Properties

Label 4006.2.a.h.1.20
Level $4006$
Weight $2$
Character 4006.1
Self dual yes
Analytic conductor $31.988$
Analytic rank $0$
Dimension $42$
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] = [4006,2,Mod(1,4006)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4006, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4006.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4006.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: \(31.9880710497\)
Analytic rank: \(0\)
Dimension: \(42\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 4006.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.340721 q^{3} +1.00000 q^{4} +1.77081 q^{5} +0.340721 q^{6} +4.18388 q^{7} -1.00000 q^{8} -2.88391 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.340721 q^{3} +1.00000 q^{4} +1.77081 q^{5} +0.340721 q^{6} +4.18388 q^{7} -1.00000 q^{8} -2.88391 q^{9} -1.77081 q^{10} +2.61292 q^{11} -0.340721 q^{12} +3.00880 q^{13} -4.18388 q^{14} -0.603351 q^{15} +1.00000 q^{16} +3.51698 q^{17} +2.88391 q^{18} +7.65680 q^{19} +1.77081 q^{20} -1.42554 q^{21} -2.61292 q^{22} +3.71765 q^{23} +0.340721 q^{24} -1.86425 q^{25} -3.00880 q^{26} +2.00477 q^{27} +4.18388 q^{28} -3.40065 q^{29} +0.603351 q^{30} +2.86692 q^{31} -1.00000 q^{32} -0.890276 q^{33} -3.51698 q^{34} +7.40884 q^{35} -2.88391 q^{36} -2.37182 q^{37} -7.65680 q^{38} -1.02516 q^{39} -1.77081 q^{40} -3.10570 q^{41} +1.42554 q^{42} -0.620433 q^{43} +2.61292 q^{44} -5.10684 q^{45} -3.71765 q^{46} +2.52070 q^{47} -0.340721 q^{48} +10.5049 q^{49} +1.86425 q^{50} -1.19831 q^{51} +3.00880 q^{52} +10.7851 q^{53} -2.00477 q^{54} +4.62697 q^{55} -4.18388 q^{56} -2.60883 q^{57} +3.40065 q^{58} +10.6124 q^{59} -0.603351 q^{60} -12.7184 q^{61} -2.86692 q^{62} -12.0659 q^{63} +1.00000 q^{64} +5.32799 q^{65} +0.890276 q^{66} -16.2208 q^{67} +3.51698 q^{68} -1.26668 q^{69} -7.40884 q^{70} +8.12622 q^{71} +2.88391 q^{72} -11.4257 q^{73} +2.37182 q^{74} +0.635189 q^{75} +7.65680 q^{76} +10.9321 q^{77} +1.02516 q^{78} -12.7445 q^{79} +1.77081 q^{80} +7.96866 q^{81} +3.10570 q^{82} +4.18060 q^{83} -1.42554 q^{84} +6.22789 q^{85} +0.620433 q^{86} +1.15867 q^{87} -2.61292 q^{88} -7.88201 q^{89} +5.10684 q^{90} +12.5885 q^{91} +3.71765 q^{92} -0.976819 q^{93} -2.52070 q^{94} +13.5587 q^{95} +0.340721 q^{96} +6.46877 q^{97} -10.5049 q^{98} -7.53542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9} - 27 q^{10} + 23 q^{11} + 15 q^{13} + 10 q^{14} + 6 q^{15} + 42 q^{16} + 14 q^{17} - 56 q^{18} - 4 q^{19} + 27 q^{20} + 26 q^{21} - 23 q^{22} + 12 q^{23} + 45 q^{25} - 15 q^{26} - 3 q^{27} - 10 q^{28} + 41 q^{29} - 6 q^{30} + 18 q^{31} - 42 q^{32} + 25 q^{33} - 14 q^{34} + 8 q^{35} + 56 q^{36} + 33 q^{37} + 4 q^{38} + 10 q^{39} - 27 q^{40} + 84 q^{41} - 26 q^{42} - 36 q^{43} + 23 q^{44} + 66 q^{45} - 12 q^{46} + 28 q^{47} + 58 q^{49} - 45 q^{50} + 17 q^{51} + 15 q^{52} + 68 q^{53} + 3 q^{54} - 28 q^{55} + 10 q^{56} - 9 q^{57} - 41 q^{58} + 59 q^{59} + 6 q^{60} + 41 q^{61} - 18 q^{62} - 28 q^{63} + 42 q^{64} + 44 q^{65} - 25 q^{66} + 14 q^{68} + 67 q^{69} - 8 q^{70} + 69 q^{71} - 56 q^{72} - 27 q^{73} - 33 q^{74} + 14 q^{75} - 4 q^{76} + 43 q^{77} - 10 q^{78} - 19 q^{79} + 27 q^{80} + 74 q^{81} - 84 q^{82} + 20 q^{83} + 26 q^{84} + 16 q^{85} + 36 q^{86} - 28 q^{87} - 23 q^{88} + 123 q^{89} - 66 q^{90} - 9 q^{91} + 12 q^{92} + 48 q^{93} - 28 q^{94} + 28 q^{95} + 10 q^{97} - 58 q^{98} + 75 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.340721 −0.196715 −0.0983577 0.995151i \(-0.531359\pi\)
−0.0983577 + 0.995151i \(0.531359\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.77081 0.791928 0.395964 0.918266i \(-0.370410\pi\)
0.395964 + 0.918266i \(0.370410\pi\)
\(6\) 0.340721 0.139099
\(7\) 4.18388 1.58136 0.790679 0.612230i \(-0.209728\pi\)
0.790679 + 0.612230i \(0.209728\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.88391 −0.961303
\(10\) −1.77081 −0.559978
\(11\) 2.61292 0.787824 0.393912 0.919148i \(-0.371121\pi\)
0.393912 + 0.919148i \(0.371121\pi\)
\(12\) −0.340721 −0.0983577
\(13\) 3.00880 0.834490 0.417245 0.908794i \(-0.362996\pi\)
0.417245 + 0.908794i \(0.362996\pi\)
\(14\) −4.18388 −1.11819
\(15\) −0.603351 −0.155784
\(16\) 1.00000 0.250000
\(17\) 3.51698 0.852993 0.426496 0.904489i \(-0.359748\pi\)
0.426496 + 0.904489i \(0.359748\pi\)
\(18\) 2.88391 0.679744
\(19\) 7.65680 1.75659 0.878295 0.478119i \(-0.158681\pi\)
0.878295 + 0.478119i \(0.158681\pi\)
\(20\) 1.77081 0.395964
\(21\) −1.42554 −0.311078
\(22\) −2.61292 −0.557076
\(23\) 3.71765 0.775184 0.387592 0.921831i \(-0.373307\pi\)
0.387592 + 0.921831i \(0.373307\pi\)
\(24\) 0.340721 0.0695494
\(25\) −1.86425 −0.372850
\(26\) −3.00880 −0.590074
\(27\) 2.00477 0.385819
\(28\) 4.18388 0.790679
\(29\) −3.40065 −0.631484 −0.315742 0.948845i \(-0.602253\pi\)
−0.315742 + 0.948845i \(0.602253\pi\)
\(30\) 0.603351 0.110156
\(31\) 2.86692 0.514913 0.257457 0.966290i \(-0.417116\pi\)
0.257457 + 0.966290i \(0.417116\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.890276 −0.154977
\(34\) −3.51698 −0.603157
\(35\) 7.40884 1.25232
\(36\) −2.88391 −0.480652
\(37\) −2.37182 −0.389925 −0.194963 0.980811i \(-0.562459\pi\)
−0.194963 + 0.980811i \(0.562459\pi\)
\(38\) −7.65680 −1.24210
\(39\) −1.02516 −0.164157
\(40\) −1.77081 −0.279989
\(41\) −3.10570 −0.485029 −0.242514 0.970148i \(-0.577972\pi\)
−0.242514 + 0.970148i \(0.577972\pi\)
\(42\) 1.42554 0.219965
\(43\) −0.620433 −0.0946151 −0.0473076 0.998880i \(-0.515064\pi\)
−0.0473076 + 0.998880i \(0.515064\pi\)
\(44\) 2.61292 0.393912
\(45\) −5.10684 −0.761283
\(46\) −3.71765 −0.548138
\(47\) 2.52070 0.367682 0.183841 0.982956i \(-0.441147\pi\)
0.183841 + 0.982956i \(0.441147\pi\)
\(48\) −0.340721 −0.0491789
\(49\) 10.5049 1.50070
\(50\) 1.86425 0.263645
\(51\) −1.19831 −0.167797
\(52\) 3.00880 0.417245
\(53\) 10.7851 1.48144 0.740721 0.671813i \(-0.234484\pi\)
0.740721 + 0.671813i \(0.234484\pi\)
\(54\) −2.00477 −0.272815
\(55\) 4.62697 0.623900
\(56\) −4.18388 −0.559095
\(57\) −2.60883 −0.345548
\(58\) 3.40065 0.446527
\(59\) 10.6124 1.38162 0.690811 0.723036i \(-0.257254\pi\)
0.690811 + 0.723036i \(0.257254\pi\)
\(60\) −0.603351 −0.0778922
\(61\) −12.7184 −1.62843 −0.814215 0.580563i \(-0.802832\pi\)
−0.814215 + 0.580563i \(0.802832\pi\)
\(62\) −2.86692 −0.364099
\(63\) −12.0659 −1.52016
\(64\) 1.00000 0.125000
\(65\) 5.32799 0.660856
\(66\) 0.890276 0.109585
\(67\) −16.2208 −1.98169 −0.990844 0.135015i \(-0.956892\pi\)
−0.990844 + 0.135015i \(0.956892\pi\)
\(68\) 3.51698 0.426496
\(69\) −1.26668 −0.152491
\(70\) −7.40884 −0.885526
\(71\) 8.12622 0.964404 0.482202 0.876060i \(-0.339837\pi\)
0.482202 + 0.876060i \(0.339837\pi\)
\(72\) 2.88391 0.339872
\(73\) −11.4257 −1.33728 −0.668640 0.743586i \(-0.733123\pi\)
−0.668640 + 0.743586i \(0.733123\pi\)
\(74\) 2.37182 0.275719
\(75\) 0.635189 0.0733453
\(76\) 7.65680 0.878295
\(77\) 10.9321 1.24583
\(78\) 1.02516 0.116077
\(79\) −12.7445 −1.43387 −0.716935 0.697140i \(-0.754456\pi\)
−0.716935 + 0.697140i \(0.754456\pi\)
\(80\) 1.77081 0.197982
\(81\) 7.96866 0.885407
\(82\) 3.10570 0.342967
\(83\) 4.18060 0.458881 0.229440 0.973323i \(-0.426310\pi\)
0.229440 + 0.973323i \(0.426310\pi\)
\(84\) −1.42554 −0.155539
\(85\) 6.22789 0.675509
\(86\) 0.620433 0.0669030
\(87\) 1.15867 0.124223
\(88\) −2.61292 −0.278538
\(89\) −7.88201 −0.835491 −0.417746 0.908564i \(-0.637180\pi\)
−0.417746 + 0.908564i \(0.637180\pi\)
\(90\) 5.10684 0.538308
\(91\) 12.5885 1.31963
\(92\) 3.71765 0.387592
\(93\) −0.976819 −0.101291
\(94\) −2.52070 −0.259990
\(95\) 13.5587 1.39109
\(96\) 0.340721 0.0347747
\(97\) 6.46877 0.656804 0.328402 0.944538i \(-0.393490\pi\)
0.328402 + 0.944538i \(0.393490\pi\)
\(98\) −10.5049 −1.06115
\(99\) −7.53542 −0.757338
\(100\) −1.86425 −0.186425
\(101\) −1.69847 −0.169004 −0.0845020 0.996423i \(-0.526930\pi\)
−0.0845020 + 0.996423i \(0.526930\pi\)
\(102\) 1.19831 0.118650
\(103\) 14.3905 1.41794 0.708970 0.705239i \(-0.249160\pi\)
0.708970 + 0.705239i \(0.249160\pi\)
\(104\) −3.00880 −0.295037
\(105\) −2.52435 −0.246351
\(106\) −10.7851 −1.04754
\(107\) 4.15309 0.401495 0.200747 0.979643i \(-0.435663\pi\)
0.200747 + 0.979643i \(0.435663\pi\)
\(108\) 2.00477 0.192909
\(109\) 2.33842 0.223980 0.111990 0.993709i \(-0.464278\pi\)
0.111990 + 0.993709i \(0.464278\pi\)
\(110\) −4.62697 −0.441164
\(111\) 0.808130 0.0767043
\(112\) 4.18388 0.395340
\(113\) −11.3894 −1.07142 −0.535712 0.844401i \(-0.679957\pi\)
−0.535712 + 0.844401i \(0.679957\pi\)
\(114\) 2.60883 0.244340
\(115\) 6.58324 0.613890
\(116\) −3.40065 −0.315742
\(117\) −8.67710 −0.802198
\(118\) −10.6124 −0.976954
\(119\) 14.7146 1.34889
\(120\) 0.603351 0.0550781
\(121\) −4.17266 −0.379333
\(122\) 12.7184 1.15147
\(123\) 1.05818 0.0954126
\(124\) 2.86692 0.257457
\(125\) −12.1552 −1.08720
\(126\) 12.0659 1.07492
\(127\) −6.97942 −0.619324 −0.309662 0.950847i \(-0.600216\pi\)
−0.309662 + 0.950847i \(0.600216\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0.211395 0.0186123
\(130\) −5.32799 −0.467296
\(131\) −2.82079 −0.246454 −0.123227 0.992379i \(-0.539324\pi\)
−0.123227 + 0.992379i \(0.539324\pi\)
\(132\) −0.890276 −0.0774886
\(133\) 32.0351 2.77780
\(134\) 16.2208 1.40126
\(135\) 3.55006 0.305541
\(136\) −3.51698 −0.301579
\(137\) 11.1347 0.951298 0.475649 0.879635i \(-0.342213\pi\)
0.475649 + 0.879635i \(0.342213\pi\)
\(138\) 1.26668 0.107827
\(139\) −7.53677 −0.639261 −0.319630 0.947542i \(-0.603559\pi\)
−0.319630 + 0.947542i \(0.603559\pi\)
\(140\) 7.40884 0.626161
\(141\) −0.858855 −0.0723286
\(142\) −8.12622 −0.681937
\(143\) 7.86174 0.657432
\(144\) −2.88391 −0.240326
\(145\) −6.02188 −0.500090
\(146\) 11.4257 0.945600
\(147\) −3.57923 −0.295210
\(148\) −2.37182 −0.194963
\(149\) −16.0103 −1.31161 −0.655807 0.754929i \(-0.727672\pi\)
−0.655807 + 0.754929i \(0.727672\pi\)
\(150\) −0.635189 −0.0518630
\(151\) 1.08999 0.0887017 0.0443509 0.999016i \(-0.485878\pi\)
0.0443509 + 0.999016i \(0.485878\pi\)
\(152\) −7.65680 −0.621048
\(153\) −10.1426 −0.819985
\(154\) −10.9321 −0.880937
\(155\) 5.07675 0.407774
\(156\) −1.02516 −0.0820786
\(157\) −18.3346 −1.46326 −0.731631 0.681701i \(-0.761240\pi\)
−0.731631 + 0.681701i \(0.761240\pi\)
\(158\) 12.7445 1.01390
\(159\) −3.67470 −0.291423
\(160\) −1.77081 −0.139994
\(161\) 15.5542 1.22584
\(162\) −7.96866 −0.626077
\(163\) −8.94236 −0.700420 −0.350210 0.936671i \(-0.613890\pi\)
−0.350210 + 0.936671i \(0.613890\pi\)
\(164\) −3.10570 −0.242514
\(165\) −1.57651 −0.122731
\(166\) −4.18060 −0.324478
\(167\) 7.95624 0.615673 0.307836 0.951439i \(-0.400395\pi\)
0.307836 + 0.951439i \(0.400395\pi\)
\(168\) 1.42554 0.109983
\(169\) −3.94714 −0.303626
\(170\) −6.22789 −0.477657
\(171\) −22.0815 −1.68862
\(172\) −0.620433 −0.0473076
\(173\) −5.52887 −0.420352 −0.210176 0.977664i \(-0.567404\pi\)
−0.210176 + 0.977664i \(0.567404\pi\)
\(174\) −1.15867 −0.0878387
\(175\) −7.79980 −0.589609
\(176\) 2.61292 0.196956
\(177\) −3.61588 −0.271786
\(178\) 7.88201 0.590782
\(179\) −4.59445 −0.343406 −0.171703 0.985149i \(-0.554927\pi\)
−0.171703 + 0.985149i \(0.554927\pi\)
\(180\) −5.10684 −0.380641
\(181\) −10.1720 −0.756081 −0.378040 0.925789i \(-0.623402\pi\)
−0.378040 + 0.925789i \(0.623402\pi\)
\(182\) −12.5885 −0.933118
\(183\) 4.33344 0.320337
\(184\) −3.71765 −0.274069
\(185\) −4.20004 −0.308793
\(186\) 0.976819 0.0716238
\(187\) 9.18958 0.672008
\(188\) 2.52070 0.183841
\(189\) 8.38773 0.610118
\(190\) −13.5587 −0.983651
\(191\) −10.5694 −0.764776 −0.382388 0.924002i \(-0.624898\pi\)
−0.382388 + 0.924002i \(0.624898\pi\)
\(192\) −0.340721 −0.0245894
\(193\) −12.4910 −0.899123 −0.449561 0.893249i \(-0.648420\pi\)
−0.449561 + 0.893249i \(0.648420\pi\)
\(194\) −6.46877 −0.464431
\(195\) −1.81536 −0.130001
\(196\) 10.5049 0.750348
\(197\) 19.3297 1.37718 0.688592 0.725149i \(-0.258229\pi\)
0.688592 + 0.725149i \(0.258229\pi\)
\(198\) 7.53542 0.535519
\(199\) −12.9496 −0.917975 −0.458988 0.888443i \(-0.651788\pi\)
−0.458988 + 0.888443i \(0.651788\pi\)
\(200\) 1.86425 0.131822
\(201\) 5.52677 0.389828
\(202\) 1.69847 0.119504
\(203\) −14.2279 −0.998603
\(204\) −1.19831 −0.0838984
\(205\) −5.49959 −0.384108
\(206\) −14.3905 −1.00263
\(207\) −10.7214 −0.745187
\(208\) 3.00880 0.208623
\(209\) 20.0066 1.38388
\(210\) 2.52435 0.174197
\(211\) 3.73521 0.257143 0.128571 0.991700i \(-0.458961\pi\)
0.128571 + 0.991700i \(0.458961\pi\)
\(212\) 10.7851 0.740721
\(213\) −2.76877 −0.189713
\(214\) −4.15309 −0.283900
\(215\) −1.09867 −0.0749284
\(216\) −2.00477 −0.136407
\(217\) 11.9948 0.814262
\(218\) −2.33842 −0.158378
\(219\) 3.89299 0.263064
\(220\) 4.62697 0.311950
\(221\) 10.5819 0.711814
\(222\) −0.808130 −0.0542382
\(223\) 9.14690 0.612521 0.306261 0.951948i \(-0.400922\pi\)
0.306261 + 0.951948i \(0.400922\pi\)
\(224\) −4.18388 −0.279547
\(225\) 5.37633 0.358422
\(226\) 11.3894 0.757611
\(227\) 6.78677 0.450454 0.225227 0.974306i \(-0.427688\pi\)
0.225227 + 0.974306i \(0.427688\pi\)
\(228\) −2.60883 −0.172774
\(229\) 23.4843 1.55189 0.775944 0.630801i \(-0.217274\pi\)
0.775944 + 0.630801i \(0.217274\pi\)
\(230\) −6.58324 −0.434086
\(231\) −3.72481 −0.245075
\(232\) 3.40065 0.223263
\(233\) 5.17551 0.339059 0.169529 0.985525i \(-0.445775\pi\)
0.169529 + 0.985525i \(0.445775\pi\)
\(234\) 8.67710 0.567240
\(235\) 4.46366 0.291177
\(236\) 10.6124 0.690811
\(237\) 4.34233 0.282064
\(238\) −14.7146 −0.953808
\(239\) 12.2861 0.794721 0.397360 0.917663i \(-0.369926\pi\)
0.397360 + 0.917663i \(0.369926\pi\)
\(240\) −0.603351 −0.0389461
\(241\) 7.08381 0.456308 0.228154 0.973625i \(-0.426731\pi\)
0.228154 + 0.973625i \(0.426731\pi\)
\(242\) 4.17266 0.268229
\(243\) −8.72941 −0.559992
\(244\) −12.7184 −0.814215
\(245\) 18.6021 1.18844
\(246\) −1.05818 −0.0674669
\(247\) 23.0378 1.46586
\(248\) −2.86692 −0.182049
\(249\) −1.42442 −0.0902690
\(250\) 12.1552 0.768765
\(251\) −30.6666 −1.93566 −0.967830 0.251605i \(-0.919042\pi\)
−0.967830 + 0.251605i \(0.919042\pi\)
\(252\) −12.0659 −0.760082
\(253\) 9.71392 0.610709
\(254\) 6.97942 0.437928
\(255\) −2.12197 −0.132883
\(256\) 1.00000 0.0625000
\(257\) 9.71507 0.606010 0.303005 0.952989i \(-0.402010\pi\)
0.303005 + 0.952989i \(0.402010\pi\)
\(258\) −0.211395 −0.0131608
\(259\) −9.92343 −0.616612
\(260\) 5.32799 0.330428
\(261\) 9.80716 0.607048
\(262\) 2.82079 0.174269
\(263\) 21.7138 1.33893 0.669464 0.742845i \(-0.266524\pi\)
0.669464 + 0.742845i \(0.266524\pi\)
\(264\) 0.890276 0.0547927
\(265\) 19.0982 1.17320
\(266\) −32.0351 −1.96420
\(267\) 2.68557 0.164354
\(268\) −16.2208 −0.990844
\(269\) −25.7944 −1.57271 −0.786355 0.617775i \(-0.788034\pi\)
−0.786355 + 0.617775i \(0.788034\pi\)
\(270\) −3.55006 −0.216050
\(271\) −18.1386 −1.10184 −0.550920 0.834558i \(-0.685723\pi\)
−0.550920 + 0.834558i \(0.685723\pi\)
\(272\) 3.51698 0.213248
\(273\) −4.28915 −0.259591
\(274\) −11.1347 −0.672669
\(275\) −4.87113 −0.293740
\(276\) −1.26668 −0.0762453
\(277\) 16.5732 0.995785 0.497893 0.867239i \(-0.334107\pi\)
0.497893 + 0.867239i \(0.334107\pi\)
\(278\) 7.53677 0.452026
\(279\) −8.26792 −0.494988
\(280\) −7.40884 −0.442763
\(281\) 11.4974 0.685875 0.342937 0.939358i \(-0.388578\pi\)
0.342937 + 0.939358i \(0.388578\pi\)
\(282\) 0.858855 0.0511441
\(283\) −4.92943 −0.293024 −0.146512 0.989209i \(-0.546805\pi\)
−0.146512 + 0.989209i \(0.546805\pi\)
\(284\) 8.12622 0.482202
\(285\) −4.61973 −0.273649
\(286\) −7.86174 −0.464874
\(287\) −12.9939 −0.767004
\(288\) 2.88391 0.169936
\(289\) −4.63085 −0.272403
\(290\) 6.02188 0.353617
\(291\) −2.20405 −0.129203
\(292\) −11.4257 −0.668640
\(293\) 17.1886 1.00417 0.502084 0.864819i \(-0.332567\pi\)
0.502084 + 0.864819i \(0.332567\pi\)
\(294\) 3.57923 0.208745
\(295\) 18.7926 1.09414
\(296\) 2.37182 0.137859
\(297\) 5.23830 0.303957
\(298\) 16.0103 0.927451
\(299\) 11.1857 0.646884
\(300\) 0.635189 0.0366727
\(301\) −2.59582 −0.149620
\(302\) −1.08999 −0.0627216
\(303\) 0.578704 0.0332457
\(304\) 7.65680 0.439147
\(305\) −22.5219 −1.28960
\(306\) 10.1426 0.579817
\(307\) 28.0058 1.59838 0.799188 0.601082i \(-0.205263\pi\)
0.799188 + 0.601082i \(0.205263\pi\)
\(308\) 10.9321 0.622916
\(309\) −4.90315 −0.278931
\(310\) −5.07675 −0.288340
\(311\) −4.19907 −0.238108 −0.119054 0.992888i \(-0.537986\pi\)
−0.119054 + 0.992888i \(0.537986\pi\)
\(312\) 1.02516 0.0580383
\(313\) 29.7464 1.68137 0.840683 0.541527i \(-0.182154\pi\)
0.840683 + 0.541527i \(0.182154\pi\)
\(314\) 18.3346 1.03468
\(315\) −21.3664 −1.20386
\(316\) −12.7445 −0.716935
\(317\) −9.87969 −0.554899 −0.277449 0.960740i \(-0.589489\pi\)
−0.277449 + 0.960740i \(0.589489\pi\)
\(318\) 3.67470 0.206067
\(319\) −8.88561 −0.497499
\(320\) 1.77081 0.0989910
\(321\) −1.41505 −0.0789802
\(322\) −15.5542 −0.866803
\(323\) 26.9288 1.49836
\(324\) 7.96866 0.442703
\(325\) −5.60915 −0.311140
\(326\) 8.94236 0.495272
\(327\) −0.796748 −0.0440603
\(328\) 3.10570 0.171484
\(329\) 10.5463 0.581436
\(330\) 1.57651 0.0867838
\(331\) 21.8518 1.20109 0.600543 0.799593i \(-0.294951\pi\)
0.600543 + 0.799593i \(0.294951\pi\)
\(332\) 4.18060 0.229440
\(333\) 6.84012 0.374836
\(334\) −7.95624 −0.435346
\(335\) −28.7239 −1.56935
\(336\) −1.42554 −0.0777694
\(337\) −24.9243 −1.35772 −0.678858 0.734270i \(-0.737525\pi\)
−0.678858 + 0.734270i \(0.737525\pi\)
\(338\) 3.94714 0.214696
\(339\) 3.88061 0.210766
\(340\) 6.22789 0.337755
\(341\) 7.49101 0.405661
\(342\) 22.0815 1.19403
\(343\) 14.6640 0.791779
\(344\) 0.620433 0.0334515
\(345\) −2.24305 −0.120762
\(346\) 5.52887 0.297234
\(347\) −14.4684 −0.776707 −0.388353 0.921511i \(-0.626956\pi\)
−0.388353 + 0.921511i \(0.626956\pi\)
\(348\) 1.15867 0.0621114
\(349\) 29.7908 1.59467 0.797334 0.603539i \(-0.206243\pi\)
0.797334 + 0.603539i \(0.206243\pi\)
\(350\) 7.79980 0.416917
\(351\) 6.03195 0.321962
\(352\) −2.61292 −0.139269
\(353\) −24.6869 −1.31395 −0.656975 0.753912i \(-0.728164\pi\)
−0.656975 + 0.753912i \(0.728164\pi\)
\(354\) 3.61588 0.192182
\(355\) 14.3899 0.763739
\(356\) −7.88201 −0.417746
\(357\) −5.01358 −0.265347
\(358\) 4.59445 0.242824
\(359\) 7.76955 0.410061 0.205031 0.978756i \(-0.434271\pi\)
0.205031 + 0.978756i \(0.434271\pi\)
\(360\) 5.10684 0.269154
\(361\) 39.6266 2.08561
\(362\) 10.1720 0.534630
\(363\) 1.42171 0.0746207
\(364\) 12.5885 0.659814
\(365\) −20.2327 −1.05903
\(366\) −4.33344 −0.226513
\(367\) 7.80521 0.407429 0.203714 0.979030i \(-0.434699\pi\)
0.203714 + 0.979030i \(0.434699\pi\)
\(368\) 3.71765 0.193796
\(369\) 8.95655 0.466259
\(370\) 4.20004 0.218350
\(371\) 45.1234 2.34269
\(372\) −0.976819 −0.0506457
\(373\) 33.6045 1.73997 0.869987 0.493075i \(-0.164127\pi\)
0.869987 + 0.493075i \(0.164127\pi\)
\(374\) −9.18958 −0.475182
\(375\) 4.14155 0.213869
\(376\) −2.52070 −0.129995
\(377\) −10.2319 −0.526967
\(378\) −8.38773 −0.431418
\(379\) −14.0025 −0.719263 −0.359631 0.933094i \(-0.617098\pi\)
−0.359631 + 0.933094i \(0.617098\pi\)
\(380\) 13.5587 0.695547
\(381\) 2.37804 0.121831
\(382\) 10.5694 0.540778
\(383\) −20.9251 −1.06922 −0.534612 0.845098i \(-0.679542\pi\)
−0.534612 + 0.845098i \(0.679542\pi\)
\(384\) 0.340721 0.0173874
\(385\) 19.3587 0.986610
\(386\) 12.4910 0.635776
\(387\) 1.78927 0.0909538
\(388\) 6.46877 0.328402
\(389\) 13.3813 0.678458 0.339229 0.940704i \(-0.389834\pi\)
0.339229 + 0.940704i \(0.389834\pi\)
\(390\) 1.81536 0.0919243
\(391\) 13.0749 0.661226
\(392\) −10.5049 −0.530576
\(393\) 0.961103 0.0484812
\(394\) −19.3297 −0.973817
\(395\) −22.5681 −1.13552
\(396\) −7.53542 −0.378669
\(397\) −34.1292 −1.71290 −0.856449 0.516232i \(-0.827334\pi\)
−0.856449 + 0.516232i \(0.827334\pi\)
\(398\) 12.9496 0.649107
\(399\) −10.9150 −0.546436
\(400\) −1.86425 −0.0932125
\(401\) 25.3458 1.26571 0.632855 0.774270i \(-0.281883\pi\)
0.632855 + 0.774270i \(0.281883\pi\)
\(402\) −5.52677 −0.275650
\(403\) 8.62597 0.429690
\(404\) −1.69847 −0.0845020
\(405\) 14.1109 0.701178
\(406\) 14.2279 0.706119
\(407\) −6.19738 −0.307193
\(408\) 1.19831 0.0593252
\(409\) 32.1256 1.58851 0.794254 0.607586i \(-0.207862\pi\)
0.794254 + 0.607586i \(0.207862\pi\)
\(410\) 5.49959 0.271605
\(411\) −3.79381 −0.187135
\(412\) 14.3905 0.708970
\(413\) 44.4012 2.18484
\(414\) 10.7214 0.526927
\(415\) 7.40303 0.363401
\(416\) −3.00880 −0.147518
\(417\) 2.56794 0.125752
\(418\) −20.0066 −0.978554
\(419\) 36.7157 1.79368 0.896839 0.442357i \(-0.145857\pi\)
0.896839 + 0.442357i \(0.145857\pi\)
\(420\) −2.52435 −0.123176
\(421\) −15.2841 −0.744902 −0.372451 0.928052i \(-0.621482\pi\)
−0.372451 + 0.928052i \(0.621482\pi\)
\(422\) −3.73521 −0.181827
\(423\) −7.26946 −0.353453
\(424\) −10.7851 −0.523769
\(425\) −6.55653 −0.318038
\(426\) 2.76877 0.134148
\(427\) −53.2125 −2.57513
\(428\) 4.15309 0.200747
\(429\) −2.67866 −0.129327
\(430\) 1.09867 0.0529824
\(431\) −9.38721 −0.452166 −0.226083 0.974108i \(-0.572592\pi\)
−0.226083 + 0.974108i \(0.572592\pi\)
\(432\) 2.00477 0.0964546
\(433\) 5.89835 0.283457 0.141728 0.989906i \(-0.454734\pi\)
0.141728 + 0.989906i \(0.454734\pi\)
\(434\) −11.9948 −0.575771
\(435\) 2.05178 0.0983755
\(436\) 2.33842 0.111990
\(437\) 28.4653 1.36168
\(438\) −3.89299 −0.186014
\(439\) 4.09609 0.195496 0.0977479 0.995211i \(-0.468836\pi\)
0.0977479 + 0.995211i \(0.468836\pi\)
\(440\) −4.62697 −0.220582
\(441\) −30.2951 −1.44262
\(442\) −10.5819 −0.503329
\(443\) 39.8409 1.89290 0.946450 0.322851i \(-0.104641\pi\)
0.946450 + 0.322851i \(0.104641\pi\)
\(444\) 0.808130 0.0383522
\(445\) −13.9575 −0.661649
\(446\) −9.14690 −0.433118
\(447\) 5.45504 0.258015
\(448\) 4.18388 0.197670
\(449\) 16.5522 0.781148 0.390574 0.920571i \(-0.372276\pi\)
0.390574 + 0.920571i \(0.372276\pi\)
\(450\) −5.37633 −0.253442
\(451\) −8.11493 −0.382117
\(452\) −11.3894 −0.535712
\(453\) −0.371381 −0.0174490
\(454\) −6.78677 −0.318519
\(455\) 22.2917 1.04505
\(456\) 2.60883 0.122170
\(457\) −35.3162 −1.65202 −0.826012 0.563653i \(-0.809395\pi\)
−0.826012 + 0.563653i \(0.809395\pi\)
\(458\) −23.4843 −1.09735
\(459\) 7.05074 0.329101
\(460\) 6.58324 0.306945
\(461\) −28.5573 −1.33004 −0.665022 0.746824i \(-0.731578\pi\)
−0.665022 + 0.746824i \(0.731578\pi\)
\(462\) 3.72481 0.173294
\(463\) 36.6034 1.70110 0.850552 0.525891i \(-0.176268\pi\)
0.850552 + 0.525891i \(0.176268\pi\)
\(464\) −3.40065 −0.157871
\(465\) −1.72976 −0.0802155
\(466\) −5.17551 −0.239751
\(467\) 12.6389 0.584859 0.292429 0.956287i \(-0.405536\pi\)
0.292429 + 0.956287i \(0.405536\pi\)
\(468\) −8.67710 −0.401099
\(469\) −67.8659 −3.13376
\(470\) −4.46366 −0.205894
\(471\) 6.24699 0.287846
\(472\) −10.6124 −0.488477
\(473\) −1.62114 −0.0745401
\(474\) −4.34233 −0.199450
\(475\) −14.2742 −0.654944
\(476\) 14.7146 0.674444
\(477\) −31.1031 −1.42411
\(478\) −12.2861 −0.561952
\(479\) 15.6466 0.714910 0.357455 0.933930i \(-0.383645\pi\)
0.357455 + 0.933930i \(0.383645\pi\)
\(480\) 0.603351 0.0275391
\(481\) −7.13634 −0.325389
\(482\) −7.08381 −0.322659
\(483\) −5.29965 −0.241142
\(484\) −4.17266 −0.189667
\(485\) 11.4549 0.520142
\(486\) 8.72941 0.395974
\(487\) −31.0411 −1.40661 −0.703303 0.710890i \(-0.748292\pi\)
−0.703303 + 0.710890i \(0.748292\pi\)
\(488\) 12.7184 0.575737
\(489\) 3.04685 0.137783
\(490\) −18.6021 −0.840356
\(491\) −20.2397 −0.913404 −0.456702 0.889620i \(-0.650969\pi\)
−0.456702 + 0.889620i \(0.650969\pi\)
\(492\) 1.05818 0.0477063
\(493\) −11.9600 −0.538652
\(494\) −23.0378 −1.03652
\(495\) −13.3438 −0.599757
\(496\) 2.86692 0.128728
\(497\) 33.9991 1.52507
\(498\) 1.42442 0.0638298
\(499\) 12.9574 0.580052 0.290026 0.957019i \(-0.406336\pi\)
0.290026 + 0.957019i \(0.406336\pi\)
\(500\) −12.1552 −0.543599
\(501\) −2.71086 −0.121112
\(502\) 30.6666 1.36872
\(503\) 4.89600 0.218302 0.109151 0.994025i \(-0.465187\pi\)
0.109151 + 0.994025i \(0.465187\pi\)
\(504\) 12.0659 0.537459
\(505\) −3.00766 −0.133839
\(506\) −9.71392 −0.431836
\(507\) 1.34487 0.0597279
\(508\) −6.97942 −0.309662
\(509\) 41.8912 1.85679 0.928397 0.371590i \(-0.121187\pi\)
0.928397 + 0.371590i \(0.121187\pi\)
\(510\) 2.12197 0.0939625
\(511\) −47.8039 −2.11472
\(512\) −1.00000 −0.0441942
\(513\) 15.3501 0.677725
\(514\) −9.71507 −0.428514
\(515\) 25.4828 1.12291
\(516\) 0.211395 0.00930613
\(517\) 6.58638 0.289668
\(518\) 9.92343 0.436010
\(519\) 1.88380 0.0826898
\(520\) −5.32799 −0.233648
\(521\) 5.44798 0.238680 0.119340 0.992853i \(-0.461922\pi\)
0.119340 + 0.992853i \(0.461922\pi\)
\(522\) −9.80716 −0.429248
\(523\) −22.8803 −1.00049 −0.500243 0.865885i \(-0.666756\pi\)
−0.500243 + 0.865885i \(0.666756\pi\)
\(524\) −2.82079 −0.123227
\(525\) 2.65756 0.115985
\(526\) −21.7138 −0.946765
\(527\) 10.0829 0.439217
\(528\) −0.890276 −0.0387443
\(529\) −9.17906 −0.399090
\(530\) −19.0982 −0.829575
\(531\) −30.6053 −1.32816
\(532\) 32.0351 1.38890
\(533\) −9.34442 −0.404752
\(534\) −2.68557 −0.116216
\(535\) 7.35432 0.317955
\(536\) 16.2208 0.700632
\(537\) 1.56543 0.0675532
\(538\) 25.7944 1.11207
\(539\) 27.4484 1.18228
\(540\) 3.55006 0.152770
\(541\) −18.0039 −0.774048 −0.387024 0.922070i \(-0.626497\pi\)
−0.387024 + 0.922070i \(0.626497\pi\)
\(542\) 18.1386 0.779119
\(543\) 3.46582 0.148733
\(544\) −3.51698 −0.150789
\(545\) 4.14088 0.177376
\(546\) 4.28915 0.183559
\(547\) 14.7664 0.631366 0.315683 0.948865i \(-0.397766\pi\)
0.315683 + 0.948865i \(0.397766\pi\)
\(548\) 11.1347 0.475649
\(549\) 36.6788 1.56541
\(550\) 4.87113 0.207706
\(551\) −26.0381 −1.10926
\(552\) 1.26668 0.0539136
\(553\) −53.3216 −2.26746
\(554\) −16.5732 −0.704126
\(555\) 1.43104 0.0607443
\(556\) −7.53677 −0.319630
\(557\) −17.1405 −0.726264 −0.363132 0.931738i \(-0.618293\pi\)
−0.363132 + 0.931738i \(0.618293\pi\)
\(558\) 8.26792 0.350009
\(559\) −1.86676 −0.0789554
\(560\) 7.40884 0.313081
\(561\) −3.13108 −0.132194
\(562\) −11.4974 −0.484987
\(563\) −34.4595 −1.45230 −0.726148 0.687538i \(-0.758691\pi\)
−0.726148 + 0.687538i \(0.758691\pi\)
\(564\) −0.858855 −0.0361643
\(565\) −20.1684 −0.848491
\(566\) 4.92943 0.207199
\(567\) 33.3399 1.40015
\(568\) −8.12622 −0.340968
\(569\) 30.0681 1.26052 0.630260 0.776384i \(-0.282948\pi\)
0.630260 + 0.776384i \(0.282948\pi\)
\(570\) 4.61973 0.193499
\(571\) 4.55022 0.190421 0.0952104 0.995457i \(-0.469648\pi\)
0.0952104 + 0.995457i \(0.469648\pi\)
\(572\) 7.86174 0.328716
\(573\) 3.60122 0.150443
\(574\) 12.9939 0.542354
\(575\) −6.93063 −0.289027
\(576\) −2.88391 −0.120163
\(577\) 8.77684 0.365384 0.182692 0.983170i \(-0.441519\pi\)
0.182692 + 0.983170i \(0.441519\pi\)
\(578\) 4.63085 0.192618
\(579\) 4.25595 0.176871
\(580\) −6.02188 −0.250045
\(581\) 17.4912 0.725655
\(582\) 2.20405 0.0913607
\(583\) 28.1805 1.16712
\(584\) 11.4257 0.472800
\(585\) −15.3655 −0.635283
\(586\) −17.1886 −0.710054
\(587\) −26.6817 −1.10127 −0.550637 0.834745i \(-0.685615\pi\)
−0.550637 + 0.834745i \(0.685615\pi\)
\(588\) −3.57923 −0.147605
\(589\) 21.9514 0.904491
\(590\) −18.7926 −0.773677
\(591\) −6.58604 −0.270914
\(592\) −2.37182 −0.0974813
\(593\) −9.35126 −0.384010 −0.192005 0.981394i \(-0.561499\pi\)
−0.192005 + 0.981394i \(0.561499\pi\)
\(594\) −5.23830 −0.214930
\(595\) 26.0567 1.06822
\(596\) −16.0103 −0.655807
\(597\) 4.41222 0.180580
\(598\) −11.1857 −0.457416
\(599\) −5.22874 −0.213640 −0.106820 0.994278i \(-0.534067\pi\)
−0.106820 + 0.994278i \(0.534067\pi\)
\(600\) −0.635189 −0.0259315
\(601\) 14.8639 0.606311 0.303155 0.952941i \(-0.401960\pi\)
0.303155 + 0.952941i \(0.401960\pi\)
\(602\) 2.59582 0.105798
\(603\) 46.7793 1.90500
\(604\) 1.08999 0.0443509
\(605\) −7.38897 −0.300404
\(606\) −0.578704 −0.0235083
\(607\) −18.5494 −0.752898 −0.376449 0.926437i \(-0.622855\pi\)
−0.376449 + 0.926437i \(0.622855\pi\)
\(608\) −7.65680 −0.310524
\(609\) 4.84775 0.196441
\(610\) 22.5219 0.911885
\(611\) 7.58427 0.306827
\(612\) −10.1426 −0.409992
\(613\) 4.32618 0.174733 0.0873663 0.996176i \(-0.472155\pi\)
0.0873663 + 0.996176i \(0.472155\pi\)
\(614\) −28.0058 −1.13022
\(615\) 1.87383 0.0755599
\(616\) −10.9321 −0.440468
\(617\) −17.8360 −0.718050 −0.359025 0.933328i \(-0.616891\pi\)
−0.359025 + 0.933328i \(0.616891\pi\)
\(618\) 4.90315 0.197234
\(619\) −13.5013 −0.542661 −0.271331 0.962486i \(-0.587464\pi\)
−0.271331 + 0.962486i \(0.587464\pi\)
\(620\) 5.07675 0.203887
\(621\) 7.45305 0.299080
\(622\) 4.19907 0.168368
\(623\) −32.9774 −1.32121
\(624\) −1.02516 −0.0410393
\(625\) −12.2033 −0.488133
\(626\) −29.7464 −1.18891
\(627\) −6.81666 −0.272231
\(628\) −18.3346 −0.731631
\(629\) −8.34165 −0.332604
\(630\) 21.3664 0.851259
\(631\) −16.1175 −0.641627 −0.320814 0.947142i \(-0.603956\pi\)
−0.320814 + 0.947142i \(0.603956\pi\)
\(632\) 12.7445 0.506950
\(633\) −1.27267 −0.0505839
\(634\) 9.87969 0.392373
\(635\) −12.3592 −0.490460
\(636\) −3.67470 −0.145711
\(637\) 31.6070 1.25232
\(638\) 8.88561 0.351785
\(639\) −23.4353 −0.927085
\(640\) −1.77081 −0.0699972
\(641\) 39.4889 1.55972 0.779858 0.625956i \(-0.215291\pi\)
0.779858 + 0.625956i \(0.215291\pi\)
\(642\) 1.41505 0.0558474
\(643\) −17.5498 −0.692097 −0.346049 0.938217i \(-0.612477\pi\)
−0.346049 + 0.938217i \(0.612477\pi\)
\(644\) 15.5542 0.612922
\(645\) 0.374339 0.0147396
\(646\) −26.9288 −1.05950
\(647\) −17.7761 −0.698851 −0.349426 0.936964i \(-0.613623\pi\)
−0.349426 + 0.936964i \(0.613623\pi\)
\(648\) −7.96866 −0.313038
\(649\) 27.7294 1.08847
\(650\) 5.60915 0.220009
\(651\) −4.08689 −0.160178
\(652\) −8.94236 −0.350210
\(653\) 17.9052 0.700685 0.350342 0.936622i \(-0.386065\pi\)
0.350342 + 0.936622i \(0.386065\pi\)
\(654\) 0.796748 0.0311553
\(655\) −4.99507 −0.195173
\(656\) −3.10570 −0.121257
\(657\) 32.9507 1.28553
\(658\) −10.5463 −0.411138
\(659\) −11.7277 −0.456845 −0.228422 0.973562i \(-0.573357\pi\)
−0.228422 + 0.973562i \(0.573357\pi\)
\(660\) −1.57651 −0.0613654
\(661\) −43.4397 −1.68961 −0.844804 0.535075i \(-0.820283\pi\)
−0.844804 + 0.535075i \(0.820283\pi\)
\(662\) −21.8518 −0.849296
\(663\) −3.60547 −0.140025
\(664\) −4.18060 −0.162239
\(665\) 56.7280 2.19982
\(666\) −6.84012 −0.265049
\(667\) −12.6424 −0.489517
\(668\) 7.95624 0.307836
\(669\) −3.11654 −0.120492
\(670\) 28.7239 1.10970
\(671\) −33.2323 −1.28292
\(672\) 1.42554 0.0549913
\(673\) 13.2021 0.508903 0.254451 0.967086i \(-0.418105\pi\)
0.254451 + 0.967086i \(0.418105\pi\)
\(674\) 24.9243 0.960050
\(675\) −3.73739 −0.143852
\(676\) −3.94714 −0.151813
\(677\) −42.6568 −1.63943 −0.819717 0.572769i \(-0.805869\pi\)
−0.819717 + 0.572769i \(0.805869\pi\)
\(678\) −3.88061 −0.149034
\(679\) 27.0646 1.03864
\(680\) −6.22789 −0.238829
\(681\) −2.31240 −0.0886112
\(682\) −7.49101 −0.286846
\(683\) 31.7498 1.21487 0.607436 0.794369i \(-0.292198\pi\)
0.607436 + 0.794369i \(0.292198\pi\)
\(684\) −22.0815 −0.844308
\(685\) 19.7173 0.753360
\(686\) −14.6640 −0.559872
\(687\) −8.00161 −0.305281
\(688\) −0.620433 −0.0236538
\(689\) 32.4501 1.23625
\(690\) 2.24305 0.0853914
\(691\) −30.9580 −1.17770 −0.588848 0.808244i \(-0.700418\pi\)
−0.588848 + 0.808244i \(0.700418\pi\)
\(692\) −5.52887 −0.210176
\(693\) −31.5273 −1.19762
\(694\) 14.4684 0.549215
\(695\) −13.3462 −0.506248
\(696\) −1.15867 −0.0439194
\(697\) −10.9227 −0.413726
\(698\) −29.7908 −1.12760
\(699\) −1.76340 −0.0666981
\(700\) −7.79980 −0.294805
\(701\) −10.2901 −0.388653 −0.194327 0.980937i \(-0.562252\pi\)
−0.194327 + 0.980937i \(0.562252\pi\)
\(702\) −6.03195 −0.227661
\(703\) −18.1606 −0.684939
\(704\) 2.61292 0.0984780
\(705\) −1.52086 −0.0572791
\(706\) 24.6869 0.929103
\(707\) −7.10619 −0.267256
\(708\) −3.61588 −0.135893
\(709\) −31.0416 −1.16579 −0.582895 0.812547i \(-0.698080\pi\)
−0.582895 + 0.812547i \(0.698080\pi\)
\(710\) −14.3899 −0.540045
\(711\) 36.7540 1.37838
\(712\) 7.88201 0.295391
\(713\) 10.6582 0.399153
\(714\) 5.01358 0.187629
\(715\) 13.9216 0.520639
\(716\) −4.59445 −0.171703
\(717\) −4.18613 −0.156334
\(718\) −7.76955 −0.289957
\(719\) 18.6707 0.696298 0.348149 0.937439i \(-0.386810\pi\)
0.348149 + 0.937439i \(0.386810\pi\)
\(720\) −5.10684 −0.190321
\(721\) 60.2082 2.24227
\(722\) −39.6266 −1.47475
\(723\) −2.41360 −0.0897628
\(724\) −10.1720 −0.378040
\(725\) 6.33965 0.235449
\(726\) −1.42171 −0.0527648
\(727\) −15.1315 −0.561196 −0.280598 0.959825i \(-0.590533\pi\)
−0.280598 + 0.959825i \(0.590533\pi\)
\(728\) −12.5885 −0.466559
\(729\) −20.9317 −0.775248
\(730\) 20.2327 0.748847
\(731\) −2.18205 −0.0807060
\(732\) 4.33344 0.160169
\(733\) −20.6092 −0.761219 −0.380610 0.924736i \(-0.624286\pi\)
−0.380610 + 0.924736i \(0.624286\pi\)
\(734\) −7.80521 −0.288096
\(735\) −6.33812 −0.233785
\(736\) −3.71765 −0.137034
\(737\) −42.3836 −1.56122
\(738\) −8.95655 −0.329695
\(739\) 12.2839 0.451869 0.225934 0.974143i \(-0.427456\pi\)
0.225934 + 0.974143i \(0.427456\pi\)
\(740\) −4.20004 −0.154396
\(741\) −7.84945 −0.288357
\(742\) −45.1234 −1.65653
\(743\) −12.9805 −0.476210 −0.238105 0.971239i \(-0.576526\pi\)
−0.238105 + 0.971239i \(0.576526\pi\)
\(744\) 0.976819 0.0358119
\(745\) −28.3511 −1.03870
\(746\) −33.6045 −1.23035
\(747\) −12.0565 −0.441124
\(748\) 9.18958 0.336004
\(749\) 17.3761 0.634907
\(750\) −4.14155 −0.151228
\(751\) 0.596847 0.0217793 0.0108896 0.999941i \(-0.496534\pi\)
0.0108896 + 0.999941i \(0.496534\pi\)
\(752\) 2.52070 0.0919204
\(753\) 10.4488 0.380774
\(754\) 10.2319 0.372622
\(755\) 1.93015 0.0702454
\(756\) 8.38773 0.305059
\(757\) −5.90667 −0.214682 −0.107341 0.994222i \(-0.534234\pi\)
−0.107341 + 0.994222i \(0.534234\pi\)
\(758\) 14.0025 0.508595
\(759\) −3.30974 −0.120136
\(760\) −13.5587 −0.491826
\(761\) 13.8249 0.501154 0.250577 0.968097i \(-0.419380\pi\)
0.250577 + 0.968097i \(0.419380\pi\)
\(762\) −2.37804 −0.0861472
\(763\) 9.78366 0.354192
\(764\) −10.5694 −0.382388
\(765\) −17.9607 −0.649369
\(766\) 20.9251 0.756055
\(767\) 31.9307 1.15295
\(768\) −0.340721 −0.0122947
\(769\) 20.3738 0.734698 0.367349 0.930083i \(-0.380266\pi\)
0.367349 + 0.930083i \(0.380266\pi\)
\(770\) −19.3587 −0.697639
\(771\) −3.31013 −0.119211
\(772\) −12.4910 −0.449561
\(773\) −0.199155 −0.00716311 −0.00358156 0.999994i \(-0.501140\pi\)
−0.00358156 + 0.999994i \(0.501140\pi\)
\(774\) −1.78927 −0.0643140
\(775\) −5.34464 −0.191985
\(776\) −6.46877 −0.232215
\(777\) 3.38112 0.121297
\(778\) −13.3813 −0.479743
\(779\) −23.7797 −0.851996
\(780\) −1.81536 −0.0650003
\(781\) 21.2331 0.759781
\(782\) −13.0749 −0.467558
\(783\) −6.81752 −0.243638
\(784\) 10.5049 0.375174
\(785\) −32.4670 −1.15880
\(786\) −0.961103 −0.0342814
\(787\) −1.42241 −0.0507036 −0.0253518 0.999679i \(-0.508071\pi\)
−0.0253518 + 0.999679i \(0.508071\pi\)
\(788\) 19.3297 0.688592
\(789\) −7.39834 −0.263388
\(790\) 22.5681 0.802936
\(791\) −47.6519 −1.69431
\(792\) 7.53542 0.267759
\(793\) −38.2672 −1.35891
\(794\) 34.1292 1.21120
\(795\) −6.50717 −0.230786
\(796\) −12.9496 −0.458988
\(797\) −37.2638 −1.31995 −0.659976 0.751286i \(-0.729434\pi\)
−0.659976 + 0.751286i \(0.729434\pi\)
\(798\) 10.9150 0.386389
\(799\) 8.86524 0.313630
\(800\) 1.86425 0.0659112
\(801\) 22.7310 0.803160
\(802\) −25.3458 −0.894992
\(803\) −29.8545 −1.05354
\(804\) 5.52677 0.194914
\(805\) 27.5435 0.970780
\(806\) −8.62597 −0.303837
\(807\) 8.78868 0.309376
\(808\) 1.69847 0.0597519
\(809\) 47.7828 1.67995 0.839977 0.542622i \(-0.182568\pi\)
0.839977 + 0.542622i \(0.182568\pi\)
\(810\) −14.1109 −0.495808
\(811\) 37.6399 1.32172 0.660858 0.750511i \(-0.270193\pi\)
0.660858 + 0.750511i \(0.270193\pi\)
\(812\) −14.2279 −0.499302
\(813\) 6.18020 0.216749
\(814\) 6.19738 0.217218
\(815\) −15.8352 −0.554682
\(816\) −1.19831 −0.0419492
\(817\) −4.75053 −0.166200
\(818\) −32.1256 −1.12324
\(819\) −36.3040 −1.26856
\(820\) −5.49959 −0.192054
\(821\) 51.9505 1.81309 0.906543 0.422115i \(-0.138712\pi\)
0.906543 + 0.422115i \(0.138712\pi\)
\(822\) 3.79381 0.132324
\(823\) 10.6966 0.372861 0.186430 0.982468i \(-0.440308\pi\)
0.186430 + 0.982468i \(0.440308\pi\)
\(824\) −14.3905 −0.501317
\(825\) 1.65970 0.0577832
\(826\) −44.4012 −1.54491
\(827\) 22.3864 0.778450 0.389225 0.921143i \(-0.372743\pi\)
0.389225 + 0.921143i \(0.372743\pi\)
\(828\) −10.7214 −0.372593
\(829\) 25.5571 0.887634 0.443817 0.896117i \(-0.353624\pi\)
0.443817 + 0.896117i \(0.353624\pi\)
\(830\) −7.40303 −0.256963
\(831\) −5.64683 −0.195886
\(832\) 3.00880 0.104311
\(833\) 36.9454 1.28008
\(834\) −2.56794 −0.0889204
\(835\) 14.0890 0.487568
\(836\) 20.0066 0.691942
\(837\) 5.74751 0.198663
\(838\) −36.7157 −1.26832
\(839\) −24.5235 −0.846645 −0.423322 0.905979i \(-0.639136\pi\)
−0.423322 + 0.905979i \(0.639136\pi\)
\(840\) 2.52435 0.0870983
\(841\) −17.4356 −0.601228
\(842\) 15.2841 0.526725
\(843\) −3.91739 −0.134922
\(844\) 3.73521 0.128571
\(845\) −6.98961 −0.240450
\(846\) 7.26946 0.249929
\(847\) −17.4579 −0.599862
\(848\) 10.7851 0.370360
\(849\) 1.67956 0.0576423
\(850\) 6.55653 0.224887
\(851\) −8.81761 −0.302264
\(852\) −2.76877 −0.0948566
\(853\) −22.0767 −0.755892 −0.377946 0.925828i \(-0.623370\pi\)
−0.377946 + 0.925828i \(0.623370\pi\)
\(854\) 53.2125 1.82089
\(855\) −39.1021 −1.33726
\(856\) −4.15309 −0.141950
\(857\) −5.33040 −0.182083 −0.0910415 0.995847i \(-0.529020\pi\)
−0.0910415 + 0.995847i \(0.529020\pi\)
\(858\) 2.67866 0.0914480
\(859\) −42.1337 −1.43758 −0.718792 0.695225i \(-0.755305\pi\)
−0.718792 + 0.695225i \(0.755305\pi\)
\(860\) −1.09867 −0.0374642
\(861\) 4.42729 0.150882
\(862\) 9.38721 0.319730
\(863\) −36.0412 −1.22686 −0.613429 0.789750i \(-0.710210\pi\)
−0.613429 + 0.789750i \(0.710210\pi\)
\(864\) −2.00477 −0.0682037
\(865\) −9.79055 −0.332889
\(866\) −5.89835 −0.200434
\(867\) 1.57783 0.0535859
\(868\) 11.9948 0.407131
\(869\) −33.3004 −1.12964
\(870\) −2.05178 −0.0695620
\(871\) −48.8051 −1.65370
\(872\) −2.33842 −0.0791888
\(873\) −18.6553 −0.631388
\(874\) −28.4653 −0.962854
\(875\) −50.8561 −1.71925
\(876\) 3.89299 0.131532
\(877\) −19.8349 −0.669778 −0.334889 0.942258i \(-0.608699\pi\)
−0.334889 + 0.942258i \(0.608699\pi\)
\(878\) −4.09609 −0.138236
\(879\) −5.85652 −0.197535
\(880\) 4.62697 0.155975
\(881\) 3.74473 0.126163 0.0630815 0.998008i \(-0.479907\pi\)
0.0630815 + 0.998008i \(0.479907\pi\)
\(882\) 30.2951 1.02009
\(883\) 20.5729 0.692332 0.346166 0.938173i \(-0.387483\pi\)
0.346166 + 0.938173i \(0.387483\pi\)
\(884\) 10.5819 0.355907
\(885\) −6.40302 −0.215235
\(886\) −39.8409 −1.33848
\(887\) −51.3374 −1.72374 −0.861871 0.507128i \(-0.830707\pi\)
−0.861871 + 0.507128i \(0.830707\pi\)
\(888\) −0.808130 −0.0271191
\(889\) −29.2011 −0.979373
\(890\) 13.9575 0.467857
\(891\) 20.8214 0.697545
\(892\) 9.14690 0.306261
\(893\) 19.3005 0.645866
\(894\) −5.45504 −0.182444
\(895\) −8.13588 −0.271953
\(896\) −4.18388 −0.139774
\(897\) −3.81119 −0.127252
\(898\) −16.5522 −0.552355
\(899\) −9.74937 −0.325160
\(900\) 5.37633 0.179211
\(901\) 37.9308 1.26366
\(902\) 8.11493 0.270198
\(903\) 0.884450 0.0294326
\(904\) 11.3894 0.378806
\(905\) −18.0127 −0.598761
\(906\) 0.371381 0.0123383
\(907\) 32.2707 1.07153 0.535765 0.844367i \(-0.320023\pi\)
0.535765 + 0.844367i \(0.320023\pi\)
\(908\) 6.78677 0.225227
\(909\) 4.89823 0.162464
\(910\) −22.2917 −0.738963
\(911\) −14.6828 −0.486464 −0.243232 0.969968i \(-0.578208\pi\)
−0.243232 + 0.969968i \(0.578208\pi\)
\(912\) −2.60883 −0.0863871
\(913\) 10.9236 0.361517
\(914\) 35.3162 1.16816
\(915\) 7.67368 0.253684
\(916\) 23.4843 0.775944
\(917\) −11.8019 −0.389731
\(918\) −7.05074 −0.232709
\(919\) 12.8944 0.425346 0.212673 0.977123i \(-0.431783\pi\)
0.212673 + 0.977123i \(0.431783\pi\)
\(920\) −6.58324 −0.217043
\(921\) −9.54217 −0.314425
\(922\) 28.5573 0.940483
\(923\) 24.4501 0.804786
\(924\) −3.72481 −0.122537
\(925\) 4.42167 0.145384
\(926\) −36.6034 −1.20286
\(927\) −41.5009 −1.36307
\(928\) 3.40065 0.111632
\(929\) −19.1605 −0.628635 −0.314318 0.949318i \(-0.601776\pi\)
−0.314318 + 0.949318i \(0.601776\pi\)
\(930\) 1.72976 0.0567209
\(931\) 80.4337 2.63611
\(932\) 5.17551 0.169529
\(933\) 1.43071 0.0468395
\(934\) −12.6389 −0.413558
\(935\) 16.2729 0.532182
\(936\) 8.67710 0.283620
\(937\) 38.5521 1.25944 0.629722 0.776821i \(-0.283169\pi\)
0.629722 + 0.776821i \(0.283169\pi\)
\(938\) 67.8659 2.21590
\(939\) −10.1352 −0.330751
\(940\) 4.46366 0.145589
\(941\) 36.2593 1.18202 0.591011 0.806664i \(-0.298729\pi\)
0.591011 + 0.806664i \(0.298729\pi\)
\(942\) −6.24699 −0.203538
\(943\) −11.5459 −0.375986
\(944\) 10.6124 0.345405
\(945\) 14.8530 0.483169
\(946\) 1.62114 0.0527078
\(947\) −30.5207 −0.991789 −0.495894 0.868383i \(-0.665160\pi\)
−0.495894 + 0.868383i \(0.665160\pi\)
\(948\) 4.34233 0.141032
\(949\) −34.3777 −1.11595
\(950\) 14.2742 0.463116
\(951\) 3.36622 0.109157
\(952\) −14.7146 −0.476904
\(953\) −57.2412 −1.85422 −0.927112 0.374786i \(-0.877716\pi\)
−0.927112 + 0.374786i \(0.877716\pi\)
\(954\) 31.1031 1.00700
\(955\) −18.7164 −0.605647
\(956\) 12.2861 0.397360
\(957\) 3.02751 0.0978657
\(958\) −15.6466 −0.505517
\(959\) 46.5861 1.50434
\(960\) −0.603351 −0.0194731
\(961\) −22.7808 −0.734864
\(962\) 7.13634 0.230085
\(963\) −11.9771 −0.385958
\(964\) 7.08381 0.228154
\(965\) −22.1191 −0.712040
\(966\) 5.29965 0.170513
\(967\) −10.2808 −0.330607 −0.165303 0.986243i \(-0.552860\pi\)
−0.165303 + 0.986243i \(0.552860\pi\)
\(968\) 4.17266 0.134114
\(969\) −9.17521 −0.294750
\(970\) −11.4549 −0.367796
\(971\) 16.8974 0.542264 0.271132 0.962542i \(-0.412602\pi\)
0.271132 + 0.962542i \(0.412602\pi\)
\(972\) −8.72941 −0.279996
\(973\) −31.5330 −1.01090
\(974\) 31.0411 0.994621
\(975\) 1.91116 0.0612060
\(976\) −12.7184 −0.407108
\(977\) −22.1223 −0.707755 −0.353878 0.935292i \(-0.615137\pi\)
−0.353878 + 0.935292i \(0.615137\pi\)
\(978\) −3.04685 −0.0974276
\(979\) −20.5950 −0.658220
\(980\) 18.6021 0.594221
\(981\) −6.74378 −0.215312
\(982\) 20.2397 0.645874
\(983\) 14.2082 0.453170 0.226585 0.973991i \(-0.427244\pi\)
0.226585 + 0.973991i \(0.427244\pi\)
\(984\) −1.05818 −0.0337335
\(985\) 34.2291 1.09063
\(986\) 11.9600 0.380884
\(987\) −3.59335 −0.114378
\(988\) 23.0378 0.732929
\(989\) −2.30655 −0.0733441
\(990\) 13.3438 0.424092
\(991\) 22.5726 0.717043 0.358521 0.933522i \(-0.383281\pi\)
0.358521 + 0.933522i \(0.383281\pi\)
\(992\) −2.86692 −0.0910247
\(993\) −7.44538 −0.236272
\(994\) −33.9991 −1.07839
\(995\) −22.9313 −0.726971
\(996\) −1.42442 −0.0451345
\(997\) −36.2712 −1.14872 −0.574360 0.818603i \(-0.694749\pi\)
−0.574360 + 0.818603i \(0.694749\pi\)
\(998\) −12.9574 −0.410158
\(999\) −4.75497 −0.150440
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 4006.2.a.h.1.20 42
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4006.2.a.h.1.20 42 1.1 even 1 trivial