
\(764\) −9.79796 −0.354478
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) 33.7980 1.22037
\(768\) 1.00000 0.0360844
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 1.10102 0.0396266
\(773\) −35.3939 −1.27303 −0.636515 0.771265i \(-0.719625\pi\)
−0.636515 + 0.771265i \(0.719625\pi\)
\(774\) −8.89898 −0.319867
\(775\) −4.00000 −0.143684
\(776\) −12.6969 −0.455794
\(777\) −29.3939 −1.05450
\(778\) 38.4949 1.38011
\(779\) 11.5959 0.415467
\(780\) −6.89898 −0.247023
\(781\) 0 0
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 17.0000 0.607143
\(785\) 1.10102 0.0392971
\(786\) 9.79796 0.349482
\(787\) 45.7980 1.63252 0.816260 0.577684i \(-0.196043\pi\)
0.816260 + 0.577684i \(0.196043\pi\)
\(788\) 13.5959 0.484335
\(789\) 8.00000 0.284808
\(790\) 5.79796 0.206282
\(791\) −38.2020 −1.35831
\(792\) 0 0
\(793\) −81.3939 −2.89038
\(794\) 1.59592 0.0566370
\(795\) 7.79796 0.276565
\(796\) −15.5959 −0.552783
\(797\) 26.0000 0.920967 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) −19.5959 −0.693688
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −7.79796 −0.275527
\(802\) 22.8990 0.808591
\(803\) 0 0
\(804\) 0.898979 0.0317046
\(805\) −19.5959 −0.690665
\(806\) 27.5959 0.972025
\(807\) −9.59592 −0.337792
\(808\) 18.8990 0.664864
\(809\) −32.6969 −1.14956 −0.574782 0.818307i \(-0.694913\pi\)
−0.574782 + 0.818307i \(0.694913\pi\)
\(810\) 1.00000 0.0351364
\(811\) −25.3939 −0.891700 −0.445850 0.895108i \(-0.647098\pi\)
−0.445850 + 0.895108i \(0.647098\pi\)
\(812\) −29.3939 −1.03152
\(813\) −1.79796 −0.0630572
\(814\) 0 0
\(815\) −2.20204 −0.0771341
\(816\) 0 0
\(817\) 35.5959 1.24534
\(818\) 17.5959 0.615227
\(819\) 33.7980 1.18100
\(820\) −2.89898 −0.101237
\(821\) −15.7980 −0.551353 −0.275676 0.961251i \(-0.588902\pi\)
−0.275676 + 0.961251i \(0.588902\pi\)
\(822\) −14.0000 −0.488306
\(823\) −20.8990 −0.728493 −0.364246 0.931303i \(-0.618673\pi\)
−0.364246 + 0.931303i \(0.618673\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −24.0000 −0.835067
\(827\) 4.00000 0.139094 0.0695468 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) 4.00000 0.139010
\(829\) 3.39388 0.117874 0.0589371 0.998262i \(-0.481229\pi\)
0.0589371 + 0.998262i \(0.481229\pi\)
\(830\) 13.7980 0.478934
\(831\) −7.79796 −0.270508
\(832\) 6.89898 0.239179
\(833\) 0 0
\(834\) 13.7980 0.477784
\(835\) 2.20204 0.0762048
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) −17.7980 −0.614820
\(839\) 7.10102 0.245154 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(840\) 4.89898 0.169031
\(841\) 7.00000 0.241379
\(842\) −14.0000 −0.482472
\(843\) 27.7980 0.957413
\(844\) 12.0000 0.413057
\(845\) −34.5959 −1.19014
\(846\) −9.79796 −0.336861
\(847\) −53.8888 −1.85164
\(848\) −7.79796 −0.267783
\(849\) −23.5959 −0.809810
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 8.89898 0.304874
\(853\) −11.3939 −0.390119 −0.195059 0.980791i \(-0.562490\pi\)
−0.195059 + 0.980791i \(0.562490\pi\)
\(854\) 57.7980 1.97781
\(855\) −4.00000 −0.136797
\(856\) 5.79796 0.198170
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 0 0
\(859\) 5.79796 0.197824 0.0989119 0.995096i \(-0.468464\pi\)
0.0989119 + 0.995096i \(0.468464\pi\)
\(860\) −8.89898 −0.303453
\(861\) 14.2020 0.484004
\(862\) −23.1010 −0.786824
\(863\) −45.3939 −1.54523 −0.772613 0.634878i \(-0.781051\pi\)
−0.772613 + 0.634878i \(0.781051\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) 35.3939 1.20273
\(867\) 0 0
\(868\) −19.5959 −0.665129
\(869\) 0 0
\(870\) −6.00000 −0.203419
\(871\) 6.20204 0.210148
\(872\) 11.7980 0.399529
\(873\) 12.6969 0.429726
\(874\) −16.0000 −0.541208
\(875\) −4.89898 −0.165616
\(876\) 10.8990 0.368242
\(877\) 31.3939 1.06010 0.530048 0.847968i \(-0.322174\pi\)
0.530048 + 0.847968i \(0.322174\pi\)
\(878\) 5.79796 0.195672
\(879\) 13.5959 0.458579
\(880\) 0 0
\(881\) −34.4949 −1.16216 −0.581081 0.813846i \(-0.697370\pi\)
−0.581081 + 0.813846i \(0.697370\pi\)
\(882\) −17.0000 −0.572420
\(883\) 26.6969 0.898424 0.449212 0.893425i \(-0.351705\pi\)
0.449212 + 0.893425i \(0.351705\pi\)
\(884\) 0 0
\(885\) −4.89898 −0.164677
\(886\) −37.7980 −1.26985
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 6.00000 0.201347
\(889\) −58.7878 −1.97168
\(890\) −7.79796 −0.261388
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) 39.1918 1.31150
\(894\) 18.8990 0.632076
\(895\) 4.89898 0.163755
\(896\) −4.89898 −0.163663
\(897\) 27.5959 0.921401
\(898\) 6.89898 0.230222
\(899\) 24.0000 0.800445
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 0 0
\(903\) 43.5959 1.45078
\(904\) 7.79796 0.259356
\(905\) −4.20204 −0.139681
\(906\) 9.79796 0.325515
\(907\) 33.3939 1.10883 0.554413 0.832242i \(-0.312943\pi\)
0.554413 + 0.832242i \(0.312943\pi\)
\(908\) 21.7980 0.723391
\(909\) −18.8990 −0.626840
\(910\) 33.7980 1.12039
\(911\) −23.1010 −0.765371 −0.382685 0.923879i \(-0.625001\pi\)
−0.382685 + 0.923879i \(0.625001\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) −16.2020 −0.535916
\(915\) 11.7980 0.390028
\(916\) −13.5959 −0.449222
\(917\) −48.0000 −1.58510
\(918\) 0 0
\(919\) 1.79796 0.0593092 0.0296546 0.999560i \(-0.490559\pi\)
0.0296546 + 0.999560i \(0.490559\pi\)
\(920\) 4.00000 0.131876
\(921\) 0.898979 0.0296224
\(922\) 28.6969 0.945083
\(923\) 61.3939 2.02080
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) −7.59592 −0.249617
\(927\) 4.00000 0.131377
\(928\) 6.00000 0.196960
\(929\) −36.2929 −1.19073 −0.595365 0.803455i \(-0.702993\pi\)
−0.595365 + 0.803455i \(0.702993\pi\)
\(930\) −4.00000 −0.131165
\(931\) 68.0000 2.22861
\(932\) −14.0000 −0.458585
\(933\) 7.10102 0.232477
\(934\) 7.59592 0.248546
\(935\) 0 0
\(936\) −6.89898 −0.225500
\(937\) 39.3939 1.28694 0.643471 0.765471i \(-0.277494\pi\)
0.643471 + 0.765471i \(0.277494\pi\)
\(938\) −4.40408 −0.143798
\(939\) −6.89898 −0.225140
\(940\) −9.79796 −0.319574
\(941\) 16.2020 0.528171 0.264086 0.964499i \(-0.414930\pi\)
0.264086 + 0.964499i \(0.414930\pi\)
\(942\) 1.10102 0.0358732
\(943\) 11.5959 0.377615
\(944\) 4.89898 0.159448
\(945\) −4.89898 −0.159364
\(946\) 0 0
\(947\) −21.7980 −0.708338 −0.354169 0.935181i \(-0.615236\pi\)
−0.354169 + 0.935181i \(0.615236\pi\)
\(948\) 5.79796 0.188309
\(949\) 75.1918 2.44083
\(950\) −4.00000 −0.129777
\(951\) −14.0000 −0.453981
\(952\) 0 0
\(953\) −41.1918 −1.33433 −0.667167 0.744908i \(-0.732493\pi\)
−0.667167 + 0.744908i \(0.732493\pi\)
\(954\) 7.79796 0.252468
\(955\) 9.79796 0.317055
\(956\) 0 0
\(957\) 0 0
\(958\) −32.8990 −1.06292
\(959\) 68.5857 2.21475
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) 41.3939 1.33459
\(963\) −5.79796 −0.186837
\(964\) −13.5959 −0.437895
\(965\) −1.10102 −0.0354431
\(966\) −19.5959 −0.630488
\(967\) −41.3939 −1.33114 −0.665569 0.746337i \(-0.731811\pi\)
−0.665569 + 0.746337i \(0.731811\pi\)
\(968\) 11.0000 0.353553
\(969\) 0 0
\(970\) 12.6969 0.407674
\(971\) 38.6969 1.24184 0.620922 0.783872i \(-0.286758\pi\)
0.620922 + 0.783872i \(0.286758\pi\)
\(972\) 1.00000 0.0320750
\(973\) −67.5959 −2.16703
\(974\) −20.8990 −0.669646
\(975\) 6.89898 0.220944
\(976\) −11.7980 −0.377643
\(977\) −29.5959 −0.946857 −0.473429 0.880832i \(-0.656984\pi\)
−0.473429 + 0.880832i \(0.656984\pi\)
\(978\) −2.20204 −0.0704135
\(979\) 0 0
\(980\) −17.0000 −0.543045
\(981\) −11.7980 −0.376680
\(982\) 1.30306 0.0415824
\(983\) −9.39388 −0.299618 −0.149809 0.988715i \(-0.547866\pi\)
−0.149809 + 0.988715i \(0.547866\pi\)
\(984\) −2.89898 −0.0924161
\(985\) −13.5959 −0.433202
\(986\) 0 0
\(987\) 48.0000 1.52786
\(988\) 27.5959 0.877943
\(989\) 35.5959 1.13188
\(990\) 0 0
\(991\) 15.5959 0.495421 0.247710 0.968834i \(-0.420322\pi\)
0.247710 + 0.968834i \(0.420322\pi\)
\(992\) 4.00000 0.127000
\(993\) −5.79796 −0.183993
\(994\) −43.5959 −1.38278
\(995\) 15.5959 0.494424
\(996\) 13.7980 0.437205
\(997\) 21.5959 0.683950 0.341975 0.939709i \(-0.388904\pi\)
0.341975 + 0.939709i \(0.388904\pi\)
\(998\) −39.5959 −1.25339
\(999\) −6.00000 −0.189832
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8670.2.a.be.1.2 2
17.16 even 2 510.2.a.h.1.1 2
51.50 odd 2 1530.2.a.s.1.1 2
68.67 odd 2 4080.2.a.bq.1.2 2
85.33 odd 4 2550.2.d.u.2449.4 4
85.67 odd 4 2550.2.d.u.2449.1 4
85.84 even 2 2550.2.a.bl.1.2 2
255.254 odd 2 7650.2.a.cu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.h.1.1 2 17.16 even 2
1530.2.a.s.1.1 2 51.50 odd 2
2550.2.a.bl.1.2 2 85.84 even 2
2550.2.d.u.2449.1 4 85.67 odd 4
2550.2.d.u.2449.4 4 85.33 odd 4
4080.2.a.bq.1.2 2 68.67 odd 2
7650.2.a.cu.1.2 2 255.254 odd 2
8670.2.a.be.1.2 2 1.1 even 1 trivial