Properties

Label 4010.2.a.h.1.3
Level $4010$
Weight $2$
Character 4010.1
Self dual yes
Analytic conductor $32.020$
Analytic rank $1$
Dimension $9$
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] = [4010,2,Mod(1,4010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4010, 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("4010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4010.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: \(32.0200112105\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 8x^{7} + 16x^{6} + 17x^{5} - 36x^{4} - 4x^{3} + 17x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.446506\) of defining polynomial
Character \(\chi\) \(=\) 4010.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.84294 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.84294 q^{6} -0.0938987 q^{7} +1.00000 q^{8} +0.396437 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.84294 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.84294 q^{6} -0.0938987 q^{7} +1.00000 q^{8} +0.396437 q^{9} +1.00000 q^{10} -3.43048 q^{11} -1.84294 q^{12} -3.36264 q^{13} -0.0938987 q^{14} -1.84294 q^{15} +1.00000 q^{16} +2.56387 q^{17} +0.396437 q^{18} +5.98130 q^{19} +1.00000 q^{20} +0.173050 q^{21} -3.43048 q^{22} -2.45511 q^{23} -1.84294 q^{24} +1.00000 q^{25} -3.36264 q^{26} +4.79822 q^{27} -0.0938987 q^{28} +5.77350 q^{29} -1.84294 q^{30} -2.23011 q^{31} +1.00000 q^{32} +6.32217 q^{33} +2.56387 q^{34} -0.0938987 q^{35} +0.396437 q^{36} +0.381184 q^{37} +5.98130 q^{38} +6.19715 q^{39} +1.00000 q^{40} -12.2166 q^{41} +0.173050 q^{42} -3.15848 q^{43} -3.43048 q^{44} +0.396437 q^{45} -2.45511 q^{46} +10.4007 q^{47} -1.84294 q^{48} -6.99118 q^{49} +1.00000 q^{50} -4.72507 q^{51} -3.36264 q^{52} -7.89272 q^{53} +4.79822 q^{54} -3.43048 q^{55} -0.0938987 q^{56} -11.0232 q^{57} +5.77350 q^{58} -12.0849 q^{59} -1.84294 q^{60} +0.457926 q^{61} -2.23011 q^{62} -0.0372249 q^{63} +1.00000 q^{64} -3.36264 q^{65} +6.32217 q^{66} -10.7784 q^{67} +2.56387 q^{68} +4.52462 q^{69} -0.0938987 q^{70} -4.78946 q^{71} +0.396437 q^{72} -15.6224 q^{73} +0.381184 q^{74} -1.84294 q^{75} +5.98130 q^{76} +0.322117 q^{77} +6.19715 q^{78} +8.50067 q^{79} +1.00000 q^{80} -10.0321 q^{81} -12.2166 q^{82} +3.35560 q^{83} +0.173050 q^{84} +2.56387 q^{85} -3.15848 q^{86} -10.6402 q^{87} -3.43048 q^{88} -7.15936 q^{89} +0.396437 q^{90} +0.315747 q^{91} -2.45511 q^{92} +4.10997 q^{93} +10.4007 q^{94} +5.98130 q^{95} -1.84294 q^{96} +6.47511 q^{97} -6.99118 q^{98} -1.35997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} - 4 q^{3} + 9 q^{4} + 9 q^{5} - 4 q^{6} - 7 q^{7} + 9 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} - 4 q^{3} + 9 q^{4} + 9 q^{5} - 4 q^{6} - 7 q^{7} + 9 q^{8} - 7 q^{9} + 9 q^{10} - 11 q^{11} - 4 q^{12} - 14 q^{13} - 7 q^{14} - 4 q^{15} + 9 q^{16} - 13 q^{17} - 7 q^{18} - 11 q^{19} + 9 q^{20} - 8 q^{21} - 11 q^{22} - 9 q^{23} - 4 q^{24} + 9 q^{25} - 14 q^{26} - 4 q^{27} - 7 q^{28} - 20 q^{29} - 4 q^{30} - 11 q^{31} + 9 q^{32} + 4 q^{33} - 13 q^{34} - 7 q^{35} - 7 q^{36} - 25 q^{37} - 11 q^{38} - 8 q^{39} + 9 q^{40} - 29 q^{41} - 8 q^{42} - 11 q^{43} - 11 q^{44} - 7 q^{45} - 9 q^{46} - 3 q^{47} - 4 q^{48} - 18 q^{49} + 9 q^{50} + q^{51} - 14 q^{52} - 9 q^{53} - 4 q^{54} - 11 q^{55} - 7 q^{56} - 17 q^{57} - 20 q^{58} - 10 q^{59} - 4 q^{60} - 10 q^{61} - 11 q^{62} + 16 q^{63} + 9 q^{64} - 14 q^{65} + 4 q^{66} - 16 q^{67} - 13 q^{68} + 5 q^{69} - 7 q^{70} - 8 q^{71} - 7 q^{72} - 22 q^{73} - 25 q^{74} - 4 q^{75} - 11 q^{76} - 15 q^{77} - 8 q^{78} - 9 q^{79} + 9 q^{80} - 15 q^{81} - 29 q^{82} + 11 q^{83} - 8 q^{84} - 13 q^{85} - 11 q^{86} + 12 q^{87} - 11 q^{88} - 28 q^{89} - 7 q^{90} - 6 q^{91} - 9 q^{92} + 16 q^{93} - 3 q^{94} - 11 q^{95} - 4 q^{96} - 28 q^{97} - 18 q^{98} + 9 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.84294 −1.06402 −0.532012 0.846737i \(-0.678564\pi\)
−0.532012 + 0.846737i \(0.678564\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.84294 −0.752378
\(7\) −0.0938987 −0.0354904 −0.0177452 0.999843i \(-0.505649\pi\)
−0.0177452 + 0.999843i \(0.505649\pi\)
\(8\) 1.00000 0.353553
\(9\) 0.396437 0.132146
\(10\) 1.00000 0.316228
\(11\) −3.43048 −1.03433 −0.517164 0.855886i \(-0.673012\pi\)
−0.517164 + 0.855886i \(0.673012\pi\)
\(12\) −1.84294 −0.532012
\(13\) −3.36264 −0.932628 −0.466314 0.884619i \(-0.654418\pi\)
−0.466314 + 0.884619i \(0.654418\pi\)
\(14\) −0.0938987 −0.0250955
\(15\) −1.84294 −0.475846
\(16\) 1.00000 0.250000
\(17\) 2.56387 0.621831 0.310915 0.950438i \(-0.399364\pi\)
0.310915 + 0.950438i \(0.399364\pi\)
\(18\) 0.396437 0.0934411
\(19\) 5.98130 1.37220 0.686102 0.727505i \(-0.259320\pi\)
0.686102 + 0.727505i \(0.259320\pi\)
\(20\) 1.00000 0.223607
\(21\) 0.173050 0.0377626
\(22\) −3.43048 −0.731380
\(23\) −2.45511 −0.511925 −0.255963 0.966687i \(-0.582392\pi\)
−0.255963 + 0.966687i \(0.582392\pi\)
\(24\) −1.84294 −0.376189
\(25\) 1.00000 0.200000
\(26\) −3.36264 −0.659468
\(27\) 4.79822 0.923417
\(28\) −0.0938987 −0.0177452
\(29\) 5.77350 1.07211 0.536056 0.844183i \(-0.319914\pi\)
0.536056 + 0.844183i \(0.319914\pi\)
\(30\) −1.84294 −0.336474
\(31\) −2.23011 −0.400540 −0.200270 0.979741i \(-0.564182\pi\)
−0.200270 + 0.979741i \(0.564182\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.32217 1.10055
\(34\) 2.56387 0.439701
\(35\) −0.0938987 −0.0158718
\(36\) 0.396437 0.0660728
\(37\) 0.381184 0.0626663 0.0313332 0.999509i \(-0.490025\pi\)
0.0313332 + 0.999509i \(0.490025\pi\)
\(38\) 5.98130 0.970295
\(39\) 6.19715 0.992338
\(40\) 1.00000 0.158114
\(41\) −12.2166 −1.90791 −0.953955 0.299949i \(-0.903030\pi\)
−0.953955 + 0.299949i \(0.903030\pi\)
\(42\) 0.173050 0.0267022
\(43\) −3.15848 −0.481663 −0.240832 0.970567i \(-0.577420\pi\)
−0.240832 + 0.970567i \(0.577420\pi\)
\(44\) −3.43048 −0.517164
\(45\) 0.396437 0.0590973
\(46\) −2.45511 −0.361986
\(47\) 10.4007 1.51710 0.758551 0.651613i \(-0.225907\pi\)
0.758551 + 0.651613i \(0.225907\pi\)
\(48\) −1.84294 −0.266006
\(49\) −6.99118 −0.998740
\(50\) 1.00000 0.141421
\(51\) −4.72507 −0.661642
\(52\) −3.36264 −0.466314
\(53\) −7.89272 −1.08415 −0.542074 0.840331i \(-0.682361\pi\)
−0.542074 + 0.840331i \(0.682361\pi\)
\(54\) 4.79822 0.652955
\(55\) −3.43048 −0.462565
\(56\) −0.0938987 −0.0125477
\(57\) −11.0232 −1.46006
\(58\) 5.77350 0.758097
\(59\) −12.0849 −1.57332 −0.786658 0.617388i \(-0.788191\pi\)
−0.786658 + 0.617388i \(0.788191\pi\)
\(60\) −1.84294 −0.237923
\(61\) 0.457926 0.0586314 0.0293157 0.999570i \(-0.490667\pi\)
0.0293157 + 0.999570i \(0.490667\pi\)
\(62\) −2.23011 −0.283225
\(63\) −0.0372249 −0.00468990
\(64\) 1.00000 0.125000
\(65\) −3.36264 −0.417084
\(66\) 6.32217 0.778205
\(67\) −10.7784 −1.31679 −0.658395 0.752672i \(-0.728764\pi\)
−0.658395 + 0.752672i \(0.728764\pi\)
\(68\) 2.56387 0.310915
\(69\) 4.52462 0.544701
\(70\) −0.0938987 −0.0112230
\(71\) −4.78946 −0.568404 −0.284202 0.958764i \(-0.591729\pi\)
−0.284202 + 0.958764i \(0.591729\pi\)
\(72\) 0.396437 0.0467205
\(73\) −15.6224 −1.82846 −0.914232 0.405190i \(-0.867205\pi\)
−0.914232 + 0.405190i \(0.867205\pi\)
\(74\) 0.381184 0.0443118
\(75\) −1.84294 −0.212805
\(76\) 5.98130 0.686102
\(77\) 0.322117 0.0367087
\(78\) 6.19715 0.701689
\(79\) 8.50067 0.956400 0.478200 0.878251i \(-0.341289\pi\)
0.478200 + 0.878251i \(0.341289\pi\)
\(80\) 1.00000 0.111803
\(81\) −10.0321 −1.11468
\(82\) −12.2166 −1.34910
\(83\) 3.35560 0.368325 0.184163 0.982896i \(-0.441043\pi\)
0.184163 + 0.982896i \(0.441043\pi\)
\(84\) 0.173050 0.0188813
\(85\) 2.56387 0.278091
\(86\) −3.15848 −0.340587
\(87\) −10.6402 −1.14075
\(88\) −3.43048 −0.365690
\(89\) −7.15936 −0.758891 −0.379446 0.925214i \(-0.623885\pi\)
−0.379446 + 0.925214i \(0.623885\pi\)
\(90\) 0.396437 0.0417881
\(91\) 0.315747 0.0330993
\(92\) −2.45511 −0.255963
\(93\) 4.10997 0.426184
\(94\) 10.4007 1.07275
\(95\) 5.98130 0.613668
\(96\) −1.84294 −0.188095
\(97\) 6.47511 0.657448 0.328724 0.944426i \(-0.393381\pi\)
0.328724 + 0.944426i \(0.393381\pi\)
\(98\) −6.99118 −0.706216
\(99\) −1.35997 −0.136682
\(100\) 1.00000 0.100000
\(101\) −2.55210 −0.253944 −0.126972 0.991906i \(-0.540526\pi\)
−0.126972 + 0.991906i \(0.540526\pi\)
\(102\) −4.72507 −0.467852
\(103\) −3.93006 −0.387240 −0.193620 0.981077i \(-0.562023\pi\)
−0.193620 + 0.981077i \(0.562023\pi\)
\(104\) −3.36264 −0.329734
\(105\) 0.173050 0.0168879
\(106\) −7.89272 −0.766608
\(107\) 18.5937 1.79752 0.898759 0.438443i \(-0.144470\pi\)
0.898759 + 0.438443i \(0.144470\pi\)
\(108\) 4.79822 0.461709
\(109\) −6.58120 −0.630365 −0.315183 0.949031i \(-0.602066\pi\)
−0.315183 + 0.949031i \(0.602066\pi\)
\(110\) −3.43048 −0.327083
\(111\) −0.702501 −0.0666784
\(112\) −0.0938987 −0.00887259
\(113\) 9.12255 0.858177 0.429089 0.903262i \(-0.358835\pi\)
0.429089 + 0.903262i \(0.358835\pi\)
\(114\) −11.0232 −1.03242
\(115\) −2.45511 −0.228940
\(116\) 5.77350 0.536056
\(117\) −1.33307 −0.123243
\(118\) −12.0849 −1.11250
\(119\) −0.240744 −0.0220690
\(120\) −1.84294 −0.168237
\(121\) 0.768168 0.0698334
\(122\) 0.457926 0.0414587
\(123\) 22.5145 2.03006
\(124\) −2.23011 −0.200270
\(125\) 1.00000 0.0894427
\(126\) −0.0372249 −0.00331626
\(127\) 21.8559 1.93940 0.969701 0.244297i \(-0.0785571\pi\)
0.969701 + 0.244297i \(0.0785571\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.82089 0.512501
\(130\) −3.36264 −0.294923
\(131\) −9.46857 −0.827273 −0.413636 0.910442i \(-0.635742\pi\)
−0.413636 + 0.910442i \(0.635742\pi\)
\(132\) 6.32217 0.550274
\(133\) −0.561636 −0.0487000
\(134\) −10.7784 −0.931112
\(135\) 4.79822 0.412965
\(136\) 2.56387 0.219850
\(137\) −15.5435 −1.32797 −0.663986 0.747745i \(-0.731137\pi\)
−0.663986 + 0.747745i \(0.731137\pi\)
\(138\) 4.52462 0.385162
\(139\) −16.9704 −1.43941 −0.719705 0.694280i \(-0.755723\pi\)
−0.719705 + 0.694280i \(0.755723\pi\)
\(140\) −0.0938987 −0.00793589
\(141\) −19.1680 −1.61423
\(142\) −4.78946 −0.401923
\(143\) 11.5355 0.964643
\(144\) 0.396437 0.0330364
\(145\) 5.77350 0.479463
\(146\) −15.6224 −1.29292
\(147\) 12.8843 1.06268
\(148\) 0.381184 0.0313332
\(149\) −4.19924 −0.344015 −0.172007 0.985096i \(-0.555025\pi\)
−0.172007 + 0.985096i \(0.555025\pi\)
\(150\) −1.84294 −0.150476
\(151\) 0.915926 0.0745370 0.0372685 0.999305i \(-0.488134\pi\)
0.0372685 + 0.999305i \(0.488134\pi\)
\(152\) 5.98130 0.485147
\(153\) 1.01641 0.0821722
\(154\) 0.322117 0.0259569
\(155\) −2.23011 −0.179127
\(156\) 6.19715 0.496169
\(157\) −18.3891 −1.46761 −0.733807 0.679358i \(-0.762258\pi\)
−0.733807 + 0.679358i \(0.762258\pi\)
\(158\) 8.50067 0.676277
\(159\) 14.5458 1.15356
\(160\) 1.00000 0.0790569
\(161\) 0.230531 0.0181684
\(162\) −10.0321 −0.788200
\(163\) −18.4734 −1.44695 −0.723473 0.690352i \(-0.757456\pi\)
−0.723473 + 0.690352i \(0.757456\pi\)
\(164\) −12.2166 −0.953955
\(165\) 6.32217 0.492180
\(166\) 3.35560 0.260445
\(167\) −21.8788 −1.69303 −0.846514 0.532366i \(-0.821303\pi\)
−0.846514 + 0.532366i \(0.821303\pi\)
\(168\) 0.173050 0.0133511
\(169\) −1.69266 −0.130205
\(170\) 2.56387 0.196640
\(171\) 2.37121 0.181331
\(172\) −3.15848 −0.240832
\(173\) 0.872010 0.0662977 0.0331488 0.999450i \(-0.489446\pi\)
0.0331488 + 0.999450i \(0.489446\pi\)
\(174\) −10.6402 −0.806633
\(175\) −0.0938987 −0.00709807
\(176\) −3.43048 −0.258582
\(177\) 22.2717 1.67405
\(178\) −7.15936 −0.536617
\(179\) −10.6354 −0.794927 −0.397463 0.917618i \(-0.630109\pi\)
−0.397463 + 0.917618i \(0.630109\pi\)
\(180\) 0.396437 0.0295487
\(181\) 6.54024 0.486132 0.243066 0.970010i \(-0.421847\pi\)
0.243066 + 0.970010i \(0.421847\pi\)
\(182\) 0.315747 0.0234047
\(183\) −0.843932 −0.0623852
\(184\) −2.45511 −0.180993
\(185\) 0.381184 0.0280252
\(186\) 4.10997 0.301358
\(187\) −8.79531 −0.643177
\(188\) 10.4007 0.758551
\(189\) −0.450546 −0.0327724
\(190\) 5.98130 0.433929
\(191\) 8.47937 0.613546 0.306773 0.951783i \(-0.400751\pi\)
0.306773 + 0.951783i \(0.400751\pi\)
\(192\) −1.84294 −0.133003
\(193\) −8.23319 −0.592638 −0.296319 0.955089i \(-0.595759\pi\)
−0.296319 + 0.955089i \(0.595759\pi\)
\(194\) 6.47511 0.464886
\(195\) 6.19715 0.443787
\(196\) −6.99118 −0.499370
\(197\) −0.175702 −0.0125183 −0.00625914 0.999980i \(-0.501992\pi\)
−0.00625914 + 0.999980i \(0.501992\pi\)
\(198\) −1.35997 −0.0966487
\(199\) 5.73441 0.406502 0.203251 0.979127i \(-0.434849\pi\)
0.203251 + 0.979127i \(0.434849\pi\)
\(200\) 1.00000 0.0707107
\(201\) 19.8640 1.40110
\(202\) −2.55210 −0.179565
\(203\) −0.542124 −0.0380496
\(204\) −4.72507 −0.330821
\(205\) −12.2166 −0.853244
\(206\) −3.93006 −0.273820
\(207\) −0.973296 −0.0676487
\(208\) −3.36264 −0.233157
\(209\) −20.5187 −1.41931
\(210\) 0.173050 0.0119416
\(211\) −1.51538 −0.104323 −0.0521616 0.998639i \(-0.516611\pi\)
−0.0521616 + 0.998639i \(0.516611\pi\)
\(212\) −7.89272 −0.542074
\(213\) 8.82670 0.604795
\(214\) 18.5937 1.27104
\(215\) −3.15848 −0.215406
\(216\) 4.79822 0.326477
\(217\) 0.209405 0.0142153
\(218\) −6.58120 −0.445735
\(219\) 28.7912 1.94553
\(220\) −3.43048 −0.231283
\(221\) −8.62138 −0.579937
\(222\) −0.702501 −0.0471488
\(223\) 15.7565 1.05514 0.527568 0.849513i \(-0.323104\pi\)
0.527568 + 0.849513i \(0.323104\pi\)
\(224\) −0.0938987 −0.00627387
\(225\) 0.396437 0.0264291
\(226\) 9.12255 0.606823
\(227\) 0.499976 0.0331846 0.0165923 0.999862i \(-0.494718\pi\)
0.0165923 + 0.999862i \(0.494718\pi\)
\(228\) −11.0232 −0.730028
\(229\) −7.06065 −0.466581 −0.233291 0.972407i \(-0.574949\pi\)
−0.233291 + 0.972407i \(0.574949\pi\)
\(230\) −2.45511 −0.161885
\(231\) −0.593643 −0.0390589
\(232\) 5.77350 0.379049
\(233\) 0.634346 0.0415574 0.0207787 0.999784i \(-0.493385\pi\)
0.0207787 + 0.999784i \(0.493385\pi\)
\(234\) −1.33307 −0.0871458
\(235\) 10.4007 0.678469
\(236\) −12.0849 −0.786658
\(237\) −15.6662 −1.01763
\(238\) −0.240744 −0.0156051
\(239\) 15.8886 1.02775 0.513876 0.857865i \(-0.328209\pi\)
0.513876 + 0.857865i \(0.328209\pi\)
\(240\) −1.84294 −0.118961
\(241\) −18.6749 −1.20295 −0.601477 0.798890i \(-0.705421\pi\)
−0.601477 + 0.798890i \(0.705421\pi\)
\(242\) 0.768168 0.0493797
\(243\) 4.09402 0.262632
\(244\) 0.457926 0.0293157
\(245\) −6.99118 −0.446650
\(246\) 22.5145 1.43547
\(247\) −20.1129 −1.27976
\(248\) −2.23011 −0.141612
\(249\) −6.18418 −0.391907
\(250\) 1.00000 0.0632456
\(251\) −4.78389 −0.301956 −0.150978 0.988537i \(-0.548242\pi\)
−0.150978 + 0.988537i \(0.548242\pi\)
\(252\) −0.0372249 −0.00234495
\(253\) 8.42219 0.529499
\(254\) 21.8559 1.37136
\(255\) −4.72507 −0.295895
\(256\) 1.00000 0.0625000
\(257\) −0.924873 −0.0576920 −0.0288460 0.999584i \(-0.509183\pi\)
−0.0288460 + 0.999584i \(0.509183\pi\)
\(258\) 5.82089 0.362393
\(259\) −0.0357927 −0.00222405
\(260\) −3.36264 −0.208542
\(261\) 2.28883 0.141675
\(262\) −9.46857 −0.584970
\(263\) −7.75351 −0.478102 −0.239051 0.971007i \(-0.576836\pi\)
−0.239051 + 0.971007i \(0.576836\pi\)
\(264\) 6.32217 0.389103
\(265\) −7.89272 −0.484846
\(266\) −0.561636 −0.0344361
\(267\) 13.1943 0.807478
\(268\) −10.7784 −0.658395
\(269\) 29.8572 1.82043 0.910213 0.414140i \(-0.135918\pi\)
0.910213 + 0.414140i \(0.135918\pi\)
\(270\) 4.79822 0.292010
\(271\) 7.22166 0.438684 0.219342 0.975648i \(-0.429609\pi\)
0.219342 + 0.975648i \(0.429609\pi\)
\(272\) 2.56387 0.155458
\(273\) −0.581904 −0.0352184
\(274\) −15.5435 −0.939019
\(275\) −3.43048 −0.206866
\(276\) 4.52462 0.272350
\(277\) −27.5920 −1.65784 −0.828921 0.559366i \(-0.811045\pi\)
−0.828921 + 0.559366i \(0.811045\pi\)
\(278\) −16.9704 −1.01782
\(279\) −0.884099 −0.0529296
\(280\) −0.0938987 −0.00561152
\(281\) −9.57593 −0.571252 −0.285626 0.958341i \(-0.592202\pi\)
−0.285626 + 0.958341i \(0.592202\pi\)
\(282\) −19.1680 −1.14144
\(283\) 4.86749 0.289342 0.144671 0.989480i \(-0.453788\pi\)
0.144671 + 0.989480i \(0.453788\pi\)
\(284\) −4.78946 −0.284202
\(285\) −11.0232 −0.652957
\(286\) 11.5355 0.682105
\(287\) 1.14712 0.0677124
\(288\) 0.396437 0.0233603
\(289\) −10.4266 −0.613326
\(290\) 5.77350 0.339031
\(291\) −11.9333 −0.699540
\(292\) −15.6224 −0.914232
\(293\) 8.86662 0.517993 0.258997 0.965878i \(-0.416608\pi\)
0.258997 + 0.965878i \(0.416608\pi\)
\(294\) 12.8843 0.751430
\(295\) −12.0849 −0.703609
\(296\) 0.381184 0.0221559
\(297\) −16.4602 −0.955116
\(298\) −4.19924 −0.243255
\(299\) 8.25564 0.477436
\(300\) −1.84294 −0.106402
\(301\) 0.296577 0.0170944
\(302\) 0.915926 0.0527056
\(303\) 4.70338 0.270202
\(304\) 5.98130 0.343051
\(305\) 0.457926 0.0262208
\(306\) 1.01641 0.0581045
\(307\) 13.1248 0.749074 0.374537 0.927212i \(-0.377802\pi\)
0.374537 + 0.927212i \(0.377802\pi\)
\(308\) 0.322117 0.0183543
\(309\) 7.24287 0.412033
\(310\) −2.23011 −0.126662
\(311\) −1.50532 −0.0853591 −0.0426795 0.999089i \(-0.513589\pi\)
−0.0426795 + 0.999089i \(0.513589\pi\)
\(312\) 6.19715 0.350844
\(313\) 2.86091 0.161708 0.0808540 0.996726i \(-0.474235\pi\)
0.0808540 + 0.996726i \(0.474235\pi\)
\(314\) −18.3891 −1.03776
\(315\) −0.0372249 −0.00209739
\(316\) 8.50067 0.478200
\(317\) −20.5102 −1.15197 −0.575984 0.817461i \(-0.695381\pi\)
−0.575984 + 0.817461i \(0.695381\pi\)
\(318\) 14.5458 0.815689
\(319\) −19.8058 −1.10891
\(320\) 1.00000 0.0559017
\(321\) −34.2671 −1.91260
\(322\) 0.230531 0.0128470
\(323\) 15.3353 0.853279
\(324\) −10.0321 −0.557342
\(325\) −3.36264 −0.186526
\(326\) −18.4734 −1.02315
\(327\) 12.1288 0.670723
\(328\) −12.2166 −0.674548
\(329\) −0.976615 −0.0538425
\(330\) 6.32217 0.348024
\(331\) −9.27471 −0.509784 −0.254892 0.966970i \(-0.582040\pi\)
−0.254892 + 0.966970i \(0.582040\pi\)
\(332\) 3.35560 0.184163
\(333\) 0.151116 0.00828108
\(334\) −21.8788 −1.19715
\(335\) −10.7784 −0.588887
\(336\) 0.173050 0.00944064
\(337\) 24.6047 1.34030 0.670152 0.742223i \(-0.266229\pi\)
0.670152 + 0.742223i \(0.266229\pi\)
\(338\) −1.69266 −0.0920688
\(339\) −16.8123 −0.913120
\(340\) 2.56387 0.139046
\(341\) 7.65035 0.414290
\(342\) 2.37121 0.128220
\(343\) 1.31375 0.0709360
\(344\) −3.15848 −0.170294
\(345\) 4.52462 0.243598
\(346\) 0.872010 0.0468795
\(347\) −3.87165 −0.207841 −0.103921 0.994586i \(-0.533139\pi\)
−0.103921 + 0.994586i \(0.533139\pi\)
\(348\) −10.6402 −0.570376
\(349\) −12.0503 −0.645036 −0.322518 0.946563i \(-0.604529\pi\)
−0.322518 + 0.946563i \(0.604529\pi\)
\(350\) −0.0938987 −0.00501910
\(351\) −16.1347 −0.861205
\(352\) −3.43048 −0.182845
\(353\) 32.0479 1.70574 0.852868 0.522126i \(-0.174861\pi\)
0.852868 + 0.522126i \(0.174861\pi\)
\(354\) 22.2717 1.18373
\(355\) −4.78946 −0.254198
\(356\) −7.15936 −0.379446
\(357\) 0.443678 0.0234819
\(358\) −10.6354 −0.562098
\(359\) −20.4761 −1.08069 −0.540344 0.841444i \(-0.681706\pi\)
−0.540344 + 0.841444i \(0.681706\pi\)
\(360\) 0.396437 0.0208941
\(361\) 16.7759 0.882944
\(362\) 6.54024 0.343747
\(363\) −1.41569 −0.0743044
\(364\) 0.315747 0.0165497
\(365\) −15.6224 −0.817714
\(366\) −0.843932 −0.0441130
\(367\) −1.25272 −0.0653917 −0.0326958 0.999465i \(-0.510409\pi\)
−0.0326958 + 0.999465i \(0.510409\pi\)
\(368\) −2.45511 −0.127981
\(369\) −4.84311 −0.252122
\(370\) 0.381184 0.0198168
\(371\) 0.741116 0.0384768
\(372\) 4.10997 0.213092
\(373\) 19.0849 0.988179 0.494090 0.869411i \(-0.335501\pi\)
0.494090 + 0.869411i \(0.335501\pi\)
\(374\) −8.79531 −0.454795
\(375\) −1.84294 −0.0951691
\(376\) 10.4007 0.536377
\(377\) −19.4142 −0.999881
\(378\) −0.450546 −0.0231736
\(379\) 19.9070 1.02255 0.511276 0.859416i \(-0.329173\pi\)
0.511276 + 0.859416i \(0.329173\pi\)
\(380\) 5.98130 0.306834
\(381\) −40.2792 −2.06357
\(382\) 8.47937 0.433842
\(383\) −22.4584 −1.14757 −0.573786 0.819005i \(-0.694526\pi\)
−0.573786 + 0.819005i \(0.694526\pi\)
\(384\) −1.84294 −0.0940473
\(385\) 0.322117 0.0164166
\(386\) −8.23319 −0.419058
\(387\) −1.25214 −0.0636497
\(388\) 6.47511 0.328724
\(389\) 24.0783 1.22082 0.610409 0.792087i \(-0.291005\pi\)
0.610409 + 0.792087i \(0.291005\pi\)
\(390\) 6.19715 0.313805
\(391\) −6.29459 −0.318331
\(392\) −6.99118 −0.353108
\(393\) 17.4500 0.880238
\(394\) −0.175702 −0.00885176
\(395\) 8.50067 0.427715
\(396\) −1.35997 −0.0683409
\(397\) 39.1825 1.96651 0.983257 0.182226i \(-0.0583303\pi\)
0.983257 + 0.182226i \(0.0583303\pi\)
\(398\) 5.73441 0.287440
\(399\) 1.03506 0.0518180
\(400\) 1.00000 0.0500000
\(401\) −1.00000 −0.0499376
\(402\) 19.8640 0.990725
\(403\) 7.49906 0.373555
\(404\) −2.55210 −0.126972
\(405\) −10.0321 −0.498501
\(406\) −0.542124 −0.0269051
\(407\) −1.30764 −0.0648175
\(408\) −4.72507 −0.233926
\(409\) −27.2597 −1.34790 −0.673952 0.738775i \(-0.735404\pi\)
−0.673952 + 0.738775i \(0.735404\pi\)
\(410\) −12.2166 −0.603334
\(411\) 28.6458 1.41299
\(412\) −3.93006 −0.193620
\(413\) 1.13475 0.0558376
\(414\) −0.973296 −0.0478349
\(415\) 3.35560 0.164720
\(416\) −3.36264 −0.164867
\(417\) 31.2755 1.53157
\(418\) −20.5187 −1.00360
\(419\) 0.625472 0.0305563 0.0152782 0.999883i \(-0.495137\pi\)
0.0152782 + 0.999883i \(0.495137\pi\)
\(420\) 0.173050 0.00844397
\(421\) 4.29032 0.209097 0.104549 0.994520i \(-0.466660\pi\)
0.104549 + 0.994520i \(0.466660\pi\)
\(422\) −1.51538 −0.0737676
\(423\) 4.12324 0.200479
\(424\) −7.89272 −0.383304
\(425\) 2.56387 0.124366
\(426\) 8.82670 0.427655
\(427\) −0.0429987 −0.00208085
\(428\) 18.5937 0.898759
\(429\) −21.2592 −1.02640
\(430\) −3.15848 −0.152315
\(431\) −29.9833 −1.44425 −0.722123 0.691764i \(-0.756834\pi\)
−0.722123 + 0.691764i \(0.756834\pi\)
\(432\) 4.79822 0.230854
\(433\) 18.0144 0.865715 0.432857 0.901462i \(-0.357505\pi\)
0.432857 + 0.901462i \(0.357505\pi\)
\(434\) 0.209405 0.0100517
\(435\) −10.6402 −0.510160
\(436\) −6.58120 −0.315183
\(437\) −14.6847 −0.702466
\(438\) 28.7912 1.37570
\(439\) 4.45552 0.212650 0.106325 0.994331i \(-0.466092\pi\)
0.106325 + 0.994331i \(0.466092\pi\)
\(440\) −3.43048 −0.163542
\(441\) −2.77156 −0.131979
\(442\) −8.62138 −0.410077
\(443\) 32.2852 1.53392 0.766958 0.641698i \(-0.221770\pi\)
0.766958 + 0.641698i \(0.221770\pi\)
\(444\) −0.702501 −0.0333392
\(445\) −7.15936 −0.339386
\(446\) 15.7565 0.746094
\(447\) 7.73895 0.366040
\(448\) −0.0938987 −0.00443630
\(449\) 17.7475 0.837555 0.418778 0.908089i \(-0.362459\pi\)
0.418778 + 0.908089i \(0.362459\pi\)
\(450\) 0.396437 0.0186882
\(451\) 41.9087 1.97340
\(452\) 9.12255 0.429089
\(453\) −1.68800 −0.0793091
\(454\) 0.499976 0.0234650
\(455\) 0.315747 0.0148025
\(456\) −11.0232 −0.516208
\(457\) −38.5645 −1.80397 −0.901986 0.431766i \(-0.857891\pi\)
−0.901986 + 0.431766i \(0.857891\pi\)
\(458\) −7.06065 −0.329923
\(459\) 12.3020 0.574209
\(460\) −2.45511 −0.114470
\(461\) 16.0662 0.748279 0.374139 0.927373i \(-0.377938\pi\)
0.374139 + 0.927373i \(0.377938\pi\)
\(462\) −0.593643 −0.0276188
\(463\) −8.15714 −0.379095 −0.189547 0.981872i \(-0.560702\pi\)
−0.189547 + 0.981872i \(0.560702\pi\)
\(464\) 5.77350 0.268028
\(465\) 4.10997 0.190595
\(466\) 0.634346 0.0293855
\(467\) 39.3652 1.82161 0.910803 0.412842i \(-0.135464\pi\)
0.910803 + 0.412842i \(0.135464\pi\)
\(468\) −1.33307 −0.0616214
\(469\) 1.01208 0.0467334
\(470\) 10.4007 0.479750
\(471\) 33.8901 1.56157
\(472\) −12.0849 −0.556252
\(473\) 10.8351 0.498197
\(474\) −15.6662 −0.719575
\(475\) 5.98130 0.274441
\(476\) −0.240744 −0.0110345
\(477\) −3.12896 −0.143265
\(478\) 15.8886 0.726730
\(479\) −34.6506 −1.58323 −0.791614 0.611021i \(-0.790759\pi\)
−0.791614 + 0.611021i \(0.790759\pi\)
\(480\) −1.84294 −0.0841184
\(481\) −1.28179 −0.0584444
\(482\) −18.6749 −0.850617
\(483\) −0.424856 −0.0193316
\(484\) 0.768168 0.0349167
\(485\) 6.47511 0.294020
\(486\) 4.09402 0.185709
\(487\) −7.33397 −0.332334 −0.166167 0.986098i \(-0.553139\pi\)
−0.166167 + 0.986098i \(0.553139\pi\)
\(488\) 0.457926 0.0207293
\(489\) 34.0454 1.53958
\(490\) −6.99118 −0.315829
\(491\) 28.8677 1.30278 0.651390 0.758743i \(-0.274186\pi\)
0.651390 + 0.758743i \(0.274186\pi\)
\(492\) 22.5145 1.01503
\(493\) 14.8025 0.666672
\(494\) −20.1129 −0.904924
\(495\) −1.35997 −0.0611260
\(496\) −2.23011 −0.100135
\(497\) 0.449724 0.0201729
\(498\) −6.18418 −0.277120
\(499\) 25.1537 1.12603 0.563017 0.826445i \(-0.309641\pi\)
0.563017 + 0.826445i \(0.309641\pi\)
\(500\) 1.00000 0.0447214
\(501\) 40.3213 1.80142
\(502\) −4.78389 −0.213515
\(503\) 41.8439 1.86573 0.932864 0.360230i \(-0.117302\pi\)
0.932864 + 0.360230i \(0.117302\pi\)
\(504\) −0.0372249 −0.00165813
\(505\) −2.55210 −0.113567
\(506\) 8.42219 0.374412
\(507\) 3.11948 0.138541
\(508\) 21.8559 0.969701
\(509\) −20.9049 −0.926595 −0.463297 0.886203i \(-0.653334\pi\)
−0.463297 + 0.886203i \(0.653334\pi\)
\(510\) −4.72507 −0.209230
\(511\) 1.46692 0.0648929
\(512\) 1.00000 0.0441942
\(513\) 28.6996 1.26712
\(514\) −0.924873 −0.0407944
\(515\) −3.93006 −0.173179
\(516\) 5.82089 0.256250
\(517\) −35.6795 −1.56918
\(518\) −0.0357927 −0.00157264
\(519\) −1.60706 −0.0705423
\(520\) −3.36264 −0.147461
\(521\) −22.4931 −0.985440 −0.492720 0.870188i \(-0.663997\pi\)
−0.492720 + 0.870188i \(0.663997\pi\)
\(522\) 2.28883 0.100179
\(523\) 28.7738 1.25819 0.629094 0.777329i \(-0.283426\pi\)
0.629094 + 0.777329i \(0.283426\pi\)
\(524\) −9.46857 −0.413636
\(525\) 0.173050 0.00755252
\(526\) −7.75351 −0.338069
\(527\) −5.71773 −0.249068
\(528\) 6.32217 0.275137
\(529\) −16.9724 −0.737932
\(530\) −7.89272 −0.342838
\(531\) −4.79089 −0.207907
\(532\) −0.561636 −0.0243500
\(533\) 41.0800 1.77937
\(534\) 13.1943 0.570973
\(535\) 18.5937 0.803874
\(536\) −10.7784 −0.465556
\(537\) 19.6004 0.845821
\(538\) 29.8572 1.28724
\(539\) 23.9831 1.03302
\(540\) 4.79822 0.206482
\(541\) −7.75971 −0.333616 −0.166808 0.985989i \(-0.553346\pi\)
−0.166808 + 0.985989i \(0.553346\pi\)
\(542\) 7.22166 0.310197
\(543\) −12.0533 −0.517256
\(544\) 2.56387 0.109925
\(545\) −6.58120 −0.281908
\(546\) −0.581904 −0.0249032
\(547\) 28.7639 1.22985 0.614927 0.788584i \(-0.289185\pi\)
0.614927 + 0.788584i \(0.289185\pi\)
\(548\) −15.5435 −0.663986
\(549\) 0.181539 0.00774789
\(550\) −3.43048 −0.146276
\(551\) 34.5330 1.47116
\(552\) 4.52462 0.192581
\(553\) −0.798202 −0.0339430
\(554\) −27.5920 −1.17227
\(555\) −0.702501 −0.0298195
\(556\) −16.9704 −0.719705
\(557\) −18.8940 −0.800563 −0.400281 0.916392i \(-0.631088\pi\)
−0.400281 + 0.916392i \(0.631088\pi\)
\(558\) −0.884099 −0.0374269
\(559\) 10.6208 0.449213
\(560\) −0.0938987 −0.00396794
\(561\) 16.2092 0.684355
\(562\) −9.57593 −0.403936
\(563\) 35.8955 1.51281 0.756407 0.654102i \(-0.226953\pi\)
0.756407 + 0.654102i \(0.226953\pi\)
\(564\) −19.1680 −0.807116
\(565\) 9.12255 0.383788
\(566\) 4.86749 0.204596
\(567\) 0.942006 0.0395605
\(568\) −4.78946 −0.200961
\(569\) −37.7436 −1.58229 −0.791146 0.611627i \(-0.790515\pi\)
−0.791146 + 0.611627i \(0.790515\pi\)
\(570\) −11.0232 −0.461711
\(571\) −23.7037 −0.991968 −0.495984 0.868332i \(-0.665193\pi\)
−0.495984 + 0.868332i \(0.665193\pi\)
\(572\) 11.5355 0.482321
\(573\) −15.6270 −0.652827
\(574\) 1.14712 0.0478799
\(575\) −2.45511 −0.102385
\(576\) 0.396437 0.0165182
\(577\) −24.8378 −1.03401 −0.517006 0.855982i \(-0.672954\pi\)
−0.517006 + 0.855982i \(0.672954\pi\)
\(578\) −10.4266 −0.433687
\(579\) 15.1733 0.630581
\(580\) 5.77350 0.239731
\(581\) −0.315087 −0.0130720
\(582\) −11.9333 −0.494649
\(583\) 27.0758 1.12136
\(584\) −15.6224 −0.646460
\(585\) −1.33307 −0.0551158
\(586\) 8.86662 0.366277
\(587\) −35.6906 −1.47311 −0.736554 0.676379i \(-0.763548\pi\)
−0.736554 + 0.676379i \(0.763548\pi\)
\(588\) 12.8843 0.531342
\(589\) −13.3390 −0.549623
\(590\) −12.0849 −0.497526
\(591\) 0.323809 0.0133197
\(592\) 0.381184 0.0156666
\(593\) 2.72379 0.111852 0.0559262 0.998435i \(-0.482189\pi\)
0.0559262 + 0.998435i \(0.482189\pi\)
\(594\) −16.4602 −0.675369
\(595\) −0.240744 −0.00986956
\(596\) −4.19924 −0.172007
\(597\) −10.5682 −0.432527
\(598\) 8.25564 0.337598
\(599\) −42.5223 −1.73741 −0.868707 0.495326i \(-0.835049\pi\)
−0.868707 + 0.495326i \(0.835049\pi\)
\(600\) −1.84294 −0.0752378
\(601\) −13.2007 −0.538467 −0.269234 0.963075i \(-0.586770\pi\)
−0.269234 + 0.963075i \(0.586770\pi\)
\(602\) 0.296577 0.0120876
\(603\) −4.27295 −0.174008
\(604\) 0.915926 0.0372685
\(605\) 0.768168 0.0312305
\(606\) 4.70338 0.191062
\(607\) −37.9231 −1.53925 −0.769625 0.638496i \(-0.779557\pi\)
−0.769625 + 0.638496i \(0.779557\pi\)
\(608\) 5.98130 0.242574
\(609\) 0.999103 0.0404857
\(610\) 0.457926 0.0185409
\(611\) −34.9739 −1.41489
\(612\) 1.01641 0.0410861
\(613\) −9.78357 −0.395155 −0.197577 0.980287i \(-0.563307\pi\)
−0.197577 + 0.980287i \(0.563307\pi\)
\(614\) 13.1248 0.529675
\(615\) 22.5145 0.907871
\(616\) 0.322117 0.0129785
\(617\) −5.73237 −0.230777 −0.115388 0.993320i \(-0.536811\pi\)
−0.115388 + 0.993320i \(0.536811\pi\)
\(618\) 7.24287 0.291351
\(619\) 23.9435 0.962370 0.481185 0.876619i \(-0.340206\pi\)
0.481185 + 0.876619i \(0.340206\pi\)
\(620\) −2.23011 −0.0895635
\(621\) −11.7801 −0.472721
\(622\) −1.50532 −0.0603580
\(623\) 0.672255 0.0269333
\(624\) 6.19715 0.248084
\(625\) 1.00000 0.0400000
\(626\) 2.86091 0.114345
\(627\) 37.8148 1.51018
\(628\) −18.3891 −0.733807
\(629\) 0.977309 0.0389678
\(630\) −0.0372249 −0.00148308
\(631\) 31.4731 1.25293 0.626463 0.779451i \(-0.284502\pi\)
0.626463 + 0.779451i \(0.284502\pi\)
\(632\) 8.50067 0.338139
\(633\) 2.79276 0.111002
\(634\) −20.5102 −0.814564
\(635\) 21.8559 0.867327
\(636\) 14.5458 0.576779
\(637\) 23.5088 0.931453
\(638\) −19.8058 −0.784121
\(639\) −1.89872 −0.0751122
\(640\) 1.00000 0.0395285
\(641\) 18.4384 0.728273 0.364136 0.931346i \(-0.381364\pi\)
0.364136 + 0.931346i \(0.381364\pi\)
\(642\) −34.2671 −1.35241
\(643\) 44.0100 1.73559 0.867793 0.496926i \(-0.165538\pi\)
0.867793 + 0.496926i \(0.165538\pi\)
\(644\) 0.230531 0.00908421
\(645\) 5.82089 0.229197
\(646\) 15.3353 0.603359
\(647\) 32.2310 1.26713 0.633565 0.773689i \(-0.281591\pi\)
0.633565 + 0.773689i \(0.281591\pi\)
\(648\) −10.0321 −0.394100
\(649\) 41.4569 1.62732
\(650\) −3.36264 −0.131894
\(651\) −0.385921 −0.0151254
\(652\) −18.4734 −0.723473
\(653\) 29.8242 1.16711 0.583556 0.812073i \(-0.301661\pi\)
0.583556 + 0.812073i \(0.301661\pi\)
\(654\) 12.1288 0.474273
\(655\) −9.46857 −0.369968
\(656\) −12.2166 −0.476978
\(657\) −6.19330 −0.241624
\(658\) −0.976615 −0.0380724
\(659\) 12.1836 0.474607 0.237303 0.971436i \(-0.423736\pi\)
0.237303 + 0.971436i \(0.423736\pi\)
\(660\) 6.32217 0.246090
\(661\) −19.1593 −0.745212 −0.372606 0.927990i \(-0.621536\pi\)
−0.372606 + 0.927990i \(0.621536\pi\)
\(662\) −9.27471 −0.360472
\(663\) 15.8887 0.617066
\(664\) 3.35560 0.130223
\(665\) −0.561636 −0.0217793
\(666\) 0.151116 0.00585561
\(667\) −14.1746 −0.548841
\(668\) −21.8788 −0.846514
\(669\) −29.0384 −1.12269
\(670\) −10.7784 −0.416406
\(671\) −1.57091 −0.0606441
\(672\) 0.173050 0.00667554
\(673\) −17.8746 −0.689013 −0.344507 0.938784i \(-0.611954\pi\)
−0.344507 + 0.938784i \(0.611954\pi\)
\(674\) 24.6047 0.947739
\(675\) 4.79822 0.184683
\(676\) −1.69266 −0.0651025
\(677\) 29.5856 1.13707 0.568533 0.822661i \(-0.307511\pi\)
0.568533 + 0.822661i \(0.307511\pi\)
\(678\) −16.8123 −0.645674
\(679\) −0.608004 −0.0233331
\(680\) 2.56387 0.0983201
\(681\) −0.921427 −0.0353091
\(682\) 7.65035 0.292947
\(683\) −47.8617 −1.83138 −0.915689 0.401887i \(-0.868355\pi\)
−0.915689 + 0.401887i \(0.868355\pi\)
\(684\) 2.37121 0.0906654
\(685\) −15.5435 −0.593888
\(686\) 1.31375 0.0501593
\(687\) 13.0124 0.496453
\(688\) −3.15848 −0.120416
\(689\) 26.5404 1.01111
\(690\) 4.52462 0.172249
\(691\) −45.0030 −1.71199 −0.855997 0.516980i \(-0.827056\pi\)
−0.855997 + 0.516980i \(0.827056\pi\)
\(692\) 0.872010 0.0331488
\(693\) 0.127699 0.00485089
\(694\) −3.87165 −0.146966
\(695\) −16.9704 −0.643724
\(696\) −10.6402 −0.403317
\(697\) −31.3218 −1.18640
\(698\) −12.0503 −0.456110
\(699\) −1.16906 −0.0442181
\(700\) −0.0938987 −0.00354904
\(701\) −25.6734 −0.969670 −0.484835 0.874606i \(-0.661120\pi\)
−0.484835 + 0.874606i \(0.661120\pi\)
\(702\) −16.1347 −0.608964
\(703\) 2.27998 0.0859910
\(704\) −3.43048 −0.129291
\(705\) −19.1680 −0.721907
\(706\) 32.0479 1.20614
\(707\) 0.239639 0.00901255
\(708\) 22.2717 0.837023
\(709\) −26.7446 −1.00441 −0.502206 0.864748i \(-0.667478\pi\)
−0.502206 + 0.864748i \(0.667478\pi\)
\(710\) −4.78946 −0.179745
\(711\) 3.36998 0.126384
\(712\) −7.15936 −0.268309
\(713\) 5.47517 0.205047
\(714\) 0.443678 0.0166042
\(715\) 11.5355 0.431401
\(716\) −10.6354 −0.397463
\(717\) −29.2819 −1.09355
\(718\) −20.4761 −0.764162
\(719\) 9.82982 0.366590 0.183295 0.983058i \(-0.441324\pi\)
0.183295 + 0.983058i \(0.441324\pi\)
\(720\) 0.396437 0.0147743
\(721\) 0.369027 0.0137433
\(722\) 16.7759 0.624335
\(723\) 34.4167 1.27997
\(724\) 6.54024 0.243066
\(725\) 5.77350 0.214422
\(726\) −1.41569 −0.0525412
\(727\) 23.7186 0.879675 0.439838 0.898077i \(-0.355036\pi\)
0.439838 + 0.898077i \(0.355036\pi\)
\(728\) 0.315747 0.0117024
\(729\) 22.5514 0.835237
\(730\) −15.6224 −0.578211
\(731\) −8.09794 −0.299513
\(732\) −0.843932 −0.0311926
\(733\) 31.9151 1.17881 0.589405 0.807838i \(-0.299362\pi\)
0.589405 + 0.807838i \(0.299362\pi\)
\(734\) −1.25272 −0.0462389
\(735\) 12.8843 0.475246
\(736\) −2.45511 −0.0904965
\(737\) 36.9750 1.36199
\(738\) −4.84311 −0.178277
\(739\) −31.6812 −1.16541 −0.582705 0.812684i \(-0.698006\pi\)
−0.582705 + 0.812684i \(0.698006\pi\)
\(740\) 0.381184 0.0140126
\(741\) 37.0670 1.36169
\(742\) 0.741116 0.0272072
\(743\) −21.1940 −0.777531 −0.388765 0.921337i \(-0.627098\pi\)
−0.388765 + 0.921337i \(0.627098\pi\)
\(744\) 4.10997 0.150679
\(745\) −4.19924 −0.153848
\(746\) 19.0849 0.698748
\(747\) 1.33029 0.0486726
\(748\) −8.79531 −0.321588
\(749\) −1.74592 −0.0637946
\(750\) −1.84294 −0.0672947
\(751\) 7.00106 0.255472 0.127736 0.991808i \(-0.459229\pi\)
0.127736 + 0.991808i \(0.459229\pi\)
\(752\) 10.4007 0.379276
\(753\) 8.81643 0.321288
\(754\) −19.4142 −0.707023
\(755\) 0.915926 0.0333339
\(756\) −0.450546 −0.0163862
\(757\) 17.4313 0.633551 0.316775 0.948501i \(-0.397400\pi\)
0.316775 + 0.948501i \(0.397400\pi\)
\(758\) 19.9070 0.723054
\(759\) −15.5216 −0.563399
\(760\) 5.98130 0.216964
\(761\) 21.8929 0.793618 0.396809 0.917901i \(-0.370118\pi\)
0.396809 + 0.917901i \(0.370118\pi\)
\(762\) −40.2792 −1.45916
\(763\) 0.617966 0.0223719
\(764\) 8.47937 0.306773
\(765\) 1.01641 0.0367485
\(766\) −22.4584 −0.811455
\(767\) 40.6371 1.46732
\(768\) −1.84294 −0.0665015
\(769\) 5.96382 0.215061 0.107530 0.994202i \(-0.465706\pi\)
0.107530 + 0.994202i \(0.465706\pi\)
\(770\) 0.322117 0.0116083
\(771\) 1.70449 0.0613856
\(772\) −8.23319 −0.296319
\(773\) 1.09252 0.0392951 0.0196476 0.999807i \(-0.493746\pi\)
0.0196476 + 0.999807i \(0.493746\pi\)
\(774\) −1.25214 −0.0450071
\(775\) −2.23011 −0.0801080
\(776\) 6.47511 0.232443
\(777\) 0.0659639 0.00236644
\(778\) 24.0783 0.863248
\(779\) −73.0711 −2.61804
\(780\) 6.19715 0.221894
\(781\) 16.4301 0.587916
\(782\) −6.29459 −0.225094
\(783\) 27.7025 0.990006
\(784\) −6.99118 −0.249685
\(785\) −18.3891 −0.656337
\(786\) 17.4500 0.622422
\(787\) 39.9819 1.42520 0.712600 0.701571i \(-0.247518\pi\)
0.712600 + 0.701571i \(0.247518\pi\)
\(788\) −0.175702 −0.00625914
\(789\) 14.2893 0.508712
\(790\) 8.50067 0.302440
\(791\) −0.856595 −0.0304570
\(792\) −1.35997 −0.0483243
\(793\) −1.53984 −0.0546813
\(794\) 39.1825 1.39053
\(795\) 14.5458 0.515887
\(796\) 5.73441 0.203251
\(797\) 32.5813 1.15409 0.577044 0.816713i \(-0.304206\pi\)
0.577044 + 0.816713i \(0.304206\pi\)
\(798\) 1.03506 0.0366408
\(799\) 26.6662 0.943381
\(800\) 1.00000 0.0353553
\(801\) −2.83824 −0.100284
\(802\) −1.00000 −0.0353112
\(803\) 53.5923 1.89123
\(804\) 19.8640 0.700548
\(805\) 0.230531 0.00812517
\(806\) 7.49906 0.264143
\(807\) −55.0251 −1.93698
\(808\) −2.55210 −0.0897826
\(809\) 21.5506 0.757680 0.378840 0.925462i \(-0.376323\pi\)
0.378840 + 0.925462i \(0.376323\pi\)
\(810\) −10.0321 −0.352494
\(811\) −11.8902 −0.417521 −0.208761 0.977967i \(-0.566943\pi\)
−0.208761 + 0.977967i \(0.566943\pi\)
\(812\) −0.542124 −0.0190248
\(813\) −13.3091 −0.466770
\(814\) −1.30764 −0.0458329
\(815\) −18.4734 −0.647094
\(816\) −4.72507 −0.165411
\(817\) −18.8918 −0.660940
\(818\) −27.2597 −0.953112
\(819\) 0.125174 0.00437393
\(820\) −12.2166 −0.426622
\(821\) 8.87452 0.309723 0.154861 0.987936i \(-0.450507\pi\)
0.154861 + 0.987936i \(0.450507\pi\)
\(822\) 28.6458 0.999138
\(823\) 35.1961 1.22686 0.613430 0.789749i \(-0.289789\pi\)
0.613430 + 0.789749i \(0.289789\pi\)
\(824\) −3.93006 −0.136910
\(825\) 6.32217 0.220110
\(826\) 1.13475 0.0394831
\(827\) 22.5418 0.783856 0.391928 0.919996i \(-0.371808\pi\)
0.391928 + 0.919996i \(0.371808\pi\)
\(828\) −0.973296 −0.0338244
\(829\) −28.9658 −1.00603 −0.503013 0.864279i \(-0.667775\pi\)
−0.503013 + 0.864279i \(0.667775\pi\)
\(830\) 3.35560 0.116475
\(831\) 50.8504 1.76398
\(832\) −3.36264 −0.116579
\(833\) −17.9245 −0.621048
\(834\) 31.2755 1.08298
\(835\) −21.8788 −0.757146
\(836\) −20.5187 −0.709654
\(837\) −10.7006 −0.369866
\(838\) 0.625472 0.0216066
\(839\) 54.4410 1.87951 0.939756 0.341847i \(-0.111052\pi\)
0.939756 + 0.341847i \(0.111052\pi\)
\(840\) 0.173050 0.00597079
\(841\) 4.33326 0.149423
\(842\) 4.29032 0.147854
\(843\) 17.6479 0.607826
\(844\) −1.51538 −0.0521616
\(845\) −1.69266 −0.0582294
\(846\) 4.12324 0.141760
\(847\) −0.0721299 −0.00247841
\(848\) −7.89272 −0.271037
\(849\) −8.97050 −0.307867
\(850\) 2.56387 0.0879402
\(851\) −0.935849 −0.0320805
\(852\) 8.82670 0.302398
\(853\) 14.9371 0.511438 0.255719 0.966751i \(-0.417688\pi\)
0.255719 + 0.966751i \(0.417688\pi\)
\(854\) −0.0429987 −0.00147138
\(855\) 2.37121 0.0810936
\(856\) 18.5937 0.635518
\(857\) −3.08893 −0.105516 −0.0527579 0.998607i \(-0.516801\pi\)
−0.0527579 + 0.998607i \(0.516801\pi\)
\(858\) −21.2592 −0.725776
\(859\) −3.15191 −0.107542 −0.0537709 0.998553i \(-0.517124\pi\)
−0.0537709 + 0.998553i \(0.517124\pi\)
\(860\) −3.15848 −0.107703
\(861\) −2.11408 −0.0720476
\(862\) −29.9833 −1.02124
\(863\) −15.5978 −0.530957 −0.265478 0.964117i \(-0.585530\pi\)
−0.265478 + 0.964117i \(0.585530\pi\)
\(864\) 4.79822 0.163239
\(865\) 0.872010 0.0296492
\(866\) 18.0144 0.612153
\(867\) 19.2155 0.652594
\(868\) 0.209405 0.00710766
\(869\) −29.1613 −0.989231
\(870\) −10.6402 −0.360737
\(871\) 36.2438 1.22808
\(872\) −6.58120 −0.222868
\(873\) 2.56697 0.0868789
\(874\) −14.6847 −0.496719
\(875\) −0.0938987 −0.00317435
\(876\) 28.7912 0.972765
\(877\) −24.1351 −0.814984 −0.407492 0.913209i \(-0.633597\pi\)
−0.407492 + 0.913209i \(0.633597\pi\)
\(878\) 4.45552 0.150367
\(879\) −16.3407 −0.551157
\(880\) −3.43048 −0.115641
\(881\) 15.7995 0.532297 0.266149 0.963932i \(-0.414249\pi\)
0.266149 + 0.963932i \(0.414249\pi\)
\(882\) −2.77156 −0.0933234
\(883\) −46.1737 −1.55387 −0.776935 0.629581i \(-0.783227\pi\)
−0.776935 + 0.629581i \(0.783227\pi\)
\(884\) −8.62138 −0.289968
\(885\) 22.2717 0.748656
\(886\) 32.2852 1.08464
\(887\) −12.5940 −0.422865 −0.211432 0.977393i \(-0.567813\pi\)
−0.211432 + 0.977393i \(0.567813\pi\)
\(888\) −0.702501 −0.0235744
\(889\) −2.05224 −0.0688301
\(890\) −7.15936 −0.239982
\(891\) 34.4150 1.15295
\(892\) 15.7565 0.527568
\(893\) 62.2099 2.08177
\(894\) 7.73895 0.258829
\(895\) −10.6354 −0.355502
\(896\) −0.0938987 −0.00313693
\(897\) −15.2147 −0.508003
\(898\) 17.7475 0.592241
\(899\) −12.8756 −0.429424
\(900\) 0.396437 0.0132146
\(901\) −20.2359 −0.674157
\(902\) 41.9087 1.39541
\(903\) −0.546574 −0.0181888
\(904\) 9.12255 0.303411
\(905\) 6.54024 0.217405
\(906\) −1.68800 −0.0560800
\(907\) 50.9973 1.69334 0.846670 0.532119i \(-0.178604\pi\)
0.846670 + 0.532119i \(0.178604\pi\)
\(908\) 0.499976 0.0165923
\(909\) −1.01175 −0.0335575
\(910\) 0.315747 0.0104669
\(911\) 18.6642 0.618374 0.309187 0.951001i \(-0.399943\pi\)
0.309187 + 0.951001i \(0.399943\pi\)
\(912\) −11.0232 −0.365014
\(913\) −11.5113 −0.380969
\(914\) −38.5645 −1.27560
\(915\) −0.843932 −0.0278995
\(916\) −7.06065 −0.233291
\(917\) 0.889086 0.0293602
\(918\) 12.3020 0.406027
\(919\) −10.8445 −0.357726 −0.178863 0.983874i \(-0.557242\pi\)
−0.178863 + 0.983874i \(0.557242\pi\)
\(920\) −2.45511 −0.0809425
\(921\) −24.1883 −0.797032
\(922\) 16.0662 0.529113
\(923\) 16.1052 0.530110
\(924\) −0.593643 −0.0195294
\(925\) 0.381184 0.0125333
\(926\) −8.15714 −0.268060
\(927\) −1.55802 −0.0511721
\(928\) 5.77350 0.189524
\(929\) −11.1726 −0.366563 −0.183281 0.983061i \(-0.558672\pi\)
−0.183281 + 0.983061i \(0.558672\pi\)
\(930\) 4.10997 0.134771
\(931\) −41.8163 −1.37048
\(932\) 0.634346 0.0207787
\(933\) 2.77423 0.0908241
\(934\) 39.3652 1.28807
\(935\) −8.79531 −0.287637
\(936\) −1.33307 −0.0435729
\(937\) 13.5690 0.443280 0.221640 0.975129i \(-0.428859\pi\)
0.221640 + 0.975129i \(0.428859\pi\)
\(938\) 1.01208 0.0330455
\(939\) −5.27249 −0.172061
\(940\) 10.4007 0.339235
\(941\) −19.2211 −0.626590 −0.313295 0.949656i \(-0.601433\pi\)
−0.313295 + 0.949656i \(0.601433\pi\)
\(942\) 33.8901 1.10420
\(943\) 29.9930 0.976708
\(944\) −12.0849 −0.393329
\(945\) −0.450546 −0.0146563
\(946\) 10.8351 0.352279
\(947\) 30.5365 0.992303 0.496152 0.868236i \(-0.334746\pi\)
0.496152 + 0.868236i \(0.334746\pi\)
\(948\) −15.6662 −0.508816
\(949\) 52.5325 1.70528
\(950\) 5.98130 0.194059
\(951\) 37.7991 1.22572
\(952\) −0.240744 −0.00780257
\(953\) −9.22331 −0.298772 −0.149386 0.988779i \(-0.547730\pi\)
−0.149386 + 0.988779i \(0.547730\pi\)
\(954\) −3.12896 −0.101304
\(955\) 8.47937 0.274386
\(956\) 15.8886 0.513876
\(957\) 36.5010 1.17991
\(958\) −34.6506 −1.11951
\(959\) 1.45952 0.0471302
\(960\) −1.84294 −0.0594807
\(961\) −26.0266 −0.839568
\(962\) −1.28179 −0.0413264
\(963\) 7.37122 0.237534
\(964\) −18.6749 −0.601477
\(965\) −8.23319 −0.265036
\(966\) −0.424856 −0.0136695
\(967\) 5.02030 0.161442 0.0807210 0.996737i \(-0.474278\pi\)
0.0807210 + 0.996737i \(0.474278\pi\)
\(968\) 0.768168 0.0246898
\(969\) −28.2621 −0.907908
\(970\) 6.47511 0.207903
\(971\) 25.2192 0.809322 0.404661 0.914467i \(-0.367389\pi\)
0.404661 + 0.914467i \(0.367389\pi\)
\(972\) 4.09402 0.131316
\(973\) 1.59350 0.0510852
\(974\) −7.33397 −0.234995
\(975\) 6.19715 0.198468
\(976\) 0.457926 0.0146579
\(977\) −26.2588 −0.840093 −0.420046 0.907503i \(-0.637986\pi\)
−0.420046 + 0.907503i \(0.637986\pi\)
\(978\) 34.0454 1.08865
\(979\) 24.5600 0.784942
\(980\) −6.99118 −0.223325
\(981\) −2.60903 −0.0833000
\(982\) 28.8677 0.921204
\(983\) 12.5753 0.401091 0.200545 0.979684i \(-0.435729\pi\)
0.200545 + 0.979684i \(0.435729\pi\)
\(984\) 22.5145 0.717735
\(985\) −0.175702 −0.00559834
\(986\) 14.8025 0.471408
\(987\) 1.79985 0.0572897
\(988\) −20.1129 −0.639878
\(989\) 7.75440 0.246576
\(990\) −1.35997 −0.0432226
\(991\) 5.72805 0.181957 0.0909787 0.995853i \(-0.471000\pi\)
0.0909787 + 0.995853i \(0.471000\pi\)
\(992\) −2.23011 −0.0708062
\(993\) 17.0928 0.542422
\(994\) 0.449724 0.0142644
\(995\) 5.73441 0.181793
\(996\) −6.18418 −0.195953
\(997\) −4.25870 −0.134874 −0.0674372 0.997724i \(-0.521482\pi\)
−0.0674372 + 0.997724i \(0.521482\pi\)
\(998\) 25.1537 0.796226
\(999\) 1.82901 0.0578672
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 4010.2.a.h.1.3 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4010.2.a.h.1.3 9 1.1 even 1 trivial