Properties

Label 8043.2.a.t.1.32
Level $8043$
Weight $2$
Character 8043.1
Self dual yes
Analytic conductor $64.224$
Analytic rank $0$
Dimension $52$
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] = [8043,2,Mod(1,8043)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8043, 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("8043.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8043 = 3 \cdot 7 \cdot 383 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8043.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: \(64.2236783457\)
Analytic rank: \(0\)
Dimension: \(52\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.32
Character \(\chi\) \(=\) 8043.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.615474 q^{2} -1.00000 q^{3} -1.62119 q^{4} -1.47525 q^{5} -0.615474 q^{6} +1.00000 q^{7} -2.22875 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.615474 q^{2} -1.00000 q^{3} -1.62119 q^{4} -1.47525 q^{5} -0.615474 q^{6} +1.00000 q^{7} -2.22875 q^{8} +1.00000 q^{9} -0.907978 q^{10} +2.65989 q^{11} +1.62119 q^{12} +3.14729 q^{13} +0.615474 q^{14} +1.47525 q^{15} +1.87064 q^{16} +6.74505 q^{17} +0.615474 q^{18} +5.14394 q^{19} +2.39166 q^{20} -1.00000 q^{21} +1.63710 q^{22} +7.40766 q^{23} +2.22875 q^{24} -2.82364 q^{25} +1.93708 q^{26} -1.00000 q^{27} -1.62119 q^{28} -2.09296 q^{29} +0.907978 q^{30} -0.306892 q^{31} +5.60883 q^{32} -2.65989 q^{33} +4.15140 q^{34} -1.47525 q^{35} -1.62119 q^{36} -7.93979 q^{37} +3.16596 q^{38} -3.14729 q^{39} +3.28796 q^{40} -1.15383 q^{41} -0.615474 q^{42} +8.15649 q^{43} -4.31220 q^{44} -1.47525 q^{45} +4.55922 q^{46} +4.92668 q^{47} -1.87064 q^{48} +1.00000 q^{49} -1.73788 q^{50} -6.74505 q^{51} -5.10236 q^{52} +1.35908 q^{53} -0.615474 q^{54} -3.92401 q^{55} -2.22875 q^{56} -5.14394 q^{57} -1.28816 q^{58} +5.14225 q^{59} -2.39166 q^{60} -3.38440 q^{61} -0.188884 q^{62} +1.00000 q^{63} -0.289193 q^{64} -4.64304 q^{65} -1.63710 q^{66} -11.8164 q^{67} -10.9350 q^{68} -7.40766 q^{69} -0.907978 q^{70} +8.99642 q^{71} -2.22875 q^{72} -11.9414 q^{73} -4.88674 q^{74} +2.82364 q^{75} -8.33931 q^{76} +2.65989 q^{77} -1.93708 q^{78} +3.67334 q^{79} -2.75967 q^{80} +1.00000 q^{81} -0.710150 q^{82} -7.07921 q^{83} +1.62119 q^{84} -9.95063 q^{85} +5.02011 q^{86} +2.09296 q^{87} -5.92824 q^{88} -15.1570 q^{89} -0.907978 q^{90} +3.14729 q^{91} -12.0092 q^{92} +0.306892 q^{93} +3.03225 q^{94} -7.58859 q^{95} -5.60883 q^{96} +16.8992 q^{97} +0.615474 q^{98} +2.65989 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.615474 0.435206 0.217603 0.976037i \(-0.430176\pi\)
0.217603 + 0.976037i \(0.430176\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.62119 −0.810596
\(5\) −1.47525 −0.659752 −0.329876 0.944024i \(-0.607007\pi\)
−0.329876 + 0.944024i \(0.607007\pi\)
\(6\) −0.615474 −0.251266
\(7\) 1.00000 0.377964
\(8\) −2.22875 −0.787982
\(9\) 1.00000 0.333333
\(10\) −0.907978 −0.287128
\(11\) 2.65989 0.801988 0.400994 0.916081i \(-0.368665\pi\)
0.400994 + 0.916081i \(0.368665\pi\)
\(12\) 1.62119 0.467998
\(13\) 3.14729 0.872901 0.436451 0.899728i \(-0.356235\pi\)
0.436451 + 0.899728i \(0.356235\pi\)
\(14\) 0.615474 0.164492
\(15\) 1.47525 0.380908
\(16\) 1.87064 0.467661
\(17\) 6.74505 1.63591 0.817957 0.575279i \(-0.195106\pi\)
0.817957 + 0.575279i \(0.195106\pi\)
\(18\) 0.615474 0.145069
\(19\) 5.14394 1.18010 0.590050 0.807367i \(-0.299108\pi\)
0.590050 + 0.807367i \(0.299108\pi\)
\(20\) 2.39166 0.534792
\(21\) −1.00000 −0.218218
\(22\) 1.63710 0.349030
\(23\) 7.40766 1.54460 0.772301 0.635256i \(-0.219106\pi\)
0.772301 + 0.635256i \(0.219106\pi\)
\(24\) 2.22875 0.454942
\(25\) −2.82364 −0.564728
\(26\) 1.93708 0.379892
\(27\) −1.00000 −0.192450
\(28\) −1.62119 −0.306376
\(29\) −2.09296 −0.388652 −0.194326 0.980937i \(-0.562252\pi\)
−0.194326 + 0.980937i \(0.562252\pi\)
\(30\) 0.907978 0.165773
\(31\) −0.306892 −0.0551193 −0.0275597 0.999620i \(-0.508774\pi\)
−0.0275597 + 0.999620i \(0.508774\pi\)
\(32\) 5.60883 0.991511
\(33\) −2.65989 −0.463028
\(34\) 4.15140 0.711960
\(35\) −1.47525 −0.249363
\(36\) −1.62119 −0.270199
\(37\) −7.93979 −1.30529 −0.652647 0.757662i \(-0.726341\pi\)
−0.652647 + 0.757662i \(0.726341\pi\)
\(38\) 3.16596 0.513587
\(39\) −3.14729 −0.503970
\(40\) 3.28796 0.519873
\(41\) −1.15383 −0.180197 −0.0900986 0.995933i \(-0.528718\pi\)
−0.0900986 + 0.995933i \(0.528718\pi\)
\(42\) −0.615474 −0.0949698
\(43\) 8.15649 1.24385 0.621927 0.783075i \(-0.286350\pi\)
0.621927 + 0.783075i \(0.286350\pi\)
\(44\) −4.31220 −0.650088
\(45\) −1.47525 −0.219917
\(46\) 4.55922 0.672221
\(47\) 4.92668 0.718631 0.359315 0.933216i \(-0.383010\pi\)
0.359315 + 0.933216i \(0.383010\pi\)
\(48\) −1.87064 −0.270004
\(49\) 1.00000 0.142857
\(50\) −1.73788 −0.245773
\(51\) −6.74505 −0.944496
\(52\) −5.10236 −0.707570
\(53\) 1.35908 0.186684 0.0933418 0.995634i \(-0.470245\pi\)
0.0933418 + 0.995634i \(0.470245\pi\)
\(54\) −0.615474 −0.0837555
\(55\) −3.92401 −0.529113
\(56\) −2.22875 −0.297829
\(57\) −5.14394 −0.681331
\(58\) −1.28816 −0.169144
\(59\) 5.14225 0.669464 0.334732 0.942313i \(-0.391354\pi\)
0.334732 + 0.942313i \(0.391354\pi\)
\(60\) −2.39166 −0.308762
\(61\) −3.38440 −0.433328 −0.216664 0.976246i \(-0.569518\pi\)
−0.216664 + 0.976246i \(0.569518\pi\)
\(62\) −0.188884 −0.0239883
\(63\) 1.00000 0.125988
\(64\) −0.289193 −0.0361492
\(65\) −4.64304 −0.575898
\(66\) −1.63710 −0.201513
\(67\) −11.8164 −1.44360 −0.721801 0.692101i \(-0.756685\pi\)
−0.721801 + 0.692101i \(0.756685\pi\)
\(68\) −10.9350 −1.32607
\(69\) −7.40766 −0.891777
\(70\) −0.907978 −0.108524
\(71\) 8.99642 1.06768 0.533839 0.845586i \(-0.320749\pi\)
0.533839 + 0.845586i \(0.320749\pi\)
\(72\) −2.22875 −0.262661
\(73\) −11.9414 −1.39763 −0.698815 0.715303i \(-0.746289\pi\)
−0.698815 + 0.715303i \(0.746289\pi\)
\(74\) −4.88674 −0.568072
\(75\) 2.82364 0.326046
\(76\) −8.33931 −0.956584
\(77\) 2.65989 0.303123
\(78\) −1.93708 −0.219331
\(79\) 3.67334 0.413283 0.206642 0.978417i \(-0.433747\pi\)
0.206642 + 0.978417i \(0.433747\pi\)
\(80\) −2.75967 −0.308540
\(81\) 1.00000 0.111111
\(82\) −0.710150 −0.0784230
\(83\) −7.07921 −0.777044 −0.388522 0.921439i \(-0.627014\pi\)
−0.388522 + 0.921439i \(0.627014\pi\)
\(84\) 1.62119 0.176886
\(85\) −9.95063 −1.07930
\(86\) 5.02011 0.541333
\(87\) 2.09296 0.224388
\(88\) −5.92824 −0.631953
\(89\) −15.1570 −1.60664 −0.803321 0.595547i \(-0.796936\pi\)
−0.803321 + 0.595547i \(0.796936\pi\)
\(90\) −0.907978 −0.0957093
\(91\) 3.14729 0.329926
\(92\) −12.0092 −1.25205
\(93\) 0.306892 0.0318232
\(94\) 3.03225 0.312752
\(95\) −7.58859 −0.778573
\(96\) −5.60883 −0.572449
\(97\) 16.8992 1.71585 0.857926 0.513773i \(-0.171753\pi\)
0.857926 + 0.513773i \(0.171753\pi\)
\(98\) 0.615474 0.0621723
\(99\) 2.65989 0.267329
\(100\) 4.57766 0.457766
\(101\) 7.70641 0.766816 0.383408 0.923579i \(-0.374750\pi\)
0.383408 + 0.923579i \(0.374750\pi\)
\(102\) −4.15140 −0.411050
\(103\) 13.1014 1.29092 0.645462 0.763792i \(-0.276665\pi\)
0.645462 + 0.763792i \(0.276665\pi\)
\(104\) −7.01452 −0.687831
\(105\) 1.47525 0.143970
\(106\) 0.836477 0.0812459
\(107\) 11.4290 1.10488 0.552442 0.833551i \(-0.313696\pi\)
0.552442 + 0.833551i \(0.313696\pi\)
\(108\) 1.62119 0.155999
\(109\) 8.24223 0.789463 0.394731 0.918797i \(-0.370838\pi\)
0.394731 + 0.918797i \(0.370838\pi\)
\(110\) −2.41513 −0.230273
\(111\) 7.93979 0.753612
\(112\) 1.87064 0.176759
\(113\) −8.02930 −0.755333 −0.377667 0.925942i \(-0.623273\pi\)
−0.377667 + 0.925942i \(0.623273\pi\)
\(114\) −3.16596 −0.296519
\(115\) −10.9281 −1.01905
\(116\) 3.39308 0.315040
\(117\) 3.14729 0.290967
\(118\) 3.16492 0.291355
\(119\) 6.74505 0.618318
\(120\) −3.28796 −0.300149
\(121\) −3.92496 −0.356815
\(122\) −2.08301 −0.188587
\(123\) 1.15383 0.104037
\(124\) 0.497530 0.0446795
\(125\) 11.5418 1.03233
\(126\) 0.615474 0.0548308
\(127\) 1.41454 0.125520 0.0627600 0.998029i \(-0.480010\pi\)
0.0627600 + 0.998029i \(0.480010\pi\)
\(128\) −11.3957 −1.00724
\(129\) −8.15649 −0.718139
\(130\) −2.85767 −0.250634
\(131\) −0.675981 −0.0590608 −0.0295304 0.999564i \(-0.509401\pi\)
−0.0295304 + 0.999564i \(0.509401\pi\)
\(132\) 4.31220 0.375329
\(133\) 5.14394 0.446036
\(134\) −7.27268 −0.628264
\(135\) 1.47525 0.126969
\(136\) −15.0330 −1.28907
\(137\) −21.5180 −1.83841 −0.919205 0.393780i \(-0.871167\pi\)
−0.919205 + 0.393780i \(0.871167\pi\)
\(138\) −4.55922 −0.388107
\(139\) 8.19044 0.694704 0.347352 0.937735i \(-0.387081\pi\)
0.347352 + 0.937735i \(0.387081\pi\)
\(140\) 2.39166 0.202132
\(141\) −4.92668 −0.414902
\(142\) 5.53706 0.464660
\(143\) 8.37146 0.700057
\(144\) 1.87064 0.155887
\(145\) 3.08763 0.256414
\(146\) −7.34960 −0.608257
\(147\) −1.00000 −0.0824786
\(148\) 12.8719 1.05807
\(149\) −9.02378 −0.739257 −0.369629 0.929180i \(-0.620515\pi\)
−0.369629 + 0.929180i \(0.620515\pi\)
\(150\) 1.73788 0.141897
\(151\) −1.10308 −0.0897674 −0.0448837 0.998992i \(-0.514292\pi\)
−0.0448837 + 0.998992i \(0.514292\pi\)
\(152\) −11.4646 −0.929898
\(153\) 6.74505 0.545305
\(154\) 1.63710 0.131921
\(155\) 0.452742 0.0363651
\(156\) 5.10236 0.408516
\(157\) −13.5279 −1.07964 −0.539821 0.841780i \(-0.681508\pi\)
−0.539821 + 0.841780i \(0.681508\pi\)
\(158\) 2.26085 0.179863
\(159\) −1.35908 −0.107782
\(160\) −8.27443 −0.654151
\(161\) 7.40766 0.583805
\(162\) 0.615474 0.0483562
\(163\) 19.5088 1.52804 0.764022 0.645190i \(-0.223222\pi\)
0.764022 + 0.645190i \(0.223222\pi\)
\(164\) 1.87057 0.146067
\(165\) 3.92401 0.305484
\(166\) −4.35707 −0.338174
\(167\) −11.5436 −0.893271 −0.446635 0.894716i \(-0.647378\pi\)
−0.446635 + 0.894716i \(0.647378\pi\)
\(168\) 2.22875 0.171952
\(169\) −3.09457 −0.238044
\(170\) −6.12436 −0.469717
\(171\) 5.14394 0.393367
\(172\) −13.2232 −1.00826
\(173\) 5.54915 0.421894 0.210947 0.977497i \(-0.432345\pi\)
0.210947 + 0.977497i \(0.432345\pi\)
\(174\) 1.28816 0.0976552
\(175\) −2.82364 −0.213447
\(176\) 4.97571 0.375059
\(177\) −5.14225 −0.386515
\(178\) −9.32876 −0.699220
\(179\) 10.1298 0.757140 0.378570 0.925573i \(-0.376416\pi\)
0.378570 + 0.925573i \(0.376416\pi\)
\(180\) 2.39166 0.178264
\(181\) −14.0856 −1.04698 −0.523489 0.852032i \(-0.675370\pi\)
−0.523489 + 0.852032i \(0.675370\pi\)
\(182\) 1.93708 0.143586
\(183\) 3.38440 0.250182
\(184\) −16.5098 −1.21712
\(185\) 11.7132 0.861170
\(186\) 0.188884 0.0138496
\(187\) 17.9411 1.31198
\(188\) −7.98710 −0.582519
\(189\) −1.00000 −0.0727393
\(190\) −4.67058 −0.338840
\(191\) −7.70439 −0.557470 −0.278735 0.960368i \(-0.589915\pi\)
−0.278735 + 0.960368i \(0.589915\pi\)
\(192\) 0.289193 0.0208707
\(193\) 12.7262 0.916053 0.458027 0.888939i \(-0.348556\pi\)
0.458027 + 0.888939i \(0.348556\pi\)
\(194\) 10.4010 0.746749
\(195\) 4.64304 0.332495
\(196\) −1.62119 −0.115799
\(197\) 14.4855 1.03205 0.516026 0.856573i \(-0.327411\pi\)
0.516026 + 0.856573i \(0.327411\pi\)
\(198\) 1.63710 0.116343
\(199\) 13.5022 0.957145 0.478573 0.878048i \(-0.341154\pi\)
0.478573 + 0.878048i \(0.341154\pi\)
\(200\) 6.29319 0.444996
\(201\) 11.8164 0.833464
\(202\) 4.74310 0.333723
\(203\) −2.09296 −0.146897
\(204\) 10.9350 0.765604
\(205\) 1.70218 0.118885
\(206\) 8.06361 0.561818
\(207\) 7.40766 0.514868
\(208\) 5.88746 0.408222
\(209\) 13.6823 0.946427
\(210\) 0.907978 0.0626565
\(211\) −2.24176 −0.154329 −0.0771647 0.997018i \(-0.524587\pi\)
−0.0771647 + 0.997018i \(0.524587\pi\)
\(212\) −2.20332 −0.151325
\(213\) −8.99642 −0.616424
\(214\) 7.03427 0.480853
\(215\) −12.0329 −0.820634
\(216\) 2.22875 0.151647
\(217\) −0.306892 −0.0208332
\(218\) 5.07288 0.343579
\(219\) 11.9414 0.806922
\(220\) 6.36157 0.428897
\(221\) 21.2286 1.42799
\(222\) 4.88674 0.327976
\(223\) −21.3518 −1.42982 −0.714910 0.699216i \(-0.753532\pi\)
−0.714910 + 0.699216i \(0.753532\pi\)
\(224\) 5.60883 0.374756
\(225\) −2.82364 −0.188243
\(226\) −4.94183 −0.328726
\(227\) 24.1739 1.60448 0.802238 0.597004i \(-0.203642\pi\)
0.802238 + 0.597004i \(0.203642\pi\)
\(228\) 8.33931 0.552284
\(229\) 26.4007 1.74461 0.872303 0.488967i \(-0.162626\pi\)
0.872303 + 0.488967i \(0.162626\pi\)
\(230\) −6.72599 −0.443499
\(231\) −2.65989 −0.175008
\(232\) 4.66468 0.306251
\(233\) −0.670690 −0.0439383 −0.0219692 0.999759i \(-0.506994\pi\)
−0.0219692 + 0.999759i \(0.506994\pi\)
\(234\) 1.93708 0.126631
\(235\) −7.26809 −0.474118
\(236\) −8.33657 −0.542664
\(237\) −3.67334 −0.238609
\(238\) 4.15140 0.269096
\(239\) 14.6314 0.946425 0.473212 0.880948i \(-0.343094\pi\)
0.473212 + 0.880948i \(0.343094\pi\)
\(240\) 2.75967 0.178136
\(241\) 22.1570 1.42726 0.713629 0.700524i \(-0.247050\pi\)
0.713629 + 0.700524i \(0.247050\pi\)
\(242\) −2.41571 −0.155288
\(243\) −1.00000 −0.0641500
\(244\) 5.48675 0.351253
\(245\) −1.47525 −0.0942502
\(246\) 0.710150 0.0452775
\(247\) 16.1895 1.03011
\(248\) 0.683985 0.0434331
\(249\) 7.07921 0.448627
\(250\) 7.10369 0.449277
\(251\) −29.0040 −1.83071 −0.915357 0.402642i \(-0.868092\pi\)
−0.915357 + 0.402642i \(0.868092\pi\)
\(252\) −1.62119 −0.102125
\(253\) 19.7036 1.23875
\(254\) 0.870612 0.0546271
\(255\) 9.95063 0.623133
\(256\) −6.43535 −0.402209
\(257\) 29.6195 1.84761 0.923807 0.382858i \(-0.125060\pi\)
0.923807 + 0.382858i \(0.125060\pi\)
\(258\) −5.02011 −0.312539
\(259\) −7.93979 −0.493355
\(260\) 7.52725 0.466820
\(261\) −2.09296 −0.129551
\(262\) −0.416049 −0.0257036
\(263\) −21.2369 −1.30952 −0.654761 0.755836i \(-0.727231\pi\)
−0.654761 + 0.755836i \(0.727231\pi\)
\(264\) 5.92824 0.364858
\(265\) −2.00498 −0.123165
\(266\) 3.16596 0.194118
\(267\) 15.1570 0.927595
\(268\) 19.1566 1.17018
\(269\) −18.5699 −1.13223 −0.566113 0.824327i \(-0.691554\pi\)
−0.566113 + 0.824327i \(0.691554\pi\)
\(270\) 0.907978 0.0552578
\(271\) 16.9097 1.02719 0.513595 0.858033i \(-0.328313\pi\)
0.513595 + 0.858033i \(0.328313\pi\)
\(272\) 12.6176 0.765053
\(273\) −3.14729 −0.190483
\(274\) −13.2438 −0.800087
\(275\) −7.51058 −0.452905
\(276\) 12.0092 0.722870
\(277\) 25.6380 1.54044 0.770218 0.637781i \(-0.220148\pi\)
0.770218 + 0.637781i \(0.220148\pi\)
\(278\) 5.04100 0.302339
\(279\) −0.306892 −0.0183731
\(280\) 3.28796 0.196493
\(281\) −2.88827 −0.172300 −0.0861499 0.996282i \(-0.527456\pi\)
−0.0861499 + 0.996282i \(0.527456\pi\)
\(282\) −3.03225 −0.180568
\(283\) 12.7424 0.757460 0.378730 0.925507i \(-0.376361\pi\)
0.378730 + 0.925507i \(0.376361\pi\)
\(284\) −14.5849 −0.865455
\(285\) 7.58859 0.449509
\(286\) 5.15242 0.304669
\(287\) −1.15383 −0.0681082
\(288\) 5.60883 0.330504
\(289\) 28.4957 1.67622
\(290\) 1.90036 0.111593
\(291\) −16.8992 −0.990647
\(292\) 19.3592 1.13291
\(293\) −27.3377 −1.59708 −0.798542 0.601940i \(-0.794395\pi\)
−0.798542 + 0.601940i \(0.794395\pi\)
\(294\) −0.615474 −0.0358952
\(295\) −7.58610 −0.441680
\(296\) 17.6958 1.02855
\(297\) −2.65989 −0.154343
\(298\) −5.55391 −0.321729
\(299\) 23.3140 1.34829
\(300\) −4.57766 −0.264291
\(301\) 8.15649 0.470132
\(302\) −0.678918 −0.0390673
\(303\) −7.70641 −0.442722
\(304\) 9.62247 0.551887
\(305\) 4.99283 0.285889
\(306\) 4.15140 0.237320
\(307\) −9.30967 −0.531331 −0.265665 0.964065i \(-0.585592\pi\)
−0.265665 + 0.964065i \(0.585592\pi\)
\(308\) −4.31220 −0.245710
\(309\) −13.1014 −0.745315
\(310\) 0.278651 0.0158263
\(311\) −4.29747 −0.243687 −0.121844 0.992549i \(-0.538881\pi\)
−0.121844 + 0.992549i \(0.538881\pi\)
\(312\) 7.01452 0.397119
\(313\) −25.4078 −1.43613 −0.718067 0.695974i \(-0.754973\pi\)
−0.718067 + 0.695974i \(0.754973\pi\)
\(314\) −8.32606 −0.469867
\(315\) −1.47525 −0.0831209
\(316\) −5.95519 −0.335006
\(317\) −0.0583539 −0.00327748 −0.00163874 0.999999i \(-0.500522\pi\)
−0.00163874 + 0.999999i \(0.500522\pi\)
\(318\) −0.836477 −0.0469073
\(319\) −5.56704 −0.311695
\(320\) 0.426632 0.0238495
\(321\) −11.4290 −0.637906
\(322\) 4.55922 0.254076
\(323\) 34.6961 1.93054
\(324\) −1.62119 −0.0900662
\(325\) −8.88681 −0.492952
\(326\) 12.0071 0.665014
\(327\) −8.24223 −0.455797
\(328\) 2.57159 0.141992
\(329\) 4.92668 0.271617
\(330\) 2.41513 0.132948
\(331\) 13.5910 0.747027 0.373513 0.927625i \(-0.378153\pi\)
0.373513 + 0.927625i \(0.378153\pi\)
\(332\) 11.4767 0.629868
\(333\) −7.93979 −0.435098
\(334\) −7.10479 −0.388757
\(335\) 17.4321 0.952418
\(336\) −1.87064 −0.102052
\(337\) 20.1155 1.09576 0.547880 0.836557i \(-0.315435\pi\)
0.547880 + 0.836557i \(0.315435\pi\)
\(338\) −1.90463 −0.103598
\(339\) 8.02930 0.436092
\(340\) 16.1319 0.874874
\(341\) −0.816299 −0.0442051
\(342\) 3.16596 0.171196
\(343\) 1.00000 0.0539949
\(344\) −18.1788 −0.980135
\(345\) 10.9281 0.588351
\(346\) 3.41536 0.183611
\(347\) −22.1518 −1.18917 −0.594584 0.804033i \(-0.702683\pi\)
−0.594584 + 0.804033i \(0.702683\pi\)
\(348\) −3.39308 −0.181888
\(349\) −0.913744 −0.0489116 −0.0244558 0.999701i \(-0.507785\pi\)
−0.0244558 + 0.999701i \(0.507785\pi\)
\(350\) −1.73788 −0.0928935
\(351\) −3.14729 −0.167990
\(352\) 14.9189 0.795180
\(353\) −21.2068 −1.12873 −0.564363 0.825527i \(-0.690878\pi\)
−0.564363 + 0.825527i \(0.690878\pi\)
\(354\) −3.16492 −0.168214
\(355\) −13.2720 −0.704402
\(356\) 24.5724 1.30234
\(357\) −6.74505 −0.356986
\(358\) 6.23466 0.329512
\(359\) −19.2109 −1.01391 −0.506956 0.861972i \(-0.669229\pi\)
−0.506956 + 0.861972i \(0.669229\pi\)
\(360\) 3.28796 0.173291
\(361\) 7.46009 0.392636
\(362\) −8.66935 −0.455651
\(363\) 3.92496 0.206007
\(364\) −5.10236 −0.267436
\(365\) 17.6165 0.922088
\(366\) 2.08301 0.108881
\(367\) −23.6552 −1.23479 −0.617396 0.786653i \(-0.711812\pi\)
−0.617396 + 0.786653i \(0.711812\pi\)
\(368\) 13.8571 0.722350
\(369\) −1.15383 −0.0600658
\(370\) 7.20916 0.374786
\(371\) 1.35908 0.0705598
\(372\) −0.497530 −0.0257957
\(373\) −7.47377 −0.386977 −0.193488 0.981103i \(-0.561980\pi\)
−0.193488 + 0.981103i \(0.561980\pi\)
\(374\) 11.0423 0.570984
\(375\) −11.5418 −0.596017
\(376\) −10.9803 −0.566268
\(377\) −6.58714 −0.339255
\(378\) −0.615474 −0.0316566
\(379\) 22.8644 1.17446 0.587232 0.809419i \(-0.300218\pi\)
0.587232 + 0.809419i \(0.300218\pi\)
\(380\) 12.3026 0.631108
\(381\) −1.41454 −0.0724690
\(382\) −4.74186 −0.242615
\(383\) −1.00000 −0.0510976
\(384\) 11.3957 0.581532
\(385\) −3.92401 −0.199986
\(386\) 7.83266 0.398672
\(387\) 8.15649 0.414618
\(388\) −27.3968 −1.39086
\(389\) 9.48152 0.480732 0.240366 0.970682i \(-0.422733\pi\)
0.240366 + 0.970682i \(0.422733\pi\)
\(390\) 2.85767 0.144704
\(391\) 49.9650 2.52684
\(392\) −2.22875 −0.112569
\(393\) 0.675981 0.0340987
\(394\) 8.91547 0.449155
\(395\) −5.41910 −0.272664
\(396\) −4.31220 −0.216696
\(397\) −3.05615 −0.153384 −0.0766919 0.997055i \(-0.524436\pi\)
−0.0766919 + 0.997055i \(0.524436\pi\)
\(398\) 8.31026 0.416555
\(399\) −5.14394 −0.257519
\(400\) −5.28202 −0.264101
\(401\) 32.1996 1.60797 0.803985 0.594650i \(-0.202709\pi\)
0.803985 + 0.594650i \(0.202709\pi\)
\(402\) 7.27268 0.362729
\(403\) −0.965877 −0.0481137
\(404\) −12.4936 −0.621578
\(405\) −1.47525 −0.0733057
\(406\) −1.28816 −0.0639304
\(407\) −21.1190 −1.04683
\(408\) 15.0330 0.744246
\(409\) 22.2879 1.10206 0.551032 0.834484i \(-0.314234\pi\)
0.551032 + 0.834484i \(0.314234\pi\)
\(410\) 1.04765 0.0517397
\(411\) 21.5180 1.06141
\(412\) −21.2400 −1.04642
\(413\) 5.14225 0.253033
\(414\) 4.55922 0.224074
\(415\) 10.4436 0.512656
\(416\) 17.6526 0.865491
\(417\) −8.19044 −0.401087
\(418\) 8.42112 0.411891
\(419\) −32.6032 −1.59277 −0.796385 0.604790i \(-0.793257\pi\)
−0.796385 + 0.604790i \(0.793257\pi\)
\(420\) −2.39166 −0.116701
\(421\) 0.957212 0.0466517 0.0233258 0.999728i \(-0.492574\pi\)
0.0233258 + 0.999728i \(0.492574\pi\)
\(422\) −1.37975 −0.0671651
\(423\) 4.92668 0.239544
\(424\) −3.02904 −0.147103
\(425\) −19.0456 −0.923847
\(426\) −5.53706 −0.268272
\(427\) −3.38440 −0.163782
\(428\) −18.5286 −0.895615
\(429\) −8.37146 −0.404178
\(430\) −7.40592 −0.357145
\(431\) 14.5142 0.699122 0.349561 0.936914i \(-0.386331\pi\)
0.349561 + 0.936914i \(0.386331\pi\)
\(432\) −1.87064 −0.0900014
\(433\) −20.1370 −0.967721 −0.483860 0.875145i \(-0.660766\pi\)
−0.483860 + 0.875145i \(0.660766\pi\)
\(434\) −0.188884 −0.00906672
\(435\) −3.08763 −0.148041
\(436\) −13.3622 −0.639935
\(437\) 38.1045 1.82279
\(438\) 7.34960 0.351177
\(439\) −36.8477 −1.75865 −0.879323 0.476226i \(-0.842004\pi\)
−0.879323 + 0.476226i \(0.842004\pi\)
\(440\) 8.74563 0.416932
\(441\) 1.00000 0.0476190
\(442\) 13.0657 0.621471
\(443\) −10.6354 −0.505304 −0.252652 0.967557i \(-0.581303\pi\)
−0.252652 + 0.967557i \(0.581303\pi\)
\(444\) −12.8719 −0.610874
\(445\) 22.3604 1.05998
\(446\) −13.1415 −0.622266
\(447\) 9.02378 0.426810
\(448\) −0.289193 −0.0136631
\(449\) −27.7051 −1.30749 −0.653743 0.756716i \(-0.726802\pi\)
−0.653743 + 0.756716i \(0.726802\pi\)
\(450\) −1.73788 −0.0819243
\(451\) −3.06905 −0.144516
\(452\) 13.0170 0.612270
\(453\) 1.10308 0.0518272
\(454\) 14.8784 0.698278
\(455\) −4.64304 −0.217669
\(456\) 11.4646 0.536877
\(457\) −7.54514 −0.352947 −0.176473 0.984305i \(-0.556469\pi\)
−0.176473 + 0.984305i \(0.556469\pi\)
\(458\) 16.2489 0.759263
\(459\) −6.74505 −0.314832
\(460\) 17.7166 0.826041
\(461\) −36.0370 −1.67841 −0.839204 0.543816i \(-0.816979\pi\)
−0.839204 + 0.543816i \(0.816979\pi\)
\(462\) −1.63710 −0.0761646
\(463\) 12.5248 0.582078 0.291039 0.956711i \(-0.405999\pi\)
0.291039 + 0.956711i \(0.405999\pi\)
\(464\) −3.91518 −0.181757
\(465\) −0.452742 −0.0209954
\(466\) −0.412792 −0.0191222
\(467\) 11.0137 0.509651 0.254826 0.966987i \(-0.417982\pi\)
0.254826 + 0.966987i \(0.417982\pi\)
\(468\) −5.10236 −0.235857
\(469\) −11.8164 −0.545630
\(470\) −4.47332 −0.206339
\(471\) 13.5279 0.623332
\(472\) −11.4608 −0.527525
\(473\) 21.6954 0.997556
\(474\) −2.26085 −0.103844
\(475\) −14.5246 −0.666435
\(476\) −10.9350 −0.501206
\(477\) 1.35908 0.0622279
\(478\) 9.00524 0.411890
\(479\) 7.73058 0.353219 0.176610 0.984281i \(-0.443487\pi\)
0.176610 + 0.984281i \(0.443487\pi\)
\(480\) 8.27443 0.377674
\(481\) −24.9888 −1.13939
\(482\) 13.6371 0.621151
\(483\) −7.40766 −0.337060
\(484\) 6.36311 0.289232
\(485\) −24.9305 −1.13204
\(486\) −0.615474 −0.0279185
\(487\) 10.6857 0.484217 0.242108 0.970249i \(-0.422161\pi\)
0.242108 + 0.970249i \(0.422161\pi\)
\(488\) 7.54298 0.341454
\(489\) −19.5088 −0.882217
\(490\) −0.907978 −0.0410183
\(491\) 32.4946 1.46646 0.733230 0.679981i \(-0.238012\pi\)
0.733230 + 0.679981i \(0.238012\pi\)
\(492\) −1.87057 −0.0843319
\(493\) −14.1171 −0.635802
\(494\) 9.96420 0.448311
\(495\) −3.92401 −0.176371
\(496\) −0.574085 −0.0257772
\(497\) 8.99642 0.403544
\(498\) 4.35707 0.195245
\(499\) −5.90323 −0.264265 −0.132133 0.991232i \(-0.542182\pi\)
−0.132133 + 0.991232i \(0.542182\pi\)
\(500\) −18.7115 −0.836804
\(501\) 11.5436 0.515730
\(502\) −17.8512 −0.796738
\(503\) −5.10199 −0.227486 −0.113743 0.993510i \(-0.536284\pi\)
−0.113743 + 0.993510i \(0.536284\pi\)
\(504\) −2.22875 −0.0992764
\(505\) −11.3689 −0.505908
\(506\) 12.1270 0.539113
\(507\) 3.09457 0.137434
\(508\) −2.29324 −0.101746
\(509\) −6.50365 −0.288269 −0.144135 0.989558i \(-0.546040\pi\)
−0.144135 + 0.989558i \(0.546040\pi\)
\(510\) 6.12436 0.271191
\(511\) −11.9414 −0.528254
\(512\) 18.8305 0.832200
\(513\) −5.14394 −0.227110
\(514\) 18.2301 0.804093
\(515\) −19.3279 −0.851689
\(516\) 13.2232 0.582120
\(517\) 13.1045 0.576333
\(518\) −4.88674 −0.214711
\(519\) −5.54915 −0.243581
\(520\) 10.3482 0.453797
\(521\) 10.9167 0.478268 0.239134 0.970987i \(-0.423137\pi\)
0.239134 + 0.970987i \(0.423137\pi\)
\(522\) −1.28816 −0.0563813
\(523\) 19.0752 0.834101 0.417050 0.908883i \(-0.363064\pi\)
0.417050 + 0.908883i \(0.363064\pi\)
\(524\) 1.09589 0.0478744
\(525\) 2.82364 0.123234
\(526\) −13.0708 −0.569912
\(527\) −2.07000 −0.0901705
\(528\) −4.97571 −0.216540
\(529\) 31.8734 1.38580
\(530\) −1.23401 −0.0536021
\(531\) 5.14225 0.223155
\(532\) −8.33931 −0.361555
\(533\) −3.63142 −0.157294
\(534\) 9.32876 0.403695
\(535\) −16.8607 −0.728950
\(536\) 26.3358 1.13753
\(537\) −10.1298 −0.437135
\(538\) −11.4293 −0.492752
\(539\) 2.65989 0.114570
\(540\) −2.39166 −0.102921
\(541\) 32.5536 1.39959 0.699794 0.714345i \(-0.253275\pi\)
0.699794 + 0.714345i \(0.253275\pi\)
\(542\) 10.4075 0.447039
\(543\) 14.0856 0.604473
\(544\) 37.8319 1.62203
\(545\) −12.1594 −0.520849
\(546\) −1.93708 −0.0828992
\(547\) −26.9108 −1.15062 −0.575312 0.817934i \(-0.695119\pi\)
−0.575312 + 0.817934i \(0.695119\pi\)
\(548\) 34.8849 1.49021
\(549\) −3.38440 −0.144443
\(550\) −4.62257 −0.197107
\(551\) −10.7660 −0.458649
\(552\) 16.5098 0.702704
\(553\) 3.67334 0.156206
\(554\) 15.7795 0.670407
\(555\) −11.7132 −0.497196
\(556\) −13.2783 −0.563124
\(557\) 42.8815 1.81695 0.908475 0.417940i \(-0.137248\pi\)
0.908475 + 0.417940i \(0.137248\pi\)
\(558\) −0.188884 −0.00799609
\(559\) 25.6708 1.08576
\(560\) −2.75967 −0.116617
\(561\) −17.9411 −0.757475
\(562\) −1.77766 −0.0749859
\(563\) 40.4360 1.70418 0.852088 0.523399i \(-0.175336\pi\)
0.852088 + 0.523399i \(0.175336\pi\)
\(564\) 7.98710 0.336317
\(565\) 11.8452 0.498332
\(566\) 7.84265 0.329651
\(567\) 1.00000 0.0419961
\(568\) −20.0508 −0.841312
\(569\) 33.2826 1.39528 0.697641 0.716448i \(-0.254233\pi\)
0.697641 + 0.716448i \(0.254233\pi\)
\(570\) 4.67058 0.195629
\(571\) 9.69896 0.405889 0.202945 0.979190i \(-0.434949\pi\)
0.202945 + 0.979190i \(0.434949\pi\)
\(572\) −13.5717 −0.567463
\(573\) 7.70439 0.321856
\(574\) −0.710150 −0.0296411
\(575\) −20.9165 −0.872280
\(576\) −0.289193 −0.0120497
\(577\) −45.1111 −1.87800 −0.939000 0.343918i \(-0.888246\pi\)
−0.939000 + 0.343918i \(0.888246\pi\)
\(578\) 17.5384 0.729500
\(579\) −12.7262 −0.528884
\(580\) −5.00564 −0.207848
\(581\) −7.07921 −0.293695
\(582\) −10.4010 −0.431136
\(583\) 3.61500 0.149718
\(584\) 26.6143 1.10131
\(585\) −4.64304 −0.191966
\(586\) −16.8256 −0.695060
\(587\) 38.7428 1.59909 0.799543 0.600609i \(-0.205075\pi\)
0.799543 + 0.600609i \(0.205075\pi\)
\(588\) 1.62119 0.0668568
\(589\) −1.57863 −0.0650463
\(590\) −4.66905 −0.192222
\(591\) −14.4855 −0.595855
\(592\) −14.8525 −0.610435
\(593\) −16.0695 −0.659897 −0.329949 0.943999i \(-0.607031\pi\)
−0.329949 + 0.943999i \(0.607031\pi\)
\(594\) −1.63710 −0.0671709
\(595\) −9.95063 −0.407936
\(596\) 14.6293 0.599239
\(597\) −13.5022 −0.552608
\(598\) 14.3492 0.586782
\(599\) −29.1502 −1.19104 −0.595522 0.803339i \(-0.703055\pi\)
−0.595522 + 0.803339i \(0.703055\pi\)
\(600\) −6.29319 −0.256918
\(601\) −10.1442 −0.413790 −0.206895 0.978363i \(-0.566336\pi\)
−0.206895 + 0.978363i \(0.566336\pi\)
\(602\) 5.02011 0.204605
\(603\) −11.8164 −0.481201
\(604\) 1.78830 0.0727651
\(605\) 5.79030 0.235409
\(606\) −4.74310 −0.192675
\(607\) −0.0952989 −0.00386806 −0.00193403 0.999998i \(-0.500616\pi\)
−0.00193403 + 0.999998i \(0.500616\pi\)
\(608\) 28.8515 1.17008
\(609\) 2.09296 0.0848109
\(610\) 3.07296 0.124420
\(611\) 15.5057 0.627294
\(612\) −10.9350 −0.442022
\(613\) −22.4925 −0.908464 −0.454232 0.890883i \(-0.650086\pi\)
−0.454232 + 0.890883i \(0.650086\pi\)
\(614\) −5.72986 −0.231238
\(615\) −1.70218 −0.0686385
\(616\) −5.92824 −0.238856
\(617\) 27.0522 1.08908 0.544541 0.838734i \(-0.316704\pi\)
0.544541 + 0.838734i \(0.316704\pi\)
\(618\) −8.06361 −0.324366
\(619\) 16.6874 0.670724 0.335362 0.942089i \(-0.391141\pi\)
0.335362 + 0.942089i \(0.391141\pi\)
\(620\) −0.733981 −0.0294774
\(621\) −7.40766 −0.297259
\(622\) −2.64498 −0.106054
\(623\) −15.1570 −0.607253
\(624\) −5.88746 −0.235687
\(625\) −2.90887 −0.116355
\(626\) −15.6379 −0.625014
\(627\) −13.6823 −0.546420
\(628\) 21.9313 0.875153
\(629\) −53.5543 −2.13535
\(630\) −0.907978 −0.0361747
\(631\) 8.82839 0.351453 0.175726 0.984439i \(-0.443773\pi\)
0.175726 + 0.984439i \(0.443773\pi\)
\(632\) −8.18696 −0.325660
\(633\) 2.24176 0.0891021
\(634\) −0.0359153 −0.00142638
\(635\) −2.08680 −0.0828120
\(636\) 2.20332 0.0873675
\(637\) 3.14729 0.124700
\(638\) −3.42637 −0.135651
\(639\) 8.99642 0.355893
\(640\) 16.8114 0.664531
\(641\) 46.9449 1.85421 0.927107 0.374797i \(-0.122288\pi\)
0.927107 + 0.374797i \(0.122288\pi\)
\(642\) −7.03427 −0.277620
\(643\) −5.38462 −0.212349 −0.106174 0.994348i \(-0.533860\pi\)
−0.106174 + 0.994348i \(0.533860\pi\)
\(644\) −12.0092 −0.473230
\(645\) 12.0329 0.473793
\(646\) 21.3546 0.840184
\(647\) −25.8377 −1.01579 −0.507893 0.861420i \(-0.669575\pi\)
−0.507893 + 0.861420i \(0.669575\pi\)
\(648\) −2.22875 −0.0875536
\(649\) 13.6778 0.536902
\(650\) −5.46960 −0.214536
\(651\) 0.306892 0.0120280
\(652\) −31.6274 −1.23863
\(653\) −16.0924 −0.629745 −0.314872 0.949134i \(-0.601962\pi\)
−0.314872 + 0.949134i \(0.601962\pi\)
\(654\) −5.07288 −0.198365
\(655\) 0.997241 0.0389654
\(656\) −2.15840 −0.0842712
\(657\) −11.9414 −0.465877
\(658\) 3.03225 0.118209
\(659\) −15.9445 −0.621108 −0.310554 0.950556i \(-0.600515\pi\)
−0.310554 + 0.950556i \(0.600515\pi\)
\(660\) −6.36157 −0.247624
\(661\) 20.9683 0.815573 0.407787 0.913077i \(-0.366301\pi\)
0.407787 + 0.913077i \(0.366301\pi\)
\(662\) 8.36489 0.325111
\(663\) −21.2286 −0.824451
\(664\) 15.7778 0.612297
\(665\) −7.58859 −0.294273
\(666\) −4.88674 −0.189357
\(667\) −15.5039 −0.600313
\(668\) 18.7144 0.724082
\(669\) 21.3518 0.825507
\(670\) 10.7290 0.414498
\(671\) −9.00214 −0.347524
\(672\) −5.60883 −0.216365
\(673\) −0.301084 −0.0116059 −0.00580297 0.999983i \(-0.501847\pi\)
−0.00580297 + 0.999983i \(0.501847\pi\)
\(674\) 12.3806 0.476881
\(675\) 2.82364 0.108682
\(676\) 5.01688 0.192957
\(677\) −2.71893 −0.104497 −0.0522484 0.998634i \(-0.516639\pi\)
−0.0522484 + 0.998634i \(0.516639\pi\)
\(678\) 4.94183 0.189790
\(679\) 16.8992 0.648531
\(680\) 22.1775 0.850467
\(681\) −24.1739 −0.926345
\(682\) −0.502411 −0.0192383
\(683\) 4.24150 0.162297 0.0811483 0.996702i \(-0.474141\pi\)
0.0811483 + 0.996702i \(0.474141\pi\)
\(684\) −8.33931 −0.318861
\(685\) 31.7445 1.21289
\(686\) 0.615474 0.0234989
\(687\) −26.4007 −1.00725
\(688\) 15.2579 0.581702
\(689\) 4.27741 0.162956
\(690\) 6.72599 0.256054
\(691\) 25.6308 0.975042 0.487521 0.873111i \(-0.337901\pi\)
0.487521 + 0.873111i \(0.337901\pi\)
\(692\) −8.99623 −0.341985
\(693\) 2.65989 0.101041
\(694\) −13.6338 −0.517534
\(695\) −12.0829 −0.458332
\(696\) −4.66468 −0.176814
\(697\) −7.78261 −0.294787
\(698\) −0.562386 −0.0212866
\(699\) 0.670690 0.0253678
\(700\) 4.57766 0.173019
\(701\) −27.4453 −1.03659 −0.518297 0.855201i \(-0.673434\pi\)
−0.518297 + 0.855201i \(0.673434\pi\)
\(702\) −1.93708 −0.0731102
\(703\) −40.8418 −1.54038
\(704\) −0.769224 −0.0289912
\(705\) 7.26809 0.273732
\(706\) −13.0523 −0.491228
\(707\) 7.70641 0.289829
\(708\) 8.33657 0.313307
\(709\) −26.7447 −1.00442 −0.502209 0.864746i \(-0.667479\pi\)
−0.502209 + 0.864746i \(0.667479\pi\)
\(710\) −8.16855 −0.306560
\(711\) 3.67334 0.137761
\(712\) 33.7812 1.26601
\(713\) −2.27335 −0.0851375
\(714\) −4.15140 −0.155362
\(715\) −12.3500 −0.461863
\(716\) −16.4224 −0.613734
\(717\) −14.6314 −0.546419
\(718\) −11.8238 −0.441261
\(719\) 20.0846 0.749028 0.374514 0.927221i \(-0.377810\pi\)
0.374514 + 0.927221i \(0.377810\pi\)
\(720\) −2.75967 −0.102847
\(721\) 13.1014 0.487923
\(722\) 4.59149 0.170878
\(723\) −22.1570 −0.824028
\(724\) 22.8355 0.848676
\(725\) 5.90975 0.219483
\(726\) 2.41571 0.0896555
\(727\) −6.01031 −0.222910 −0.111455 0.993769i \(-0.535551\pi\)
−0.111455 + 0.993769i \(0.535551\pi\)
\(728\) −7.01452 −0.259976
\(729\) 1.00000 0.0370370
\(730\) 10.8425 0.401298
\(731\) 55.0159 2.03484
\(732\) −5.48675 −0.202796
\(733\) −38.0210 −1.40434 −0.702169 0.712010i \(-0.747785\pi\)
−0.702169 + 0.712010i \(0.747785\pi\)
\(734\) −14.5592 −0.537389
\(735\) 1.47525 0.0544154
\(736\) 41.5483 1.53149
\(737\) −31.4303 −1.15775
\(738\) −0.710150 −0.0261410
\(739\) −2.83293 −0.104211 −0.0521056 0.998642i \(-0.516593\pi\)
−0.0521056 + 0.998642i \(0.516593\pi\)
\(740\) −18.9893 −0.698060
\(741\) −16.1895 −0.594735
\(742\) 0.836477 0.0307080
\(743\) −32.7158 −1.20023 −0.600114 0.799915i \(-0.704878\pi\)
−0.600114 + 0.799915i \(0.704878\pi\)
\(744\) −0.683985 −0.0250761
\(745\) 13.3123 0.487726
\(746\) −4.59991 −0.168415
\(747\) −7.07921 −0.259015
\(748\) −29.0860 −1.06349
\(749\) 11.4290 0.417607
\(750\) −7.10369 −0.259390
\(751\) 41.5043 1.51451 0.757256 0.653118i \(-0.226539\pi\)
0.757256 + 0.653118i \(0.226539\pi\)
\(752\) 9.21607 0.336075
\(753\) 29.0040 1.05696
\(754\) −4.05422 −0.147646
\(755\) 1.62732 0.0592242
\(756\) 1.62119 0.0589622
\(757\) 45.4278 1.65110 0.825551 0.564327i \(-0.190864\pi\)
0.825551 + 0.564327i \(0.190864\pi\)
\(758\) 14.0724 0.511134
\(759\) −19.7036 −0.715195
\(760\) 16.9131 0.613502
\(761\) −16.5363 −0.599439 −0.299720 0.954027i \(-0.596893\pi\)
−0.299720 + 0.954027i \(0.596893\pi\)
\(762\) −0.870612 −0.0315389
\(763\) 8.24223 0.298389
\(764\) 12.4903 0.451883
\(765\) −9.95063 −0.359766
\(766\) −0.615474 −0.0222380
\(767\) 16.1841 0.584376
\(768\) 6.43535 0.232216
\(769\) 26.2505 0.946618 0.473309 0.880897i \(-0.343059\pi\)
0.473309 + 0.880897i \(0.343059\pi\)
\(770\) −2.41513 −0.0870351
\(771\) −29.6195 −1.06672
\(772\) −20.6316 −0.742549
\(773\) 25.5123 0.917612 0.458806 0.888536i \(-0.348277\pi\)
0.458806 + 0.888536i \(0.348277\pi\)
\(774\) 5.02011 0.180444
\(775\) 0.866551 0.0311274
\(776\) −37.6641 −1.35206
\(777\) 7.93979 0.284838
\(778\) 5.83563 0.209217
\(779\) −5.93521 −0.212651
\(780\) −7.52725 −0.269519
\(781\) 23.9295 0.856266
\(782\) 30.7522 1.09970
\(783\) 2.09296 0.0747962
\(784\) 1.87064 0.0668087
\(785\) 19.9570 0.712296
\(786\) 0.416049 0.0148400
\(787\) −16.1606 −0.576062 −0.288031 0.957621i \(-0.593001\pi\)
−0.288031 + 0.957621i \(0.593001\pi\)
\(788\) −23.4838 −0.836576
\(789\) 21.2369 0.756053
\(790\) −3.33531 −0.118665
\(791\) −8.02930 −0.285489
\(792\) −5.92824 −0.210651
\(793\) −10.6517 −0.378252
\(794\) −1.88098 −0.0667536
\(795\) 2.00498 0.0711092
\(796\) −21.8896 −0.775858
\(797\) 36.2983 1.28575 0.642876 0.765971i \(-0.277741\pi\)
0.642876 + 0.765971i \(0.277741\pi\)
\(798\) −3.16596 −0.112074
\(799\) 33.2307 1.17562
\(800\) −15.8373 −0.559934
\(801\) −15.1570 −0.535547
\(802\) 19.8180 0.699798
\(803\) −31.7627 −1.12088
\(804\) −19.1566 −0.675602
\(805\) −10.9281 −0.385166
\(806\) −0.594472 −0.0209394
\(807\) 18.5699 0.653691
\(808\) −17.1757 −0.604238
\(809\) −13.1186 −0.461225 −0.230613 0.973046i \(-0.574073\pi\)
−0.230613 + 0.973046i \(0.574073\pi\)
\(810\) −0.907978 −0.0319031
\(811\) −2.48481 −0.0872533 −0.0436267 0.999048i \(-0.513891\pi\)
−0.0436267 + 0.999048i \(0.513891\pi\)
\(812\) 3.39308 0.119074
\(813\) −16.9097 −0.593048
\(814\) −12.9982 −0.455587
\(815\) −28.7803 −1.00813
\(816\) −12.6176 −0.441704
\(817\) 41.9565 1.46787
\(818\) 13.7176 0.479625
\(819\) 3.14729 0.109975
\(820\) −2.75956 −0.0963680
\(821\) −6.05824 −0.211434 −0.105717 0.994396i \(-0.533714\pi\)
−0.105717 + 0.994396i \(0.533714\pi\)
\(822\) 13.2438 0.461931
\(823\) 37.7463 1.31575 0.657876 0.753126i \(-0.271455\pi\)
0.657876 + 0.753126i \(0.271455\pi\)
\(824\) −29.1999 −1.01723
\(825\) 7.51058 0.261485
\(826\) 3.16492 0.110122
\(827\) −35.4685 −1.23336 −0.616680 0.787214i \(-0.711523\pi\)
−0.616680 + 0.787214i \(0.711523\pi\)
\(828\) −12.0092 −0.417349
\(829\) 48.9929 1.70159 0.850797 0.525495i \(-0.176120\pi\)
0.850797 + 0.525495i \(0.176120\pi\)
\(830\) 6.42777 0.223111
\(831\) −25.6380 −0.889371
\(832\) −0.910175 −0.0315546
\(833\) 6.74505 0.233702
\(834\) −5.04100 −0.174556
\(835\) 17.0297 0.589337
\(836\) −22.1817 −0.767169
\(837\) 0.306892 0.0106077
\(838\) −20.0664 −0.693183
\(839\) −6.83910 −0.236112 −0.118056 0.993007i \(-0.537666\pi\)
−0.118056 + 0.993007i \(0.537666\pi\)
\(840\) −3.28796 −0.113445
\(841\) −24.6195 −0.848949
\(842\) 0.589140 0.0203031
\(843\) 2.88827 0.0994773
\(844\) 3.63433 0.125099
\(845\) 4.56526 0.157050
\(846\) 3.03225 0.104251
\(847\) −3.92496 −0.134863
\(848\) 2.54235 0.0873046
\(849\) −12.7424 −0.437320
\(850\) −11.7221 −0.402064
\(851\) −58.8152 −2.01616
\(852\) 14.5849 0.499671
\(853\) 9.49202 0.325001 0.162500 0.986708i \(-0.448044\pi\)
0.162500 + 0.986708i \(0.448044\pi\)
\(854\) −2.08301 −0.0712791
\(855\) −7.58859 −0.259524
\(856\) −25.4724 −0.870630
\(857\) −1.40803 −0.0480972 −0.0240486 0.999711i \(-0.507656\pi\)
−0.0240486 + 0.999711i \(0.507656\pi\)
\(858\) −5.15242 −0.175901
\(859\) −46.5823 −1.58937 −0.794683 0.607024i \(-0.792363\pi\)
−0.794683 + 0.607024i \(0.792363\pi\)
\(860\) 19.5076 0.665203
\(861\) 1.15383 0.0393223
\(862\) 8.93309 0.304262
\(863\) 20.4439 0.695918 0.347959 0.937510i \(-0.386875\pi\)
0.347959 + 0.937510i \(0.386875\pi\)
\(864\) −5.60883 −0.190816
\(865\) −8.18638 −0.278345
\(866\) −12.3938 −0.421158
\(867\) −28.4957 −0.967764
\(868\) 0.497530 0.0168873
\(869\) 9.77070 0.331448
\(870\) −1.90036 −0.0644282
\(871\) −37.1896 −1.26012
\(872\) −18.3699 −0.622083
\(873\) 16.8992 0.571951
\(874\) 23.4524 0.793288
\(875\) 11.5418 0.390185
\(876\) −19.3592 −0.654087
\(877\) 49.0726 1.65707 0.828533 0.559941i \(-0.189176\pi\)
0.828533 + 0.559941i \(0.189176\pi\)
\(878\) −22.6788 −0.765373
\(879\) 27.3377 0.922076
\(880\) −7.34042 −0.247446
\(881\) −13.7137 −0.462026 −0.231013 0.972951i \(-0.574204\pi\)
−0.231013 + 0.972951i \(0.574204\pi\)
\(882\) 0.615474 0.0207241
\(883\) 12.0308 0.404869 0.202435 0.979296i \(-0.435115\pi\)
0.202435 + 0.979296i \(0.435115\pi\)
\(884\) −34.4157 −1.15752
\(885\) 7.58610 0.255004
\(886\) −6.54583 −0.219912
\(887\) 39.6259 1.33051 0.665253 0.746618i \(-0.268324\pi\)
0.665253 + 0.746618i \(0.268324\pi\)
\(888\) −17.6958 −0.593833
\(889\) 1.41454 0.0474421
\(890\) 13.7623 0.461312
\(891\) 2.65989 0.0891098
\(892\) 34.6153 1.15901
\(893\) 25.3425 0.848056
\(894\) 5.55391 0.185750
\(895\) −14.9440 −0.499524
\(896\) −11.3957 −0.380702
\(897\) −23.3140 −0.778433
\(898\) −17.0518 −0.569026
\(899\) 0.642311 0.0214223
\(900\) 4.57766 0.152589
\(901\) 9.16704 0.305398
\(902\) −1.88892 −0.0628943
\(903\) −8.15649 −0.271431
\(904\) 17.8953 0.595189
\(905\) 20.7798 0.690745
\(906\) 0.678918 0.0225555
\(907\) 3.14956 0.104579 0.0522897 0.998632i \(-0.483348\pi\)
0.0522897 + 0.998632i \(0.483348\pi\)
\(908\) −39.1905 −1.30058
\(909\) 7.70641 0.255605
\(910\) −2.85767 −0.0947309
\(911\) 14.0435 0.465281 0.232641 0.972563i \(-0.425263\pi\)
0.232641 + 0.972563i \(0.425263\pi\)
\(912\) −9.62247 −0.318632
\(913\) −18.8299 −0.623180
\(914\) −4.64384 −0.153605
\(915\) −4.99283 −0.165058
\(916\) −42.8005 −1.41417
\(917\) −0.675981 −0.0223229
\(918\) −4.15140 −0.137017
\(919\) 45.4700 1.49992 0.749959 0.661485i \(-0.230073\pi\)
0.749959 + 0.661485i \(0.230073\pi\)
\(920\) 24.3561 0.802997
\(921\) 9.30967 0.306764
\(922\) −22.1798 −0.730454
\(923\) 28.3143 0.931978
\(924\) 4.31220 0.141861
\(925\) 22.4191 0.737136
\(926\) 7.70872 0.253324
\(927\) 13.1014 0.430308
\(928\) −11.7390 −0.385353
\(929\) 6.63884 0.217813 0.108907 0.994052i \(-0.465265\pi\)
0.108907 + 0.994052i \(0.465265\pi\)
\(930\) −0.278651 −0.00913732
\(931\) 5.14394 0.168586
\(932\) 1.08732 0.0356162
\(933\) 4.29747 0.140693
\(934\) 6.77862 0.221803
\(935\) −26.4676 −0.865584
\(936\) −7.01452 −0.229277
\(937\) 31.9585 1.04404 0.522020 0.852933i \(-0.325179\pi\)
0.522020 + 0.852933i \(0.325179\pi\)
\(938\) −7.27268 −0.237462
\(939\) 25.4078 0.829152
\(940\) 11.7830 0.384318
\(941\) −16.6734 −0.543537 −0.271769 0.962363i \(-0.587609\pi\)
−0.271769 + 0.962363i \(0.587609\pi\)
\(942\) 8.32606 0.271278
\(943\) −8.54714 −0.278333
\(944\) 9.61931 0.313082
\(945\) 1.47525 0.0479899
\(946\) 13.3530 0.434142
\(947\) 3.91223 0.127130 0.0635651 0.997978i \(-0.479753\pi\)
0.0635651 + 0.997978i \(0.479753\pi\)
\(948\) 5.95519 0.193416
\(949\) −37.5829 −1.21999
\(950\) −8.93953 −0.290037
\(951\) 0.0583539 0.00189225
\(952\) −15.0330 −0.487223
\(953\) −27.6278 −0.894952 −0.447476 0.894296i \(-0.647677\pi\)
−0.447476 + 0.894296i \(0.647677\pi\)
\(954\) 0.836477 0.0270820
\(955\) 11.3659 0.367792
\(956\) −23.7203 −0.767168
\(957\) 5.56704 0.179957
\(958\) 4.75797 0.153723
\(959\) −21.5180 −0.694854
\(960\) −0.426632 −0.0137695
\(961\) −30.9058 −0.996962
\(962\) −15.3800 −0.495871
\(963\) 11.4290 0.368295
\(964\) −35.9207 −1.15693
\(965\) −18.7743 −0.604367
\(966\) −4.55922 −0.146691
\(967\) −16.6158 −0.534329 −0.267164 0.963651i \(-0.586087\pi\)
−0.267164 + 0.963651i \(0.586087\pi\)
\(968\) 8.74776 0.281164
\(969\) −34.6961 −1.11460
\(970\) −15.3441 −0.492669
\(971\) 50.4062 1.61761 0.808806 0.588076i \(-0.200114\pi\)
0.808806 + 0.588076i \(0.200114\pi\)
\(972\) 1.62119 0.0519997
\(973\) 8.19044 0.262573
\(974\) 6.57680 0.210734
\(975\) 8.88681 0.284606
\(976\) −6.33100 −0.202650
\(977\) −20.5800 −0.658413 −0.329207 0.944258i \(-0.606781\pi\)
−0.329207 + 0.944258i \(0.606781\pi\)
\(978\) −12.0071 −0.383946
\(979\) −40.3161 −1.28851
\(980\) 2.39166 0.0763988
\(981\) 8.24223 0.263154
\(982\) 19.9996 0.638212
\(983\) −15.4553 −0.492946 −0.246473 0.969150i \(-0.579272\pi\)
−0.246473 + 0.969150i \(0.579272\pi\)
\(984\) −2.57159 −0.0819793
\(985\) −21.3698 −0.680898
\(986\) −8.68871 −0.276705
\(987\) −4.92668 −0.156818
\(988\) −26.2462 −0.835003
\(989\) 60.4205 1.92126
\(990\) −2.41513 −0.0767578
\(991\) −11.6290 −0.369407 −0.184704 0.982794i \(-0.559133\pi\)
−0.184704 + 0.982794i \(0.559133\pi\)
\(992\) −1.72130 −0.0546515
\(993\) −13.5910 −0.431296
\(994\) 5.53706 0.175625
\(995\) −19.9191 −0.631478
\(996\) −11.4767 −0.363655
\(997\) 47.2683 1.49700 0.748501 0.663133i \(-0.230774\pi\)
0.748501 + 0.663133i \(0.230774\pi\)
\(998\) −3.63329 −0.115010
\(999\) 7.93979 0.251204
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 8043.2.a.t.1.32 52
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8043.2.a.t.1.32 52 1.1 even 1 trivial