Properties

Label 6038.2.a.d.1.16
Level $6038$
Weight $2$
Character 6038.1
Self dual yes
Analytic conductor $48.214$
Analytic rank $0$
Dimension $69$
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] = [6038,2,Mod(1,6038)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6038, 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("6038.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6038 = 2 \cdot 3019 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6038.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: \(48.2136727404\)
Analytic rank: \(0\)
Dimension: \(69\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) \(=\) 6038.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.10090 q^{3} +1.00000 q^{4} -3.50554 q^{5} +2.10090 q^{6} +3.00259 q^{7} -1.00000 q^{8} +1.41376 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.10090 q^{3} +1.00000 q^{4} -3.50554 q^{5} +2.10090 q^{6} +3.00259 q^{7} -1.00000 q^{8} +1.41376 q^{9} +3.50554 q^{10} +1.00020 q^{11} -2.10090 q^{12} +0.205336 q^{13} -3.00259 q^{14} +7.36478 q^{15} +1.00000 q^{16} -6.81052 q^{17} -1.41376 q^{18} -1.24499 q^{19} -3.50554 q^{20} -6.30813 q^{21} -1.00020 q^{22} -4.38251 q^{23} +2.10090 q^{24} +7.28882 q^{25} -0.205336 q^{26} +3.33252 q^{27} +3.00259 q^{28} -5.03372 q^{29} -7.36478 q^{30} +5.72537 q^{31} -1.00000 q^{32} -2.10131 q^{33} +6.81052 q^{34} -10.5257 q^{35} +1.41376 q^{36} +7.15265 q^{37} +1.24499 q^{38} -0.431391 q^{39} +3.50554 q^{40} +1.01826 q^{41} +6.30813 q^{42} -7.58576 q^{43} +1.00020 q^{44} -4.95601 q^{45} +4.38251 q^{46} +5.84435 q^{47} -2.10090 q^{48} +2.01556 q^{49} -7.28882 q^{50} +14.3082 q^{51} +0.205336 q^{52} -6.56170 q^{53} -3.33252 q^{54} -3.50624 q^{55} -3.00259 q^{56} +2.61559 q^{57} +5.03372 q^{58} +3.16104 q^{59} +7.36478 q^{60} -4.70986 q^{61} -5.72537 q^{62} +4.24496 q^{63} +1.00000 q^{64} -0.719815 q^{65} +2.10131 q^{66} +15.4818 q^{67} -6.81052 q^{68} +9.20721 q^{69} +10.5257 q^{70} -12.0465 q^{71} -1.41376 q^{72} -7.76330 q^{73} -7.15265 q^{74} -15.3130 q^{75} -1.24499 q^{76} +3.00319 q^{77} +0.431391 q^{78} +0.496972 q^{79} -3.50554 q^{80} -11.2426 q^{81} -1.01826 q^{82} +8.02145 q^{83} -6.30813 q^{84} +23.8746 q^{85} +7.58576 q^{86} +10.5753 q^{87} -1.00020 q^{88} -8.27457 q^{89} +4.95601 q^{90} +0.616542 q^{91} -4.38251 q^{92} -12.0284 q^{93} -5.84435 q^{94} +4.36436 q^{95} +2.10090 q^{96} -5.09387 q^{97} -2.01556 q^{98} +1.41405 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 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
\(3\) −2.10090 −1.21295 −0.606476 0.795101i \(-0.707418\pi\)
−0.606476 + 0.795101i \(0.707418\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.50554 −1.56773 −0.783863 0.620934i \(-0.786753\pi\)
−0.783863 + 0.620934i \(0.786753\pi\)
\(6\) 2.10090 0.857687
\(7\) 3.00259 1.13487 0.567437 0.823417i \(-0.307935\pi\)
0.567437 + 0.823417i \(0.307935\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.41376 0.471255
\(10\) 3.50554 1.10855
\(11\) 1.00020 0.301571 0.150786 0.988566i \(-0.451820\pi\)
0.150786 + 0.988566i \(0.451820\pi\)
\(12\) −2.10090 −0.606476
\(13\) 0.205336 0.0569501 0.0284750 0.999595i \(-0.490935\pi\)
0.0284750 + 0.999595i \(0.490935\pi\)
\(14\) −3.00259 −0.802476
\(15\) 7.36478 1.90158
\(16\) 1.00000 0.250000
\(17\) −6.81052 −1.65179 −0.825897 0.563821i \(-0.809331\pi\)
−0.825897 + 0.563821i \(0.809331\pi\)
\(18\) −1.41376 −0.333227
\(19\) −1.24499 −0.285620 −0.142810 0.989750i \(-0.545614\pi\)
−0.142810 + 0.989750i \(0.545614\pi\)
\(20\) −3.50554 −0.783863
\(21\) −6.30813 −1.37655
\(22\) −1.00020 −0.213243
\(23\) −4.38251 −0.913817 −0.456909 0.889514i \(-0.651043\pi\)
−0.456909 + 0.889514i \(0.651043\pi\)
\(24\) 2.10090 0.428844
\(25\) 7.28882 1.45776
\(26\) −0.205336 −0.0402698
\(27\) 3.33252 0.641343
\(28\) 3.00259 0.567437
\(29\) −5.03372 −0.934739 −0.467370 0.884062i \(-0.654798\pi\)
−0.467370 + 0.884062i \(0.654798\pi\)
\(30\) −7.36478 −1.34462
\(31\) 5.72537 1.02831 0.514154 0.857698i \(-0.328106\pi\)
0.514154 + 0.857698i \(0.328106\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.10131 −0.365792
\(34\) 6.81052 1.16799
\(35\) −10.5257 −1.77917
\(36\) 1.41376 0.235627
\(37\) 7.15265 1.17589 0.587945 0.808901i \(-0.299937\pi\)
0.587945 + 0.808901i \(0.299937\pi\)
\(38\) 1.24499 0.201964
\(39\) −0.431391 −0.0690778
\(40\) 3.50554 0.554275
\(41\) 1.01826 0.159026 0.0795129 0.996834i \(-0.474664\pi\)
0.0795129 + 0.996834i \(0.474664\pi\)
\(42\) 6.30813 0.973366
\(43\) −7.58576 −1.15682 −0.578409 0.815747i \(-0.696326\pi\)
−0.578409 + 0.815747i \(0.696326\pi\)
\(44\) 1.00020 0.150786
\(45\) −4.95601 −0.738798
\(46\) 4.38251 0.646166
\(47\) 5.84435 0.852486 0.426243 0.904609i \(-0.359837\pi\)
0.426243 + 0.904609i \(0.359837\pi\)
\(48\) −2.10090 −0.303238
\(49\) 2.01556 0.287937
\(50\) −7.28882 −1.03079
\(51\) 14.3082 2.00355
\(52\) 0.205336 0.0284750
\(53\) −6.56170 −0.901319 −0.450660 0.892696i \(-0.648811\pi\)
−0.450660 + 0.892696i \(0.648811\pi\)
\(54\) −3.33252 −0.453498
\(55\) −3.50624 −0.472781
\(56\) −3.00259 −0.401238
\(57\) 2.61559 0.346443
\(58\) 5.03372 0.660960
\(59\) 3.16104 0.411533 0.205766 0.978601i \(-0.434031\pi\)
0.205766 + 0.978601i \(0.434031\pi\)
\(60\) 7.36478 0.950789
\(61\) −4.70986 −0.603036 −0.301518 0.953461i \(-0.597493\pi\)
−0.301518 + 0.953461i \(0.597493\pi\)
\(62\) −5.72537 −0.727123
\(63\) 4.24496 0.534814
\(64\) 1.00000 0.125000
\(65\) −0.719815 −0.0892821
\(66\) 2.10131 0.258654
\(67\) 15.4818 1.89141 0.945703 0.325033i \(-0.105376\pi\)
0.945703 + 0.325033i \(0.105376\pi\)
\(68\) −6.81052 −0.825897
\(69\) 9.20721 1.10842
\(70\) 10.5257 1.25806
\(71\) −12.0465 −1.42965 −0.714827 0.699302i \(-0.753494\pi\)
−0.714827 + 0.699302i \(0.753494\pi\)
\(72\) −1.41376 −0.166614
\(73\) −7.76330 −0.908626 −0.454313 0.890842i \(-0.650115\pi\)
−0.454313 + 0.890842i \(0.650115\pi\)
\(74\) −7.15265 −0.831479
\(75\) −15.3130 −1.76820
\(76\) −1.24499 −0.142810
\(77\) 3.00319 0.342245
\(78\) 0.431391 0.0488454
\(79\) 0.496972 0.0559137 0.0279569 0.999609i \(-0.491100\pi\)
0.0279569 + 0.999609i \(0.491100\pi\)
\(80\) −3.50554 −0.391931
\(81\) −11.2426 −1.24917
\(82\) −1.01826 −0.112448
\(83\) 8.02145 0.880469 0.440234 0.897883i \(-0.354895\pi\)
0.440234 + 0.897883i \(0.354895\pi\)
\(84\) −6.30813 −0.688274
\(85\) 23.8746 2.58956
\(86\) 7.58576 0.817993
\(87\) 10.5753 1.13379
\(88\) −1.00020 −0.106622
\(89\) −8.27457 −0.877103 −0.438551 0.898706i \(-0.644508\pi\)
−0.438551 + 0.898706i \(0.644508\pi\)
\(90\) 4.95601 0.522409
\(91\) 0.616542 0.0646311
\(92\) −4.38251 −0.456909
\(93\) −12.0284 −1.24729
\(94\) −5.84435 −0.602799
\(95\) 4.36436 0.447773
\(96\) 2.10090 0.214422
\(97\) −5.09387 −0.517204 −0.258602 0.965984i \(-0.583262\pi\)
−0.258602 + 0.965984i \(0.583262\pi\)
\(98\) −2.01556 −0.203602
\(99\) 1.41405 0.142117
\(100\) 7.28882 0.728882
\(101\) 3.26281 0.324661 0.162331 0.986736i \(-0.448099\pi\)
0.162331 + 0.986736i \(0.448099\pi\)
\(102\) −14.3082 −1.41672
\(103\) −16.2738 −1.60350 −0.801751 0.597658i \(-0.796098\pi\)
−0.801751 + 0.597658i \(0.796098\pi\)
\(104\) −0.205336 −0.0201349
\(105\) 22.1134 2.15805
\(106\) 6.56170 0.637329
\(107\) 14.4097 1.39304 0.696521 0.717536i \(-0.254730\pi\)
0.696521 + 0.717536i \(0.254730\pi\)
\(108\) 3.33252 0.320671
\(109\) −17.3981 −1.66643 −0.833217 0.552946i \(-0.813504\pi\)
−0.833217 + 0.552946i \(0.813504\pi\)
\(110\) 3.50624 0.334307
\(111\) −15.0270 −1.42630
\(112\) 3.00259 0.283718
\(113\) −17.8859 −1.68257 −0.841284 0.540593i \(-0.818200\pi\)
−0.841284 + 0.540593i \(0.818200\pi\)
\(114\) −2.61559 −0.244972
\(115\) 15.3631 1.43261
\(116\) −5.03372 −0.467370
\(117\) 0.290297 0.0268380
\(118\) −3.16104 −0.290998
\(119\) −20.4492 −1.87458
\(120\) −7.36478 −0.672309
\(121\) −9.99960 −0.909055
\(122\) 4.70986 0.426411
\(123\) −2.13926 −0.192891
\(124\) 5.72537 0.514154
\(125\) −8.02354 −0.717647
\(126\) −4.24496 −0.378171
\(127\) 17.5216 1.55479 0.777395 0.629013i \(-0.216541\pi\)
0.777395 + 0.629013i \(0.216541\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 15.9369 1.40316
\(130\) 0.719815 0.0631320
\(131\) −7.20210 −0.629250 −0.314625 0.949216i \(-0.601879\pi\)
−0.314625 + 0.949216i \(0.601879\pi\)
\(132\) −2.10131 −0.182896
\(133\) −3.73819 −0.324142
\(134\) −15.4818 −1.33743
\(135\) −11.6823 −1.00545
\(136\) 6.81052 0.583997
\(137\) −12.4217 −1.06126 −0.530629 0.847604i \(-0.678044\pi\)
−0.530629 + 0.847604i \(0.678044\pi\)
\(138\) −9.20721 −0.783769
\(139\) 5.91539 0.501737 0.250868 0.968021i \(-0.419284\pi\)
0.250868 + 0.968021i \(0.419284\pi\)
\(140\) −10.5257 −0.889585
\(141\) −12.2784 −1.03403
\(142\) 12.0465 1.01092
\(143\) 0.205377 0.0171745
\(144\) 1.41376 0.117814
\(145\) 17.6459 1.46541
\(146\) 7.76330 0.642495
\(147\) −4.23448 −0.349254
\(148\) 7.15265 0.587945
\(149\) −2.03614 −0.166807 −0.0834036 0.996516i \(-0.526579\pi\)
−0.0834036 + 0.996516i \(0.526579\pi\)
\(150\) 15.3130 1.25031
\(151\) −19.3814 −1.57724 −0.788620 0.614881i \(-0.789204\pi\)
−0.788620 + 0.614881i \(0.789204\pi\)
\(152\) 1.24499 0.100982
\(153\) −9.62847 −0.778416
\(154\) −3.00319 −0.242004
\(155\) −20.0705 −1.61210
\(156\) −0.431391 −0.0345389
\(157\) −7.21945 −0.576175 −0.288088 0.957604i \(-0.593019\pi\)
−0.288088 + 0.957604i \(0.593019\pi\)
\(158\) −0.496972 −0.0395370
\(159\) 13.7855 1.09326
\(160\) 3.50554 0.277137
\(161\) −13.1589 −1.03707
\(162\) 11.2426 0.883299
\(163\) −1.52785 −0.119670 −0.0598352 0.998208i \(-0.519058\pi\)
−0.0598352 + 0.998208i \(0.519058\pi\)
\(164\) 1.01826 0.0795129
\(165\) 7.36624 0.573461
\(166\) −8.02145 −0.622585
\(167\) 8.32014 0.643831 0.321916 0.946768i \(-0.395673\pi\)
0.321916 + 0.946768i \(0.395673\pi\)
\(168\) 6.30813 0.486683
\(169\) −12.9578 −0.996757
\(170\) −23.8746 −1.83109
\(171\) −1.76012 −0.134600
\(172\) −7.58576 −0.578409
\(173\) 3.17957 0.241738 0.120869 0.992668i \(-0.461432\pi\)
0.120869 + 0.992668i \(0.461432\pi\)
\(174\) −10.5753 −0.801714
\(175\) 21.8853 1.65438
\(176\) 1.00020 0.0753928
\(177\) −6.64102 −0.499170
\(178\) 8.27457 0.620205
\(179\) −12.5332 −0.936777 −0.468389 0.883522i \(-0.655165\pi\)
−0.468389 + 0.883522i \(0.655165\pi\)
\(180\) −4.95601 −0.369399
\(181\) 14.3766 1.06861 0.534303 0.845293i \(-0.320574\pi\)
0.534303 + 0.845293i \(0.320574\pi\)
\(182\) −0.616542 −0.0457011
\(183\) 9.89493 0.731454
\(184\) 4.38251 0.323083
\(185\) −25.0739 −1.84347
\(186\) 12.0284 0.881966
\(187\) −6.81188 −0.498134
\(188\) 5.84435 0.426243
\(189\) 10.0062 0.727843
\(190\) −4.36436 −0.316624
\(191\) 21.3554 1.54522 0.772612 0.634879i \(-0.218950\pi\)
0.772612 + 0.634879i \(0.218950\pi\)
\(192\) −2.10090 −0.151619
\(193\) −0.779713 −0.0561250 −0.0280625 0.999606i \(-0.508934\pi\)
−0.0280625 + 0.999606i \(0.508934\pi\)
\(194\) 5.09387 0.365718
\(195\) 1.51226 0.108295
\(196\) 2.01556 0.143968
\(197\) 2.04711 0.145850 0.0729252 0.997337i \(-0.476767\pi\)
0.0729252 + 0.997337i \(0.476767\pi\)
\(198\) −1.41405 −0.100492
\(199\) −14.5252 −1.02967 −0.514833 0.857290i \(-0.672146\pi\)
−0.514833 + 0.857290i \(0.672146\pi\)
\(200\) −7.28882 −0.515397
\(201\) −32.5257 −2.29419
\(202\) −3.26281 −0.229570
\(203\) −15.1142 −1.06081
\(204\) 14.3082 1.00177
\(205\) −3.56956 −0.249309
\(206\) 16.2738 1.13385
\(207\) −6.19584 −0.430641
\(208\) 0.205336 0.0142375
\(209\) −1.24524 −0.0861347
\(210\) −22.1134 −1.52597
\(211\) 24.7555 1.70424 0.852121 0.523345i \(-0.175316\pi\)
0.852121 + 0.523345i \(0.175316\pi\)
\(212\) −6.56170 −0.450660
\(213\) 25.3084 1.73410
\(214\) −14.4097 −0.985029
\(215\) 26.5922 1.81357
\(216\) −3.33252 −0.226749
\(217\) 17.1910 1.16700
\(218\) 17.3981 1.17835
\(219\) 16.3099 1.10212
\(220\) −3.50624 −0.236391
\(221\) −1.39845 −0.0940698
\(222\) 15.0270 1.00855
\(223\) 12.9017 0.863963 0.431982 0.901882i \(-0.357815\pi\)
0.431982 + 0.901882i \(0.357815\pi\)
\(224\) −3.00259 −0.200619
\(225\) 10.3047 0.686978
\(226\) 17.8859 1.18976
\(227\) −12.4118 −0.823798 −0.411899 0.911230i \(-0.635134\pi\)
−0.411899 + 0.911230i \(0.635134\pi\)
\(228\) 2.61559 0.173222
\(229\) −21.0661 −1.39209 −0.696044 0.717999i \(-0.745058\pi\)
−0.696044 + 0.717999i \(0.745058\pi\)
\(230\) −15.3631 −1.01301
\(231\) −6.30939 −0.415127
\(232\) 5.03372 0.330480
\(233\) 13.5088 0.884989 0.442495 0.896771i \(-0.354094\pi\)
0.442495 + 0.896771i \(0.354094\pi\)
\(234\) −0.290297 −0.0189773
\(235\) −20.4876 −1.33646
\(236\) 3.16104 0.205766
\(237\) −1.04409 −0.0678207
\(238\) 20.4492 1.32553
\(239\) −10.5268 −0.680919 −0.340460 0.940259i \(-0.610583\pi\)
−0.340460 + 0.940259i \(0.610583\pi\)
\(240\) 7.36478 0.475394
\(241\) −11.8794 −0.765219 −0.382609 0.923910i \(-0.624974\pi\)
−0.382609 + 0.923910i \(0.624974\pi\)
\(242\) 9.99960 0.642799
\(243\) 13.6219 0.873846
\(244\) −4.70986 −0.301518
\(245\) −7.06562 −0.451406
\(246\) 2.13926 0.136394
\(247\) −0.255641 −0.0162661
\(248\) −5.72537 −0.363561
\(249\) −16.8522 −1.06797
\(250\) 8.02354 0.507453
\(251\) −5.87461 −0.370802 −0.185401 0.982663i \(-0.559358\pi\)
−0.185401 + 0.982663i \(0.559358\pi\)
\(252\) 4.24496 0.267407
\(253\) −4.38339 −0.275581
\(254\) −17.5216 −1.09940
\(255\) −50.1580 −3.14101
\(256\) 1.00000 0.0625000
\(257\) 19.2762 1.20242 0.601208 0.799092i \(-0.294686\pi\)
0.601208 + 0.799092i \(0.294686\pi\)
\(258\) −15.9369 −0.992187
\(259\) 21.4765 1.33448
\(260\) −0.719815 −0.0446411
\(261\) −7.11650 −0.440500
\(262\) 7.20210 0.444947
\(263\) −17.1743 −1.05901 −0.529505 0.848307i \(-0.677622\pi\)
−0.529505 + 0.848307i \(0.677622\pi\)
\(264\) 2.10131 0.129327
\(265\) 23.0023 1.41302
\(266\) 3.73819 0.229203
\(267\) 17.3840 1.06388
\(268\) 15.4818 0.945703
\(269\) 25.2632 1.54033 0.770163 0.637848i \(-0.220175\pi\)
0.770163 + 0.637848i \(0.220175\pi\)
\(270\) 11.6823 0.710960
\(271\) 27.1284 1.64793 0.823967 0.566638i \(-0.191756\pi\)
0.823967 + 0.566638i \(0.191756\pi\)
\(272\) −6.81052 −0.412948
\(273\) −1.29529 −0.0783945
\(274\) 12.4217 0.750423
\(275\) 7.29027 0.439620
\(276\) 9.20721 0.554209
\(277\) −16.9599 −1.01902 −0.509510 0.860465i \(-0.670173\pi\)
−0.509510 + 0.860465i \(0.670173\pi\)
\(278\) −5.91539 −0.354781
\(279\) 8.09433 0.484595
\(280\) 10.5257 0.629031
\(281\) 19.2714 1.14964 0.574819 0.818281i \(-0.305073\pi\)
0.574819 + 0.818281i \(0.305073\pi\)
\(282\) 12.2784 0.731167
\(283\) 19.8000 1.17699 0.588494 0.808501i \(-0.299721\pi\)
0.588494 + 0.808501i \(0.299721\pi\)
\(284\) −12.0465 −0.714827
\(285\) −9.16906 −0.543128
\(286\) −0.205377 −0.0121442
\(287\) 3.05743 0.180474
\(288\) −1.41376 −0.0833069
\(289\) 29.3832 1.72842
\(290\) −17.6459 −1.03620
\(291\) 10.7017 0.627344
\(292\) −7.76330 −0.454313
\(293\) 33.4163 1.95220 0.976101 0.217316i \(-0.0697302\pi\)
0.976101 + 0.217316i \(0.0697302\pi\)
\(294\) 4.23448 0.246960
\(295\) −11.0812 −0.645170
\(296\) −7.15265 −0.415740
\(297\) 3.33318 0.193411
\(298\) 2.03614 0.117950
\(299\) −0.899890 −0.0520420
\(300\) −15.3130 −0.884099
\(301\) −22.7769 −1.31284
\(302\) 19.3814 1.11528
\(303\) −6.85482 −0.393799
\(304\) −1.24499 −0.0714049
\(305\) 16.5106 0.945394
\(306\) 9.62847 0.550423
\(307\) 23.9060 1.36438 0.682192 0.731173i \(-0.261027\pi\)
0.682192 + 0.731173i \(0.261027\pi\)
\(308\) 3.00319 0.171123
\(309\) 34.1895 1.94497
\(310\) 20.0705 1.13993
\(311\) 24.8251 1.40770 0.703852 0.710346i \(-0.251462\pi\)
0.703852 + 0.710346i \(0.251462\pi\)
\(312\) 0.431391 0.0244227
\(313\) −17.6837 −0.999542 −0.499771 0.866157i \(-0.666583\pi\)
−0.499771 + 0.866157i \(0.666583\pi\)
\(314\) 7.21945 0.407417
\(315\) −14.8809 −0.838442
\(316\) 0.496972 0.0279569
\(317\) 8.17786 0.459314 0.229657 0.973272i \(-0.426239\pi\)
0.229657 + 0.973272i \(0.426239\pi\)
\(318\) −13.7855 −0.773050
\(319\) −5.03473 −0.281891
\(320\) −3.50554 −0.195966
\(321\) −30.2734 −1.68969
\(322\) 13.1589 0.733317
\(323\) 8.47901 0.471785
\(324\) −11.2426 −0.624587
\(325\) 1.49666 0.0830198
\(326\) 1.52785 0.0846198
\(327\) 36.5516 2.02131
\(328\) −1.01826 −0.0562241
\(329\) 17.5482 0.967464
\(330\) −7.36624 −0.405498
\(331\) −4.88685 −0.268606 −0.134303 0.990940i \(-0.542879\pi\)
−0.134303 + 0.990940i \(0.542879\pi\)
\(332\) 8.02145 0.440234
\(333\) 10.1122 0.554143
\(334\) −8.32014 −0.455258
\(335\) −54.2721 −2.96520
\(336\) −6.30813 −0.344137
\(337\) 16.9881 0.925400 0.462700 0.886515i \(-0.346881\pi\)
0.462700 + 0.886515i \(0.346881\pi\)
\(338\) 12.9578 0.704813
\(339\) 37.5765 2.04088
\(340\) 23.8746 1.29478
\(341\) 5.72651 0.310108
\(342\) 1.76012 0.0951764
\(343\) −14.9662 −0.808101
\(344\) 7.58576 0.408997
\(345\) −32.2762 −1.73769
\(346\) −3.17957 −0.170935
\(347\) 24.4040 1.31008 0.655038 0.755596i \(-0.272652\pi\)
0.655038 + 0.755596i \(0.272652\pi\)
\(348\) 10.5753 0.566897
\(349\) −17.6222 −0.943296 −0.471648 0.881787i \(-0.656341\pi\)
−0.471648 + 0.881787i \(0.656341\pi\)
\(350\) −21.8853 −1.16982
\(351\) 0.684287 0.0365245
\(352\) −1.00020 −0.0533108
\(353\) −5.06883 −0.269787 −0.134893 0.990860i \(-0.543069\pi\)
−0.134893 + 0.990860i \(0.543069\pi\)
\(354\) 6.64102 0.352966
\(355\) 42.2294 2.24130
\(356\) −8.27457 −0.438551
\(357\) 42.9617 2.27377
\(358\) 12.5332 0.662402
\(359\) −9.47904 −0.500285 −0.250142 0.968209i \(-0.580477\pi\)
−0.250142 + 0.968209i \(0.580477\pi\)
\(360\) 4.95601 0.261205
\(361\) −17.4500 −0.918421
\(362\) −14.3766 −0.755619
\(363\) 21.0081 1.10264
\(364\) 0.616542 0.0323156
\(365\) 27.2146 1.42448
\(366\) −9.89493 −0.517216
\(367\) −4.88793 −0.255148 −0.127574 0.991829i \(-0.540719\pi\)
−0.127574 + 0.991829i \(0.540719\pi\)
\(368\) −4.38251 −0.228454
\(369\) 1.43958 0.0749417
\(370\) 25.0739 1.30353
\(371\) −19.7021 −1.02288
\(372\) −12.0284 −0.623644
\(373\) −35.4079 −1.83335 −0.916676 0.399631i \(-0.869138\pi\)
−0.916676 + 0.399631i \(0.869138\pi\)
\(374\) 6.81188 0.352234
\(375\) 16.8566 0.870472
\(376\) −5.84435 −0.301399
\(377\) −1.03361 −0.0532335
\(378\) −10.0062 −0.514663
\(379\) 8.19626 0.421014 0.210507 0.977592i \(-0.432489\pi\)
0.210507 + 0.977592i \(0.432489\pi\)
\(380\) 4.36436 0.223887
\(381\) −36.8110 −1.88589
\(382\) −21.3554 −1.09264
\(383\) 1.07701 0.0550327 0.0275163 0.999621i \(-0.491240\pi\)
0.0275163 + 0.999621i \(0.491240\pi\)
\(384\) 2.10090 0.107211
\(385\) −10.5278 −0.536547
\(386\) 0.779713 0.0396864
\(387\) −10.7245 −0.545156
\(388\) −5.09387 −0.258602
\(389\) 37.1713 1.88466 0.942331 0.334683i \(-0.108629\pi\)
0.942331 + 0.334683i \(0.108629\pi\)
\(390\) −1.51226 −0.0765761
\(391\) 29.8472 1.50944
\(392\) −2.01556 −0.101801
\(393\) 15.1309 0.763251
\(394\) −2.04711 −0.103132
\(395\) −1.74216 −0.0876574
\(396\) 1.41405 0.0710585
\(397\) −6.57009 −0.329743 −0.164872 0.986315i \(-0.552721\pi\)
−0.164872 + 0.986315i \(0.552721\pi\)
\(398\) 14.5252 0.728084
\(399\) 7.85355 0.393169
\(400\) 7.28882 0.364441
\(401\) −15.0532 −0.751722 −0.375861 0.926676i \(-0.622653\pi\)
−0.375861 + 0.926676i \(0.622653\pi\)
\(402\) 32.5257 1.62223
\(403\) 1.17563 0.0585622
\(404\) 3.26281 0.162331
\(405\) 39.4113 1.95836
\(406\) 15.1142 0.750106
\(407\) 7.15408 0.354614
\(408\) −14.3082 −0.708361
\(409\) 17.5973 0.870131 0.435066 0.900399i \(-0.356725\pi\)
0.435066 + 0.900399i \(0.356725\pi\)
\(410\) 3.56956 0.176288
\(411\) 26.0967 1.28726
\(412\) −16.2738 −0.801751
\(413\) 9.49132 0.467037
\(414\) 6.19584 0.304509
\(415\) −28.1195 −1.38033
\(416\) −0.205336 −0.0100674
\(417\) −12.4276 −0.608583
\(418\) 1.24524 0.0609065
\(419\) 2.96510 0.144855 0.0724273 0.997374i \(-0.476925\pi\)
0.0724273 + 0.997374i \(0.476925\pi\)
\(420\) 22.1134 1.07902
\(421\) 1.01231 0.0493369 0.0246685 0.999696i \(-0.492147\pi\)
0.0246685 + 0.999696i \(0.492147\pi\)
\(422\) −24.7555 −1.20508
\(423\) 8.26254 0.401738
\(424\) 6.56170 0.318665
\(425\) −49.6406 −2.40792
\(426\) −25.3084 −1.22620
\(427\) −14.1418 −0.684369
\(428\) 14.4097 0.696521
\(429\) −0.431476 −0.0208319
\(430\) −26.5922 −1.28239
\(431\) 2.33478 0.112462 0.0562311 0.998418i \(-0.482092\pi\)
0.0562311 + 0.998418i \(0.482092\pi\)
\(432\) 3.33252 0.160336
\(433\) 40.4137 1.94216 0.971079 0.238757i \(-0.0767399\pi\)
0.971079 + 0.238757i \(0.0767399\pi\)
\(434\) −17.1910 −0.825192
\(435\) −37.0723 −1.77748
\(436\) −17.3981 −0.833217
\(437\) 5.45618 0.261004
\(438\) −16.3099 −0.779317
\(439\) −16.8983 −0.806512 −0.403256 0.915087i \(-0.632122\pi\)
−0.403256 + 0.915087i \(0.632122\pi\)
\(440\) 3.50624 0.167153
\(441\) 2.84952 0.135692
\(442\) 1.39845 0.0665174
\(443\) 26.7150 1.26927 0.634635 0.772812i \(-0.281150\pi\)
0.634635 + 0.772812i \(0.281150\pi\)
\(444\) −15.0270 −0.713149
\(445\) 29.0068 1.37506
\(446\) −12.9017 −0.610914
\(447\) 4.27772 0.202329
\(448\) 3.00259 0.141859
\(449\) −36.9537 −1.74395 −0.871976 0.489548i \(-0.837162\pi\)
−0.871976 + 0.489548i \(0.837162\pi\)
\(450\) −10.3047 −0.485767
\(451\) 1.01847 0.0479576
\(452\) −17.8859 −0.841284
\(453\) 40.7184 1.91312
\(454\) 12.4118 0.582513
\(455\) −2.16131 −0.101324
\(456\) −2.61559 −0.122486
\(457\) −38.3405 −1.79349 −0.896747 0.442544i \(-0.854076\pi\)
−0.896747 + 0.442544i \(0.854076\pi\)
\(458\) 21.0661 0.984354
\(459\) −22.6962 −1.05937
\(460\) 15.3631 0.716307
\(461\) 29.7056 1.38353 0.691764 0.722124i \(-0.256834\pi\)
0.691764 + 0.722124i \(0.256834\pi\)
\(462\) 6.30939 0.293539
\(463\) 28.9956 1.34754 0.673771 0.738940i \(-0.264674\pi\)
0.673771 + 0.738940i \(0.264674\pi\)
\(464\) −5.03372 −0.233685
\(465\) 42.1661 1.95541
\(466\) −13.5088 −0.625782
\(467\) 12.6310 0.584493 0.292247 0.956343i \(-0.405597\pi\)
0.292247 + 0.956343i \(0.405597\pi\)
\(468\) 0.290297 0.0134190
\(469\) 46.4856 2.14650
\(470\) 20.4876 0.945023
\(471\) 15.1673 0.698873
\(472\) −3.16104 −0.145499
\(473\) −7.58727 −0.348863
\(474\) 1.04409 0.0479565
\(475\) −9.07449 −0.416366
\(476\) −20.4492 −0.937288
\(477\) −9.27670 −0.424751
\(478\) 10.5268 0.481483
\(479\) 30.4992 1.39355 0.696773 0.717292i \(-0.254618\pi\)
0.696773 + 0.717292i \(0.254618\pi\)
\(480\) −7.36478 −0.336155
\(481\) 1.46870 0.0669670
\(482\) 11.8794 0.541091
\(483\) 27.6455 1.25791
\(484\) −9.99960 −0.454527
\(485\) 17.8568 0.810834
\(486\) −13.6219 −0.617902
\(487\) −17.6744 −0.800902 −0.400451 0.916318i \(-0.631147\pi\)
−0.400451 + 0.916318i \(0.631147\pi\)
\(488\) 4.70986 0.213205
\(489\) 3.20985 0.145155
\(490\) 7.06562 0.319192
\(491\) −42.6650 −1.92544 −0.962722 0.270493i \(-0.912813\pi\)
−0.962722 + 0.270493i \(0.912813\pi\)
\(492\) −2.13926 −0.0964454
\(493\) 34.2823 1.54400
\(494\) 0.255641 0.0115018
\(495\) −4.95700 −0.222800
\(496\) 5.72537 0.257077
\(497\) −36.1707 −1.62248
\(498\) 16.8522 0.755167
\(499\) 4.97906 0.222894 0.111447 0.993770i \(-0.464452\pi\)
0.111447 + 0.993770i \(0.464452\pi\)
\(500\) −8.02354 −0.358824
\(501\) −17.4797 −0.780937
\(502\) 5.87461 0.262196
\(503\) −19.0518 −0.849476 −0.424738 0.905316i \(-0.639634\pi\)
−0.424738 + 0.905316i \(0.639634\pi\)
\(504\) −4.24496 −0.189085
\(505\) −11.4379 −0.508980
\(506\) 4.38339 0.194865
\(507\) 27.2231 1.20902
\(508\) 17.5216 0.777395
\(509\) −5.41836 −0.240165 −0.120082 0.992764i \(-0.538316\pi\)
−0.120082 + 0.992764i \(0.538316\pi\)
\(510\) 50.1580 2.22103
\(511\) −23.3100 −1.03117
\(512\) −1.00000 −0.0441942
\(513\) −4.14894 −0.183180
\(514\) −19.2762 −0.850237
\(515\) 57.0483 2.51385
\(516\) 15.9369 0.701582
\(517\) 5.84552 0.257086
\(518\) −21.4765 −0.943623
\(519\) −6.67995 −0.293217
\(520\) 0.719815 0.0315660
\(521\) 15.0102 0.657608 0.328804 0.944398i \(-0.393355\pi\)
0.328804 + 0.944398i \(0.393355\pi\)
\(522\) 7.11650 0.311481
\(523\) 25.4656 1.11353 0.556766 0.830670i \(-0.312042\pi\)
0.556766 + 0.830670i \(0.312042\pi\)
\(524\) −7.20210 −0.314625
\(525\) −45.9788 −2.00668
\(526\) 17.1743 0.748833
\(527\) −38.9928 −1.69855
\(528\) −2.10131 −0.0914480
\(529\) −3.79357 −0.164938
\(530\) −23.0023 −0.999157
\(531\) 4.46897 0.193937
\(532\) −3.73819 −0.162071
\(533\) 0.209086 0.00905654
\(534\) −17.3840 −0.752280
\(535\) −50.5139 −2.18391
\(536\) −15.4818 −0.668713
\(537\) 26.3310 1.13627
\(538\) −25.2632 −1.08917
\(539\) 2.01596 0.0868335
\(540\) −11.6823 −0.502725
\(541\) −14.3607 −0.617416 −0.308708 0.951157i \(-0.599897\pi\)
−0.308708 + 0.951157i \(0.599897\pi\)
\(542\) −27.1284 −1.16526
\(543\) −30.2038 −1.29617
\(544\) 6.81052 0.291999
\(545\) 60.9897 2.61251
\(546\) 1.29529 0.0554333
\(547\) −5.74147 −0.245488 −0.122744 0.992438i \(-0.539169\pi\)
−0.122744 + 0.992438i \(0.539169\pi\)
\(548\) −12.4217 −0.530629
\(549\) −6.65863 −0.284183
\(550\) −7.29027 −0.310858
\(551\) 6.26692 0.266980
\(552\) −9.20721 −0.391885
\(553\) 1.49220 0.0634550
\(554\) 16.9599 0.720556
\(555\) 52.6777 2.23604
\(556\) 5.91539 0.250868
\(557\) 3.26352 0.138280 0.0691399 0.997607i \(-0.477975\pi\)
0.0691399 + 0.997607i \(0.477975\pi\)
\(558\) −8.09433 −0.342660
\(559\) −1.55763 −0.0658808
\(560\) −10.5257 −0.444792
\(561\) 14.3110 0.604213
\(562\) −19.2714 −0.812916
\(563\) −45.4495 −1.91547 −0.957735 0.287653i \(-0.907125\pi\)
−0.957735 + 0.287653i \(0.907125\pi\)
\(564\) −12.2784 −0.517013
\(565\) 62.6999 2.63781
\(566\) −19.8000 −0.832257
\(567\) −33.7568 −1.41765
\(568\) 12.0465 0.505459
\(569\) 7.18806 0.301339 0.150670 0.988584i \(-0.451857\pi\)
0.150670 + 0.988584i \(0.451857\pi\)
\(570\) 9.16906 0.384050
\(571\) 19.5525 0.818249 0.409124 0.912479i \(-0.365834\pi\)
0.409124 + 0.912479i \(0.365834\pi\)
\(572\) 0.205377 0.00858726
\(573\) −44.8655 −1.87428
\(574\) −3.05743 −0.127615
\(575\) −31.9433 −1.33213
\(576\) 1.41376 0.0589069
\(577\) 11.9721 0.498406 0.249203 0.968451i \(-0.419831\pi\)
0.249203 + 0.968451i \(0.419831\pi\)
\(578\) −29.3832 −1.22218
\(579\) 1.63810 0.0680770
\(580\) 17.6459 0.732707
\(581\) 24.0851 0.999220
\(582\) −10.7017 −0.443599
\(583\) −6.56301 −0.271812
\(584\) 7.76330 0.321248
\(585\) −1.01765 −0.0420746
\(586\) −33.4163 −1.38042
\(587\) −15.7956 −0.651955 −0.325978 0.945377i \(-0.605693\pi\)
−0.325978 + 0.945377i \(0.605693\pi\)
\(588\) −4.23448 −0.174627
\(589\) −7.12802 −0.293705
\(590\) 11.0812 0.456204
\(591\) −4.30076 −0.176910
\(592\) 7.15265 0.293972
\(593\) 20.1155 0.826044 0.413022 0.910721i \(-0.364473\pi\)
0.413022 + 0.910721i \(0.364473\pi\)
\(594\) −3.33318 −0.136762
\(595\) 71.6855 2.93882
\(596\) −2.03614 −0.0834036
\(597\) 30.5160 1.24894
\(598\) 0.899890 0.0367992
\(599\) 0.909511 0.0371616 0.0185808 0.999827i \(-0.494085\pi\)
0.0185808 + 0.999827i \(0.494085\pi\)
\(600\) 15.3130 0.625153
\(601\) 35.8855 1.46380 0.731899 0.681413i \(-0.238634\pi\)
0.731899 + 0.681413i \(0.238634\pi\)
\(602\) 22.7769 0.928318
\(603\) 21.8876 0.891334
\(604\) −19.3814 −0.788620
\(605\) 35.0540 1.42515
\(606\) 6.85482 0.278458
\(607\) 32.2208 1.30780 0.653900 0.756581i \(-0.273132\pi\)
0.653900 + 0.756581i \(0.273132\pi\)
\(608\) 1.24499 0.0504909
\(609\) 31.7534 1.28671
\(610\) −16.5106 −0.668495
\(611\) 1.20006 0.0485492
\(612\) −9.62847 −0.389208
\(613\) 34.6671 1.40019 0.700096 0.714049i \(-0.253140\pi\)
0.700096 + 0.714049i \(0.253140\pi\)
\(614\) −23.9060 −0.964766
\(615\) 7.49928 0.302400
\(616\) −3.00319 −0.121002
\(617\) 10.4351 0.420103 0.210051 0.977690i \(-0.432637\pi\)
0.210051 + 0.977690i \(0.432637\pi\)
\(618\) −34.1895 −1.37530
\(619\) 16.1814 0.650385 0.325192 0.945648i \(-0.394571\pi\)
0.325192 + 0.945648i \(0.394571\pi\)
\(620\) −20.0705 −0.806052
\(621\) −14.6048 −0.586070
\(622\) −24.8251 −0.995397
\(623\) −24.8452 −0.995400
\(624\) −0.431391 −0.0172694
\(625\) −8.31723 −0.332689
\(626\) 17.6837 0.706783
\(627\) 2.61611 0.104477
\(628\) −7.21945 −0.288088
\(629\) −48.7133 −1.94233
\(630\) 14.8809 0.592868
\(631\) −42.7927 −1.70355 −0.851776 0.523906i \(-0.824474\pi\)
−0.851776 + 0.523906i \(0.824474\pi\)
\(632\) −0.496972 −0.0197685
\(633\) −52.0088 −2.06717
\(634\) −8.17786 −0.324784
\(635\) −61.4226 −2.43748
\(636\) 13.7855 0.546629
\(637\) 0.413868 0.0163980
\(638\) 5.03473 0.199327
\(639\) −17.0309 −0.673731
\(640\) 3.50554 0.138569
\(641\) 45.2345 1.78665 0.893327 0.449407i \(-0.148365\pi\)
0.893327 + 0.449407i \(0.148365\pi\)
\(642\) 30.2734 1.19479
\(643\) 20.5811 0.811639 0.405820 0.913953i \(-0.366986\pi\)
0.405820 + 0.913953i \(0.366986\pi\)
\(644\) −13.1589 −0.518533
\(645\) −55.8674 −2.19978
\(646\) −8.47901 −0.333602
\(647\) 24.7330 0.972354 0.486177 0.873860i \(-0.338391\pi\)
0.486177 + 0.873860i \(0.338391\pi\)
\(648\) 11.2426 0.441650
\(649\) 3.16167 0.124107
\(650\) −1.49666 −0.0587038
\(651\) −36.1164 −1.41551
\(652\) −1.52785 −0.0598352
\(653\) 6.66382 0.260775 0.130388 0.991463i \(-0.458378\pi\)
0.130388 + 0.991463i \(0.458378\pi\)
\(654\) −36.5516 −1.42928
\(655\) 25.2473 0.986492
\(656\) 1.01826 0.0397565
\(657\) −10.9755 −0.428194
\(658\) −17.5482 −0.684100
\(659\) 33.5356 1.30636 0.653181 0.757202i \(-0.273434\pi\)
0.653181 + 0.757202i \(0.273434\pi\)
\(660\) 7.36624 0.286731
\(661\) 45.7292 1.77866 0.889329 0.457267i \(-0.151172\pi\)
0.889329 + 0.457267i \(0.151172\pi\)
\(662\) 4.88685 0.189933
\(663\) 2.93799 0.114102
\(664\) −8.02145 −0.311293
\(665\) 13.1044 0.508166
\(666\) −10.1122 −0.391839
\(667\) 22.0604 0.854181
\(668\) 8.32014 0.321916
\(669\) −27.1052 −1.04795
\(670\) 54.2721 2.09672
\(671\) −4.71080 −0.181858
\(672\) 6.30813 0.243342
\(673\) 36.7109 1.41510 0.707551 0.706663i \(-0.249800\pi\)
0.707551 + 0.706663i \(0.249800\pi\)
\(674\) −16.9881 −0.654357
\(675\) 24.2901 0.934926
\(676\) −12.9578 −0.498378
\(677\) −19.6727 −0.756083 −0.378042 0.925789i \(-0.623402\pi\)
−0.378042 + 0.925789i \(0.623402\pi\)
\(678\) −37.5765 −1.44312
\(679\) −15.2948 −0.586961
\(680\) −23.8746 −0.915547
\(681\) 26.0758 0.999228
\(682\) −5.72651 −0.219279
\(683\) 29.3426 1.12276 0.561382 0.827557i \(-0.310270\pi\)
0.561382 + 0.827557i \(0.310270\pi\)
\(684\) −1.76012 −0.0672998
\(685\) 43.5448 1.66376
\(686\) 14.9662 0.571414
\(687\) 44.2577 1.68854
\(688\) −7.58576 −0.289204
\(689\) −1.34736 −0.0513302
\(690\) 32.2762 1.22874
\(691\) −5.83284 −0.221892 −0.110946 0.993826i \(-0.535388\pi\)
−0.110946 + 0.993826i \(0.535388\pi\)
\(692\) 3.17957 0.120869
\(693\) 4.24580 0.161285
\(694\) −24.4040 −0.926364
\(695\) −20.7366 −0.786586
\(696\) −10.5753 −0.400857
\(697\) −6.93490 −0.262678
\(698\) 17.6222 0.667011
\(699\) −28.3805 −1.07345
\(700\) 21.8853 0.827188
\(701\) 44.7647 1.69074 0.845371 0.534180i \(-0.179380\pi\)
0.845371 + 0.534180i \(0.179380\pi\)
\(702\) −0.684287 −0.0258268
\(703\) −8.90497 −0.335857
\(704\) 1.00020 0.0376964
\(705\) 43.0424 1.62107
\(706\) 5.06883 0.190768
\(707\) 9.79688 0.368449
\(708\) −6.64102 −0.249585
\(709\) 42.0611 1.57964 0.789819 0.613340i \(-0.210175\pi\)
0.789819 + 0.613340i \(0.210175\pi\)
\(710\) −42.2294 −1.58484
\(711\) 0.702602 0.0263496
\(712\) 8.27457 0.310103
\(713\) −25.0915 −0.939685
\(714\) −42.9617 −1.60780
\(715\) −0.719959 −0.0269249
\(716\) −12.5332 −0.468389
\(717\) 22.1156 0.825923
\(718\) 9.47904 0.353755
\(719\) −49.3625 −1.84091 −0.920455 0.390848i \(-0.872182\pi\)
−0.920455 + 0.390848i \(0.872182\pi\)
\(720\) −4.95601 −0.184700
\(721\) −48.8635 −1.81977
\(722\) 17.4500 0.649422
\(723\) 24.9574 0.928174
\(724\) 14.3766 0.534303
\(725\) −36.6899 −1.36263
\(726\) −21.0081 −0.779685
\(727\) 19.5188 0.723911 0.361956 0.932195i \(-0.382109\pi\)
0.361956 + 0.932195i \(0.382109\pi\)
\(728\) −0.616542 −0.0228506
\(729\) 5.10947 0.189240
\(730\) −27.2146 −1.00726
\(731\) 51.6629 1.91082
\(732\) 9.89493 0.365727
\(733\) −45.2232 −1.67036 −0.835178 0.549979i \(-0.814636\pi\)
−0.835178 + 0.549979i \(0.814636\pi\)
\(734\) 4.88793 0.180417
\(735\) 14.8441 0.547534
\(736\) 4.38251 0.161542
\(737\) 15.4849 0.570394
\(738\) −1.43958 −0.0529918
\(739\) −12.1355 −0.446412 −0.223206 0.974771i \(-0.571652\pi\)
−0.223206 + 0.974771i \(0.571652\pi\)
\(740\) −25.0739 −0.921736
\(741\) 0.537076 0.0197300
\(742\) 19.7021 0.723288
\(743\) 15.6314 0.573462 0.286731 0.958011i \(-0.407431\pi\)
0.286731 + 0.958011i \(0.407431\pi\)
\(744\) 12.0284 0.440983
\(745\) 7.13778 0.261508
\(746\) 35.4079 1.29638
\(747\) 11.3404 0.414925
\(748\) −6.81188 −0.249067
\(749\) 43.2666 1.58093
\(750\) −16.8566 −0.615517
\(751\) −21.4060 −0.781116 −0.390558 0.920578i \(-0.627718\pi\)
−0.390558 + 0.920578i \(0.627718\pi\)
\(752\) 5.84435 0.213122
\(753\) 12.3419 0.449765
\(754\) 1.03361 0.0376417
\(755\) 67.9424 2.47268
\(756\) 10.0062 0.363921
\(757\) −7.29388 −0.265100 −0.132550 0.991176i \(-0.542317\pi\)
−0.132550 + 0.991176i \(0.542317\pi\)
\(758\) −8.19626 −0.297702
\(759\) 9.20904 0.334267
\(760\) −4.36436 −0.158312
\(761\) 8.77669 0.318155 0.159077 0.987266i \(-0.449148\pi\)
0.159077 + 0.987266i \(0.449148\pi\)
\(762\) 36.8110 1.33352
\(763\) −52.2393 −1.89119
\(764\) 21.3554 0.772612
\(765\) 33.7530 1.22034
\(766\) −1.07701 −0.0389140
\(767\) 0.649077 0.0234368
\(768\) −2.10090 −0.0758096
\(769\) 14.8253 0.534614 0.267307 0.963611i \(-0.413866\pi\)
0.267307 + 0.963611i \(0.413866\pi\)
\(770\) 10.5278 0.379396
\(771\) −40.4973 −1.45847
\(772\) −0.779713 −0.0280625
\(773\) 25.9202 0.932284 0.466142 0.884710i \(-0.345644\pi\)
0.466142 + 0.884710i \(0.345644\pi\)
\(774\) 10.7245 0.385483
\(775\) 41.7312 1.49903
\(776\) 5.09387 0.182859
\(777\) −45.1199 −1.61867
\(778\) −37.1713 −1.33266
\(779\) −1.26772 −0.0454209
\(780\) 1.51226 0.0541475
\(781\) −12.0489 −0.431143
\(782\) −29.8472 −1.06733
\(783\) −16.7750 −0.599488
\(784\) 2.01556 0.0719842
\(785\) 25.3081 0.903284
\(786\) −15.1309 −0.539700
\(787\) 21.5205 0.767124 0.383562 0.923515i \(-0.374697\pi\)
0.383562 + 0.923515i \(0.374697\pi\)
\(788\) 2.04711 0.0729252
\(789\) 36.0813 1.28453
\(790\) 1.74216 0.0619831
\(791\) −53.7042 −1.90950
\(792\) −1.41405 −0.0502459
\(793\) −0.967106 −0.0343429
\(794\) 6.57009 0.233164
\(795\) −48.3255 −1.71393
\(796\) −14.5252 −0.514833
\(797\) −3.15998 −0.111932 −0.0559662 0.998433i \(-0.517824\pi\)
−0.0559662 + 0.998433i \(0.517824\pi\)
\(798\) −7.85355 −0.278013
\(799\) −39.8031 −1.40813
\(800\) −7.28882 −0.257699
\(801\) −11.6983 −0.413339
\(802\) 15.0532 0.531547
\(803\) −7.76485 −0.274016
\(804\) −32.5257 −1.14709
\(805\) 46.1291 1.62584
\(806\) −1.17563 −0.0414097
\(807\) −53.0754 −1.86834
\(808\) −3.26281 −0.114785
\(809\) −28.2805 −0.994291 −0.497145 0.867667i \(-0.665618\pi\)
−0.497145 + 0.867667i \(0.665618\pi\)
\(810\) −39.4113 −1.38477
\(811\) −20.0316 −0.703406 −0.351703 0.936112i \(-0.614397\pi\)
−0.351703 + 0.936112i \(0.614397\pi\)
\(812\) −15.1142 −0.530405
\(813\) −56.9940 −1.99887
\(814\) −7.15408 −0.250750
\(815\) 5.35594 0.187610
\(816\) 14.3082 0.500887
\(817\) 9.44417 0.330410
\(818\) −17.5973 −0.615276
\(819\) 0.871645 0.0304577
\(820\) −3.56956 −0.124654
\(821\) −36.6075 −1.27761 −0.638805 0.769369i \(-0.720571\pi\)
−0.638805 + 0.769369i \(0.720571\pi\)
\(822\) −26.0967 −0.910228
\(823\) −2.39587 −0.0835146 −0.0417573 0.999128i \(-0.513296\pi\)
−0.0417573 + 0.999128i \(0.513296\pi\)
\(824\) 16.2738 0.566923
\(825\) −15.3161 −0.533238
\(826\) −9.49132 −0.330245
\(827\) −24.9206 −0.866573 −0.433287 0.901256i \(-0.642646\pi\)
−0.433287 + 0.901256i \(0.642646\pi\)
\(828\) −6.19584 −0.215320
\(829\) −32.4953 −1.12861 −0.564305 0.825566i \(-0.690856\pi\)
−0.564305 + 0.825566i \(0.690856\pi\)
\(830\) 28.1195 0.976043
\(831\) 35.6309 1.23602
\(832\) 0.205336 0.00711876
\(833\) −13.7270 −0.475612
\(834\) 12.4276 0.430333
\(835\) −29.1666 −1.00935
\(836\) −1.24524 −0.0430674
\(837\) 19.0799 0.659498
\(838\) −2.96510 −0.102428
\(839\) −32.3198 −1.11580 −0.557902 0.829907i \(-0.688394\pi\)
−0.557902 + 0.829907i \(0.688394\pi\)
\(840\) −22.1134 −0.762985
\(841\) −3.66162 −0.126263
\(842\) −1.01231 −0.0348865
\(843\) −40.4873 −1.39446
\(844\) 24.7555 0.852121
\(845\) 45.4242 1.56264
\(846\) −8.26254 −0.284072
\(847\) −30.0247 −1.03166
\(848\) −6.56170 −0.225330
\(849\) −41.5978 −1.42763
\(850\) 49.6406 1.70266
\(851\) −31.3466 −1.07455
\(852\) 25.3084 0.867051
\(853\) −9.31247 −0.318853 −0.159426 0.987210i \(-0.550964\pi\)
−0.159426 + 0.987210i \(0.550964\pi\)
\(854\) 14.1418 0.483922
\(855\) 6.17017 0.211015
\(856\) −14.4097 −0.492515
\(857\) 36.2038 1.23670 0.618350 0.785903i \(-0.287801\pi\)
0.618350 + 0.785903i \(0.287801\pi\)
\(858\) 0.431476 0.0147304
\(859\) −31.8269 −1.08592 −0.542960 0.839759i \(-0.682696\pi\)
−0.542960 + 0.839759i \(0.682696\pi\)
\(860\) 26.5922 0.906786
\(861\) −6.42334 −0.218907
\(862\) −2.33478 −0.0795228
\(863\) 33.1246 1.12758 0.563788 0.825920i \(-0.309344\pi\)
0.563788 + 0.825920i \(0.309344\pi\)
\(864\) −3.33252 −0.113374
\(865\) −11.1461 −0.378979
\(866\) −40.4137 −1.37331
\(867\) −61.7310 −2.09649
\(868\) 17.1910 0.583499
\(869\) 0.497071 0.0168620
\(870\) 37.0723 1.25687
\(871\) 3.17898 0.107716
\(872\) 17.3981 0.589173
\(873\) −7.20153 −0.243735
\(874\) −5.45618 −0.184558
\(875\) −24.0914 −0.814439
\(876\) 16.3099 0.551060
\(877\) −33.0020 −1.11440 −0.557199 0.830379i \(-0.688124\pi\)
−0.557199 + 0.830379i \(0.688124\pi\)
\(878\) 16.8983 0.570290
\(879\) −70.2042 −2.36793
\(880\) −3.50624 −0.118195
\(881\) 18.0075 0.606688 0.303344 0.952881i \(-0.401897\pi\)
0.303344 + 0.952881i \(0.401897\pi\)
\(882\) −2.84952 −0.0959485
\(883\) 30.0777 1.01220 0.506098 0.862476i \(-0.331087\pi\)
0.506098 + 0.862476i \(0.331087\pi\)
\(884\) −1.39845 −0.0470349
\(885\) 23.2804 0.782561
\(886\) −26.7150 −0.897509
\(887\) −38.6262 −1.29694 −0.648470 0.761240i \(-0.724591\pi\)
−0.648470 + 0.761240i \(0.724591\pi\)
\(888\) 15.0270 0.504273
\(889\) 52.6102 1.76449
\(890\) −29.0068 −0.972311
\(891\) −11.2448 −0.376715
\(892\) 12.9017 0.431982
\(893\) −7.27615 −0.243487
\(894\) −4.27772 −0.143068
\(895\) 43.9357 1.46861
\(896\) −3.00259 −0.100310
\(897\) 1.89058 0.0631245
\(898\) 36.9537 1.23316
\(899\) −28.8199 −0.961199
\(900\) 10.3047 0.343489
\(901\) 44.6886 1.48879
\(902\) −1.01847 −0.0339112
\(903\) 47.8520 1.59241
\(904\) 17.8859 0.594878
\(905\) −50.3979 −1.67528
\(906\) −40.7184 −1.35278
\(907\) −10.1871 −0.338259 −0.169129 0.985594i \(-0.554096\pi\)
−0.169129 + 0.985594i \(0.554096\pi\)
\(908\) −12.4118 −0.411899
\(909\) 4.61284 0.152998
\(910\) 2.16131 0.0716468
\(911\) 32.2951 1.06998 0.534992 0.844857i \(-0.320315\pi\)
0.534992 + 0.844857i \(0.320315\pi\)
\(912\) 2.61559 0.0866108
\(913\) 8.02305 0.265524
\(914\) 38.3405 1.26819
\(915\) −34.6871 −1.14672
\(916\) −21.0661 −0.696044
\(917\) −21.6250 −0.714119
\(918\) 22.6962 0.749085
\(919\) 10.5091 0.346663 0.173332 0.984864i \(-0.444547\pi\)
0.173332 + 0.984864i \(0.444547\pi\)
\(920\) −15.3631 −0.506506
\(921\) −50.2239 −1.65493
\(922\) −29.7056 −0.978302
\(923\) −2.47358 −0.0814189
\(924\) −6.30939 −0.207564
\(925\) 52.1344 1.71417
\(926\) −28.9956 −0.952856
\(927\) −23.0073 −0.755658
\(928\) 5.03372 0.165240
\(929\) −25.1154 −0.824009 −0.412004 0.911182i \(-0.635171\pi\)
−0.412004 + 0.911182i \(0.635171\pi\)
\(930\) −42.1661 −1.38268
\(931\) −2.50934 −0.0822405
\(932\) 13.5088 0.442495
\(933\) −52.1550 −1.70748
\(934\) −12.6310 −0.413299
\(935\) 23.8793 0.780937
\(936\) −0.290297 −0.00948867
\(937\) 18.2646 0.596679 0.298340 0.954460i \(-0.403567\pi\)
0.298340 + 0.954460i \(0.403567\pi\)
\(938\) −46.4856 −1.51781
\(939\) 37.1516 1.21240
\(940\) −20.4876 −0.668232
\(941\) 25.9106 0.844660 0.422330 0.906442i \(-0.361212\pi\)
0.422330 + 0.906442i \(0.361212\pi\)
\(942\) −15.1673 −0.494178
\(943\) −4.46255 −0.145321
\(944\) 3.16104 0.102883
\(945\) −35.0771 −1.14106
\(946\) 7.58727 0.246683
\(947\) 31.0640 1.00944 0.504721 0.863282i \(-0.331595\pi\)
0.504721 + 0.863282i \(0.331595\pi\)
\(948\) −1.04409 −0.0339104
\(949\) −1.59409 −0.0517463
\(950\) 9.07449 0.294415
\(951\) −17.1808 −0.557127
\(952\) 20.4492 0.662763
\(953\) 30.4592 0.986670 0.493335 0.869839i \(-0.335778\pi\)
0.493335 + 0.869839i \(0.335778\pi\)
\(954\) 9.27670 0.300344
\(955\) −74.8623 −2.42249
\(956\) −10.5268 −0.340460
\(957\) 10.5774 0.341920
\(958\) −30.4992 −0.985386
\(959\) −37.2973 −1.20439
\(960\) 7.36478 0.237697
\(961\) 1.77988 0.0574153
\(962\) −1.46870 −0.0473528
\(963\) 20.3720 0.656478
\(964\) −11.8794 −0.382609
\(965\) 2.73332 0.0879886
\(966\) −27.6455 −0.889479
\(967\) −31.7794 −1.02196 −0.510978 0.859594i \(-0.670717\pi\)
−0.510978 + 0.859594i \(0.670717\pi\)
\(968\) 9.99960 0.321399
\(969\) −17.8135 −0.572253
\(970\) −17.8568 −0.573346
\(971\) −28.9408 −0.928754 −0.464377 0.885638i \(-0.653722\pi\)
−0.464377 + 0.885638i \(0.653722\pi\)
\(972\) 13.6219 0.436923
\(973\) 17.7615 0.569408
\(974\) 17.6744 0.566323
\(975\) −3.14433 −0.100699
\(976\) −4.70986 −0.150759
\(977\) 3.49262 0.111739 0.0558694 0.998438i \(-0.482207\pi\)
0.0558694 + 0.998438i \(0.482207\pi\)
\(978\) −3.20985 −0.102640
\(979\) −8.27622 −0.264509
\(980\) −7.06562 −0.225703
\(981\) −24.5968 −0.785315
\(982\) 42.6650 1.36149
\(983\) 15.0208 0.479090 0.239545 0.970885i \(-0.423002\pi\)
0.239545 + 0.970885i \(0.423002\pi\)
\(984\) 2.13926 0.0681972
\(985\) −7.17622 −0.228653
\(986\) −34.2823 −1.09177
\(987\) −36.8670 −1.17349
\(988\) −0.255641 −0.00813304
\(989\) 33.2447 1.05712
\(990\) 4.95700 0.157544
\(991\) 26.3069 0.835667 0.417833 0.908524i \(-0.362790\pi\)
0.417833 + 0.908524i \(0.362790\pi\)
\(992\) −5.72537 −0.181781
\(993\) 10.2668 0.325806
\(994\) 36.1707 1.14726
\(995\) 50.9188 1.61423
\(996\) −16.8522 −0.533984
\(997\) −35.8211 −1.13447 −0.567233 0.823558i \(-0.691986\pi\)
−0.567233 + 0.823558i \(0.691986\pi\)
\(998\) −4.97906 −0.157610
\(999\) 23.8363 0.754148
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 6038.2.a.d.1.16 69
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6038.2.a.d.1.16 69 1.1 even 1 trivial