Properties

Label 847.2.a.l.1.3
Level $847$
Weight $2$
Character 847.1
Self dual yes
Analytic conductor $6.763$
Analytic rank $1$
Dimension $4$
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] = [847,2,Mod(1,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.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: \(6.76332905120\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2525.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.77748\) of defining polynomial
Character \(\chi\) \(=\) 847.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.77748 q^{2} +0.618034 q^{3} +1.15945 q^{4} -2.77748 q^{5} +1.09855 q^{6} -1.00000 q^{7} -1.49406 q^{8} -2.61803 q^{9} +O(q^{10})\) \(q+1.77748 q^{2} +0.618034 q^{3} +1.15945 q^{4} -2.77748 q^{5} +1.09855 q^{6} -1.00000 q^{7} -1.49406 q^{8} -2.61803 q^{9} -4.93693 q^{10} +0.716580 q^{12} -4.29348 q^{13} -1.77748 q^{14} -1.71658 q^{15} -4.97458 q^{16} -2.75556 q^{17} -4.65351 q^{18} +1.93910 q^{19} -3.22035 q^{20} -0.618034 q^{21} +4.37009 q^{23} -0.923382 q^{24} +2.71442 q^{25} -7.63159 q^{26} -3.47214 q^{27} -1.15945 q^{28} +8.62809 q^{29} -3.05119 q^{30} -0.200588 q^{31} -5.85410 q^{32} -4.89796 q^{34} +2.77748 q^{35} -3.03548 q^{36} +1.03548 q^{37} +3.44671 q^{38} -2.65351 q^{39} +4.14974 q^{40} +9.60616 q^{41} -1.09855 q^{42} -4.70820 q^{43} +7.27155 q^{45} +7.76777 q^{46} -13.0455 q^{47} -3.07446 q^{48} +1.00000 q^{49} +4.82484 q^{50} -1.70303 q^{51} -4.97807 q^{52} -3.90012 q^{53} -6.17167 q^{54} +1.49406 q^{56} +1.19843 q^{57} +15.3363 q^{58} -8.55713 q^{59} -1.99029 q^{60} -0.988609 q^{61} -0.356542 q^{62} +2.61803 q^{63} -0.456423 q^{64} +11.9251 q^{65} -5.41745 q^{67} -3.19493 q^{68} +2.70087 q^{69} +4.93693 q^{70} -2.01705 q^{71} +3.91151 q^{72} -9.97108 q^{73} +1.84055 q^{74} +1.67760 q^{75} +2.24828 q^{76} -4.71658 q^{78} +6.29348 q^{79} +13.8168 q^{80} +5.70820 q^{81} +17.0748 q^{82} -1.72146 q^{83} -0.716580 q^{84} +7.65351 q^{85} -8.36876 q^{86} +5.33245 q^{87} -15.3035 q^{89} +12.9251 q^{90} +4.29348 q^{91} +5.06691 q^{92} -0.123970 q^{93} -23.1882 q^{94} -5.38581 q^{95} -3.61803 q^{96} +11.6162 q^{97} +1.77748 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} - q^{6} - 4 q^{7} + 9 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} - q^{6} - 4 q^{7} + 9 q^{8} - 6 q^{9} - 14 q^{10} - 7 q^{12} - 2 q^{14} + 3 q^{15} - 4 q^{16} - 3 q^{17} - 3 q^{18} + 3 q^{19} - 17 q^{20} + 2 q^{21} - 8 q^{23} - 12 q^{24} - 12 q^{26} + 4 q^{27} - 4 q^{28} + 3 q^{29} + 12 q^{30} - 3 q^{31} - 10 q^{32} - 12 q^{34} + 6 q^{35} - q^{36} - 7 q^{37} - 20 q^{38} + 5 q^{39} - 13 q^{40} + 4 q^{41} + q^{42} + 8 q^{43} + 9 q^{45} - 3 q^{46} - 14 q^{47} - 3 q^{48} + 4 q^{49} + 33 q^{50} - 11 q^{51} - 17 q^{52} - 9 q^{53} + 2 q^{54} - 9 q^{56} + 6 q^{57} + 3 q^{58} - 25 q^{59} + 21 q^{60} - 19 q^{61} + 10 q^{62} + 6 q^{63} + 3 q^{64} + 12 q^{65} - 15 q^{67} - q^{68} + 14 q^{69} + 14 q^{70} - 7 q^{71} - 6 q^{72} - 11 q^{73} + 8 q^{74} - 5 q^{75} - 26 q^{76} - 9 q^{78} + 8 q^{79} - 4 q^{80} - 4 q^{81} + 3 q^{82} - q^{83} + 7 q^{84} + 15 q^{85} + 4 q^{86} + 6 q^{87} - 17 q^{89} + 16 q^{90} - 17 q^{92} - 11 q^{93} - 20 q^{94} + 17 q^{95} - 10 q^{96} - 15 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.77748 1.25687 0.628436 0.777862i \(-0.283696\pi\)
0.628436 + 0.777862i \(0.283696\pi\)
\(3\) 0.618034 0.356822 0.178411 0.983956i \(-0.442904\pi\)
0.178411 + 0.983956i \(0.442904\pi\)
\(4\) 1.15945 0.579725
\(5\) −2.77748 −1.24213 −0.621064 0.783760i \(-0.713299\pi\)
−0.621064 + 0.783760i \(0.713299\pi\)
\(6\) 1.09855 0.448479
\(7\) −1.00000 −0.377964
\(8\) −1.49406 −0.528231
\(9\) −2.61803 −0.872678
\(10\) −4.93693 −1.56120
\(11\) 0 0
\(12\) 0.716580 0.206859
\(13\) −4.29348 −1.19080 −0.595398 0.803431i \(-0.703006\pi\)
−0.595398 + 0.803431i \(0.703006\pi\)
\(14\) −1.77748 −0.475053
\(15\) −1.71658 −0.443219
\(16\) −4.97458 −1.24364
\(17\) −2.75556 −0.668321 −0.334160 0.942516i \(-0.608453\pi\)
−0.334160 + 0.942516i \(0.608453\pi\)
\(18\) −4.65351 −1.09684
\(19\) 1.93910 0.444859 0.222429 0.974949i \(-0.428601\pi\)
0.222429 + 0.974949i \(0.428601\pi\)
\(20\) −3.22035 −0.720093
\(21\) −0.618034 −0.134866
\(22\) 0 0
\(23\) 4.37009 0.911228 0.455614 0.890178i \(-0.349420\pi\)
0.455614 + 0.890178i \(0.349420\pi\)
\(24\) −0.923382 −0.188485
\(25\) 2.71442 0.542884
\(26\) −7.63159 −1.49668
\(27\) −3.47214 −0.668213
\(28\) −1.15945 −0.219116
\(29\) 8.62809 1.60220 0.801098 0.598533i \(-0.204250\pi\)
0.801098 + 0.598533i \(0.204250\pi\)
\(30\) −3.05119 −0.557069
\(31\) −0.200588 −0.0360266 −0.0180133 0.999838i \(-0.505734\pi\)
−0.0180133 + 0.999838i \(0.505734\pi\)
\(32\) −5.85410 −1.03487
\(33\) 0 0
\(34\) −4.89796 −0.839993
\(35\) 2.77748 0.469481
\(36\) −3.03548 −0.505913
\(37\) 1.03548 0.170232 0.0851159 0.996371i \(-0.472874\pi\)
0.0851159 + 0.996371i \(0.472874\pi\)
\(38\) 3.44671 0.559130
\(39\) −2.65351 −0.424902
\(40\) 4.14974 0.656131
\(41\) 9.60616 1.50023 0.750115 0.661307i \(-0.229998\pi\)
0.750115 + 0.661307i \(0.229998\pi\)
\(42\) −1.09855 −0.169509
\(43\) −4.70820 −0.717994 −0.358997 0.933339i \(-0.616881\pi\)
−0.358997 + 0.933339i \(0.616881\pi\)
\(44\) 0 0
\(45\) 7.27155 1.08398
\(46\) 7.76777 1.14530
\(47\) −13.0455 −1.90289 −0.951443 0.307823i \(-0.900399\pi\)
−0.951443 + 0.307823i \(0.900399\pi\)
\(48\) −3.07446 −0.443760
\(49\) 1.00000 0.142857
\(50\) 4.82484 0.682335
\(51\) −1.70303 −0.238472
\(52\) −4.97807 −0.690334
\(53\) −3.90012 −0.535723 −0.267861 0.963457i \(-0.586317\pi\)
−0.267861 + 0.963457i \(0.586317\pi\)
\(54\) −6.17167 −0.839857
\(55\) 0 0
\(56\) 1.49406 0.199653
\(57\) 1.19843 0.158736
\(58\) 15.3363 2.01375
\(59\) −8.55713 −1.11404 −0.557022 0.830498i \(-0.688056\pi\)
−0.557022 + 0.830498i \(0.688056\pi\)
\(60\) −1.99029 −0.256945
\(61\) −0.988609 −0.126578 −0.0632892 0.997995i \(-0.520159\pi\)
−0.0632892 + 0.997995i \(0.520159\pi\)
\(62\) −0.356542 −0.0452808
\(63\) 2.61803 0.329841
\(64\) −0.456423 −0.0570529
\(65\) 11.9251 1.47912
\(66\) 0 0
\(67\) −5.41745 −0.661846 −0.330923 0.943658i \(-0.607360\pi\)
−0.330923 + 0.943658i \(0.607360\pi\)
\(68\) −3.19493 −0.387442
\(69\) 2.70087 0.325146
\(70\) 4.93693 0.590077
\(71\) −2.01705 −0.239380 −0.119690 0.992811i \(-0.538190\pi\)
−0.119690 + 0.992811i \(0.538190\pi\)
\(72\) 3.91151 0.460976
\(73\) −9.97108 −1.16703 −0.583513 0.812104i \(-0.698322\pi\)
−0.583513 + 0.812104i \(0.698322\pi\)
\(74\) 1.84055 0.213960
\(75\) 1.67760 0.193713
\(76\) 2.24828 0.257896
\(77\) 0 0
\(78\) −4.71658 −0.534047
\(79\) 6.29348 0.708071 0.354036 0.935232i \(-0.384809\pi\)
0.354036 + 0.935232i \(0.384809\pi\)
\(80\) 13.8168 1.54477
\(81\) 5.70820 0.634245
\(82\) 17.0748 1.88560
\(83\) −1.72146 −0.188955 −0.0944773 0.995527i \(-0.530118\pi\)
−0.0944773 + 0.995527i \(0.530118\pi\)
\(84\) −0.716580 −0.0781853
\(85\) 7.65351 0.830140
\(86\) −8.36876 −0.902426
\(87\) 5.33245 0.571699
\(88\) 0 0
\(89\) −15.3035 −1.62217 −0.811086 0.584928i \(-0.801123\pi\)
−0.811086 + 0.584928i \(0.801123\pi\)
\(90\) 12.9251 1.36242
\(91\) 4.29348 0.450079
\(92\) 5.06691 0.528262
\(93\) −0.123970 −0.0128551
\(94\) −23.1882 −2.39168
\(95\) −5.38581 −0.552572
\(96\) −3.61803 −0.369264
\(97\) 11.6162 1.17945 0.589724 0.807605i \(-0.299236\pi\)
0.589724 + 0.807605i \(0.299236\pi\)
\(98\) 1.77748 0.179553
\(99\) 0 0
\(100\) 3.14723 0.314723
\(101\) 3.41179 0.339486 0.169743 0.985488i \(-0.445706\pi\)
0.169743 + 0.985488i \(0.445706\pi\)
\(102\) −3.02710 −0.299728
\(103\) −18.1826 −1.79158 −0.895791 0.444475i \(-0.853390\pi\)
−0.895791 + 0.444475i \(0.853390\pi\)
\(104\) 6.41473 0.629016
\(105\) 1.71658 0.167521
\(106\) −6.93240 −0.673334
\(107\) 3.24746 0.313944 0.156972 0.987603i \(-0.449827\pi\)
0.156972 + 0.987603i \(0.449827\pi\)
\(108\) −4.02577 −0.387380
\(109\) 12.6912 1.21559 0.607796 0.794093i \(-0.292054\pi\)
0.607796 + 0.794093i \(0.292054\pi\)
\(110\) 0 0
\(111\) 0.639962 0.0607425
\(112\) 4.97458 0.470053
\(113\) −18.4480 −1.73545 −0.867723 0.497048i \(-0.834417\pi\)
−0.867723 + 0.497048i \(0.834417\pi\)
\(114\) 2.13018 0.199510
\(115\) −12.1379 −1.13186
\(116\) 10.0038 0.928833
\(117\) 11.2405 1.03918
\(118\) −15.2102 −1.40021
\(119\) 2.75556 0.252601
\(120\) 2.56468 0.234122
\(121\) 0 0
\(122\) −1.75724 −0.159093
\(123\) 5.93693 0.535315
\(124\) −0.232572 −0.0208855
\(125\) 6.34817 0.567797
\(126\) 4.65351 0.414568
\(127\) 19.5093 1.73117 0.865585 0.500762i \(-0.166947\pi\)
0.865585 + 0.500762i \(0.166947\pi\)
\(128\) 10.8969 0.963161
\(129\) −2.90983 −0.256196
\(130\) 21.1966 1.85907
\(131\) −6.89796 −0.602677 −0.301339 0.953517i \(-0.597433\pi\)
−0.301339 + 0.953517i \(0.597433\pi\)
\(132\) 0 0
\(133\) −1.93910 −0.168141
\(134\) −9.62943 −0.831856
\(135\) 9.64380 0.830006
\(136\) 4.11698 0.353028
\(137\) −2.61070 −0.223047 −0.111523 0.993762i \(-0.535573\pi\)
−0.111523 + 0.993762i \(0.535573\pi\)
\(138\) 4.80075 0.408667
\(139\) −1.00134 −0.0849322 −0.0424661 0.999098i \(-0.513521\pi\)
−0.0424661 + 0.999098i \(0.513521\pi\)
\(140\) 3.22035 0.272170
\(141\) −8.06258 −0.678992
\(142\) −3.58527 −0.300869
\(143\) 0 0
\(144\) 13.0236 1.08530
\(145\) −23.9644 −1.99013
\(146\) −17.7234 −1.46680
\(147\) 0.618034 0.0509746
\(148\) 1.20059 0.0986877
\(149\) −14.8948 −1.22023 −0.610113 0.792314i \(-0.708876\pi\)
−0.610113 + 0.792314i \(0.708876\pi\)
\(150\) 2.98191 0.243472
\(151\) 15.8046 1.28616 0.643080 0.765799i \(-0.277656\pi\)
0.643080 + 0.765799i \(0.277656\pi\)
\(152\) −2.89713 −0.234988
\(153\) 7.21414 0.583229
\(154\) 0 0
\(155\) 0.557129 0.0447497
\(156\) −3.07662 −0.246327
\(157\) 4.59123 0.366420 0.183210 0.983074i \(-0.441351\pi\)
0.183210 + 0.983074i \(0.441351\pi\)
\(158\) 11.1866 0.889955
\(159\) −2.41041 −0.191158
\(160\) 16.2597 1.28544
\(161\) −4.37009 −0.344412
\(162\) 10.1462 0.797164
\(163\) 8.03764 0.629557 0.314778 0.949165i \(-0.398070\pi\)
0.314778 + 0.949165i \(0.398070\pi\)
\(164\) 11.1379 0.869721
\(165\) 0 0
\(166\) −3.05987 −0.237492
\(167\) −13.4069 −1.03746 −0.518729 0.854939i \(-0.673595\pi\)
−0.518729 + 0.854939i \(0.673595\pi\)
\(168\) 0.923382 0.0712405
\(169\) 5.43394 0.417995
\(170\) 13.6040 1.04338
\(171\) −5.07662 −0.388219
\(172\) −5.45893 −0.416239
\(173\) −20.5821 −1.56483 −0.782413 0.622760i \(-0.786011\pi\)
−0.782413 + 0.622760i \(0.786011\pi\)
\(174\) 9.47835 0.718552
\(175\) −2.71442 −0.205191
\(176\) 0 0
\(177\) −5.28860 −0.397515
\(178\) −27.2018 −2.03886
\(179\) −3.71520 −0.277687 −0.138843 0.990314i \(-0.544338\pi\)
−0.138843 + 0.990314i \(0.544338\pi\)
\(180\) 8.43100 0.628410
\(181\) 4.77183 0.354687 0.177344 0.984149i \(-0.443250\pi\)
0.177344 + 0.984149i \(0.443250\pi\)
\(182\) 7.63159 0.565691
\(183\) −0.610994 −0.0451660
\(184\) −6.52920 −0.481339
\(185\) −2.87603 −0.211450
\(186\) −0.220355 −0.0161572
\(187\) 0 0
\(188\) −15.1257 −1.10315
\(189\) 3.47214 0.252561
\(190\) −9.57319 −0.694512
\(191\) −0.829158 −0.0599958 −0.0299979 0.999550i \(-0.509550\pi\)
−0.0299979 + 0.999550i \(0.509550\pi\)
\(192\) −0.282085 −0.0203577
\(193\) −6.73803 −0.485014 −0.242507 0.970150i \(-0.577970\pi\)
−0.242507 + 0.970150i \(0.577970\pi\)
\(194\) 20.6476 1.48241
\(195\) 7.37009 0.527783
\(196\) 1.15945 0.0828179
\(197\) −10.9216 −0.778129 −0.389065 0.921210i \(-0.627202\pi\)
−0.389065 + 0.921210i \(0.627202\pi\)
\(198\) 0 0
\(199\) 20.9746 1.48685 0.743424 0.668820i \(-0.233200\pi\)
0.743424 + 0.668820i \(0.233200\pi\)
\(200\) −4.05552 −0.286768
\(201\) −3.34817 −0.236161
\(202\) 6.06440 0.426690
\(203\) −8.62809 −0.605573
\(204\) −1.97458 −0.138248
\(205\) −26.6810 −1.86348
\(206\) −32.3192 −2.25179
\(207\) −11.4411 −0.795208
\(208\) 21.3582 1.48093
\(209\) 0 0
\(210\) 3.05119 0.210552
\(211\) −6.02361 −0.414682 −0.207341 0.978269i \(-0.566481\pi\)
−0.207341 + 0.978269i \(0.566481\pi\)
\(212\) −4.52199 −0.310572
\(213\) −1.24660 −0.0854159
\(214\) 5.77231 0.394587
\(215\) 13.0770 0.891841
\(216\) 5.18759 0.352971
\(217\) 0.200588 0.0136168
\(218\) 22.5583 1.52784
\(219\) −6.16247 −0.416421
\(220\) 0 0
\(221\) 11.8309 0.795833
\(222\) 1.13752 0.0763455
\(223\) −4.62507 −0.309718 −0.154859 0.987937i \(-0.549492\pi\)
−0.154859 + 0.987937i \(0.549492\pi\)
\(224\) 5.85410 0.391144
\(225\) −7.10644 −0.473763
\(226\) −32.7911 −2.18123
\(227\) 17.3003 1.14826 0.574132 0.818763i \(-0.305340\pi\)
0.574132 + 0.818763i \(0.305340\pi\)
\(228\) 1.38952 0.0920230
\(229\) −4.42612 −0.292486 −0.146243 0.989249i \(-0.546718\pi\)
−0.146243 + 0.989249i \(0.546718\pi\)
\(230\) −21.5749 −1.42260
\(231\) 0 0
\(232\) −12.8909 −0.846330
\(233\) −10.5330 −0.690042 −0.345021 0.938595i \(-0.612128\pi\)
−0.345021 + 0.938595i \(0.612128\pi\)
\(234\) 19.9798 1.30612
\(235\) 36.2338 2.36363
\(236\) −9.92157 −0.645839
\(237\) 3.88958 0.252656
\(238\) 4.89796 0.317487
\(239\) −9.75646 −0.631093 −0.315546 0.948910i \(-0.602188\pi\)
−0.315546 + 0.948910i \(0.602188\pi\)
\(240\) 8.53926 0.551207
\(241\) 12.5501 0.808422 0.404211 0.914666i \(-0.367546\pi\)
0.404211 + 0.914666i \(0.367546\pi\)
\(242\) 0 0
\(243\) 13.9443 0.894525
\(244\) −1.14624 −0.0733807
\(245\) −2.77748 −0.177447
\(246\) 10.5528 0.672822
\(247\) −8.32546 −0.529736
\(248\) 0.299691 0.0190304
\(249\) −1.06392 −0.0674232
\(250\) 11.2838 0.713648
\(251\) −10.9912 −0.693758 −0.346879 0.937910i \(-0.612759\pi\)
−0.346879 + 0.937910i \(0.612759\pi\)
\(252\) 3.03548 0.191217
\(253\) 0 0
\(254\) 34.6775 2.17586
\(255\) 4.73013 0.296212
\(256\) 20.2819 1.26762
\(257\) 5.33280 0.332651 0.166325 0.986071i \(-0.446810\pi\)
0.166325 + 0.986071i \(0.446810\pi\)
\(258\) −5.17218 −0.322006
\(259\) −1.03548 −0.0643416
\(260\) 13.8265 0.857484
\(261\) −22.5886 −1.39820
\(262\) −12.2610 −0.757488
\(263\) −8.18034 −0.504421 −0.252211 0.967672i \(-0.581158\pi\)
−0.252211 + 0.967672i \(0.581158\pi\)
\(264\) 0 0
\(265\) 10.8325 0.665436
\(266\) −3.44671 −0.211331
\(267\) −9.45810 −0.578826
\(268\) −6.28126 −0.383689
\(269\) 12.8439 0.783107 0.391554 0.920155i \(-0.371938\pi\)
0.391554 + 0.920155i \(0.371938\pi\)
\(270\) 17.1417 1.04321
\(271\) −22.0472 −1.33927 −0.669636 0.742689i \(-0.733550\pi\)
−0.669636 + 0.742689i \(0.733550\pi\)
\(272\) 13.7077 0.831153
\(273\) 2.65351 0.160598
\(274\) −4.64047 −0.280341
\(275\) 0 0
\(276\) 3.13152 0.188495
\(277\) 24.0770 1.44664 0.723322 0.690511i \(-0.242614\pi\)
0.723322 + 0.690511i \(0.242614\pi\)
\(278\) −1.77986 −0.106749
\(279\) 0.525146 0.0314396
\(280\) −4.14974 −0.247994
\(281\) 15.4418 0.921182 0.460591 0.887612i \(-0.347637\pi\)
0.460591 + 0.887612i \(0.347637\pi\)
\(282\) −14.3311 −0.853406
\(283\) −3.35732 −0.199572 −0.0997860 0.995009i \(-0.531816\pi\)
−0.0997860 + 0.995009i \(0.531816\pi\)
\(284\) −2.33867 −0.138774
\(285\) −3.32861 −0.197170
\(286\) 0 0
\(287\) −9.60616 −0.567034
\(288\) 15.3262 0.903107
\(289\) −9.40691 −0.553348
\(290\) −42.5963 −2.50134
\(291\) 7.17922 0.420853
\(292\) −11.5610 −0.676555
\(293\) 6.83705 0.399425 0.199712 0.979855i \(-0.435999\pi\)
0.199712 + 0.979855i \(0.435999\pi\)
\(294\) 1.09855 0.0640685
\(295\) 23.7673 1.38379
\(296\) −1.54707 −0.0899218
\(297\) 0 0
\(298\) −26.4752 −1.53367
\(299\) −18.7629 −1.08509
\(300\) 1.94510 0.112300
\(301\) 4.70820 0.271376
\(302\) 28.0924 1.61654
\(303\) 2.10860 0.121136
\(304\) −9.64618 −0.553246
\(305\) 2.74585 0.157227
\(306\) 12.8230 0.733043
\(307\) −11.7970 −0.673293 −0.336646 0.941631i \(-0.609293\pi\)
−0.336646 + 0.941631i \(0.609293\pi\)
\(308\) 0 0
\(309\) −11.2375 −0.639276
\(310\) 0.990289 0.0562446
\(311\) 25.3139 1.43542 0.717709 0.696343i \(-0.245191\pi\)
0.717709 + 0.696343i \(0.245191\pi\)
\(312\) 3.96452 0.224447
\(313\) −19.6543 −1.11093 −0.555464 0.831540i \(-0.687460\pi\)
−0.555464 + 0.831540i \(0.687460\pi\)
\(314\) 8.16083 0.460542
\(315\) −7.27155 −0.409705
\(316\) 7.29697 0.410487
\(317\) 3.61056 0.202789 0.101395 0.994846i \(-0.467670\pi\)
0.101395 + 0.994846i \(0.467670\pi\)
\(318\) −4.28446 −0.240261
\(319\) 0 0
\(320\) 1.26771 0.0708670
\(321\) 2.00704 0.112022
\(322\) −7.76777 −0.432881
\(323\) −5.34329 −0.297308
\(324\) 6.61838 0.367688
\(325\) −11.6543 −0.646464
\(326\) 14.2868 0.791272
\(327\) 7.84357 0.433750
\(328\) −14.3522 −0.792469
\(329\) 13.0455 0.719224
\(330\) 0 0
\(331\) −26.5335 −1.45841 −0.729205 0.684295i \(-0.760110\pi\)
−0.729205 + 0.684295i \(0.760110\pi\)
\(332\) −1.99595 −0.109542
\(333\) −2.71092 −0.148558
\(334\) −23.8306 −1.30395
\(335\) 15.0469 0.822098
\(336\) 3.07446 0.167725
\(337\) −0.685979 −0.0373676 −0.0186838 0.999825i \(-0.505948\pi\)
−0.0186838 + 0.999825i \(0.505948\pi\)
\(338\) 9.65874 0.525366
\(339\) −11.4015 −0.619246
\(340\) 8.87387 0.481253
\(341\) 0 0
\(342\) −9.02361 −0.487941
\(343\) −1.00000 −0.0539949
\(344\) 7.03436 0.379267
\(345\) −7.50161 −0.403873
\(346\) −36.5843 −1.96678
\(347\) 21.5055 1.15447 0.577237 0.816577i \(-0.304131\pi\)
0.577237 + 0.816577i \(0.304131\pi\)
\(348\) 6.18271 0.331428
\(349\) 19.4429 1.04075 0.520377 0.853937i \(-0.325792\pi\)
0.520377 + 0.853937i \(0.325792\pi\)
\(350\) −4.82484 −0.257898
\(351\) 14.9075 0.795705
\(352\) 0 0
\(353\) −20.9307 −1.11403 −0.557015 0.830502i \(-0.688053\pi\)
−0.557015 + 0.830502i \(0.688053\pi\)
\(354\) −9.40040 −0.499625
\(355\) 5.60232 0.297340
\(356\) −17.7437 −0.940413
\(357\) 1.70303 0.0901338
\(358\) −6.60370 −0.349017
\(359\) 9.77127 0.515708 0.257854 0.966184i \(-0.416985\pi\)
0.257854 + 0.966184i \(0.416985\pi\)
\(360\) −10.8642 −0.572591
\(361\) −15.2399 −0.802100
\(362\) 8.48185 0.445796
\(363\) 0 0
\(364\) 4.97807 0.260922
\(365\) 27.6945 1.44960
\(366\) −1.08603 −0.0567678
\(367\) 9.89969 0.516759 0.258380 0.966043i \(-0.416811\pi\)
0.258380 + 0.966043i \(0.416811\pi\)
\(368\) −21.7394 −1.13324
\(369\) −25.1493 −1.30922
\(370\) −5.11210 −0.265765
\(371\) 3.90012 0.202484
\(372\) −0.143737 −0.00745242
\(373\) −4.27475 −0.221338 −0.110669 0.993857i \(-0.535299\pi\)
−0.110669 + 0.993857i \(0.535299\pi\)
\(374\) 0 0
\(375\) 3.92338 0.202603
\(376\) 19.4909 1.00516
\(377\) −37.0445 −1.90789
\(378\) 6.17167 0.317436
\(379\) 4.32594 0.222209 0.111104 0.993809i \(-0.464561\pi\)
0.111104 + 0.993809i \(0.464561\pi\)
\(380\) −6.24458 −0.320340
\(381\) 12.0574 0.617720
\(382\) −1.47382 −0.0754070
\(383\) −1.32210 −0.0675561 −0.0337781 0.999429i \(-0.510754\pi\)
−0.0337781 + 0.999429i \(0.510754\pi\)
\(384\) 6.73467 0.343677
\(385\) 0 0
\(386\) −11.9767 −0.609600
\(387\) 12.3262 0.626578
\(388\) 13.4684 0.683756
\(389\) −38.5092 −1.95249 −0.976246 0.216665i \(-0.930482\pi\)
−0.976246 + 0.216665i \(0.930482\pi\)
\(390\) 13.1002 0.663356
\(391\) −12.0420 −0.608992
\(392\) −1.49406 −0.0754616
\(393\) −4.26317 −0.215049
\(394\) −19.4129 −0.978008
\(395\) −17.4800 −0.879516
\(396\) 0 0
\(397\) −0.410109 −0.0205828 −0.0102914 0.999947i \(-0.503276\pi\)
−0.0102914 + 0.999947i \(0.503276\pi\)
\(398\) 37.2820 1.86878
\(399\) −1.19843 −0.0599964
\(400\) −13.5031 −0.675154
\(401\) −1.56684 −0.0782443 −0.0391221 0.999234i \(-0.512456\pi\)
−0.0391221 + 0.999234i \(0.512456\pi\)
\(402\) −5.95131 −0.296824
\(403\) 0.861219 0.0429004
\(404\) 3.95580 0.196808
\(405\) −15.8544 −0.787814
\(406\) −15.3363 −0.761127
\(407\) 0 0
\(408\) 2.54443 0.125968
\(409\) −6.77852 −0.335176 −0.167588 0.985857i \(-0.553598\pi\)
−0.167588 + 0.985857i \(0.553598\pi\)
\(410\) −47.4250 −2.34215
\(411\) −1.61350 −0.0795880
\(412\) −21.0818 −1.03863
\(413\) 8.55713 0.421069
\(414\) −20.3363 −0.999474
\(415\) 4.78133 0.234706
\(416\) 25.1344 1.23232
\(417\) −0.618859 −0.0303057
\(418\) 0 0
\(419\) 28.7218 1.40315 0.701577 0.712594i \(-0.252480\pi\)
0.701577 + 0.712594i \(0.252480\pi\)
\(420\) 1.99029 0.0971161
\(421\) 12.2256 0.595838 0.297919 0.954591i \(-0.403707\pi\)
0.297919 + 0.954591i \(0.403707\pi\)
\(422\) −10.7069 −0.521202
\(423\) 34.1537 1.66061
\(424\) 5.82703 0.282985
\(425\) −7.47973 −0.362820
\(426\) −2.21582 −0.107357
\(427\) 0.988609 0.0478421
\(428\) 3.76527 0.182001
\(429\) 0 0
\(430\) 23.2441 1.12093
\(431\) 3.69129 0.177803 0.0889016 0.996040i \(-0.471664\pi\)
0.0889016 + 0.996040i \(0.471664\pi\)
\(432\) 17.2724 0.831019
\(433\) 29.0749 1.39725 0.698625 0.715488i \(-0.253796\pi\)
0.698625 + 0.715488i \(0.253796\pi\)
\(434\) 0.356542 0.0171145
\(435\) −14.8108 −0.710124
\(436\) 14.7148 0.704709
\(437\) 8.47403 0.405368
\(438\) −10.9537 −0.523387
\(439\) −14.2017 −0.677811 −0.338905 0.940820i \(-0.610057\pi\)
−0.338905 + 0.940820i \(0.610057\pi\)
\(440\) 0 0
\(441\) −2.61803 −0.124668
\(442\) 21.0293 1.00026
\(443\) 27.2356 1.29400 0.647001 0.762489i \(-0.276023\pi\)
0.647001 + 0.762489i \(0.276023\pi\)
\(444\) 0.742004 0.0352140
\(445\) 42.5053 2.01495
\(446\) −8.22100 −0.389275
\(447\) −9.20547 −0.435404
\(448\) 0.456423 0.0215640
\(449\) −41.9159 −1.97813 −0.989067 0.147467i \(-0.952888\pi\)
−0.989067 + 0.147467i \(0.952888\pi\)
\(450\) −12.6316 −0.595459
\(451\) 0 0
\(452\) −21.3896 −1.00608
\(453\) 9.76777 0.458930
\(454\) 30.7511 1.44322
\(455\) −11.9251 −0.559056
\(456\) −1.79053 −0.0838491
\(457\) 19.4977 0.912063 0.456032 0.889964i \(-0.349270\pi\)
0.456032 + 0.889964i \(0.349270\pi\)
\(458\) −7.86736 −0.367617
\(459\) 9.56767 0.446580
\(460\) −14.0733 −0.656169
\(461\) 12.2251 0.569380 0.284690 0.958620i \(-0.408109\pi\)
0.284690 + 0.958620i \(0.408109\pi\)
\(462\) 0 0
\(463\) 13.8550 0.643894 0.321947 0.946758i \(-0.395663\pi\)
0.321947 + 0.946758i \(0.395663\pi\)
\(464\) −42.9211 −1.99256
\(465\) 0.344325 0.0159677
\(466\) −18.7223 −0.867294
\(467\) −16.4207 −0.759861 −0.379930 0.925015i \(-0.624052\pi\)
−0.379930 + 0.925015i \(0.624052\pi\)
\(468\) 13.0328 0.602440
\(469\) 5.41745 0.250154
\(470\) 64.4050 2.97078
\(471\) 2.83753 0.130747
\(472\) 12.7849 0.588473
\(473\) 0 0
\(474\) 6.91367 0.317555
\(475\) 5.26352 0.241507
\(476\) 3.19493 0.146439
\(477\) 10.2106 0.467513
\(478\) −17.3420 −0.793202
\(479\) −24.7914 −1.13275 −0.566374 0.824149i \(-0.691654\pi\)
−0.566374 + 0.824149i \(0.691654\pi\)
\(480\) 10.0490 0.458673
\(481\) −4.44581 −0.202711
\(482\) 22.3076 1.01608
\(483\) −2.70087 −0.122894
\(484\) 0 0
\(485\) −32.2639 −1.46503
\(486\) 24.7857 1.12430
\(487\) −14.3342 −0.649544 −0.324772 0.945792i \(-0.605288\pi\)
−0.324772 + 0.945792i \(0.605288\pi\)
\(488\) 1.47704 0.0668627
\(489\) 4.96754 0.224640
\(490\) −4.93693 −0.223028
\(491\) −22.0193 −0.993719 −0.496859 0.867831i \(-0.665514\pi\)
−0.496859 + 0.867831i \(0.665514\pi\)
\(492\) 6.88358 0.310336
\(493\) −23.7752 −1.07078
\(494\) −14.7984 −0.665810
\(495\) 0 0
\(496\) 0.997839 0.0448043
\(497\) 2.01705 0.0904770
\(498\) −1.89110 −0.0847423
\(499\) −25.3117 −1.13311 −0.566554 0.824024i \(-0.691724\pi\)
−0.566554 + 0.824024i \(0.691724\pi\)
\(500\) 7.36038 0.329166
\(501\) −8.28593 −0.370188
\(502\) −19.5367 −0.871964
\(503\) −26.0214 −1.16024 −0.580119 0.814531i \(-0.696994\pi\)
−0.580119 + 0.814531i \(0.696994\pi\)
\(504\) −3.91151 −0.174233
\(505\) −9.47619 −0.421685
\(506\) 0 0
\(507\) 3.35836 0.149150
\(508\) 22.6201 1.00360
\(509\) 38.1269 1.68994 0.844972 0.534810i \(-0.179617\pi\)
0.844972 + 0.534810i \(0.179617\pi\)
\(510\) 8.40773 0.372301
\(511\) 9.97108 0.441095
\(512\) 14.2570 0.630077
\(513\) −6.73280 −0.297261
\(514\) 9.47896 0.418099
\(515\) 50.5018 2.22538
\(516\) −3.37380 −0.148523
\(517\) 0 0
\(518\) −1.84055 −0.0808691
\(519\) −12.7204 −0.558364
\(520\) −17.8168 −0.781319
\(521\) −26.0420 −1.14092 −0.570460 0.821325i \(-0.693235\pi\)
−0.570460 + 0.821325i \(0.693235\pi\)
\(522\) −40.1509 −1.75736
\(523\) 19.1230 0.836189 0.418095 0.908403i \(-0.362698\pi\)
0.418095 + 0.908403i \(0.362698\pi\)
\(524\) −7.99784 −0.349387
\(525\) −1.67760 −0.0732166
\(526\) −14.5404 −0.633993
\(527\) 0.552731 0.0240773
\(528\) 0 0
\(529\) −3.90228 −0.169664
\(530\) 19.2546 0.836368
\(531\) 22.4029 0.972201
\(532\) −2.24828 −0.0974755
\(533\) −41.2438 −1.78647
\(534\) −16.8116 −0.727510
\(535\) −9.01977 −0.389959
\(536\) 8.09401 0.349608
\(537\) −2.29612 −0.0990848
\(538\) 22.8298 0.984265
\(539\) 0 0
\(540\) 11.1815 0.481176
\(541\) 10.8860 0.468027 0.234014 0.972233i \(-0.424814\pi\)
0.234014 + 0.972233i \(0.424814\pi\)
\(542\) −39.1886 −1.68329
\(543\) 2.94915 0.126560
\(544\) 16.1313 0.691624
\(545\) −35.2495 −1.50992
\(546\) 4.71658 0.201851
\(547\) −12.9091 −0.551951 −0.275976 0.961165i \(-0.589001\pi\)
−0.275976 + 0.961165i \(0.589001\pi\)
\(548\) −3.02697 −0.129306
\(549\) 2.58821 0.110462
\(550\) 0 0
\(551\) 16.7307 0.712751
\(552\) −4.03527 −0.171752
\(553\) −6.29348 −0.267626
\(554\) 42.7964 1.81825
\(555\) −1.77748 −0.0754500
\(556\) −1.16100 −0.0492373
\(557\) −0.762626 −0.0323135 −0.0161567 0.999869i \(-0.505143\pi\)
−0.0161567 + 0.999869i \(0.505143\pi\)
\(558\) 0.933438 0.0395156
\(559\) 20.2146 0.854985
\(560\) −13.8168 −0.583867
\(561\) 0 0
\(562\) 27.4476 1.15781
\(563\) −22.2281 −0.936803 −0.468402 0.883516i \(-0.655170\pi\)
−0.468402 + 0.883516i \(0.655170\pi\)
\(564\) −9.34817 −0.393629
\(565\) 51.2392 2.15565
\(566\) −5.96758 −0.250836
\(567\) −5.70820 −0.239722
\(568\) 3.01360 0.126448
\(569\) −20.5131 −0.859955 −0.429978 0.902839i \(-0.641479\pi\)
−0.429978 + 0.902839i \(0.641479\pi\)
\(570\) −5.91656 −0.247817
\(571\) −19.5654 −0.818785 −0.409393 0.912358i \(-0.634259\pi\)
−0.409393 + 0.912358i \(0.634259\pi\)
\(572\) 0 0
\(573\) −0.512448 −0.0214078
\(574\) −17.0748 −0.712688
\(575\) 11.8623 0.494691
\(576\) 1.19493 0.0497888
\(577\) −19.5716 −0.814776 −0.407388 0.913255i \(-0.633560\pi\)
−0.407388 + 0.913255i \(0.633560\pi\)
\(578\) −16.7206 −0.695487
\(579\) −4.16433 −0.173064
\(580\) −27.7855 −1.15373
\(581\) 1.72146 0.0714182
\(582\) 12.7609 0.528958
\(583\) 0 0
\(584\) 14.8974 0.616460
\(585\) −31.2202 −1.29080
\(586\) 12.1528 0.502026
\(587\) 1.57068 0.0648290 0.0324145 0.999475i \(-0.489680\pi\)
0.0324145 + 0.999475i \(0.489680\pi\)
\(588\) 0.716580 0.0295512
\(589\) −0.388959 −0.0160268
\(590\) 42.2460 1.73924
\(591\) −6.74990 −0.277654
\(592\) −5.15107 −0.211708
\(593\) −30.1230 −1.23700 −0.618502 0.785783i \(-0.712260\pi\)
−0.618502 + 0.785783i \(0.712260\pi\)
\(594\) 0 0
\(595\) −7.65351 −0.313763
\(596\) −17.2697 −0.707396
\(597\) 12.9630 0.530540
\(598\) −33.3507 −1.36381
\(599\) 5.77792 0.236079 0.118040 0.993009i \(-0.462339\pi\)
0.118040 + 0.993009i \(0.462339\pi\)
\(600\) −2.50645 −0.102325
\(601\) 45.5645 1.85862 0.929308 0.369305i \(-0.120404\pi\)
0.929308 + 0.369305i \(0.120404\pi\)
\(602\) 8.36876 0.341085
\(603\) 14.1831 0.577579
\(604\) 18.3246 0.745619
\(605\) 0 0
\(606\) 3.74801 0.152252
\(607\) 34.7211 1.40928 0.704642 0.709563i \(-0.251108\pi\)
0.704642 + 0.709563i \(0.251108\pi\)
\(608\) −11.3517 −0.460371
\(609\) −5.33245 −0.216082
\(610\) 4.88070 0.197614
\(611\) 56.0107 2.26595
\(612\) 8.36444 0.338112
\(613\) −24.1423 −0.975097 −0.487548 0.873096i \(-0.662109\pi\)
−0.487548 + 0.873096i \(0.662109\pi\)
\(614\) −20.9690 −0.846242
\(615\) −16.4897 −0.664931
\(616\) 0 0
\(617\) −13.4967 −0.543358 −0.271679 0.962388i \(-0.587579\pi\)
−0.271679 + 0.962388i \(0.587579\pi\)
\(618\) −19.9744 −0.803488
\(619\) −43.4856 −1.74783 −0.873917 0.486074i \(-0.838428\pi\)
−0.873917 + 0.486074i \(0.838428\pi\)
\(620\) 0.645964 0.0259425
\(621\) −15.1736 −0.608894
\(622\) 44.9950 1.80414
\(623\) 15.3035 0.613123
\(624\) 13.2001 0.528427
\(625\) −31.2040 −1.24816
\(626\) −34.9353 −1.39629
\(627\) 0 0
\(628\) 5.32330 0.212423
\(629\) −2.85332 −0.113769
\(630\) −12.9251 −0.514947
\(631\) 6.42012 0.255581 0.127790 0.991801i \(-0.459212\pi\)
0.127790 + 0.991801i \(0.459212\pi\)
\(632\) −9.40286 −0.374026
\(633\) −3.72279 −0.147968
\(634\) 6.41771 0.254880
\(635\) −54.1868 −2.15034
\(636\) −2.79475 −0.110819
\(637\) −4.29348 −0.170114
\(638\) 0 0
\(639\) 5.28070 0.208901
\(640\) −30.2660 −1.19637
\(641\) −6.39600 −0.252627 −0.126313 0.991990i \(-0.540314\pi\)
−0.126313 + 0.991990i \(0.540314\pi\)
\(642\) 3.56748 0.140797
\(643\) 0.652660 0.0257384 0.0128692 0.999917i \(-0.495903\pi\)
0.0128692 + 0.999917i \(0.495903\pi\)
\(644\) −5.06691 −0.199664
\(645\) 8.08201 0.318229
\(646\) −9.49761 −0.373678
\(647\) 17.9621 0.706165 0.353082 0.935592i \(-0.385134\pi\)
0.353082 + 0.935592i \(0.385134\pi\)
\(648\) −8.52842 −0.335028
\(649\) 0 0
\(650\) −20.7153 −0.812522
\(651\) 0.123970 0.00485877
\(652\) 9.31925 0.364970
\(653\) −16.9181 −0.662055 −0.331028 0.943621i \(-0.607395\pi\)
−0.331028 + 0.943621i \(0.607395\pi\)
\(654\) 13.9418 0.545168
\(655\) 19.1590 0.748603
\(656\) −47.7866 −1.86575
\(657\) 26.1046 1.01844
\(658\) 23.1882 0.903972
\(659\) 23.6249 0.920297 0.460148 0.887842i \(-0.347796\pi\)
0.460148 + 0.887842i \(0.347796\pi\)
\(660\) 0 0
\(661\) −20.9819 −0.816103 −0.408051 0.912959i \(-0.633792\pi\)
−0.408051 + 0.912959i \(0.633792\pi\)
\(662\) −47.1628 −1.83303
\(663\) 7.31191 0.283971
\(664\) 2.57197 0.0998118
\(665\) 5.38581 0.208853
\(666\) −4.81862 −0.186718
\(667\) 37.7056 1.45997
\(668\) −15.5446 −0.601440
\(669\) −2.85845 −0.110514
\(670\) 26.7456 1.03327
\(671\) 0 0
\(672\) 3.61803 0.139569
\(673\) 12.0986 0.466368 0.233184 0.972433i \(-0.425086\pi\)
0.233184 + 0.972433i \(0.425086\pi\)
\(674\) −1.21932 −0.0469663
\(675\) −9.42483 −0.362762
\(676\) 6.30038 0.242322
\(677\) −9.42988 −0.362420 −0.181210 0.983444i \(-0.558001\pi\)
−0.181210 + 0.983444i \(0.558001\pi\)
\(678\) −20.2660 −0.778312
\(679\) −11.6162 −0.445790
\(680\) −11.4348 −0.438506
\(681\) 10.6922 0.409726
\(682\) 0 0
\(683\) −15.2986 −0.585385 −0.292692 0.956207i \(-0.594551\pi\)
−0.292692 + 0.956207i \(0.594551\pi\)
\(684\) −5.88609 −0.225060
\(685\) 7.25117 0.277053
\(686\) −1.77748 −0.0678647
\(687\) −2.73549 −0.104366
\(688\) 23.4213 0.892929
\(689\) 16.7451 0.637936
\(690\) −13.3340 −0.507617
\(691\) 22.8193 0.868086 0.434043 0.900892i \(-0.357087\pi\)
0.434043 + 0.900892i \(0.357087\pi\)
\(692\) −23.8639 −0.907169
\(693\) 0 0
\(694\) 38.2256 1.45102
\(695\) 2.78119 0.105497
\(696\) −7.96703 −0.301989
\(697\) −26.4703 −1.00263
\(698\) 34.5594 1.30809
\(699\) −6.50978 −0.246222
\(700\) −3.14723 −0.118954
\(701\) −32.3242 −1.22087 −0.610433 0.792068i \(-0.709005\pi\)
−0.610433 + 0.792068i \(0.709005\pi\)
\(702\) 26.4979 1.00010
\(703\) 2.00789 0.0757292
\(704\) 0 0
\(705\) 22.3937 0.843396
\(706\) −37.2040 −1.40019
\(707\) −3.41179 −0.128314
\(708\) −6.13186 −0.230450
\(709\) 14.5598 0.546804 0.273402 0.961900i \(-0.411851\pi\)
0.273402 + 0.961900i \(0.411851\pi\)
\(710\) 9.95804 0.373718
\(711\) −16.4765 −0.617918
\(712\) 22.8645 0.856882
\(713\) −0.876587 −0.0328285
\(714\) 3.02710 0.113287
\(715\) 0 0
\(716\) −4.30759 −0.160982
\(717\) −6.02982 −0.225188
\(718\) 17.3683 0.648178
\(719\) 44.8602 1.67300 0.836501 0.547965i \(-0.184597\pi\)
0.836501 + 0.547965i \(0.184597\pi\)
\(720\) −36.1729 −1.34808
\(721\) 18.1826 0.677155
\(722\) −27.0887 −1.00814
\(723\) 7.75638 0.288463
\(724\) 5.53270 0.205621
\(725\) 23.4202 0.869806
\(726\) 0 0
\(727\) 28.3582 1.05175 0.525874 0.850562i \(-0.323738\pi\)
0.525874 + 0.850562i \(0.323738\pi\)
\(728\) −6.41473 −0.237746
\(729\) −8.50658 −0.315058
\(730\) 49.2266 1.82196
\(731\) 12.9737 0.479850
\(732\) −0.708417 −0.0261838
\(733\) 6.25696 0.231106 0.115553 0.993301i \(-0.463136\pi\)
0.115553 + 0.993301i \(0.463136\pi\)
\(734\) 17.5965 0.649500
\(735\) −1.71658 −0.0633170
\(736\) −25.5830 −0.943001
\(737\) 0 0
\(738\) −44.7024 −1.64552
\(739\) 9.35201 0.344019 0.172010 0.985095i \(-0.444974\pi\)
0.172010 + 0.985095i \(0.444974\pi\)
\(740\) −3.33461 −0.122583
\(741\) −5.14542 −0.189022
\(742\) 6.93240 0.254496
\(743\) 25.2066 0.924740 0.462370 0.886687i \(-0.346999\pi\)
0.462370 + 0.886687i \(0.346999\pi\)
\(744\) 0.185219 0.00679047
\(745\) 41.3700 1.51568
\(746\) −7.59830 −0.278193
\(747\) 4.50684 0.164897
\(748\) 0 0
\(749\) −3.24746 −0.118660
\(750\) 6.97375 0.254645
\(751\) −34.7493 −1.26802 −0.634010 0.773325i \(-0.718592\pi\)
−0.634010 + 0.773325i \(0.718592\pi\)
\(752\) 64.8960 2.36651
\(753\) −6.79293 −0.247548
\(754\) −65.8460 −2.39797
\(755\) −43.8970 −1.59758
\(756\) 4.02577 0.146416
\(757\) 34.7960 1.26468 0.632341 0.774690i \(-0.282094\pi\)
0.632341 + 0.774690i \(0.282094\pi\)
\(758\) 7.68929 0.279288
\(759\) 0 0
\(760\) 8.04674 0.291886
\(761\) −12.2255 −0.443172 −0.221586 0.975141i \(-0.571123\pi\)
−0.221586 + 0.975141i \(0.571123\pi\)
\(762\) 21.4319 0.776394
\(763\) −12.6912 −0.459451
\(764\) −0.961368 −0.0347811
\(765\) −20.0372 −0.724445
\(766\) −2.35001 −0.0849094
\(767\) 36.7398 1.32660
\(768\) 12.5349 0.452315
\(769\) −2.61946 −0.0944603 −0.0472301 0.998884i \(-0.515039\pi\)
−0.0472301 + 0.998884i \(0.515039\pi\)
\(770\) 0 0
\(771\) 3.29585 0.118697
\(772\) −7.81241 −0.281175
\(773\) −0.212804 −0.00765405 −0.00382702 0.999993i \(-0.501218\pi\)
−0.00382702 + 0.999993i \(0.501218\pi\)
\(774\) 21.9097 0.787528
\(775\) −0.544479 −0.0195583
\(776\) −17.3554 −0.623022
\(777\) −0.639962 −0.0229585
\(778\) −68.4494 −2.45403
\(779\) 18.6273 0.667391
\(780\) 8.54526 0.305969
\(781\) 0 0
\(782\) −21.4045 −0.765425
\(783\) −29.9579 −1.07061
\(784\) −4.97458 −0.177663
\(785\) −12.7521 −0.455141
\(786\) −7.57772 −0.270288
\(787\) 29.1375 1.03864 0.519320 0.854580i \(-0.326185\pi\)
0.519320 + 0.854580i \(0.326185\pi\)
\(788\) −12.6630 −0.451101
\(789\) −5.05573 −0.179989
\(790\) −31.0705 −1.10544
\(791\) 18.4480 0.655937
\(792\) 0 0
\(793\) 4.24457 0.150729
\(794\) −0.728962 −0.0258699
\(795\) 6.69486 0.237442
\(796\) 24.3190 0.861963
\(797\) 32.2284 1.14159 0.570794 0.821093i \(-0.306635\pi\)
0.570794 + 0.821093i \(0.306635\pi\)
\(798\) −2.13018 −0.0754077
\(799\) 35.9477 1.27174
\(800\) −15.8905 −0.561813
\(801\) 40.0652 1.41563
\(802\) −2.78503 −0.0983430
\(803\) 0 0
\(804\) −3.88203 −0.136909
\(805\) 12.1379 0.427804
\(806\) 1.53080 0.0539202
\(807\) 7.93797 0.279430
\(808\) −5.09743 −0.179327
\(809\) −29.3461 −1.03175 −0.515877 0.856663i \(-0.672534\pi\)
−0.515877 + 0.856663i \(0.672534\pi\)
\(810\) −28.1810 −0.990180
\(811\) 1.03940 0.0364982 0.0182491 0.999833i \(-0.494191\pi\)
0.0182491 + 0.999833i \(0.494191\pi\)
\(812\) −10.0038 −0.351066
\(813\) −13.6259 −0.477882
\(814\) 0 0
\(815\) −22.3244 −0.781990
\(816\) 8.47184 0.296574
\(817\) −9.12966 −0.319406
\(818\) −12.0487 −0.421274
\(819\) −11.2405 −0.392774
\(820\) −30.9352 −1.08031
\(821\) 28.5323 0.995786 0.497893 0.867239i \(-0.334107\pi\)
0.497893 + 0.867239i \(0.334107\pi\)
\(822\) −2.86797 −0.100032
\(823\) 26.6577 0.929227 0.464614 0.885513i \(-0.346193\pi\)
0.464614 + 0.885513i \(0.346193\pi\)
\(824\) 27.1659 0.946370
\(825\) 0 0
\(826\) 15.2102 0.529229
\(827\) 1.71964 0.0597978 0.0298989 0.999553i \(-0.490481\pi\)
0.0298989 + 0.999553i \(0.490481\pi\)
\(828\) −13.2653 −0.461002
\(829\) 27.9605 0.971107 0.485554 0.874207i \(-0.338618\pi\)
0.485554 + 0.874207i \(0.338618\pi\)
\(830\) 8.49873 0.294995
\(831\) 14.8804 0.516195
\(832\) 1.95964 0.0679383
\(833\) −2.75556 −0.0954744
\(834\) −1.10001 −0.0380903
\(835\) 37.2375 1.28866
\(836\) 0 0
\(837\) 0.696468 0.0240735
\(838\) 51.0526 1.76358
\(839\) 35.8624 1.23811 0.619054 0.785348i \(-0.287516\pi\)
0.619054 + 0.785348i \(0.287516\pi\)
\(840\) −2.56468 −0.0884899
\(841\) 45.4439 1.56703
\(842\) 21.7308 0.748892
\(843\) 9.54358 0.328698
\(844\) −6.98407 −0.240402
\(845\) −15.0927 −0.519204
\(846\) 60.7076 2.08717
\(847\) 0 0
\(848\) 19.4014 0.666248
\(849\) −2.07494 −0.0712117
\(850\) −13.2951 −0.456018
\(851\) 4.52515 0.155120
\(852\) −1.44538 −0.0495178
\(853\) −15.1427 −0.518478 −0.259239 0.965813i \(-0.583472\pi\)
−0.259239 + 0.965813i \(0.583472\pi\)
\(854\) 1.75724 0.0601314
\(855\) 14.1002 0.482218
\(856\) −4.85191 −0.165835
\(857\) 25.3267 0.865142 0.432571 0.901600i \(-0.357606\pi\)
0.432571 + 0.901600i \(0.357606\pi\)
\(858\) 0 0
\(859\) 41.5291 1.41696 0.708478 0.705733i \(-0.249382\pi\)
0.708478 + 0.705733i \(0.249382\pi\)
\(860\) 15.1621 0.517023
\(861\) −5.93693 −0.202330
\(862\) 6.56121 0.223476
\(863\) −24.0504 −0.818684 −0.409342 0.912381i \(-0.634242\pi\)
−0.409342 + 0.912381i \(0.634242\pi\)
\(864\) 20.3262 0.691513
\(865\) 57.1664 1.94372
\(866\) 51.6801 1.75616
\(867\) −5.81379 −0.197447
\(868\) 0.232572 0.00789399
\(869\) 0 0
\(870\) −26.3260 −0.892534
\(871\) 23.2597 0.788124
\(872\) −18.9614 −0.642114
\(873\) −30.4117 −1.02928
\(874\) 15.0625 0.509495
\(875\) −6.34817 −0.214607
\(876\) −7.14507 −0.241410
\(877\) 33.8052 1.14152 0.570760 0.821117i \(-0.306649\pi\)
0.570760 + 0.821117i \(0.306649\pi\)
\(878\) −25.2433 −0.851921
\(879\) 4.22553 0.142524
\(880\) 0 0
\(881\) −36.8296 −1.24082 −0.620410 0.784278i \(-0.713034\pi\)
−0.620410 + 0.784278i \(0.713034\pi\)
\(882\) −4.65351 −0.156692
\(883\) 53.4103 1.79740 0.898701 0.438563i \(-0.144512\pi\)
0.898701 + 0.438563i \(0.144512\pi\)
\(884\) 13.7174 0.461365
\(885\) 14.6890 0.493765
\(886\) 48.4108 1.62639
\(887\) 11.4616 0.384843 0.192421 0.981312i \(-0.438366\pi\)
0.192421 + 0.981312i \(0.438366\pi\)
\(888\) −0.956144 −0.0320861
\(889\) −19.5093 −0.654321
\(890\) 75.5525 2.53253
\(891\) 0 0
\(892\) −5.36254 −0.179551
\(893\) −25.2965 −0.846516
\(894\) −16.3626 −0.547246
\(895\) 10.3189 0.344923
\(896\) −10.8969 −0.364041
\(897\) −11.5961 −0.387183
\(898\) −74.5049 −2.48626
\(899\) −1.73069 −0.0577217
\(900\) −8.23956 −0.274652
\(901\) 10.7470 0.358034
\(902\) 0 0
\(903\) 2.90983 0.0968331
\(904\) 27.5626 0.916717
\(905\) −13.2537 −0.440567
\(906\) 17.3621 0.576816
\(907\) 57.0582 1.89459 0.947293 0.320369i \(-0.103807\pi\)
0.947293 + 0.320369i \(0.103807\pi\)
\(908\) 20.0589 0.665677
\(909\) −8.93218 −0.296262
\(910\) −21.1966 −0.702661
\(911\) 6.67566 0.221175 0.110587 0.993866i \(-0.464727\pi\)
0.110587 + 0.993866i \(0.464727\pi\)
\(912\) −5.96167 −0.197410
\(913\) 0 0
\(914\) 34.6568 1.14635
\(915\) 1.69703 0.0561019
\(916\) −5.13186 −0.169562
\(917\) 6.89796 0.227791
\(918\) 17.0064 0.561294
\(919\) −29.1339 −0.961038 −0.480519 0.876984i \(-0.659552\pi\)
−0.480519 + 0.876984i \(0.659552\pi\)
\(920\) 18.1347 0.597885
\(921\) −7.29097 −0.240246
\(922\) 21.7299 0.715637
\(923\) 8.66015 0.285052
\(924\) 0 0
\(925\) 2.81073 0.0924161
\(926\) 24.6270 0.809292
\(927\) 47.6026 1.56347
\(928\) −50.5097 −1.65806
\(929\) 2.34472 0.0769277 0.0384638 0.999260i \(-0.487754\pi\)
0.0384638 + 0.999260i \(0.487754\pi\)
\(930\) 0.612032 0.0200693
\(931\) 1.93910 0.0635513
\(932\) −12.2125 −0.400035
\(933\) 15.6448 0.512189
\(934\) −29.1876 −0.955047
\(935\) 0 0
\(936\) −16.7940 −0.548928
\(937\) 8.78780 0.287085 0.143542 0.989644i \(-0.454151\pi\)
0.143542 + 0.989644i \(0.454151\pi\)
\(938\) 9.62943 0.314412
\(939\) −12.1470 −0.396404
\(940\) 42.0113 1.37026
\(941\) 8.88871 0.289764 0.144882 0.989449i \(-0.453720\pi\)
0.144882 + 0.989449i \(0.453720\pi\)
\(942\) 5.04367 0.164332
\(943\) 41.9798 1.36705
\(944\) 42.5681 1.38547
\(945\) −9.64380 −0.313713
\(946\) 0 0
\(947\) −7.86275 −0.255505 −0.127752 0.991806i \(-0.540776\pi\)
−0.127752 + 0.991806i \(0.540776\pi\)
\(948\) 4.50978 0.146471
\(949\) 42.8106 1.38969
\(950\) 9.35582 0.303543
\(951\) 2.23145 0.0723597
\(952\) −4.11698 −0.133432
\(953\) −17.4644 −0.565727 −0.282864 0.959160i \(-0.591284\pi\)
−0.282864 + 0.959160i \(0.591284\pi\)
\(954\) 18.1493 0.587604
\(955\) 2.30297 0.0745225
\(956\) −11.3121 −0.365860
\(957\) 0 0
\(958\) −44.0663 −1.42372
\(959\) 2.61070 0.0843038
\(960\) 0.783486 0.0252869
\(961\) −30.9598 −0.998702
\(962\) −7.90236 −0.254782
\(963\) −8.50196 −0.273972
\(964\) 14.5512 0.468663
\(965\) 18.7148 0.602450
\(966\) −4.80075 −0.154462
\(967\) −45.6122 −1.46679 −0.733395 0.679802i \(-0.762066\pi\)
−0.733395 + 0.679802i \(0.762066\pi\)
\(968\) 0 0
\(969\) −3.30233 −0.106086
\(970\) −57.3485 −1.84135
\(971\) 2.38378 0.0764992 0.0382496 0.999268i \(-0.487822\pi\)
0.0382496 + 0.999268i \(0.487822\pi\)
\(972\) 16.1677 0.518579
\(973\) 1.00134 0.0321013
\(974\) −25.4788 −0.816393
\(975\) −7.20275 −0.230673
\(976\) 4.91791 0.157418
\(977\) −19.2662 −0.616380 −0.308190 0.951325i \(-0.599723\pi\)
−0.308190 + 0.951325i \(0.599723\pi\)
\(978\) 8.82972 0.282343
\(979\) 0 0
\(980\) −3.22035 −0.102870
\(981\) −33.2259 −1.06082
\(982\) −39.1390 −1.24898
\(983\) −16.2256 −0.517517 −0.258759 0.965942i \(-0.583313\pi\)
−0.258759 + 0.965942i \(0.583313\pi\)
\(984\) −8.87016 −0.282770
\(985\) 30.3345 0.966537
\(986\) −42.2600 −1.34583
\(987\) 8.06258 0.256635
\(988\) −9.65296 −0.307101
\(989\) −20.5753 −0.654256
\(990\) 0 0
\(991\) −50.5214 −1.60487 −0.802433 0.596743i \(-0.796461\pi\)
−0.802433 + 0.596743i \(0.796461\pi\)
\(992\) 1.17426 0.0372828
\(993\) −16.3986 −0.520393
\(994\) 3.58527 0.113718
\(995\) −58.2566 −1.84686
\(996\) −1.23356 −0.0390869
\(997\) −45.0206 −1.42582 −0.712909 0.701257i \(-0.752623\pi\)
−0.712909 + 0.701257i \(0.752623\pi\)
\(998\) −44.9912 −1.42417
\(999\) −3.59533 −0.113751
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 847.2.a.l.1.3 4
3.2 odd 2 7623.2.a.ch.1.2 4
7.6 odd 2 5929.2.a.bi.1.3 4
11.2 odd 10 847.2.f.s.323.2 8
11.3 even 5 77.2.f.a.64.2 8
11.4 even 5 77.2.f.a.71.2 yes 8
11.5 even 5 847.2.f.p.729.1 8
11.6 odd 10 847.2.f.s.729.2 8
11.7 odd 10 847.2.f.q.148.1 8
11.8 odd 10 847.2.f.q.372.1 8
11.9 even 5 847.2.f.p.323.1 8
11.10 odd 2 847.2.a.k.1.2 4
33.14 odd 10 693.2.m.g.64.1 8
33.26 odd 10 693.2.m.g.379.1 8
33.32 even 2 7623.2.a.co.1.3 4
77.3 odd 30 539.2.q.b.471.2 16
77.4 even 15 539.2.q.c.324.1 16
77.25 even 15 539.2.q.c.471.2 16
77.26 odd 30 539.2.q.b.214.2 16
77.37 even 15 539.2.q.c.214.2 16
77.47 odd 30 539.2.q.b.361.1 16
77.48 odd 10 539.2.f.d.148.2 8
77.58 even 15 539.2.q.c.361.1 16
77.59 odd 30 539.2.q.b.324.1 16
77.69 odd 10 539.2.f.d.295.2 8
77.76 even 2 5929.2.a.bb.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.a.64.2 8 11.3 even 5
77.2.f.a.71.2 yes 8 11.4 even 5
539.2.f.d.148.2 8 77.48 odd 10
539.2.f.d.295.2 8 77.69 odd 10
539.2.q.b.214.2 16 77.26 odd 30
539.2.q.b.324.1 16 77.59 odd 30
539.2.q.b.361.1 16 77.47 odd 30
539.2.q.b.471.2 16 77.3 odd 30
539.2.q.c.214.2 16 77.37 even 15
539.2.q.c.324.1 16 77.4 even 15
539.2.q.c.361.1 16 77.58 even 15
539.2.q.c.471.2 16 77.25 even 15
693.2.m.g.64.1 8 33.14 odd 10
693.2.m.g.379.1 8 33.26 odd 10
847.2.a.k.1.2 4 11.10 odd 2
847.2.a.l.1.3 4 1.1 even 1 trivial
847.2.f.p.323.1 8 11.9 even 5
847.2.f.p.729.1 8 11.5 even 5
847.2.f.q.148.1 8 11.7 odd 10
847.2.f.q.372.1 8 11.8 odd 10
847.2.f.s.323.2 8 11.2 odd 10
847.2.f.s.729.2 8 11.6 odd 10
5929.2.a.bb.1.2 4 77.76 even 2
5929.2.a.bi.1.3 4 7.6 odd 2
7623.2.a.ch.1.2 4 3.2 odd 2
7623.2.a.co.1.3 4 33.32 even 2