Properties

Label 4031.2.a.c.1.38
Level $4031$
Weight $2$
Character 4031.1
Self dual yes
Analytic conductor $32.188$
Analytic rank $1$
Dimension $61$
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] = [4031,2,Mod(1,4031)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4031, 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("4031.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4031 = 29 \cdot 139 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4031.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: \(32.1876970548\)
Analytic rank: \(1\)
Dimension: \(61\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.38
Character \(\chi\) \(=\) 4031.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.663063 q^{2} -2.54639 q^{3} -1.56035 q^{4} -2.36647 q^{5} -1.68842 q^{6} -2.65587 q^{7} -2.36073 q^{8} +3.48413 q^{9} +O(q^{10})\) \(q+0.663063 q^{2} -2.54639 q^{3} -1.56035 q^{4} -2.36647 q^{5} -1.68842 q^{6} -2.65587 q^{7} -2.36073 q^{8} +3.48413 q^{9} -1.56912 q^{10} -4.08058 q^{11} +3.97326 q^{12} +4.38872 q^{13} -1.76101 q^{14} +6.02597 q^{15} +1.55538 q^{16} -3.17585 q^{17} +2.31020 q^{18} -2.20707 q^{19} +3.69252 q^{20} +6.76289 q^{21} -2.70568 q^{22} +4.41820 q^{23} +6.01136 q^{24} +0.600182 q^{25} +2.91000 q^{26} -1.23278 q^{27} +4.14408 q^{28} -1.00000 q^{29} +3.99560 q^{30} +8.62167 q^{31} +5.75278 q^{32} +10.3908 q^{33} -2.10579 q^{34} +6.28503 q^{35} -5.43645 q^{36} +3.48283 q^{37} -1.46343 q^{38} -11.1754 q^{39} +5.58661 q^{40} -1.04873 q^{41} +4.48422 q^{42} -2.02748 q^{43} +6.36712 q^{44} -8.24508 q^{45} +2.92955 q^{46} -1.01270 q^{47} -3.96061 q^{48} +0.0536357 q^{49} +0.397958 q^{50} +8.08697 q^{51} -6.84793 q^{52} -3.40628 q^{53} -0.817409 q^{54} +9.65657 q^{55} +6.26980 q^{56} +5.62007 q^{57} -0.663063 q^{58} +11.4130 q^{59} -9.40260 q^{60} +3.09165 q^{61} +5.71671 q^{62} -9.25338 q^{63} +0.703698 q^{64} -10.3858 q^{65} +6.88973 q^{66} +4.87423 q^{67} +4.95543 q^{68} -11.2505 q^{69} +4.16737 q^{70} +8.87674 q^{71} -8.22510 q^{72} -7.51540 q^{73} +2.30934 q^{74} -1.52830 q^{75} +3.44380 q^{76} +10.8375 q^{77} -7.41000 q^{78} -4.88878 q^{79} -3.68076 q^{80} -7.31324 q^{81} -0.695372 q^{82} -8.13823 q^{83} -10.5525 q^{84} +7.51556 q^{85} -1.34434 q^{86} +2.54639 q^{87} +9.63317 q^{88} +10.3070 q^{89} -5.46701 q^{90} -11.6559 q^{91} -6.89393 q^{92} -21.9542 q^{93} -0.671485 q^{94} +5.22297 q^{95} -14.6489 q^{96} -19.4873 q^{97} +0.0355639 q^{98} -14.2173 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.663063 0.468856 0.234428 0.972133i \(-0.424678\pi\)
0.234428 + 0.972133i \(0.424678\pi\)
\(3\) −2.54639 −1.47016 −0.735081 0.677979i \(-0.762856\pi\)
−0.735081 + 0.677979i \(0.762856\pi\)
\(4\) −1.56035 −0.780174
\(5\) −2.36647 −1.05832 −0.529159 0.848523i \(-0.677492\pi\)
−0.529159 + 0.848523i \(0.677492\pi\)
\(6\) −1.68842 −0.689295
\(7\) −2.65587 −1.00382 −0.501912 0.864919i \(-0.667370\pi\)
−0.501912 + 0.864919i \(0.667370\pi\)
\(8\) −2.36073 −0.834646
\(9\) 3.48413 1.16138
\(10\) −1.56912 −0.496199
\(11\) −4.08058 −1.23034 −0.615171 0.788394i \(-0.710913\pi\)
−0.615171 + 0.788394i \(0.710913\pi\)
\(12\) 3.97326 1.14698
\(13\) 4.38872 1.21721 0.608606 0.793473i \(-0.291729\pi\)
0.608606 + 0.793473i \(0.291729\pi\)
\(14\) −1.76101 −0.470649
\(15\) 6.02597 1.55590
\(16\) 1.55538 0.388845
\(17\) −3.17585 −0.770257 −0.385128 0.922863i \(-0.625843\pi\)
−0.385128 + 0.922863i \(0.625843\pi\)
\(18\) 2.31020 0.544518
\(19\) −2.20707 −0.506337 −0.253168 0.967422i \(-0.581473\pi\)
−0.253168 + 0.967422i \(0.581473\pi\)
\(20\) 3.69252 0.825672
\(21\) 6.76289 1.47578
\(22\) −2.70568 −0.576853
\(23\) 4.41820 0.921259 0.460629 0.887593i \(-0.347624\pi\)
0.460629 + 0.887593i \(0.347624\pi\)
\(24\) 6.01136 1.22706
\(25\) 0.600182 0.120036
\(26\) 2.91000 0.570697
\(27\) −1.23278 −0.237248
\(28\) 4.14408 0.783157
\(29\) −1.00000 −0.185695
\(30\) 3.99560 0.729493
\(31\) 8.62167 1.54850 0.774249 0.632881i \(-0.218128\pi\)
0.774249 + 0.632881i \(0.218128\pi\)
\(32\) 5.75278 1.01696
\(33\) 10.3908 1.80880
\(34\) −2.10579 −0.361140
\(35\) 6.28503 1.06236
\(36\) −5.43645 −0.906075
\(37\) 3.48283 0.572574 0.286287 0.958144i \(-0.407579\pi\)
0.286287 + 0.958144i \(0.407579\pi\)
\(38\) −1.46343 −0.237399
\(39\) −11.1754 −1.78950
\(40\) 5.58661 0.883320
\(41\) −1.04873 −0.163784 −0.0818918 0.996641i \(-0.526096\pi\)
−0.0818918 + 0.996641i \(0.526096\pi\)
\(42\) 4.48422 0.691930
\(43\) −2.02748 −0.309187 −0.154594 0.987978i \(-0.549407\pi\)
−0.154594 + 0.987978i \(0.549407\pi\)
\(44\) 6.36712 0.959880
\(45\) −8.24508 −1.22910
\(46\) 2.92955 0.431938
\(47\) −1.01270 −0.147718 −0.0738589 0.997269i \(-0.523531\pi\)
−0.0738589 + 0.997269i \(0.523531\pi\)
\(48\) −3.96061 −0.571665
\(49\) 0.0536357 0.00766225
\(50\) 0.397958 0.0562798
\(51\) 8.08697 1.13240
\(52\) −6.84793 −0.949637
\(53\) −3.40628 −0.467888 −0.233944 0.972250i \(-0.575163\pi\)
−0.233944 + 0.972250i \(0.575163\pi\)
\(54\) −0.817409 −0.111235
\(55\) 9.65657 1.30209
\(56\) 6.26980 0.837837
\(57\) 5.62007 0.744397
\(58\) −0.663063 −0.0870644
\(59\) 11.4130 1.48585 0.742924 0.669376i \(-0.233439\pi\)
0.742924 + 0.669376i \(0.233439\pi\)
\(60\) −9.40260 −1.21387
\(61\) 3.09165 0.395845 0.197922 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(62\) 5.71671 0.726023
\(63\) −9.25338 −1.16582
\(64\) 0.703698 0.0879623
\(65\) −10.3858 −1.28820
\(66\) 6.88973 0.848068
\(67\) 4.87423 0.595482 0.297741 0.954647i \(-0.403767\pi\)
0.297741 + 0.954647i \(0.403767\pi\)
\(68\) 4.95543 0.600934
\(69\) −11.2505 −1.35440
\(70\) 4.16737 0.498096
\(71\) 8.87674 1.05347 0.526737 0.850028i \(-0.323415\pi\)
0.526737 + 0.850028i \(0.323415\pi\)
\(72\) −8.22510 −0.969337
\(73\) −7.51540 −0.879611 −0.439806 0.898093i \(-0.644953\pi\)
−0.439806 + 0.898093i \(0.644953\pi\)
\(74\) 2.30934 0.268455
\(75\) −1.52830 −0.176473
\(76\) 3.44380 0.395030
\(77\) 10.8375 1.23505
\(78\) −7.41000 −0.839017
\(79\) −4.88878 −0.550030 −0.275015 0.961440i \(-0.588683\pi\)
−0.275015 + 0.961440i \(0.588683\pi\)
\(80\) −3.68076 −0.411521
\(81\) −7.31324 −0.812582
\(82\) −0.695372 −0.0767910
\(83\) −8.13823 −0.893287 −0.446644 0.894712i \(-0.647381\pi\)
−0.446644 + 0.894712i \(0.647381\pi\)
\(84\) −10.5525 −1.15137
\(85\) 7.51556 0.815176
\(86\) −1.34434 −0.144964
\(87\) 2.54639 0.273002
\(88\) 9.63317 1.02690
\(89\) 10.3070 1.09254 0.546269 0.837610i \(-0.316048\pi\)
0.546269 + 0.837610i \(0.316048\pi\)
\(90\) −5.46701 −0.576273
\(91\) −11.6559 −1.22187
\(92\) −6.89393 −0.718742
\(93\) −21.9542 −2.27654
\(94\) −0.671485 −0.0692584
\(95\) 5.22297 0.535865
\(96\) −14.6489 −1.49509
\(97\) −19.4873 −1.97863 −0.989317 0.145781i \(-0.953431\pi\)
−0.989317 + 0.145781i \(0.953431\pi\)
\(98\) 0.0355639 0.00359249
\(99\) −14.2173 −1.42889
\(100\) −0.936492 −0.0936492
\(101\) −2.76890 −0.275516 −0.137758 0.990466i \(-0.543990\pi\)
−0.137758 + 0.990466i \(0.543990\pi\)
\(102\) 5.36217 0.530934
\(103\) −4.29238 −0.422941 −0.211470 0.977384i \(-0.567825\pi\)
−0.211470 + 0.977384i \(0.567825\pi\)
\(104\) −10.3606 −1.01594
\(105\) −16.0042 −1.56185
\(106\) −2.25858 −0.219372
\(107\) 16.5070 1.59579 0.797897 0.602794i \(-0.205946\pi\)
0.797897 + 0.602794i \(0.205946\pi\)
\(108\) 1.92356 0.185095
\(109\) −8.42699 −0.807159 −0.403579 0.914945i \(-0.632234\pi\)
−0.403579 + 0.914945i \(0.632234\pi\)
\(110\) 6.40292 0.610494
\(111\) −8.86866 −0.841776
\(112\) −4.13088 −0.390332
\(113\) 14.8571 1.39763 0.698817 0.715300i \(-0.253710\pi\)
0.698817 + 0.715300i \(0.253710\pi\)
\(114\) 3.72646 0.349015
\(115\) −10.4555 −0.974985
\(116\) 1.56035 0.144875
\(117\) 15.2909 1.41364
\(118\) 7.56754 0.696649
\(119\) 8.43464 0.773202
\(120\) −14.2257 −1.29862
\(121\) 5.65114 0.513740
\(122\) 2.04996 0.185594
\(123\) 2.67047 0.240788
\(124\) −13.4528 −1.20810
\(125\) 10.4120 0.931281
\(126\) −6.13557 −0.546600
\(127\) 20.5921 1.82725 0.913626 0.406556i \(-0.133270\pi\)
0.913626 + 0.406556i \(0.133270\pi\)
\(128\) −11.0390 −0.975716
\(129\) 5.16276 0.454555
\(130\) −6.88642 −0.603979
\(131\) 4.77071 0.416819 0.208410 0.978042i \(-0.433171\pi\)
0.208410 + 0.978042i \(0.433171\pi\)
\(132\) −16.2132 −1.41118
\(133\) 5.86169 0.508273
\(134\) 3.23192 0.279195
\(135\) 2.91733 0.251084
\(136\) 7.49734 0.642892
\(137\) 4.90344 0.418929 0.209465 0.977816i \(-0.432828\pi\)
0.209465 + 0.977816i \(0.432828\pi\)
\(138\) −7.45978 −0.635019
\(139\) −1.00000 −0.0848189
\(140\) −9.80684 −0.828829
\(141\) 2.57874 0.217169
\(142\) 5.88583 0.493928
\(143\) −17.9085 −1.49759
\(144\) 5.41914 0.451595
\(145\) 2.36647 0.196525
\(146\) −4.98319 −0.412411
\(147\) −0.136578 −0.0112647
\(148\) −5.43442 −0.446707
\(149\) −6.27975 −0.514457 −0.257229 0.966351i \(-0.582809\pi\)
−0.257229 + 0.966351i \(0.582809\pi\)
\(150\) −1.01336 −0.0827404
\(151\) −18.0807 −1.47138 −0.735692 0.677316i \(-0.763143\pi\)
−0.735692 + 0.677316i \(0.763143\pi\)
\(152\) 5.21031 0.422612
\(153\) −11.0651 −0.894558
\(154\) 7.18593 0.579059
\(155\) −20.4029 −1.63880
\(156\) 17.4375 1.39612
\(157\) −18.8883 −1.50745 −0.753725 0.657191i \(-0.771745\pi\)
−0.753725 + 0.657191i \(0.771745\pi\)
\(158\) −3.24157 −0.257885
\(159\) 8.67373 0.687872
\(160\) −13.6138 −1.07626
\(161\) −11.7342 −0.924782
\(162\) −4.84914 −0.380984
\(163\) 7.52863 0.589688 0.294844 0.955545i \(-0.404732\pi\)
0.294844 + 0.955545i \(0.404732\pi\)
\(164\) 1.63638 0.127780
\(165\) −24.5894 −1.91429
\(166\) −5.39616 −0.418823
\(167\) 1.85660 0.143668 0.0718342 0.997417i \(-0.477115\pi\)
0.0718342 + 0.997417i \(0.477115\pi\)
\(168\) −15.9654 −1.23176
\(169\) 6.26086 0.481605
\(170\) 4.98329 0.382201
\(171\) −7.68971 −0.588047
\(172\) 3.16357 0.241220
\(173\) 9.94817 0.756346 0.378173 0.925735i \(-0.376553\pi\)
0.378173 + 0.925735i \(0.376553\pi\)
\(174\) 1.68842 0.127999
\(175\) −1.59400 −0.120495
\(176\) −6.34685 −0.478412
\(177\) −29.0620 −2.18444
\(178\) 6.83418 0.512244
\(179\) −0.550522 −0.0411480 −0.0205740 0.999788i \(-0.506549\pi\)
−0.0205740 + 0.999788i \(0.506549\pi\)
\(180\) 12.8652 0.958915
\(181\) −9.00173 −0.669093 −0.334547 0.942379i \(-0.608583\pi\)
−0.334547 + 0.942379i \(0.608583\pi\)
\(182\) −7.72857 −0.572880
\(183\) −7.87255 −0.581956
\(184\) −10.4302 −0.768925
\(185\) −8.24201 −0.605965
\(186\) −14.5570 −1.06737
\(187\) 12.9593 0.947679
\(188\) 1.58017 0.115246
\(189\) 3.27409 0.238155
\(190\) 3.46315 0.251244
\(191\) −2.12697 −0.153902 −0.0769510 0.997035i \(-0.524518\pi\)
−0.0769510 + 0.997035i \(0.524518\pi\)
\(192\) −1.79189 −0.129319
\(193\) 1.52641 0.109873 0.0549366 0.998490i \(-0.482504\pi\)
0.0549366 + 0.998490i \(0.482504\pi\)
\(194\) −12.9213 −0.927695
\(195\) 26.4463 1.89386
\(196\) −0.0836904 −0.00597788
\(197\) −8.93752 −0.636772 −0.318386 0.947961i \(-0.603141\pi\)
−0.318386 + 0.947961i \(0.603141\pi\)
\(198\) −9.42694 −0.669943
\(199\) −8.85206 −0.627506 −0.313753 0.949505i \(-0.601586\pi\)
−0.313753 + 0.949505i \(0.601586\pi\)
\(200\) −1.41687 −0.100188
\(201\) −12.4117 −0.875454
\(202\) −1.83595 −0.129177
\(203\) 2.65587 0.186405
\(204\) −12.6185 −0.883471
\(205\) 2.48178 0.173335
\(206\) −2.84612 −0.198299
\(207\) 15.3936 1.06993
\(208\) 6.82612 0.473307
\(209\) 9.00613 0.622967
\(210\) −10.6118 −0.732282
\(211\) 20.0135 1.37778 0.688892 0.724864i \(-0.258098\pi\)
0.688892 + 0.724864i \(0.258098\pi\)
\(212\) 5.31498 0.365034
\(213\) −22.6037 −1.54878
\(214\) 10.9452 0.748198
\(215\) 4.79796 0.327218
\(216\) 2.91026 0.198018
\(217\) −22.8980 −1.55442
\(218\) −5.58762 −0.378441
\(219\) 19.1372 1.29317
\(220\) −15.0676 −1.01586
\(221\) −13.9379 −0.937566
\(222\) −5.88048 −0.394672
\(223\) −17.0648 −1.14275 −0.571373 0.820690i \(-0.693589\pi\)
−0.571373 + 0.820690i \(0.693589\pi\)
\(224\) −15.2786 −1.02085
\(225\) 2.09111 0.139407
\(226\) 9.85116 0.655290
\(227\) 6.03701 0.400690 0.200345 0.979725i \(-0.435794\pi\)
0.200345 + 0.979725i \(0.435794\pi\)
\(228\) −8.76926 −0.580759
\(229\) −10.0352 −0.663147 −0.331573 0.943429i \(-0.607579\pi\)
−0.331573 + 0.943429i \(0.607579\pi\)
\(230\) −6.93268 −0.457128
\(231\) −27.5965 −1.81572
\(232\) 2.36073 0.154990
\(233\) 2.41989 0.158532 0.0792661 0.996853i \(-0.474742\pi\)
0.0792661 + 0.996853i \(0.474742\pi\)
\(234\) 10.1388 0.662794
\(235\) 2.39653 0.156332
\(236\) −17.8083 −1.15922
\(237\) 12.4488 0.808633
\(238\) 5.59270 0.362521
\(239\) −11.1870 −0.723628 −0.361814 0.932250i \(-0.617843\pi\)
−0.361814 + 0.932250i \(0.617843\pi\)
\(240\) 9.37267 0.605003
\(241\) −13.4475 −0.866231 −0.433116 0.901338i \(-0.642586\pi\)
−0.433116 + 0.901338i \(0.642586\pi\)
\(242\) 3.74706 0.240870
\(243\) 22.3207 1.43188
\(244\) −4.82404 −0.308828
\(245\) −0.126927 −0.00810909
\(246\) 1.77069 0.112895
\(247\) −9.68621 −0.616319
\(248\) −20.3535 −1.29245
\(249\) 20.7232 1.31328
\(250\) 6.90384 0.436637
\(251\) 25.7559 1.62570 0.812849 0.582475i \(-0.197915\pi\)
0.812849 + 0.582475i \(0.197915\pi\)
\(252\) 14.4385 0.909539
\(253\) −18.0288 −1.13346
\(254\) 13.6538 0.856719
\(255\) −19.1376 −1.19844
\(256\) −8.72693 −0.545433
\(257\) 9.59008 0.598213 0.299106 0.954220i \(-0.403311\pi\)
0.299106 + 0.954220i \(0.403311\pi\)
\(258\) 3.42323 0.213121
\(259\) −9.24994 −0.574763
\(260\) 16.2054 1.00502
\(261\) −3.48413 −0.215662
\(262\) 3.16328 0.195428
\(263\) −3.65688 −0.225493 −0.112746 0.993624i \(-0.535965\pi\)
−0.112746 + 0.993624i \(0.535965\pi\)
\(264\) −24.5298 −1.50971
\(265\) 8.06086 0.495175
\(266\) 3.88667 0.238307
\(267\) −26.2457 −1.60621
\(268\) −7.60549 −0.464579
\(269\) 10.2364 0.624122 0.312061 0.950062i \(-0.398981\pi\)
0.312061 + 0.950062i \(0.398981\pi\)
\(270\) 1.93437 0.117722
\(271\) 20.5581 1.24881 0.624407 0.781099i \(-0.285341\pi\)
0.624407 + 0.781099i \(0.285341\pi\)
\(272\) −4.93965 −0.299510
\(273\) 29.6804 1.79634
\(274\) 3.25129 0.196418
\(275\) −2.44909 −0.147686
\(276\) 17.5547 1.05667
\(277\) −17.1215 −1.02873 −0.514364 0.857572i \(-0.671972\pi\)
−0.514364 + 0.857572i \(0.671972\pi\)
\(278\) −0.663063 −0.0397679
\(279\) 30.0390 1.79839
\(280\) −14.8373 −0.886698
\(281\) −15.2045 −0.907025 −0.453512 0.891250i \(-0.649829\pi\)
−0.453512 + 0.891250i \(0.649829\pi\)
\(282\) 1.70987 0.101821
\(283\) 13.6131 0.809214 0.404607 0.914491i \(-0.367408\pi\)
0.404607 + 0.914491i \(0.367408\pi\)
\(284\) −13.8508 −0.821893
\(285\) −13.2997 −0.787808
\(286\) −11.8745 −0.702153
\(287\) 2.78528 0.164410
\(288\) 20.0434 1.18107
\(289\) −6.91397 −0.406704
\(290\) 1.56912 0.0921418
\(291\) 49.6223 2.90891
\(292\) 11.7266 0.686250
\(293\) 22.2224 1.29825 0.649125 0.760682i \(-0.275135\pi\)
0.649125 + 0.760682i \(0.275135\pi\)
\(294\) −0.0905596 −0.00528154
\(295\) −27.0085 −1.57250
\(296\) −8.22204 −0.477896
\(297\) 5.03045 0.291896
\(298\) −4.16387 −0.241207
\(299\) 19.3903 1.12137
\(300\) 2.38468 0.137679
\(301\) 5.38471 0.310370
\(302\) −11.9886 −0.689868
\(303\) 7.05071 0.405053
\(304\) −3.43283 −0.196886
\(305\) −7.31629 −0.418929
\(306\) −7.33683 −0.419419
\(307\) 18.7197 1.06839 0.534194 0.845362i \(-0.320615\pi\)
0.534194 + 0.845362i \(0.320615\pi\)
\(308\) −16.9102 −0.963551
\(309\) 10.9301 0.621792
\(310\) −13.5284 −0.768363
\(311\) 24.2299 1.37395 0.686977 0.726679i \(-0.258937\pi\)
0.686977 + 0.726679i \(0.258937\pi\)
\(312\) 26.3822 1.49360
\(313\) −19.8799 −1.12368 −0.561838 0.827247i \(-0.689906\pi\)
−0.561838 + 0.827247i \(0.689906\pi\)
\(314\) −12.5241 −0.706777
\(315\) 21.8979 1.23380
\(316\) 7.62819 0.429119
\(317\) 19.8499 1.11488 0.557439 0.830218i \(-0.311784\pi\)
0.557439 + 0.830218i \(0.311784\pi\)
\(318\) 5.75123 0.322513
\(319\) 4.08058 0.228469
\(320\) −1.66528 −0.0930921
\(321\) −42.0334 −2.34607
\(322\) −7.78049 −0.433590
\(323\) 7.00932 0.390009
\(324\) 11.4112 0.633955
\(325\) 2.63403 0.146110
\(326\) 4.99195 0.276479
\(327\) 21.4584 1.18665
\(328\) 2.47577 0.136701
\(329\) 2.68960 0.148283
\(330\) −16.3044 −0.897525
\(331\) 4.38849 0.241213 0.120607 0.992700i \(-0.461516\pi\)
0.120607 + 0.992700i \(0.461516\pi\)
\(332\) 12.6985 0.696919
\(333\) 12.1346 0.664973
\(334\) 1.23105 0.0673598
\(335\) −11.5347 −0.630209
\(336\) 10.5189 0.573851
\(337\) −20.8314 −1.13476 −0.567381 0.823456i \(-0.692043\pi\)
−0.567381 + 0.823456i \(0.692043\pi\)
\(338\) 4.15134 0.225803
\(339\) −37.8319 −2.05475
\(340\) −11.7269 −0.635979
\(341\) −35.1814 −1.90518
\(342\) −5.09876 −0.275709
\(343\) 18.4486 0.996132
\(344\) 4.78633 0.258062
\(345\) 26.6239 1.43339
\(346\) 6.59627 0.354617
\(347\) −6.10675 −0.327828 −0.163914 0.986475i \(-0.552412\pi\)
−0.163914 + 0.986475i \(0.552412\pi\)
\(348\) −3.97326 −0.212989
\(349\) 29.0866 1.55697 0.778484 0.627664i \(-0.215989\pi\)
0.778484 + 0.627664i \(0.215989\pi\)
\(350\) −1.05692 −0.0564950
\(351\) −5.41031 −0.288781
\(352\) −23.4747 −1.25121
\(353\) −19.1357 −1.01849 −0.509246 0.860621i \(-0.670076\pi\)
−0.509246 + 0.860621i \(0.670076\pi\)
\(354\) −19.2700 −1.02419
\(355\) −21.0065 −1.11491
\(356\) −16.0825 −0.852370
\(357\) −21.4779 −1.13673
\(358\) −0.365031 −0.0192925
\(359\) 11.8779 0.626890 0.313445 0.949606i \(-0.398517\pi\)
0.313445 + 0.949606i \(0.398517\pi\)
\(360\) 19.4644 1.02587
\(361\) −14.1288 −0.743623
\(362\) −5.96871 −0.313709
\(363\) −14.3900 −0.755281
\(364\) 18.1872 0.953268
\(365\) 17.7850 0.930908
\(366\) −5.22000 −0.272854
\(367\) −9.80599 −0.511869 −0.255934 0.966694i \(-0.582383\pi\)
−0.255934 + 0.966694i \(0.582383\pi\)
\(368\) 6.87198 0.358227
\(369\) −3.65390 −0.190214
\(370\) −5.46497 −0.284110
\(371\) 9.04663 0.469677
\(372\) 34.2562 1.77610
\(373\) 17.0149 0.880996 0.440498 0.897753i \(-0.354802\pi\)
0.440498 + 0.897753i \(0.354802\pi\)
\(374\) 8.59284 0.444325
\(375\) −26.5132 −1.36913
\(376\) 2.39072 0.123292
\(377\) −4.38872 −0.226031
\(378\) 2.17093 0.111661
\(379\) −10.7945 −0.554478 −0.277239 0.960801i \(-0.589419\pi\)
−0.277239 + 0.960801i \(0.589419\pi\)
\(380\) −8.14964 −0.418068
\(381\) −52.4356 −2.68636
\(382\) −1.41031 −0.0721579
\(383\) 8.73640 0.446409 0.223205 0.974772i \(-0.428348\pi\)
0.223205 + 0.974772i \(0.428348\pi\)
\(384\) 28.1096 1.43446
\(385\) −25.6466 −1.30707
\(386\) 1.01210 0.0515147
\(387\) −7.06399 −0.359083
\(388\) 30.4069 1.54368
\(389\) 24.5119 1.24280 0.621402 0.783492i \(-0.286563\pi\)
0.621402 + 0.783492i \(0.286563\pi\)
\(390\) 17.5355 0.887947
\(391\) −14.0316 −0.709606
\(392\) −0.126620 −0.00639526
\(393\) −12.1481 −0.612791
\(394\) −5.92614 −0.298555
\(395\) 11.5691 0.582107
\(396\) 22.1839 1.11478
\(397\) −26.6509 −1.33757 −0.668785 0.743456i \(-0.733185\pi\)
−0.668785 + 0.743456i \(0.733185\pi\)
\(398\) −5.86947 −0.294210
\(399\) −14.9262 −0.747243
\(400\) 0.933510 0.0466755
\(401\) 16.0327 0.800634 0.400317 0.916377i \(-0.368900\pi\)
0.400317 + 0.916377i \(0.368900\pi\)
\(402\) −8.22974 −0.410462
\(403\) 37.8381 1.88485
\(404\) 4.32044 0.214950
\(405\) 17.3066 0.859970
\(406\) 1.76101 0.0873973
\(407\) −14.2120 −0.704461
\(408\) −19.0912 −0.945155
\(409\) 17.8344 0.881855 0.440928 0.897543i \(-0.354650\pi\)
0.440928 + 0.897543i \(0.354650\pi\)
\(410\) 1.64558 0.0812692
\(411\) −12.4861 −0.615894
\(412\) 6.69761 0.329967
\(413\) −30.3114 −1.49153
\(414\) 10.2069 0.501642
\(415\) 19.2589 0.945382
\(416\) 25.2474 1.23785
\(417\) 2.54639 0.124697
\(418\) 5.97163 0.292082
\(419\) −13.7739 −0.672898 −0.336449 0.941702i \(-0.609226\pi\)
−0.336449 + 0.941702i \(0.609226\pi\)
\(420\) 24.9721 1.21851
\(421\) −27.9889 −1.36409 −0.682047 0.731308i \(-0.738910\pi\)
−0.682047 + 0.731308i \(0.738910\pi\)
\(422\) 13.2702 0.645982
\(423\) −3.52838 −0.171556
\(424\) 8.04132 0.390521
\(425\) −1.90609 −0.0924588
\(426\) −14.9877 −0.726154
\(427\) −8.21100 −0.397358
\(428\) −25.7567 −1.24500
\(429\) 45.6022 2.20169
\(430\) 3.18135 0.153418
\(431\) 24.9272 1.20070 0.600350 0.799737i \(-0.295028\pi\)
0.600350 + 0.799737i \(0.295028\pi\)
\(432\) −1.91744 −0.0922527
\(433\) 17.4130 0.836814 0.418407 0.908260i \(-0.362589\pi\)
0.418407 + 0.908260i \(0.362589\pi\)
\(434\) −15.1828 −0.728799
\(435\) −6.02597 −0.288923
\(436\) 13.1490 0.629724
\(437\) −9.75128 −0.466467
\(438\) 12.6892 0.606311
\(439\) 7.70376 0.367680 0.183840 0.982956i \(-0.441147\pi\)
0.183840 + 0.982956i \(0.441147\pi\)
\(440\) −22.7966 −1.08679
\(441\) 0.186874 0.00889874
\(442\) −9.24172 −0.439584
\(443\) 6.02754 0.286377 0.143189 0.989695i \(-0.454264\pi\)
0.143189 + 0.989695i \(0.454264\pi\)
\(444\) 13.8382 0.656731
\(445\) −24.3912 −1.15625
\(446\) −11.3151 −0.535784
\(447\) 15.9907 0.756335
\(448\) −1.86893 −0.0882987
\(449\) 8.54161 0.403104 0.201552 0.979478i \(-0.435402\pi\)
0.201552 + 0.979478i \(0.435402\pi\)
\(450\) 1.38654 0.0653620
\(451\) 4.27941 0.201510
\(452\) −23.1822 −1.09040
\(453\) 46.0405 2.16317
\(454\) 4.00291 0.187866
\(455\) 27.5832 1.29312
\(456\) −13.2675 −0.621307
\(457\) −1.29272 −0.0604709 −0.0302355 0.999543i \(-0.509626\pi\)
−0.0302355 + 0.999543i \(0.509626\pi\)
\(458\) −6.65399 −0.310920
\(459\) 3.91512 0.182742
\(460\) 16.3143 0.760657
\(461\) −10.2785 −0.478719 −0.239359 0.970931i \(-0.576937\pi\)
−0.239359 + 0.970931i \(0.576937\pi\)
\(462\) −18.2982 −0.851311
\(463\) −32.2980 −1.50102 −0.750508 0.660862i \(-0.770191\pi\)
−0.750508 + 0.660862i \(0.770191\pi\)
\(464\) −1.55538 −0.0722067
\(465\) 51.9539 2.40931
\(466\) 1.60454 0.0743289
\(467\) −7.11434 −0.329213 −0.164606 0.986359i \(-0.552635\pi\)
−0.164606 + 0.986359i \(0.552635\pi\)
\(468\) −23.8590 −1.10288
\(469\) −12.9453 −0.597759
\(470\) 1.58905 0.0732974
\(471\) 48.0970 2.21619
\(472\) −26.9431 −1.24016
\(473\) 8.27328 0.380406
\(474\) 8.25431 0.379133
\(475\) −1.32464 −0.0607788
\(476\) −13.1610 −0.603232
\(477\) −11.8679 −0.543394
\(478\) −7.41770 −0.339278
\(479\) −32.6050 −1.48976 −0.744880 0.667199i \(-0.767493\pi\)
−0.744880 + 0.667199i \(0.767493\pi\)
\(480\) 34.6661 1.58228
\(481\) 15.2852 0.696943
\(482\) −8.91655 −0.406138
\(483\) 29.8798 1.35958
\(484\) −8.81774 −0.400806
\(485\) 46.1161 2.09402
\(486\) 14.8001 0.671344
\(487\) 17.2987 0.783880 0.391940 0.919991i \(-0.371804\pi\)
0.391940 + 0.919991i \(0.371804\pi\)
\(488\) −7.29855 −0.330390
\(489\) −19.1709 −0.866936
\(490\) −0.0841608 −0.00380200
\(491\) −25.0616 −1.13101 −0.565506 0.824744i \(-0.691319\pi\)
−0.565506 + 0.824744i \(0.691319\pi\)
\(492\) −4.16687 −0.187857
\(493\) 3.17585 0.143033
\(494\) −6.42257 −0.288965
\(495\) 33.6447 1.51222
\(496\) 13.4100 0.602126
\(497\) −23.5754 −1.05750
\(498\) 13.7408 0.615738
\(499\) 0.857549 0.0383891 0.0191946 0.999816i \(-0.493890\pi\)
0.0191946 + 0.999816i \(0.493890\pi\)
\(500\) −16.2464 −0.726561
\(501\) −4.72765 −0.211216
\(502\) 17.0778 0.762219
\(503\) −28.9728 −1.29183 −0.645916 0.763408i \(-0.723525\pi\)
−0.645916 + 0.763408i \(0.723525\pi\)
\(504\) 21.8448 0.973044
\(505\) 6.55252 0.291583
\(506\) −11.9543 −0.531431
\(507\) −15.9426 −0.708037
\(508\) −32.1308 −1.42557
\(509\) −19.6979 −0.873096 −0.436548 0.899681i \(-0.643799\pi\)
−0.436548 + 0.899681i \(0.643799\pi\)
\(510\) −12.6894 −0.561897
\(511\) 19.9599 0.882975
\(512\) 16.2914 0.719987
\(513\) 2.72083 0.120127
\(514\) 6.35883 0.280476
\(515\) 10.1578 0.447606
\(516\) −8.05569 −0.354632
\(517\) 4.13241 0.181743
\(518\) −6.13329 −0.269481
\(519\) −25.3320 −1.11195
\(520\) 24.5181 1.07519
\(521\) 27.3820 1.19963 0.599813 0.800140i \(-0.295242\pi\)
0.599813 + 0.800140i \(0.295242\pi\)
\(522\) −2.31020 −0.101114
\(523\) −9.37070 −0.409752 −0.204876 0.978788i \(-0.565679\pi\)
−0.204876 + 0.978788i \(0.565679\pi\)
\(524\) −7.44397 −0.325191
\(525\) 4.05896 0.177148
\(526\) −2.42474 −0.105724
\(527\) −27.3811 −1.19274
\(528\) 16.1616 0.703343
\(529\) −3.47948 −0.151282
\(530\) 5.34486 0.232166
\(531\) 39.7644 1.72563
\(532\) −9.14627 −0.396541
\(533\) −4.60257 −0.199359
\(534\) −17.4025 −0.753081
\(535\) −39.0634 −1.68886
\(536\) −11.5068 −0.497016
\(537\) 1.40185 0.0604942
\(538\) 6.78736 0.292624
\(539\) −0.218865 −0.00942718
\(540\) −4.55205 −0.195889
\(541\) 42.1264 1.81115 0.905577 0.424181i \(-0.139438\pi\)
0.905577 + 0.424181i \(0.139438\pi\)
\(542\) 13.6313 0.585515
\(543\) 22.9220 0.983675
\(544\) −18.2700 −0.783319
\(545\) 19.9422 0.854230
\(546\) 19.6800 0.842226
\(547\) −13.3623 −0.571330 −0.285665 0.958330i \(-0.592214\pi\)
−0.285665 + 0.958330i \(0.592214\pi\)
\(548\) −7.65107 −0.326838
\(549\) 10.7717 0.459724
\(550\) −1.62390 −0.0692434
\(551\) 2.20707 0.0940243
\(552\) 26.5594 1.13044
\(553\) 12.9839 0.552133
\(554\) −11.3526 −0.482326
\(555\) 20.9874 0.890866
\(556\) 1.56035 0.0661735
\(557\) 7.70532 0.326485 0.163242 0.986586i \(-0.447805\pi\)
0.163242 + 0.986586i \(0.447805\pi\)
\(558\) 19.9178 0.843186
\(559\) −8.89803 −0.376346
\(560\) 9.77561 0.413095
\(561\) −32.9995 −1.39324
\(562\) −10.0815 −0.425264
\(563\) −12.5779 −0.530094 −0.265047 0.964235i \(-0.585387\pi\)
−0.265047 + 0.964235i \(0.585387\pi\)
\(564\) −4.02373 −0.169430
\(565\) −35.1588 −1.47914
\(566\) 9.02633 0.379405
\(567\) 19.4230 0.815690
\(568\) −20.9556 −0.879278
\(569\) −16.4727 −0.690573 −0.345287 0.938497i \(-0.612218\pi\)
−0.345287 + 0.938497i \(0.612218\pi\)
\(570\) −8.81856 −0.369369
\(571\) 7.09721 0.297009 0.148505 0.988912i \(-0.452554\pi\)
0.148505 + 0.988912i \(0.452554\pi\)
\(572\) 27.9435 1.16838
\(573\) 5.41610 0.226261
\(574\) 1.84682 0.0770846
\(575\) 2.65172 0.110585
\(576\) 2.45177 0.102157
\(577\) 4.04803 0.168522 0.0842609 0.996444i \(-0.473147\pi\)
0.0842609 + 0.996444i \(0.473147\pi\)
\(578\) −4.58440 −0.190686
\(579\) −3.88684 −0.161531
\(580\) −3.69252 −0.153323
\(581\) 21.6141 0.896703
\(582\) 32.9027 1.36386
\(583\) 13.8996 0.575662
\(584\) 17.7419 0.734164
\(585\) −36.1854 −1.49608
\(586\) 14.7349 0.608692
\(587\) −2.48820 −0.102699 −0.0513494 0.998681i \(-0.516352\pi\)
−0.0513494 + 0.998681i \(0.516352\pi\)
\(588\) 0.213109 0.00878846
\(589\) −19.0286 −0.784061
\(590\) −17.9084 −0.737276
\(591\) 22.7585 0.936158
\(592\) 5.41712 0.222642
\(593\) 37.0980 1.52343 0.761717 0.647910i \(-0.224357\pi\)
0.761717 + 0.647910i \(0.224357\pi\)
\(594\) 3.33550 0.136857
\(595\) −19.9603 −0.818294
\(596\) 9.79859 0.401366
\(597\) 22.5408 0.922535
\(598\) 12.8570 0.525760
\(599\) −29.8212 −1.21846 −0.609230 0.792993i \(-0.708522\pi\)
−0.609230 + 0.792993i \(0.708522\pi\)
\(600\) 3.60791 0.147292
\(601\) 39.2623 1.60154 0.800771 0.598971i \(-0.204423\pi\)
0.800771 + 0.598971i \(0.204423\pi\)
\(602\) 3.57040 0.145519
\(603\) 16.9824 0.691578
\(604\) 28.2121 1.14794
\(605\) −13.3733 −0.543700
\(606\) 4.67506 0.189912
\(607\) 31.3026 1.27053 0.635267 0.772292i \(-0.280890\pi\)
0.635267 + 0.772292i \(0.280890\pi\)
\(608\) −12.6968 −0.514923
\(609\) −6.76289 −0.274046
\(610\) −4.85116 −0.196418
\(611\) −4.44447 −0.179804
\(612\) 17.2653 0.697910
\(613\) −29.6426 −1.19725 −0.598626 0.801029i \(-0.704286\pi\)
−0.598626 + 0.801029i \(0.704286\pi\)
\(614\) 12.4123 0.500921
\(615\) −6.31959 −0.254831
\(616\) −25.5844 −1.03083
\(617\) 18.2502 0.734727 0.367363 0.930078i \(-0.380261\pi\)
0.367363 + 0.930078i \(0.380261\pi\)
\(618\) 7.24734 0.291531
\(619\) 2.48820 0.100009 0.0500046 0.998749i \(-0.484076\pi\)
0.0500046 + 0.998749i \(0.484076\pi\)
\(620\) 31.8357 1.27855
\(621\) −5.44666 −0.218567
\(622\) 16.0660 0.644187
\(623\) −27.3740 −1.09672
\(624\) −17.3820 −0.695837
\(625\) −27.6407 −1.10563
\(626\) −13.1816 −0.526842
\(627\) −22.9332 −0.915862
\(628\) 29.4723 1.17607
\(629\) −11.0609 −0.441029
\(630\) 14.5197 0.578477
\(631\) −41.1710 −1.63899 −0.819496 0.573085i \(-0.805746\pi\)
−0.819496 + 0.573085i \(0.805746\pi\)
\(632\) 11.5411 0.459080
\(633\) −50.9622 −2.02556
\(634\) 13.1617 0.522718
\(635\) −48.7306 −1.93381
\(636\) −13.5340 −0.536659
\(637\) 0.235392 0.00932658
\(638\) 2.70568 0.107119
\(639\) 30.9277 1.22348
\(640\) 26.1234 1.03262
\(641\) −15.8374 −0.625541 −0.312770 0.949829i \(-0.601257\pi\)
−0.312770 + 0.949829i \(0.601257\pi\)
\(642\) −27.8708 −1.09997
\(643\) 15.2114 0.599878 0.299939 0.953958i \(-0.403034\pi\)
0.299939 + 0.953958i \(0.403034\pi\)
\(644\) 18.3094 0.721490
\(645\) −12.2175 −0.481064
\(646\) 4.64762 0.182858
\(647\) −28.2478 −1.11053 −0.555267 0.831672i \(-0.687384\pi\)
−0.555267 + 0.831672i \(0.687384\pi\)
\(648\) 17.2646 0.678218
\(649\) −46.5717 −1.82810
\(650\) 1.74653 0.0685044
\(651\) 58.3074 2.28525
\(652\) −11.7473 −0.460059
\(653\) −3.92079 −0.153432 −0.0767162 0.997053i \(-0.524444\pi\)
−0.0767162 + 0.997053i \(0.524444\pi\)
\(654\) 14.2283 0.556370
\(655\) −11.2897 −0.441127
\(656\) −1.63117 −0.0636864
\(657\) −26.1846 −1.02156
\(658\) 1.78338 0.0695233
\(659\) −38.9914 −1.51889 −0.759444 0.650573i \(-0.774529\pi\)
−0.759444 + 0.650573i \(0.774529\pi\)
\(660\) 38.3681 1.49348
\(661\) 15.0467 0.585249 0.292624 0.956227i \(-0.405471\pi\)
0.292624 + 0.956227i \(0.405471\pi\)
\(662\) 2.90984 0.113094
\(663\) 35.4914 1.37837
\(664\) 19.2122 0.745578
\(665\) −13.8715 −0.537914
\(666\) 8.04602 0.311777
\(667\) −4.41820 −0.171073
\(668\) −2.89695 −0.112086
\(669\) 43.4538 1.68002
\(670\) −7.64824 −0.295477
\(671\) −12.6157 −0.487024
\(672\) 38.9054 1.50081
\(673\) 43.1013 1.66143 0.830717 0.556696i \(-0.187931\pi\)
0.830717 + 0.556696i \(0.187931\pi\)
\(674\) −13.8126 −0.532040
\(675\) −0.739890 −0.0284784
\(676\) −9.76912 −0.375735
\(677\) 29.0666 1.11712 0.558559 0.829465i \(-0.311354\pi\)
0.558559 + 0.829465i \(0.311354\pi\)
\(678\) −25.0849 −0.963382
\(679\) 51.7557 1.98620
\(680\) −17.7422 −0.680384
\(681\) −15.3726 −0.589079
\(682\) −23.3275 −0.893256
\(683\) 29.1695 1.11614 0.558069 0.829794i \(-0.311542\pi\)
0.558069 + 0.829794i \(0.311542\pi\)
\(684\) 11.9986 0.458779
\(685\) −11.6038 −0.443360
\(686\) 12.2326 0.467043
\(687\) 25.5537 0.974933
\(688\) −3.15350 −0.120226
\(689\) −14.9492 −0.569519
\(690\) 17.6534 0.672052
\(691\) 12.2564 0.466257 0.233128 0.972446i \(-0.425104\pi\)
0.233128 + 0.972446i \(0.425104\pi\)
\(692\) −15.5226 −0.590081
\(693\) 37.7592 1.43435
\(694\) −4.04916 −0.153704
\(695\) 2.36647 0.0897653
\(696\) −6.01136 −0.227860
\(697\) 3.33060 0.126155
\(698\) 19.2862 0.729995
\(699\) −6.16200 −0.233068
\(700\) 2.48720 0.0940073
\(701\) −31.2340 −1.17969 −0.589846 0.807516i \(-0.700812\pi\)
−0.589846 + 0.807516i \(0.700812\pi\)
\(702\) −3.58738 −0.135397
\(703\) −7.68685 −0.289915
\(704\) −2.87150 −0.108224
\(705\) −6.10251 −0.229834
\(706\) −12.6882 −0.477526
\(707\) 7.35383 0.276569
\(708\) 45.3469 1.70424
\(709\) −40.7262 −1.52951 −0.764753 0.644324i \(-0.777139\pi\)
−0.764753 + 0.644324i \(0.777139\pi\)
\(710\) −13.9287 −0.522733
\(711\) −17.0331 −0.638792
\(712\) −24.3321 −0.911883
\(713\) 38.0923 1.42657
\(714\) −14.2412 −0.532964
\(715\) 42.3800 1.58492
\(716\) 0.859006 0.0321026
\(717\) 28.4866 1.06385
\(718\) 7.87577 0.293921
\(719\) 27.3143 1.01865 0.509327 0.860573i \(-0.329895\pi\)
0.509327 + 0.860573i \(0.329895\pi\)
\(720\) −12.8242 −0.477931
\(721\) 11.4000 0.424558
\(722\) −9.36831 −0.348652
\(723\) 34.2427 1.27350
\(724\) 14.0458 0.522009
\(725\) −0.600182 −0.0222902
\(726\) −9.54150 −0.354118
\(727\) −2.45544 −0.0910673 −0.0455337 0.998963i \(-0.514499\pi\)
−0.0455337 + 0.998963i \(0.514499\pi\)
\(728\) 27.5164 1.01983
\(729\) −34.8977 −1.29251
\(730\) 11.7926 0.436462
\(731\) 6.43896 0.238154
\(732\) 12.2839 0.454027
\(733\) 40.6914 1.50297 0.751486 0.659750i \(-0.229338\pi\)
0.751486 + 0.659750i \(0.229338\pi\)
\(734\) −6.50199 −0.239993
\(735\) 0.323207 0.0119217
\(736\) 25.4170 0.936882
\(737\) −19.8897 −0.732646
\(738\) −2.42276 −0.0891832
\(739\) −6.18246 −0.227426 −0.113713 0.993514i \(-0.536274\pi\)
−0.113713 + 0.993514i \(0.536274\pi\)
\(740\) 12.8604 0.472758
\(741\) 24.6649 0.906088
\(742\) 5.99848 0.220211
\(743\) 1.18751 0.0435654 0.0217827 0.999763i \(-0.493066\pi\)
0.0217827 + 0.999763i \(0.493066\pi\)
\(744\) 51.8280 1.90011
\(745\) 14.8608 0.544459
\(746\) 11.2819 0.413061
\(747\) −28.3546 −1.03744
\(748\) −20.2210 −0.739354
\(749\) −43.8405 −1.60190
\(750\) −17.5799 −0.641927
\(751\) −50.8223 −1.85453 −0.927265 0.374405i \(-0.877847\pi\)
−0.927265 + 0.374405i \(0.877847\pi\)
\(752\) −1.57514 −0.0574393
\(753\) −65.5847 −2.39004
\(754\) −2.91000 −0.105976
\(755\) 42.7874 1.55719
\(756\) −5.10873 −0.185803
\(757\) −44.3162 −1.61070 −0.805350 0.592799i \(-0.798023\pi\)
−0.805350 + 0.592799i \(0.798023\pi\)
\(758\) −7.15745 −0.259970
\(759\) 45.9085 1.66637
\(760\) −12.3300 −0.447257
\(761\) 6.00843 0.217806 0.108903 0.994052i \(-0.465266\pi\)
0.108903 + 0.994052i \(0.465266\pi\)
\(762\) −34.7681 −1.25951
\(763\) 22.3810 0.810245
\(764\) 3.31881 0.120070
\(765\) 26.1851 0.946726
\(766\) 5.79278 0.209302
\(767\) 50.0885 1.80859
\(768\) 22.2222 0.801875
\(769\) −40.7642 −1.46999 −0.734997 0.678071i \(-0.762816\pi\)
−0.734997 + 0.678071i \(0.762816\pi\)
\(770\) −17.0053 −0.612828
\(771\) −24.4201 −0.879470
\(772\) −2.38173 −0.0857202
\(773\) −1.98713 −0.0714721 −0.0357360 0.999361i \(-0.511378\pi\)
−0.0357360 + 0.999361i \(0.511378\pi\)
\(774\) −4.68387 −0.168358
\(775\) 5.17457 0.185876
\(776\) 46.0043 1.65146
\(777\) 23.5540 0.844995
\(778\) 16.2530 0.582697
\(779\) 2.31461 0.0829296
\(780\) −41.2654 −1.47754
\(781\) −36.2222 −1.29613
\(782\) −9.30380 −0.332703
\(783\) 1.23278 0.0440559
\(784\) 0.0834239 0.00297943
\(785\) 44.6986 1.59536
\(786\) −8.05497 −0.287311
\(787\) −15.7847 −0.562665 −0.281332 0.959610i \(-0.590776\pi\)
−0.281332 + 0.959610i \(0.590776\pi\)
\(788\) 13.9456 0.496793
\(789\) 9.31186 0.331511
\(790\) 7.67107 0.272924
\(791\) −39.4584 −1.40298
\(792\) 33.5632 1.19262
\(793\) 13.5684 0.481827
\(794\) −17.6712 −0.627128
\(795\) −20.5261 −0.727987
\(796\) 13.8123 0.489564
\(797\) 2.84443 0.100755 0.0503774 0.998730i \(-0.483958\pi\)
0.0503774 + 0.998730i \(0.483958\pi\)
\(798\) −9.89699 −0.350350
\(799\) 3.21619 0.113781
\(800\) 3.45272 0.122072
\(801\) 35.9108 1.26885
\(802\) 10.6307 0.375383
\(803\) 30.6672 1.08222
\(804\) 19.3666 0.683007
\(805\) 27.7686 0.978713
\(806\) 25.0890 0.883724
\(807\) −26.0658 −0.917561
\(808\) 6.53664 0.229958
\(809\) −23.9498 −0.842031 −0.421016 0.907053i \(-0.638326\pi\)
−0.421016 + 0.907053i \(0.638326\pi\)
\(810\) 11.4753 0.403203
\(811\) 46.9938 1.65017 0.825087 0.565006i \(-0.191126\pi\)
0.825087 + 0.565006i \(0.191126\pi\)
\(812\) −4.14408 −0.145429
\(813\) −52.3490 −1.83596
\(814\) −9.42343 −0.330291
\(815\) −17.8163 −0.624077
\(816\) 12.5783 0.440329
\(817\) 4.47478 0.156553
\(818\) 11.8253 0.413463
\(819\) −40.6105 −1.41905
\(820\) −3.87244 −0.135231
\(821\) −10.1990 −0.355946 −0.177973 0.984035i \(-0.556954\pi\)
−0.177973 + 0.984035i \(0.556954\pi\)
\(822\) −8.27907 −0.288766
\(823\) 33.9216 1.18243 0.591216 0.806513i \(-0.298648\pi\)
0.591216 + 0.806513i \(0.298648\pi\)
\(824\) 10.1332 0.353006
\(825\) 6.23635 0.217122
\(826\) −20.0984 −0.699313
\(827\) −38.7238 −1.34656 −0.673280 0.739388i \(-0.735115\pi\)
−0.673280 + 0.739388i \(0.735115\pi\)
\(828\) −24.0193 −0.834729
\(829\) −30.1736 −1.04797 −0.523986 0.851727i \(-0.675556\pi\)
−0.523986 + 0.851727i \(0.675556\pi\)
\(830\) 12.7699 0.443248
\(831\) 43.5980 1.51240
\(832\) 3.08833 0.107069
\(833\) −0.170339 −0.00590190
\(834\) 1.68842 0.0584652
\(835\) −4.39360 −0.152047
\(836\) −14.0527 −0.486022
\(837\) −10.6286 −0.367378
\(838\) −9.13295 −0.315493
\(839\) −36.5864 −1.26310 −0.631551 0.775335i \(-0.717581\pi\)
−0.631551 + 0.775335i \(0.717581\pi\)
\(840\) 37.7816 1.30359
\(841\) 1.00000 0.0344828
\(842\) −18.5584 −0.639564
\(843\) 38.7167 1.33347
\(844\) −31.2280 −1.07491
\(845\) −14.8161 −0.509691
\(846\) −2.33954 −0.0804350
\(847\) −15.0087 −0.515704
\(848\) −5.29806 −0.181936
\(849\) −34.6643 −1.18968
\(850\) −1.26386 −0.0433499
\(851\) 15.3878 0.527489
\(852\) 35.2696 1.20832
\(853\) 21.4996 0.736131 0.368065 0.929800i \(-0.380020\pi\)
0.368065 + 0.929800i \(0.380020\pi\)
\(854\) −5.44441 −0.186304
\(855\) 18.1975 0.622340
\(856\) −38.9687 −1.33192
\(857\) −19.5159 −0.666649 −0.333325 0.942812i \(-0.608171\pi\)
−0.333325 + 0.942812i \(0.608171\pi\)
\(858\) 30.2371 1.03228
\(859\) 36.0006 1.22833 0.614163 0.789179i \(-0.289494\pi\)
0.614163 + 0.789179i \(0.289494\pi\)
\(860\) −7.48649 −0.255287
\(861\) −7.09242 −0.241709
\(862\) 16.5283 0.562956
\(863\) −6.68443 −0.227541 −0.113770 0.993507i \(-0.536293\pi\)
−0.113770 + 0.993507i \(0.536293\pi\)
\(864\) −7.09190 −0.241271
\(865\) −23.5421 −0.800454
\(866\) 11.5459 0.392345
\(867\) 17.6057 0.597921
\(868\) 35.7289 1.21272
\(869\) 19.9490 0.676725
\(870\) −3.99560 −0.135463
\(871\) 21.3916 0.724827
\(872\) 19.8939 0.673692
\(873\) −67.8962 −2.29794
\(874\) −6.46571 −0.218706
\(875\) −27.6530 −0.934842
\(876\) −29.8607 −1.00890
\(877\) −0.430584 −0.0145398 −0.00726989 0.999974i \(-0.502314\pi\)
−0.00726989 + 0.999974i \(0.502314\pi\)
\(878\) 5.10808 0.172389
\(879\) −56.5871 −1.90864
\(880\) 15.0196 0.506312
\(881\) −5.64082 −0.190044 −0.0950220 0.995475i \(-0.530292\pi\)
−0.0950220 + 0.995475i \(0.530292\pi\)
\(882\) 0.123909 0.00417223
\(883\) −17.5655 −0.591126 −0.295563 0.955323i \(-0.595507\pi\)
−0.295563 + 0.955323i \(0.595507\pi\)
\(884\) 21.7480 0.731464
\(885\) 68.7744 2.31183
\(886\) 3.99664 0.134270
\(887\) −3.06399 −0.102879 −0.0514393 0.998676i \(-0.516381\pi\)
−0.0514393 + 0.998676i \(0.516381\pi\)
\(888\) 20.9365 0.702585
\(889\) −54.6899 −1.83424
\(890\) −16.1729 −0.542116
\(891\) 29.8423 0.999754
\(892\) 26.6271 0.891541
\(893\) 2.23510 0.0747949
\(894\) 10.6029 0.354613
\(895\) 1.30279 0.0435476
\(896\) 29.3181 0.979447
\(897\) −49.3752 −1.64859
\(898\) 5.66363 0.188998
\(899\) −8.62167 −0.287549
\(900\) −3.26286 −0.108762
\(901\) 10.8178 0.360394
\(902\) 2.83752 0.0944791
\(903\) −13.7116 −0.456293
\(904\) −35.0736 −1.16653
\(905\) 21.3023 0.708113
\(906\) 30.5278 1.01422
\(907\) 27.7618 0.921815 0.460907 0.887448i \(-0.347524\pi\)
0.460907 + 0.887448i \(0.347524\pi\)
\(908\) −9.41983 −0.312608
\(909\) −9.64720 −0.319977
\(910\) 18.2894 0.606289
\(911\) −48.8713 −1.61918 −0.809590 0.586996i \(-0.800310\pi\)
−0.809590 + 0.586996i \(0.800310\pi\)
\(912\) 8.74134 0.289455
\(913\) 33.2087 1.09905
\(914\) −0.857155 −0.0283522
\(915\) 18.6302 0.615894
\(916\) 15.6584 0.517370
\(917\) −12.6704 −0.418413
\(918\) 2.59597 0.0856797
\(919\) −54.5708 −1.80012 −0.900062 0.435762i \(-0.856479\pi\)
−0.900062 + 0.435762i \(0.856479\pi\)
\(920\) 24.6828 0.813767
\(921\) −47.6677 −1.57070
\(922\) −6.81531 −0.224450
\(923\) 38.9575 1.28230
\(924\) 43.0602 1.41658
\(925\) 2.09033 0.0687296
\(926\) −21.4156 −0.703760
\(927\) −14.9552 −0.491193
\(928\) −5.75278 −0.188844
\(929\) −46.0806 −1.51186 −0.755928 0.654655i \(-0.772814\pi\)
−0.755928 + 0.654655i \(0.772814\pi\)
\(930\) 34.4487 1.12962
\(931\) −0.118378 −0.00387967
\(932\) −3.77587 −0.123683
\(933\) −61.6990 −2.01993
\(934\) −4.71726 −0.154353
\(935\) −30.6678 −1.00295
\(936\) −36.0976 −1.17989
\(937\) 47.0754 1.53789 0.768944 0.639317i \(-0.220783\pi\)
0.768944 + 0.639317i \(0.220783\pi\)
\(938\) −8.58355 −0.280263
\(939\) 50.6220 1.65198
\(940\) −3.73942 −0.121966
\(941\) 47.5842 1.55120 0.775600 0.631224i \(-0.217447\pi\)
0.775600 + 0.631224i \(0.217447\pi\)
\(942\) 31.8914 1.03908
\(943\) −4.63349 −0.150887
\(944\) 17.7516 0.577764
\(945\) −7.74805 −0.252044
\(946\) 5.48571 0.178356
\(947\) −39.4413 −1.28167 −0.640835 0.767678i \(-0.721412\pi\)
−0.640835 + 0.767678i \(0.721412\pi\)
\(948\) −19.4244 −0.630875
\(949\) −32.9830 −1.07067
\(950\) −0.878322 −0.0284965
\(951\) −50.5456 −1.63905
\(952\) −19.9119 −0.645350
\(953\) −25.0439 −0.811250 −0.405625 0.914039i \(-0.632946\pi\)
−0.405625 + 0.914039i \(0.632946\pi\)
\(954\) −7.86917 −0.254774
\(955\) 5.03340 0.162877
\(956\) 17.4556 0.564556
\(957\) −10.3908 −0.335886
\(958\) −21.6192 −0.698483
\(959\) −13.0229 −0.420531
\(960\) 4.24046 0.136860
\(961\) 43.3333 1.39785
\(962\) 10.1350 0.326766
\(963\) 57.5125 1.85332
\(964\) 20.9828 0.675811
\(965\) −3.61220 −0.116281
\(966\) 19.8122 0.637447
\(967\) −10.8112 −0.347665 −0.173832 0.984775i \(-0.555615\pi\)
−0.173832 + 0.984775i \(0.555615\pi\)
\(968\) −13.3408 −0.428791
\(969\) −17.8485 −0.573377
\(970\) 30.5779 0.981796
\(971\) −54.2227 −1.74009 −0.870045 0.492973i \(-0.835910\pi\)
−0.870045 + 0.492973i \(0.835910\pi\)
\(972\) −34.8281 −1.11711
\(973\) 2.65587 0.0851432
\(974\) 11.4701 0.367527
\(975\) −6.70728 −0.214805
\(976\) 4.80868 0.153922
\(977\) 6.94094 0.222060 0.111030 0.993817i \(-0.464585\pi\)
0.111030 + 0.993817i \(0.464585\pi\)
\(978\) −12.7115 −0.406468
\(979\) −42.0585 −1.34420
\(980\) 0.198051 0.00632650
\(981\) −29.3607 −0.937414
\(982\) −16.6174 −0.530283
\(983\) −29.7102 −0.947607 −0.473804 0.880631i \(-0.657119\pi\)
−0.473804 + 0.880631i \(0.657119\pi\)
\(984\) −6.30428 −0.200973
\(985\) 21.1504 0.673907
\(986\) 2.10579 0.0670620
\(987\) −6.84879 −0.217999
\(988\) 15.1139 0.480836
\(989\) −8.95780 −0.284842
\(990\) 22.3086 0.709013
\(991\) −6.54877 −0.208028 −0.104014 0.994576i \(-0.533169\pi\)
−0.104014 + 0.994576i \(0.533169\pi\)
\(992\) 49.5986 1.57476
\(993\) −11.1748 −0.354622
\(994\) −15.6320 −0.495817
\(995\) 20.9481 0.664101
\(996\) −32.3353 −1.02458
\(997\) −7.99707 −0.253270 −0.126635 0.991949i \(-0.540418\pi\)
−0.126635 + 0.991949i \(0.540418\pi\)
\(998\) 0.568609 0.0179990
\(999\) −4.29355 −0.135842
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 4031.2.a.c.1.38 61
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4031.2.a.c.1.38 61 1.1 even 1 trivial