Properties

Label 6046.2.a.f.1.16
Level $6046$
Weight $2$
Character 6046.1
Self dual yes
Analytic conductor $48.278$
Analytic rank $0$
Dimension $67$
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] = [6046,2,Mod(1,6046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6046, 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("6046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6046 = 2 \cdot 3023 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6046.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: \(48.2775530621\)
Analytic rank: \(0\)
Dimension: \(67\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) \(=\) 6046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.49577 q^{3} +1.00000 q^{4} -3.91198 q^{5} -1.49577 q^{6} +2.85048 q^{7} +1.00000 q^{8} -0.762675 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.49577 q^{3} +1.00000 q^{4} -3.91198 q^{5} -1.49577 q^{6} +2.85048 q^{7} +1.00000 q^{8} -0.762675 q^{9} -3.91198 q^{10} +5.45305 q^{11} -1.49577 q^{12} +2.90708 q^{13} +2.85048 q^{14} +5.85141 q^{15} +1.00000 q^{16} -6.18307 q^{17} -0.762675 q^{18} -0.516450 q^{19} -3.91198 q^{20} -4.26366 q^{21} +5.45305 q^{22} +2.30990 q^{23} -1.49577 q^{24} +10.3036 q^{25} +2.90708 q^{26} +5.62809 q^{27} +2.85048 q^{28} +1.70467 q^{29} +5.85141 q^{30} +1.93749 q^{31} +1.00000 q^{32} -8.15651 q^{33} -6.18307 q^{34} -11.1510 q^{35} -0.762675 q^{36} -5.17117 q^{37} -0.516450 q^{38} -4.34832 q^{39} -3.91198 q^{40} +8.84244 q^{41} -4.26366 q^{42} -12.6113 q^{43} +5.45305 q^{44} +2.98356 q^{45} +2.30990 q^{46} -6.22673 q^{47} -1.49577 q^{48} +1.12525 q^{49} +10.3036 q^{50} +9.24844 q^{51} +2.90708 q^{52} +1.75975 q^{53} +5.62809 q^{54} -21.3322 q^{55} +2.85048 q^{56} +0.772490 q^{57} +1.70467 q^{58} +10.1312 q^{59} +5.85141 q^{60} -10.6073 q^{61} +1.93749 q^{62} -2.17399 q^{63} +1.00000 q^{64} -11.3724 q^{65} -8.15651 q^{66} +7.97994 q^{67} -6.18307 q^{68} -3.45508 q^{69} -11.1510 q^{70} +0.744429 q^{71} -0.762675 q^{72} -10.5711 q^{73} -5.17117 q^{74} -15.4117 q^{75} -0.516450 q^{76} +15.5438 q^{77} -4.34832 q^{78} +11.6973 q^{79} -3.91198 q^{80} -6.13030 q^{81} +8.84244 q^{82} -9.52116 q^{83} -4.26366 q^{84} +24.1880 q^{85} -12.6113 q^{86} -2.54979 q^{87} +5.45305 q^{88} +18.7254 q^{89} +2.98356 q^{90} +8.28659 q^{91} +2.30990 q^{92} -2.89804 q^{93} -6.22673 q^{94} +2.02034 q^{95} -1.49577 q^{96} -12.3870 q^{97} +1.12525 q^{98} -4.15890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 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
\(3\) −1.49577 −0.863583 −0.431791 0.901974i \(-0.642118\pi\)
−0.431791 + 0.901974i \(0.642118\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.91198 −1.74949 −0.874744 0.484585i \(-0.838971\pi\)
−0.874744 + 0.484585i \(0.838971\pi\)
\(6\) −1.49577 −0.610645
\(7\) 2.85048 1.07738 0.538691 0.842504i \(-0.318919\pi\)
0.538691 + 0.842504i \(0.318919\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.762675 −0.254225
\(10\) −3.91198 −1.23708
\(11\) 5.45305 1.64416 0.822078 0.569374i \(-0.192814\pi\)
0.822078 + 0.569374i \(0.192814\pi\)
\(12\) −1.49577 −0.431791
\(13\) 2.90708 0.806280 0.403140 0.915138i \(-0.367919\pi\)
0.403140 + 0.915138i \(0.367919\pi\)
\(14\) 2.85048 0.761824
\(15\) 5.85141 1.51083
\(16\) 1.00000 0.250000
\(17\) −6.18307 −1.49961 −0.749807 0.661656i \(-0.769854\pi\)
−0.749807 + 0.661656i \(0.769854\pi\)
\(18\) −0.762675 −0.179764
\(19\) −0.516450 −0.118482 −0.0592409 0.998244i \(-0.518868\pi\)
−0.0592409 + 0.998244i \(0.518868\pi\)
\(20\) −3.91198 −0.874744
\(21\) −4.26366 −0.930408
\(22\) 5.45305 1.16259
\(23\) 2.30990 0.481648 0.240824 0.970569i \(-0.422582\pi\)
0.240824 + 0.970569i \(0.422582\pi\)
\(24\) −1.49577 −0.305323
\(25\) 10.3036 2.06071
\(26\) 2.90708 0.570126
\(27\) 5.62809 1.08313
\(28\) 2.85048 0.538691
\(29\) 1.70467 0.316549 0.158274 0.987395i \(-0.449407\pi\)
0.158274 + 0.987395i \(0.449407\pi\)
\(30\) 5.85141 1.06832
\(31\) 1.93749 0.347983 0.173992 0.984747i \(-0.444333\pi\)
0.173992 + 0.984747i \(0.444333\pi\)
\(32\) 1.00000 0.176777
\(33\) −8.15651 −1.41987
\(34\) −6.18307 −1.06039
\(35\) −11.1510 −1.88487
\(36\) −0.762675 −0.127112
\(37\) −5.17117 −0.850135 −0.425067 0.905162i \(-0.639750\pi\)
−0.425067 + 0.905162i \(0.639750\pi\)
\(38\) −0.516450 −0.0837793
\(39\) −4.34832 −0.696289
\(40\) −3.91198 −0.618538
\(41\) 8.84244 1.38096 0.690478 0.723353i \(-0.257400\pi\)
0.690478 + 0.723353i \(0.257400\pi\)
\(42\) −4.26366 −0.657898
\(43\) −12.6113 −1.92320 −0.961600 0.274454i \(-0.911503\pi\)
−0.961600 + 0.274454i \(0.911503\pi\)
\(44\) 5.45305 0.822078
\(45\) 2.98356 0.444764
\(46\) 2.30990 0.340577
\(47\) −6.22673 −0.908262 −0.454131 0.890935i \(-0.650050\pi\)
−0.454131 + 0.890935i \(0.650050\pi\)
\(48\) −1.49577 −0.215896
\(49\) 1.12525 0.160750
\(50\) 10.3036 1.45714
\(51\) 9.24844 1.29504
\(52\) 2.90708 0.403140
\(53\) 1.75975 0.241720 0.120860 0.992670i \(-0.461435\pi\)
0.120860 + 0.992670i \(0.461435\pi\)
\(54\) 5.62809 0.765886
\(55\) −21.3322 −2.87643
\(56\) 2.85048 0.380912
\(57\) 0.772490 0.102319
\(58\) 1.70467 0.223834
\(59\) 10.1312 1.31897 0.659487 0.751716i \(-0.270773\pi\)
0.659487 + 0.751716i \(0.270773\pi\)
\(60\) 5.85141 0.755414
\(61\) −10.6073 −1.35812 −0.679060 0.734083i \(-0.737612\pi\)
−0.679060 + 0.734083i \(0.737612\pi\)
\(62\) 1.93749 0.246061
\(63\) −2.17399 −0.273897
\(64\) 1.00000 0.125000
\(65\) −11.3724 −1.41058
\(66\) −8.15651 −1.00400
\(67\) 7.97994 0.974905 0.487453 0.873149i \(-0.337926\pi\)
0.487453 + 0.873149i \(0.337926\pi\)
\(68\) −6.18307 −0.749807
\(69\) −3.45508 −0.415943
\(70\) −11.1510 −1.33280
\(71\) 0.744429 0.0883475 0.0441737 0.999024i \(-0.485934\pi\)
0.0441737 + 0.999024i \(0.485934\pi\)
\(72\) −0.762675 −0.0898821
\(73\) −10.5711 −1.23726 −0.618629 0.785683i \(-0.712312\pi\)
−0.618629 + 0.785683i \(0.712312\pi\)
\(74\) −5.17117 −0.601136
\(75\) −15.4117 −1.77959
\(76\) −0.516450 −0.0592409
\(77\) 15.5438 1.77138
\(78\) −4.34832 −0.492351
\(79\) 11.6973 1.31604 0.658022 0.752998i \(-0.271393\pi\)
0.658022 + 0.752998i \(0.271393\pi\)
\(80\) −3.91198 −0.437372
\(81\) −6.13030 −0.681145
\(82\) 8.84244 0.976484
\(83\) −9.52116 −1.04508 −0.522542 0.852614i \(-0.675016\pi\)
−0.522542 + 0.852614i \(0.675016\pi\)
\(84\) −4.26366 −0.465204
\(85\) 24.1880 2.62356
\(86\) −12.6113 −1.35991
\(87\) −2.54979 −0.273366
\(88\) 5.45305 0.581297
\(89\) 18.7254 1.98489 0.992443 0.122709i \(-0.0391583\pi\)
0.992443 + 0.122709i \(0.0391583\pi\)
\(90\) 2.98356 0.314495
\(91\) 8.28659 0.868670
\(92\) 2.30990 0.240824
\(93\) −2.89804 −0.300512
\(94\) −6.22673 −0.642238
\(95\) 2.02034 0.207283
\(96\) −1.49577 −0.152661
\(97\) −12.3870 −1.25771 −0.628856 0.777522i \(-0.716477\pi\)
−0.628856 + 0.777522i \(0.716477\pi\)
\(98\) 1.12525 0.113668
\(99\) −4.15890 −0.417986
\(100\) 10.3036 1.03036
\(101\) 5.84555 0.581654 0.290827 0.956776i \(-0.406070\pi\)
0.290827 + 0.956776i \(0.406070\pi\)
\(102\) 9.24844 0.915732
\(103\) −10.3675 −1.02154 −0.510771 0.859717i \(-0.670640\pi\)
−0.510771 + 0.859717i \(0.670640\pi\)
\(104\) 2.90708 0.285063
\(105\) 16.6793 1.62774
\(106\) 1.75975 0.170922
\(107\) −14.4937 −1.40116 −0.700580 0.713574i \(-0.747075\pi\)
−0.700580 + 0.713574i \(0.747075\pi\)
\(108\) 5.62809 0.541563
\(109\) 16.7151 1.60102 0.800509 0.599321i \(-0.204563\pi\)
0.800509 + 0.599321i \(0.204563\pi\)
\(110\) −21.3322 −2.03395
\(111\) 7.73487 0.734162
\(112\) 2.85048 0.269345
\(113\) 4.78584 0.450214 0.225107 0.974334i \(-0.427727\pi\)
0.225107 + 0.974334i \(0.427727\pi\)
\(114\) 0.772490 0.0723503
\(115\) −9.03628 −0.842638
\(116\) 1.70467 0.158274
\(117\) −2.21716 −0.204976
\(118\) 10.1312 0.932656
\(119\) −17.6247 −1.61566
\(120\) 5.85141 0.534158
\(121\) 18.7358 1.70325
\(122\) −10.6073 −0.960336
\(123\) −13.2262 −1.19257
\(124\) 1.93749 0.173992
\(125\) −20.7474 −1.85570
\(126\) −2.17399 −0.193674
\(127\) −2.61522 −0.232063 −0.116032 0.993246i \(-0.537017\pi\)
−0.116032 + 0.993246i \(0.537017\pi\)
\(128\) 1.00000 0.0883883
\(129\) 18.8635 1.66084
\(130\) −11.3724 −0.997429
\(131\) 21.6185 1.88882 0.944408 0.328777i \(-0.106636\pi\)
0.944408 + 0.328777i \(0.106636\pi\)
\(132\) −8.15651 −0.709933
\(133\) −1.47213 −0.127650
\(134\) 7.97994 0.689362
\(135\) −22.0170 −1.89492
\(136\) −6.18307 −0.530194
\(137\) 16.7400 1.43020 0.715098 0.699024i \(-0.246382\pi\)
0.715098 + 0.699024i \(0.246382\pi\)
\(138\) −3.45508 −0.294116
\(139\) 17.2878 1.46633 0.733165 0.680050i \(-0.238042\pi\)
0.733165 + 0.680050i \(0.238042\pi\)
\(140\) −11.1510 −0.942433
\(141\) 9.31375 0.784359
\(142\) 0.744429 0.0624711
\(143\) 15.8525 1.32565
\(144\) −0.762675 −0.0635562
\(145\) −6.66862 −0.553798
\(146\) −10.5711 −0.874874
\(147\) −1.68312 −0.138821
\(148\) −5.17117 −0.425067
\(149\) 20.6188 1.68916 0.844579 0.535431i \(-0.179851\pi\)
0.844579 + 0.535431i \(0.179851\pi\)
\(150\) −15.4117 −1.25836
\(151\) −1.64808 −0.134119 −0.0670593 0.997749i \(-0.521362\pi\)
−0.0670593 + 0.997749i \(0.521362\pi\)
\(152\) −0.516450 −0.0418896
\(153\) 4.71567 0.381239
\(154\) 15.5438 1.25256
\(155\) −7.57941 −0.608793
\(156\) −4.34832 −0.348145
\(157\) 21.7783 1.73809 0.869047 0.494729i \(-0.164733\pi\)
0.869047 + 0.494729i \(0.164733\pi\)
\(158\) 11.6973 0.930584
\(159\) −2.63218 −0.208746
\(160\) −3.91198 −0.309269
\(161\) 6.58434 0.518919
\(162\) −6.13030 −0.481642
\(163\) −4.33889 −0.339848 −0.169924 0.985457i \(-0.554352\pi\)
−0.169924 + 0.985457i \(0.554352\pi\)
\(164\) 8.84244 0.690478
\(165\) 31.9081 2.48404
\(166\) −9.52116 −0.738986
\(167\) −2.53407 −0.196092 −0.0980462 0.995182i \(-0.531259\pi\)
−0.0980462 + 0.995182i \(0.531259\pi\)
\(168\) −4.26366 −0.328949
\(169\) −4.54887 −0.349913
\(170\) 24.1880 1.85514
\(171\) 0.393883 0.0301210
\(172\) −12.6113 −0.961600
\(173\) 12.2243 0.929396 0.464698 0.885469i \(-0.346163\pi\)
0.464698 + 0.885469i \(0.346163\pi\)
\(174\) −2.54979 −0.193299
\(175\) 29.3701 2.22017
\(176\) 5.45305 0.411039
\(177\) −15.1540 −1.13904
\(178\) 18.7254 1.40353
\(179\) 14.2787 1.06724 0.533619 0.845725i \(-0.320832\pi\)
0.533619 + 0.845725i \(0.320832\pi\)
\(180\) 2.98356 0.222382
\(181\) −19.5444 −1.45272 −0.726362 0.687313i \(-0.758790\pi\)
−0.726362 + 0.687313i \(0.758790\pi\)
\(182\) 8.28659 0.614243
\(183\) 15.8660 1.17285
\(184\) 2.30990 0.170288
\(185\) 20.2295 1.48730
\(186\) −2.89804 −0.212494
\(187\) −33.7166 −2.46560
\(188\) −6.22673 −0.454131
\(189\) 16.0428 1.16694
\(190\) 2.02034 0.146571
\(191\) −0.858528 −0.0621209 −0.0310605 0.999518i \(-0.509888\pi\)
−0.0310605 + 0.999518i \(0.509888\pi\)
\(192\) −1.49577 −0.107948
\(193\) 7.52385 0.541578 0.270789 0.962639i \(-0.412715\pi\)
0.270789 + 0.962639i \(0.412715\pi\)
\(194\) −12.3870 −0.889337
\(195\) 17.0105 1.21815
\(196\) 1.12525 0.0803751
\(197\) −6.22799 −0.443726 −0.221863 0.975078i \(-0.571214\pi\)
−0.221863 + 0.975078i \(0.571214\pi\)
\(198\) −4.15890 −0.295560
\(199\) 8.36632 0.593072 0.296536 0.955022i \(-0.404168\pi\)
0.296536 + 0.955022i \(0.404168\pi\)
\(200\) 10.3036 0.728571
\(201\) −11.9362 −0.841911
\(202\) 5.84555 0.411292
\(203\) 4.85912 0.341044
\(204\) 9.24844 0.647521
\(205\) −34.5914 −2.41597
\(206\) −10.3675 −0.722339
\(207\) −1.76170 −0.122447
\(208\) 2.90708 0.201570
\(209\) −2.81623 −0.194803
\(210\) 16.6793 1.15098
\(211\) −6.91765 −0.476231 −0.238115 0.971237i \(-0.576530\pi\)
−0.238115 + 0.971237i \(0.576530\pi\)
\(212\) 1.75975 0.120860
\(213\) −1.11349 −0.0762954
\(214\) −14.4937 −0.990769
\(215\) 49.3350 3.36462
\(216\) 5.62809 0.382943
\(217\) 5.52278 0.374911
\(218\) 16.7151 1.13209
\(219\) 15.8120 1.06847
\(220\) −21.3322 −1.43822
\(221\) −17.9747 −1.20911
\(222\) 7.73487 0.519131
\(223\) −4.84079 −0.324163 −0.162082 0.986777i \(-0.551821\pi\)
−0.162082 + 0.986777i \(0.551821\pi\)
\(224\) 2.85048 0.190456
\(225\) −7.85826 −0.523884
\(226\) 4.78584 0.318349
\(227\) −12.7637 −0.847159 −0.423580 0.905859i \(-0.639227\pi\)
−0.423580 + 0.905859i \(0.639227\pi\)
\(228\) 0.772490 0.0511594
\(229\) −15.3170 −1.01217 −0.506086 0.862483i \(-0.668908\pi\)
−0.506086 + 0.862483i \(0.668908\pi\)
\(230\) −9.03628 −0.595835
\(231\) −23.2500 −1.52974
\(232\) 1.70467 0.111917
\(233\) 0.614597 0.0402636 0.0201318 0.999797i \(-0.493591\pi\)
0.0201318 + 0.999797i \(0.493591\pi\)
\(234\) −2.21716 −0.144940
\(235\) 24.3588 1.58899
\(236\) 10.1312 0.659487
\(237\) −17.4964 −1.13651
\(238\) −17.6247 −1.14244
\(239\) 9.56775 0.618886 0.309443 0.950918i \(-0.399857\pi\)
0.309443 + 0.950918i \(0.399857\pi\)
\(240\) 5.85141 0.377707
\(241\) −11.1574 −0.718710 −0.359355 0.933201i \(-0.617003\pi\)
−0.359355 + 0.933201i \(0.617003\pi\)
\(242\) 18.7358 1.20438
\(243\) −7.71476 −0.494902
\(244\) −10.6073 −0.679060
\(245\) −4.40195 −0.281231
\(246\) −13.2262 −0.843275
\(247\) −1.50136 −0.0955295
\(248\) 1.93749 0.123031
\(249\) 14.2415 0.902516
\(250\) −20.7474 −1.31218
\(251\) −9.61550 −0.606925 −0.303462 0.952843i \(-0.598143\pi\)
−0.303462 + 0.952843i \(0.598143\pi\)
\(252\) −2.17399 −0.136949
\(253\) 12.5960 0.791905
\(254\) −2.61522 −0.164094
\(255\) −36.1797 −2.26566
\(256\) 1.00000 0.0625000
\(257\) 22.0832 1.37751 0.688757 0.724992i \(-0.258157\pi\)
0.688757 + 0.724992i \(0.258157\pi\)
\(258\) 18.8635 1.17439
\(259\) −14.7403 −0.915919
\(260\) −11.3724 −0.705288
\(261\) −1.30011 −0.0804746
\(262\) 21.6185 1.33559
\(263\) 5.60551 0.345651 0.172825 0.984952i \(-0.444710\pi\)
0.172825 + 0.984952i \(0.444710\pi\)
\(264\) −8.15651 −0.501998
\(265\) −6.88410 −0.422887
\(266\) −1.47213 −0.0902622
\(267\) −28.0088 −1.71411
\(268\) 7.97994 0.487453
\(269\) 1.51483 0.0923608 0.0461804 0.998933i \(-0.485295\pi\)
0.0461804 + 0.998933i \(0.485295\pi\)
\(270\) −22.0170 −1.33991
\(271\) 9.34971 0.567955 0.283977 0.958831i \(-0.408346\pi\)
0.283977 + 0.958831i \(0.408346\pi\)
\(272\) −6.18307 −0.374904
\(273\) −12.3948 −0.750169
\(274\) 16.7400 1.01130
\(275\) 56.1858 3.38813
\(276\) −3.45508 −0.207971
\(277\) 28.5493 1.71536 0.857679 0.514185i \(-0.171905\pi\)
0.857679 + 0.514185i \(0.171905\pi\)
\(278\) 17.2878 1.03685
\(279\) −1.47767 −0.0884660
\(280\) −11.1510 −0.666401
\(281\) 13.1826 0.786409 0.393205 0.919451i \(-0.371366\pi\)
0.393205 + 0.919451i \(0.371366\pi\)
\(282\) 9.31375 0.554626
\(283\) −26.9503 −1.60203 −0.801013 0.598647i \(-0.795705\pi\)
−0.801013 + 0.598647i \(0.795705\pi\)
\(284\) 0.744429 0.0441737
\(285\) −3.02196 −0.179006
\(286\) 15.8525 0.937376
\(287\) 25.2052 1.48782
\(288\) −0.762675 −0.0449410
\(289\) 21.2303 1.24884
\(290\) −6.66862 −0.391595
\(291\) 18.5281 1.08614
\(292\) −10.5711 −0.618629
\(293\) 23.8893 1.39563 0.697815 0.716278i \(-0.254156\pi\)
0.697815 + 0.716278i \(0.254156\pi\)
\(294\) −1.68312 −0.0981613
\(295\) −39.6332 −2.30753
\(296\) −5.17117 −0.300568
\(297\) 30.6903 1.78083
\(298\) 20.6188 1.19442
\(299\) 6.71508 0.388343
\(300\) −15.4117 −0.889797
\(301\) −35.9482 −2.07202
\(302\) −1.64808 −0.0948362
\(303\) −8.74360 −0.502306
\(304\) −0.516450 −0.0296205
\(305\) 41.4953 2.37602
\(306\) 4.71567 0.269577
\(307\) 4.79854 0.273867 0.136933 0.990580i \(-0.456275\pi\)
0.136933 + 0.990580i \(0.456275\pi\)
\(308\) 15.5438 0.885692
\(309\) 15.5074 0.882186
\(310\) −7.57941 −0.430482
\(311\) −0.139042 −0.00788433 −0.00394217 0.999992i \(-0.501255\pi\)
−0.00394217 + 0.999992i \(0.501255\pi\)
\(312\) −4.34832 −0.246175
\(313\) 24.3189 1.37458 0.687292 0.726381i \(-0.258799\pi\)
0.687292 + 0.726381i \(0.258799\pi\)
\(314\) 21.7783 1.22902
\(315\) 8.50460 0.479180
\(316\) 11.6973 0.658022
\(317\) 17.7111 0.994753 0.497377 0.867535i \(-0.334297\pi\)
0.497377 + 0.867535i \(0.334297\pi\)
\(318\) −2.63218 −0.147605
\(319\) 9.29564 0.520456
\(320\) −3.91198 −0.218686
\(321\) 21.6792 1.21002
\(322\) 6.58434 0.366931
\(323\) 3.19325 0.177677
\(324\) −6.13030 −0.340572
\(325\) 29.9533 1.66151
\(326\) −4.33889 −0.240309
\(327\) −25.0019 −1.38261
\(328\) 8.84244 0.488242
\(329\) −17.7492 −0.978544
\(330\) 31.9081 1.75648
\(331\) 29.3181 1.61147 0.805734 0.592277i \(-0.201771\pi\)
0.805734 + 0.592277i \(0.201771\pi\)
\(332\) −9.52116 −0.522542
\(333\) 3.94392 0.216125
\(334\) −2.53407 −0.138658
\(335\) −31.2173 −1.70559
\(336\) −4.26366 −0.232602
\(337\) 15.7130 0.855944 0.427972 0.903792i \(-0.359228\pi\)
0.427972 + 0.903792i \(0.359228\pi\)
\(338\) −4.54887 −0.247426
\(339\) −7.15851 −0.388797
\(340\) 24.1880 1.31178
\(341\) 10.5652 0.572139
\(342\) 0.393883 0.0212988
\(343\) −16.7459 −0.904192
\(344\) −12.6113 −0.679954
\(345\) 13.5162 0.727687
\(346\) 12.2243 0.657183
\(347\) 22.4086 1.20296 0.601479 0.798888i \(-0.294578\pi\)
0.601479 + 0.798888i \(0.294578\pi\)
\(348\) −2.54979 −0.136683
\(349\) −6.38298 −0.341673 −0.170837 0.985299i \(-0.554647\pi\)
−0.170837 + 0.985299i \(0.554647\pi\)
\(350\) 29.3701 1.56990
\(351\) 16.3613 0.873303
\(352\) 5.45305 0.290649
\(353\) 9.69691 0.516114 0.258057 0.966130i \(-0.416918\pi\)
0.258057 + 0.966130i \(0.416918\pi\)
\(354\) −15.1540 −0.805426
\(355\) −2.91219 −0.154563
\(356\) 18.7254 0.992443
\(357\) 26.3625 1.39525
\(358\) 14.2787 0.754651
\(359\) 5.84467 0.308470 0.154235 0.988034i \(-0.450709\pi\)
0.154235 + 0.988034i \(0.450709\pi\)
\(360\) 2.98356 0.157248
\(361\) −18.7333 −0.985962
\(362\) −19.5444 −1.02723
\(363\) −28.0244 −1.47090
\(364\) 8.28659 0.434335
\(365\) 41.3540 2.16457
\(366\) 15.8660 0.829330
\(367\) −0.873000 −0.0455702 −0.0227851 0.999740i \(-0.507253\pi\)
−0.0227851 + 0.999740i \(0.507253\pi\)
\(368\) 2.30990 0.120412
\(369\) −6.74390 −0.351074
\(370\) 20.2295 1.05168
\(371\) 5.01614 0.260425
\(372\) −2.89804 −0.150256
\(373\) 37.6523 1.94956 0.974780 0.223169i \(-0.0716401\pi\)
0.974780 + 0.223169i \(0.0716401\pi\)
\(374\) −33.7166 −1.74344
\(375\) 31.0333 1.60255
\(376\) −6.22673 −0.321119
\(377\) 4.95561 0.255227
\(378\) 16.0428 0.825152
\(379\) 1.50114 0.0771084 0.0385542 0.999257i \(-0.487725\pi\)
0.0385542 + 0.999257i \(0.487725\pi\)
\(380\) 2.02034 0.103641
\(381\) 3.91177 0.200406
\(382\) −0.858528 −0.0439261
\(383\) 13.4312 0.686302 0.343151 0.939280i \(-0.388506\pi\)
0.343151 + 0.939280i \(0.388506\pi\)
\(384\) −1.49577 −0.0763307
\(385\) −60.8071 −3.09902
\(386\) 7.52385 0.382954
\(387\) 9.61829 0.488925
\(388\) −12.3870 −0.628856
\(389\) −16.5486 −0.839047 −0.419523 0.907745i \(-0.637803\pi\)
−0.419523 + 0.907745i \(0.637803\pi\)
\(390\) 17.0105 0.861362
\(391\) −14.2823 −0.722286
\(392\) 1.12525 0.0568338
\(393\) −32.3363 −1.63115
\(394\) −6.22799 −0.313762
\(395\) −45.7594 −2.30241
\(396\) −4.15890 −0.208993
\(397\) 28.3813 1.42442 0.712208 0.701968i \(-0.247695\pi\)
0.712208 + 0.701968i \(0.247695\pi\)
\(398\) 8.36632 0.419366
\(399\) 2.20197 0.110236
\(400\) 10.3036 0.515178
\(401\) 23.6405 1.18055 0.590275 0.807202i \(-0.299019\pi\)
0.590275 + 0.807202i \(0.299019\pi\)
\(402\) −11.9362 −0.595321
\(403\) 5.63244 0.280572
\(404\) 5.84555 0.290827
\(405\) 23.9816 1.19166
\(406\) 4.85912 0.241154
\(407\) −28.1986 −1.39775
\(408\) 9.24844 0.457866
\(409\) −6.01198 −0.297273 −0.148637 0.988892i \(-0.547488\pi\)
−0.148637 + 0.988892i \(0.547488\pi\)
\(410\) −34.5914 −1.70835
\(411\) −25.0392 −1.23509
\(412\) −10.3675 −0.510771
\(413\) 28.8789 1.42104
\(414\) −1.76170 −0.0865830
\(415\) 37.2466 1.82836
\(416\) 2.90708 0.142531
\(417\) −25.8585 −1.26630
\(418\) −2.81623 −0.137746
\(419\) −16.8076 −0.821107 −0.410554 0.911837i \(-0.634665\pi\)
−0.410554 + 0.911837i \(0.634665\pi\)
\(420\) 16.6793 0.813869
\(421\) −17.2956 −0.842937 −0.421468 0.906843i \(-0.638485\pi\)
−0.421468 + 0.906843i \(0.638485\pi\)
\(422\) −6.91765 −0.336746
\(423\) 4.74897 0.230903
\(424\) 1.75975 0.0854611
\(425\) −63.7076 −3.09027
\(426\) −1.11349 −0.0539490
\(427\) −30.2358 −1.46321
\(428\) −14.4937 −0.700580
\(429\) −23.7116 −1.14481
\(430\) 49.3350 2.37914
\(431\) 19.9251 0.959758 0.479879 0.877335i \(-0.340681\pi\)
0.479879 + 0.877335i \(0.340681\pi\)
\(432\) 5.62809 0.270782
\(433\) −33.0782 −1.58964 −0.794819 0.606847i \(-0.792434\pi\)
−0.794819 + 0.606847i \(0.792434\pi\)
\(434\) 5.52278 0.265102
\(435\) 9.97471 0.478251
\(436\) 16.7151 0.800509
\(437\) −1.19295 −0.0570665
\(438\) 15.8120 0.755526
\(439\) −2.33731 −0.111554 −0.0557768 0.998443i \(-0.517764\pi\)
−0.0557768 + 0.998443i \(0.517764\pi\)
\(440\) −21.3322 −1.01697
\(441\) −0.858200 −0.0408667
\(442\) −17.9747 −0.854969
\(443\) −38.6638 −1.83697 −0.918486 0.395454i \(-0.870587\pi\)
−0.918486 + 0.395454i \(0.870587\pi\)
\(444\) 7.73487 0.367081
\(445\) −73.2532 −3.47253
\(446\) −4.84079 −0.229218
\(447\) −30.8410 −1.45873
\(448\) 2.85048 0.134673
\(449\) −27.9726 −1.32011 −0.660054 0.751218i \(-0.729467\pi\)
−0.660054 + 0.751218i \(0.729467\pi\)
\(450\) −7.85826 −0.370442
\(451\) 48.2183 2.27051
\(452\) 4.78584 0.225107
\(453\) 2.46514 0.115823
\(454\) −12.7637 −0.599032
\(455\) −32.4169 −1.51973
\(456\) 0.772490 0.0361752
\(457\) 5.37327 0.251351 0.125675 0.992071i \(-0.459890\pi\)
0.125675 + 0.992071i \(0.459890\pi\)
\(458\) −15.3170 −0.715714
\(459\) −34.7989 −1.62427
\(460\) −9.03628 −0.421319
\(461\) 15.0988 0.703222 0.351611 0.936146i \(-0.385634\pi\)
0.351611 + 0.936146i \(0.385634\pi\)
\(462\) −23.2500 −1.08169
\(463\) 14.1717 0.658616 0.329308 0.944223i \(-0.393185\pi\)
0.329308 + 0.944223i \(0.393185\pi\)
\(464\) 1.70467 0.0791372
\(465\) 11.3370 0.525743
\(466\) 0.614597 0.0284707
\(467\) 31.2076 1.44411 0.722057 0.691833i \(-0.243197\pi\)
0.722057 + 0.691833i \(0.243197\pi\)
\(468\) −2.21716 −0.102488
\(469\) 22.7467 1.05034
\(470\) 24.3588 1.12359
\(471\) −32.5753 −1.50099
\(472\) 10.1312 0.466328
\(473\) −68.7699 −3.16204
\(474\) −17.4964 −0.803636
\(475\) −5.32127 −0.244157
\(476\) −17.6247 −0.807828
\(477\) −1.34212 −0.0614513
\(478\) 9.56775 0.437619
\(479\) −24.4964 −1.11927 −0.559634 0.828740i \(-0.689058\pi\)
−0.559634 + 0.828740i \(0.689058\pi\)
\(480\) 5.85141 0.267079
\(481\) −15.0330 −0.685446
\(482\) −11.1574 −0.508205
\(483\) −9.84865 −0.448129
\(484\) 18.7358 0.851626
\(485\) 48.4578 2.20035
\(486\) −7.71476 −0.349949
\(487\) 7.59090 0.343976 0.171988 0.985099i \(-0.444981\pi\)
0.171988 + 0.985099i \(0.444981\pi\)
\(488\) −10.6073 −0.480168
\(489\) 6.48998 0.293487
\(490\) −4.40195 −0.198860
\(491\) 22.3729 1.00968 0.504838 0.863214i \(-0.331552\pi\)
0.504838 + 0.863214i \(0.331552\pi\)
\(492\) −13.2262 −0.596285
\(493\) −10.5401 −0.474701
\(494\) −1.50136 −0.0675495
\(495\) 16.2695 0.731261
\(496\) 1.93749 0.0869958
\(497\) 2.12198 0.0951839
\(498\) 14.2415 0.638175
\(499\) 44.1792 1.97773 0.988866 0.148812i \(-0.0475449\pi\)
0.988866 + 0.148812i \(0.0475449\pi\)
\(500\) −20.7474 −0.927850
\(501\) 3.79039 0.169342
\(502\) −9.61550 −0.429161
\(503\) 4.78572 0.213385 0.106692 0.994292i \(-0.465974\pi\)
0.106692 + 0.994292i \(0.465974\pi\)
\(504\) −2.17399 −0.0968372
\(505\) −22.8677 −1.01760
\(506\) 12.5960 0.559961
\(507\) 6.80406 0.302179
\(508\) −2.61522 −0.116032
\(509\) −22.4385 −0.994569 −0.497285 0.867587i \(-0.665670\pi\)
−0.497285 + 0.867587i \(0.665670\pi\)
\(510\) −36.1797 −1.60206
\(511\) −30.1328 −1.33300
\(512\) 1.00000 0.0441942
\(513\) −2.90663 −0.128331
\(514\) 22.0832 0.974050
\(515\) 40.5575 1.78718
\(516\) 18.8635 0.830421
\(517\) −33.9547 −1.49332
\(518\) −14.7403 −0.647653
\(519\) −18.2847 −0.802611
\(520\) −11.3724 −0.498714
\(521\) −38.5387 −1.68841 −0.844205 0.536020i \(-0.819927\pi\)
−0.844205 + 0.536020i \(0.819927\pi\)
\(522\) −1.30011 −0.0569041
\(523\) −24.4090 −1.06733 −0.533665 0.845696i \(-0.679185\pi\)
−0.533665 + 0.845696i \(0.679185\pi\)
\(524\) 21.6185 0.944408
\(525\) −43.9309 −1.91730
\(526\) 5.60551 0.244412
\(527\) −11.9796 −0.521841
\(528\) −8.15651 −0.354966
\(529\) −17.6643 −0.768015
\(530\) −6.88410 −0.299026
\(531\) −7.72684 −0.335316
\(532\) −1.47213 −0.0638250
\(533\) 25.7057 1.11344
\(534\) −28.0088 −1.21206
\(535\) 56.6990 2.45131
\(536\) 7.97994 0.344681
\(537\) −21.3576 −0.921648
\(538\) 1.51483 0.0653090
\(539\) 6.13605 0.264298
\(540\) −22.0170 −0.947459
\(541\) 7.64893 0.328853 0.164426 0.986389i \(-0.447423\pi\)
0.164426 + 0.986389i \(0.447423\pi\)
\(542\) 9.34971 0.401605
\(543\) 29.2339 1.25455
\(544\) −6.18307 −0.265097
\(545\) −65.3891 −2.80096
\(546\) −12.3948 −0.530449
\(547\) 11.2499 0.481010 0.240505 0.970648i \(-0.422687\pi\)
0.240505 + 0.970648i \(0.422687\pi\)
\(548\) 16.7400 0.715098
\(549\) 8.08989 0.345268
\(550\) 56.1858 2.39577
\(551\) −0.880376 −0.0375053
\(552\) −3.45508 −0.147058
\(553\) 33.3428 1.41788
\(554\) 28.5493 1.21294
\(555\) −30.2586 −1.28441
\(556\) 17.2878 0.733165
\(557\) −20.1901 −0.855481 −0.427740 0.903902i \(-0.640690\pi\)
−0.427740 + 0.903902i \(0.640690\pi\)
\(558\) −1.47767 −0.0625549
\(559\) −36.6620 −1.55064
\(560\) −11.1510 −0.471217
\(561\) 50.4322 2.12925
\(562\) 13.1826 0.556075
\(563\) 24.1740 1.01881 0.509406 0.860526i \(-0.329865\pi\)
0.509406 + 0.860526i \(0.329865\pi\)
\(564\) 9.31375 0.392179
\(565\) −18.7221 −0.787644
\(566\) −26.9503 −1.13280
\(567\) −17.4743 −0.733853
\(568\) 0.744429 0.0312356
\(569\) 14.4340 0.605106 0.302553 0.953133i \(-0.402161\pi\)
0.302553 + 0.953133i \(0.402161\pi\)
\(570\) −3.02196 −0.126576
\(571\) 7.30129 0.305550 0.152775 0.988261i \(-0.451179\pi\)
0.152775 + 0.988261i \(0.451179\pi\)
\(572\) 15.8525 0.662825
\(573\) 1.28416 0.0536465
\(574\) 25.2052 1.05205
\(575\) 23.8002 0.992537
\(576\) −0.762675 −0.0317781
\(577\) −7.34658 −0.305842 −0.152921 0.988238i \(-0.548868\pi\)
−0.152921 + 0.988238i \(0.548868\pi\)
\(578\) 21.2303 0.883065
\(579\) −11.2539 −0.467698
\(580\) −6.66862 −0.276899
\(581\) −27.1399 −1.12595
\(582\) 18.5281 0.768016
\(583\) 9.59601 0.397426
\(584\) −10.5711 −0.437437
\(585\) 8.67347 0.358604
\(586\) 23.8893 0.986860
\(587\) −5.64225 −0.232881 −0.116440 0.993198i \(-0.537148\pi\)
−0.116440 + 0.993198i \(0.537148\pi\)
\(588\) −1.68312 −0.0694105
\(589\) −1.00062 −0.0412297
\(590\) −39.6332 −1.63167
\(591\) 9.31564 0.383194
\(592\) −5.17117 −0.212534
\(593\) −26.9550 −1.10691 −0.553455 0.832879i \(-0.686691\pi\)
−0.553455 + 0.832879i \(0.686691\pi\)
\(594\) 30.6903 1.25924
\(595\) 68.9475 2.82657
\(596\) 20.6188 0.844579
\(597\) −12.5141 −0.512167
\(598\) 6.71508 0.274600
\(599\) −28.5194 −1.16527 −0.582635 0.812734i \(-0.697978\pi\)
−0.582635 + 0.812734i \(0.697978\pi\)
\(600\) −15.4117 −0.629181
\(601\) −14.9903 −0.611466 −0.305733 0.952117i \(-0.598901\pi\)
−0.305733 + 0.952117i \(0.598901\pi\)
\(602\) −35.9482 −1.46514
\(603\) −6.08610 −0.247845
\(604\) −1.64808 −0.0670593
\(605\) −73.2939 −2.97982
\(606\) −8.74360 −0.355184
\(607\) −31.4329 −1.27582 −0.637911 0.770110i \(-0.720201\pi\)
−0.637911 + 0.770110i \(0.720201\pi\)
\(608\) −0.516450 −0.0209448
\(609\) −7.26813 −0.294519
\(610\) 41.4953 1.68010
\(611\) −18.1016 −0.732313
\(612\) 4.71567 0.190620
\(613\) −30.9411 −1.24970 −0.624851 0.780744i \(-0.714840\pi\)
−0.624851 + 0.780744i \(0.714840\pi\)
\(614\) 4.79854 0.193653
\(615\) 51.7407 2.08639
\(616\) 15.5438 0.626279
\(617\) 27.2781 1.09817 0.549087 0.835765i \(-0.314976\pi\)
0.549087 + 0.835765i \(0.314976\pi\)
\(618\) 15.5074 0.623800
\(619\) 21.1983 0.852032 0.426016 0.904716i \(-0.359917\pi\)
0.426016 + 0.904716i \(0.359917\pi\)
\(620\) −7.57941 −0.304396
\(621\) 13.0003 0.521686
\(622\) −0.139042 −0.00557506
\(623\) 53.3763 2.13848
\(624\) −4.34832 −0.174072
\(625\) 29.6454 1.18582
\(626\) 24.3189 0.971978
\(627\) 4.21243 0.168228
\(628\) 21.7783 0.869047
\(629\) 31.9737 1.27487
\(630\) 8.50460 0.338831
\(631\) −17.4526 −0.694776 −0.347388 0.937722i \(-0.612931\pi\)
−0.347388 + 0.937722i \(0.612931\pi\)
\(632\) 11.6973 0.465292
\(633\) 10.3472 0.411265
\(634\) 17.7111 0.703397
\(635\) 10.2307 0.405992
\(636\) −2.63218 −0.104373
\(637\) 3.27120 0.129610
\(638\) 9.29564 0.368018
\(639\) −0.567757 −0.0224601
\(640\) −3.91198 −0.154634
\(641\) −23.0179 −0.909153 −0.454576 0.890708i \(-0.650209\pi\)
−0.454576 + 0.890708i \(0.650209\pi\)
\(642\) 21.6792 0.855611
\(643\) −11.7524 −0.463469 −0.231734 0.972779i \(-0.574440\pi\)
−0.231734 + 0.972779i \(0.574440\pi\)
\(644\) 6.58434 0.259459
\(645\) −73.7937 −2.90563
\(646\) 3.19325 0.125637
\(647\) −38.4497 −1.51162 −0.755808 0.654794i \(-0.772756\pi\)
−0.755808 + 0.654794i \(0.772756\pi\)
\(648\) −6.13030 −0.240821
\(649\) 55.2462 2.16860
\(650\) 29.9533 1.17486
\(651\) −8.26080 −0.323766
\(652\) −4.33889 −0.169924
\(653\) 32.6070 1.27601 0.638005 0.770032i \(-0.279760\pi\)
0.638005 + 0.770032i \(0.279760\pi\)
\(654\) −25.0019 −0.977653
\(655\) −84.5710 −3.30446
\(656\) 8.84244 0.345239
\(657\) 8.06234 0.314542
\(658\) −17.7492 −0.691935
\(659\) 32.0876 1.24996 0.624978 0.780642i \(-0.285108\pi\)
0.624978 + 0.780642i \(0.285108\pi\)
\(660\) 31.9081 1.24202
\(661\) 3.37129 0.131128 0.0655640 0.997848i \(-0.479115\pi\)
0.0655640 + 0.997848i \(0.479115\pi\)
\(662\) 29.3181 1.13948
\(663\) 26.8860 1.04417
\(664\) −9.52116 −0.369493
\(665\) 5.75895 0.223322
\(666\) 3.94392 0.152824
\(667\) 3.93762 0.152465
\(668\) −2.53407 −0.0980462
\(669\) 7.24070 0.279942
\(670\) −31.2173 −1.20603
\(671\) −57.8419 −2.23296
\(672\) −4.26366 −0.164474
\(673\) −0.769850 −0.0296755 −0.0148378 0.999890i \(-0.504723\pi\)
−0.0148378 + 0.999890i \(0.504723\pi\)
\(674\) 15.7130 0.605244
\(675\) 57.9893 2.23201
\(676\) −4.54887 −0.174957
\(677\) −16.1994 −0.622592 −0.311296 0.950313i \(-0.600763\pi\)
−0.311296 + 0.950313i \(0.600763\pi\)
\(678\) −7.15851 −0.274921
\(679\) −35.3090 −1.35504
\(680\) 24.1880 0.927568
\(681\) 19.0916 0.731592
\(682\) 10.5652 0.404564
\(683\) 12.0081 0.459476 0.229738 0.973252i \(-0.426213\pi\)
0.229738 + 0.973252i \(0.426213\pi\)
\(684\) 0.393883 0.0150605
\(685\) −65.4865 −2.50211
\(686\) −16.7459 −0.639360
\(687\) 22.9106 0.874095
\(688\) −12.6113 −0.480800
\(689\) 5.11574 0.194894
\(690\) 13.5162 0.514553
\(691\) 35.2406 1.34062 0.670308 0.742083i \(-0.266162\pi\)
0.670308 + 0.742083i \(0.266162\pi\)
\(692\) 12.2243 0.464698
\(693\) −11.8549 −0.450330
\(694\) 22.4086 0.850620
\(695\) −67.6294 −2.56533
\(696\) −2.54979 −0.0966495
\(697\) −54.6734 −2.07090
\(698\) −6.38298 −0.241599
\(699\) −0.919296 −0.0347710
\(700\) 29.3701 1.11009
\(701\) 21.5333 0.813300 0.406650 0.913584i \(-0.366697\pi\)
0.406650 + 0.913584i \(0.366697\pi\)
\(702\) 16.3613 0.617519
\(703\) 2.67065 0.100725
\(704\) 5.45305 0.205520
\(705\) −36.4352 −1.37223
\(706\) 9.69691 0.364948
\(707\) 16.6626 0.626663
\(708\) −15.1540 −0.569522
\(709\) −27.9947 −1.05136 −0.525681 0.850682i \(-0.676190\pi\)
−0.525681 + 0.850682i \(0.676190\pi\)
\(710\) −2.91219 −0.109293
\(711\) −8.92121 −0.334571
\(712\) 18.7254 0.701763
\(713\) 4.47541 0.167606
\(714\) 26.3625 0.986593
\(715\) −62.0145 −2.31921
\(716\) 14.2787 0.533619
\(717\) −14.3111 −0.534460
\(718\) 5.84467 0.218121
\(719\) −4.08740 −0.152434 −0.0762171 0.997091i \(-0.524284\pi\)
−0.0762171 + 0.997091i \(0.524284\pi\)
\(720\) 2.98356 0.111191
\(721\) −29.5524 −1.10059
\(722\) −18.7333 −0.697180
\(723\) 16.6889 0.620666
\(724\) −19.5444 −0.726362
\(725\) 17.5641 0.652315
\(726\) −28.0244 −1.04008
\(727\) 17.2234 0.638782 0.319391 0.947623i \(-0.396522\pi\)
0.319391 + 0.947623i \(0.396522\pi\)
\(728\) 8.28659 0.307121
\(729\) 29.9304 1.10853
\(730\) 41.3540 1.53058
\(731\) 77.9763 2.88406
\(732\) 15.8660 0.586425
\(733\) 25.1996 0.930769 0.465385 0.885109i \(-0.345916\pi\)
0.465385 + 0.885109i \(0.345916\pi\)
\(734\) −0.873000 −0.0322230
\(735\) 6.58431 0.242866
\(736\) 2.30990 0.0851441
\(737\) 43.5150 1.60290
\(738\) −6.74390 −0.248246
\(739\) −2.38421 −0.0877046 −0.0438523 0.999038i \(-0.513963\pi\)
−0.0438523 + 0.999038i \(0.513963\pi\)
\(740\) 20.2295 0.743650
\(741\) 2.24569 0.0824976
\(742\) 5.01614 0.184148
\(743\) 32.8316 1.20448 0.602238 0.798317i \(-0.294276\pi\)
0.602238 + 0.798317i \(0.294276\pi\)
\(744\) −2.89804 −0.106247
\(745\) −80.6603 −2.95516
\(746\) 37.6523 1.37855
\(747\) 7.26155 0.265686
\(748\) −33.7166 −1.23280
\(749\) −41.3140 −1.50958
\(750\) 31.0333 1.13317
\(751\) −11.0473 −0.403121 −0.201561 0.979476i \(-0.564601\pi\)
−0.201561 + 0.979476i \(0.564601\pi\)
\(752\) −6.22673 −0.227065
\(753\) 14.3826 0.524130
\(754\) 4.95561 0.180473
\(755\) 6.44724 0.234639
\(756\) 16.0428 0.583470
\(757\) −18.6842 −0.679087 −0.339544 0.940590i \(-0.610273\pi\)
−0.339544 + 0.940590i \(0.610273\pi\)
\(758\) 1.50114 0.0545239
\(759\) −18.8407 −0.683875
\(760\) 2.02034 0.0732855
\(761\) −7.65353 −0.277440 −0.138720 0.990332i \(-0.544299\pi\)
−0.138720 + 0.990332i \(0.544299\pi\)
\(762\) 3.91177 0.141708
\(763\) 47.6461 1.72491
\(764\) −0.858528 −0.0310605
\(765\) −18.4476 −0.666974
\(766\) 13.4312 0.485289
\(767\) 29.4523 1.06346
\(768\) −1.49577 −0.0539739
\(769\) 38.4120 1.38517 0.692587 0.721334i \(-0.256471\pi\)
0.692587 + 0.721334i \(0.256471\pi\)
\(770\) −60.8071 −2.19133
\(771\) −33.0314 −1.18960
\(772\) 7.52385 0.270789
\(773\) −46.0469 −1.65619 −0.828096 0.560586i \(-0.810576\pi\)
−0.828096 + 0.560586i \(0.810576\pi\)
\(774\) 9.61829 0.345722
\(775\) 19.9630 0.717093
\(776\) −12.3870 −0.444668
\(777\) 22.0481 0.790972
\(778\) −16.5486 −0.593296
\(779\) −4.56668 −0.163618
\(780\) 17.0105 0.609075
\(781\) 4.05941 0.145257
\(782\) −14.2823 −0.510734
\(783\) 9.59402 0.342862
\(784\) 1.12525 0.0401875
\(785\) −85.1960 −3.04078
\(786\) −32.3363 −1.15340
\(787\) −21.7091 −0.773845 −0.386922 0.922112i \(-0.626462\pi\)
−0.386922 + 0.922112i \(0.626462\pi\)
\(788\) −6.22799 −0.221863
\(789\) −8.38456 −0.298498
\(790\) −45.7594 −1.62805
\(791\) 13.6419 0.485052
\(792\) −4.15890 −0.147780
\(793\) −30.8362 −1.09502
\(794\) 28.3813 1.00721
\(795\) 10.2970 0.365198
\(796\) 8.36632 0.296536
\(797\) 24.6514 0.873197 0.436598 0.899657i \(-0.356183\pi\)
0.436598 + 0.899657i \(0.356183\pi\)
\(798\) 2.20197 0.0779489
\(799\) 38.5003 1.36204
\(800\) 10.3036 0.364286
\(801\) −14.2814 −0.504607
\(802\) 23.6405 0.834775
\(803\) −57.6449 −2.03425
\(804\) −11.9362 −0.420956
\(805\) −25.7578 −0.907842
\(806\) 5.63244 0.198394
\(807\) −2.26584 −0.0797612
\(808\) 5.84555 0.205646
\(809\) −5.27338 −0.185402 −0.0927011 0.995694i \(-0.529550\pi\)
−0.0927011 + 0.995694i \(0.529550\pi\)
\(810\) 23.9816 0.842627
\(811\) −12.8916 −0.452685 −0.226342 0.974048i \(-0.572677\pi\)
−0.226342 + 0.974048i \(0.572677\pi\)
\(812\) 4.85912 0.170522
\(813\) −13.9850 −0.490476
\(814\) −28.1986 −0.988362
\(815\) 16.9736 0.594561
\(816\) 9.24844 0.323760
\(817\) 6.51309 0.227864
\(818\) −6.01198 −0.210204
\(819\) −6.31997 −0.220838
\(820\) −34.5914 −1.20798
\(821\) −4.37698 −0.152758 −0.0763789 0.997079i \(-0.524336\pi\)
−0.0763789 + 0.997079i \(0.524336\pi\)
\(822\) −25.0392 −0.873342
\(823\) 27.3019 0.951685 0.475843 0.879530i \(-0.342143\pi\)
0.475843 + 0.879530i \(0.342143\pi\)
\(824\) −10.3675 −0.361170
\(825\) −84.0410 −2.92593
\(826\) 28.8789 1.00483
\(827\) −4.45600 −0.154950 −0.0774751 0.996994i \(-0.524686\pi\)
−0.0774751 + 0.996994i \(0.524686\pi\)
\(828\) −1.76170 −0.0612235
\(829\) 9.83165 0.341467 0.170734 0.985317i \(-0.445386\pi\)
0.170734 + 0.985317i \(0.445386\pi\)
\(830\) 37.2466 1.29285
\(831\) −42.7031 −1.48135
\(832\) 2.90708 0.100785
\(833\) −6.95750 −0.241063
\(834\) −25.8585 −0.895408
\(835\) 9.91323 0.343061
\(836\) −2.81623 −0.0974013
\(837\) 10.9044 0.376910
\(838\) −16.8076 −0.580610
\(839\) −33.1175 −1.14334 −0.571672 0.820483i \(-0.693705\pi\)
−0.571672 + 0.820483i \(0.693705\pi\)
\(840\) 16.6793 0.575492
\(841\) −26.0941 −0.899797
\(842\) −17.2956 −0.596046
\(843\) −19.7182 −0.679130
\(844\) −6.91765 −0.238115
\(845\) 17.7951 0.612169
\(846\) 4.74897 0.163273
\(847\) 53.4060 1.83505
\(848\) 1.75975 0.0604301
\(849\) 40.3114 1.38348
\(850\) −63.7076 −2.18515
\(851\) −11.9449 −0.409466
\(852\) −1.11349 −0.0381477
\(853\) −54.7531 −1.87471 −0.937354 0.348377i \(-0.886733\pi\)
−0.937354 + 0.348377i \(0.886733\pi\)
\(854\) −30.2358 −1.03465
\(855\) −1.54086 −0.0526964
\(856\) −14.4937 −0.495385
\(857\) −26.4355 −0.903020 −0.451510 0.892266i \(-0.649114\pi\)
−0.451510 + 0.892266i \(0.649114\pi\)
\(858\) −23.7116 −0.809502
\(859\) 22.4718 0.766729 0.383364 0.923597i \(-0.374765\pi\)
0.383364 + 0.923597i \(0.374765\pi\)
\(860\) 49.3350 1.68231
\(861\) −37.7012 −1.28485
\(862\) 19.9251 0.678652
\(863\) −6.70965 −0.228399 −0.114200 0.993458i \(-0.536430\pi\)
−0.114200 + 0.993458i \(0.536430\pi\)
\(864\) 5.62809 0.191472
\(865\) −47.8212 −1.62597
\(866\) −33.0782 −1.12404
\(867\) −31.7557 −1.07848
\(868\) 5.52278 0.187455
\(869\) 63.7858 2.16378
\(870\) 9.97471 0.338174
\(871\) 23.1984 0.786046
\(872\) 16.7151 0.566045
\(873\) 9.44727 0.319742
\(874\) −1.19295 −0.0403521
\(875\) −59.1400 −1.99930
\(876\) 15.8120 0.534237
\(877\) −41.1863 −1.39076 −0.695381 0.718641i \(-0.744765\pi\)
−0.695381 + 0.718641i \(0.744765\pi\)
\(878\) −2.33731 −0.0788803
\(879\) −35.7329 −1.20524
\(880\) −21.3322 −0.719108
\(881\) −7.02626 −0.236721 −0.118360 0.992971i \(-0.537764\pi\)
−0.118360 + 0.992971i \(0.537764\pi\)
\(882\) −0.858200 −0.0288971
\(883\) 18.3987 0.619166 0.309583 0.950872i \(-0.399811\pi\)
0.309583 + 0.950872i \(0.399811\pi\)
\(884\) −17.9747 −0.604554
\(885\) 59.2821 1.99274
\(886\) −38.6638 −1.29893
\(887\) −39.3363 −1.32078 −0.660392 0.750921i \(-0.729610\pi\)
−0.660392 + 0.750921i \(0.729610\pi\)
\(888\) 7.73487 0.259565
\(889\) −7.45465 −0.250021
\(890\) −73.2532 −2.45545
\(891\) −33.4289 −1.11991
\(892\) −4.84079 −0.162082
\(893\) 3.21579 0.107612
\(894\) −30.8410 −1.03148
\(895\) −55.8578 −1.86712
\(896\) 2.85048 0.0952279
\(897\) −10.0442 −0.335366
\(898\) −27.9726 −0.933457
\(899\) 3.30277 0.110154
\(900\) −7.85826 −0.261942
\(901\) −10.8807 −0.362487
\(902\) 48.2183 1.60549
\(903\) 53.7702 1.78936
\(904\) 4.78584 0.159175
\(905\) 76.4572 2.54152
\(906\) 2.46514 0.0818989
\(907\) 28.3747 0.942166 0.471083 0.882089i \(-0.343863\pi\)
0.471083 + 0.882089i \(0.343863\pi\)
\(908\) −12.7637 −0.423580
\(909\) −4.45825 −0.147871
\(910\) −32.4169 −1.07461
\(911\) −30.6980 −1.01707 −0.508536 0.861041i \(-0.669813\pi\)
−0.508536 + 0.861041i \(0.669813\pi\)
\(912\) 0.772490 0.0255797
\(913\) −51.9194 −1.71828
\(914\) 5.37327 0.177732
\(915\) −62.0674 −2.05189
\(916\) −15.3170 −0.506086
\(917\) 61.6231 2.03497
\(918\) −34.7989 −1.14853
\(919\) −55.5616 −1.83281 −0.916404 0.400254i \(-0.868922\pi\)
−0.916404 + 0.400254i \(0.868922\pi\)
\(920\) −9.03628 −0.297917
\(921\) −7.17750 −0.236507
\(922\) 15.0988 0.497253
\(923\) 2.16412 0.0712328
\(924\) −23.2500 −0.764868
\(925\) −53.2814 −1.75188
\(926\) 14.1717 0.465711
\(927\) 7.90704 0.259701
\(928\) 1.70467 0.0559584
\(929\) 21.2688 0.697808 0.348904 0.937158i \(-0.386554\pi\)
0.348904 + 0.937158i \(0.386554\pi\)
\(930\) 11.3370 0.371756
\(931\) −0.581136 −0.0190460
\(932\) 0.614597 0.0201318
\(933\) 0.207974 0.00680877
\(934\) 31.2076 1.02114
\(935\) 131.898 4.31354
\(936\) −2.21716 −0.0724701
\(937\) 5.45972 0.178361 0.0891807 0.996015i \(-0.471575\pi\)
0.0891807 + 0.996015i \(0.471575\pi\)
\(938\) 22.7467 0.742706
\(939\) −36.3754 −1.18707
\(940\) 24.3588 0.794497
\(941\) 53.2700 1.73655 0.868277 0.496080i \(-0.165228\pi\)
0.868277 + 0.496080i \(0.165228\pi\)
\(942\) −32.5753 −1.06136
\(943\) 20.4252 0.665135
\(944\) 10.1312 0.329744
\(945\) −62.7590 −2.04155
\(946\) −68.7699 −2.23590
\(947\) 3.57908 0.116304 0.0581522 0.998308i \(-0.481479\pi\)
0.0581522 + 0.998308i \(0.481479\pi\)
\(948\) −17.4964 −0.568257
\(949\) −30.7312 −0.997576
\(950\) −5.32127 −0.172645
\(951\) −26.4917 −0.859052
\(952\) −17.6247 −0.571221
\(953\) −23.1390 −0.749544 −0.374772 0.927117i \(-0.622279\pi\)
−0.374772 + 0.927117i \(0.622279\pi\)
\(954\) −1.34212 −0.0434527
\(955\) 3.35854 0.108680
\(956\) 9.56775 0.309443
\(957\) −13.9041 −0.449457
\(958\) −24.4964 −0.791442
\(959\) 47.7171 1.54087
\(960\) 5.85141 0.188854
\(961\) −27.2461 −0.878908
\(962\) −15.0330 −0.484684
\(963\) 11.0540 0.356210
\(964\) −11.1574 −0.359355
\(965\) −29.4331 −0.947485
\(966\) −9.84865 −0.316875
\(967\) 49.9955 1.60775 0.803874 0.594800i \(-0.202769\pi\)
0.803874 + 0.594800i \(0.202769\pi\)
\(968\) 18.7358 0.602190
\(969\) −4.77636 −0.153439
\(970\) 48.4578 1.55588
\(971\) 2.75022 0.0882587 0.0441294 0.999026i \(-0.485949\pi\)
0.0441294 + 0.999026i \(0.485949\pi\)
\(972\) −7.71476 −0.247451
\(973\) 49.2785 1.57980
\(974\) 7.59090 0.243228
\(975\) −44.8032 −1.43485
\(976\) −10.6073 −0.339530
\(977\) −54.1191 −1.73142 −0.865712 0.500543i \(-0.833134\pi\)
−0.865712 + 0.500543i \(0.833134\pi\)
\(978\) 6.48998 0.207527
\(979\) 102.110 3.26346
\(980\) −4.40195 −0.140615
\(981\) −12.7482 −0.407018
\(982\) 22.3729 0.713949
\(983\) 32.1045 1.02397 0.511987 0.858993i \(-0.328910\pi\)
0.511987 + 0.858993i \(0.328910\pi\)
\(984\) −13.2262 −0.421637
\(985\) 24.3638 0.776294
\(986\) −10.5401 −0.335664
\(987\) 26.5487 0.845054
\(988\) −1.50136 −0.0477647
\(989\) −29.1308 −0.926306
\(990\) 16.2695 0.517080
\(991\) 2.29004 0.0727455 0.0363728 0.999338i \(-0.488420\pi\)
0.0363728 + 0.999338i \(0.488420\pi\)
\(992\) 1.93749 0.0615153
\(993\) −43.8531 −1.39164
\(994\) 2.12198 0.0673052
\(995\) −32.7288 −1.03757
\(996\) 14.2415 0.451258
\(997\) −0.679662 −0.0215251 −0.0107626 0.999942i \(-0.503426\pi\)
−0.0107626 + 0.999942i \(0.503426\pi\)
\(998\) 44.1792 1.39847
\(999\) −29.1038 −0.920804
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 6046.2.a.f.1.16 67
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6046.2.a.f.1.16 67 1.1 even 1 trivial