Properties

Label 6002.2.a.b.1.15
Level $6002$
Weight $2$
Character 6002.1
Self dual yes
Analytic conductor $47.926$
Analytic rank $1$
Dimension $56$
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] = [6002,2,Mod(1,6002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6002, 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("6002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6002 = 2 \cdot 3001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6002.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: \(47.9262112932\)
Analytic rank: \(1\)
Dimension: \(56\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) \(=\) 6002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.01653 q^{3} +1.00000 q^{4} -1.99229 q^{5} +2.01653 q^{6} -2.17859 q^{7} -1.00000 q^{8} +1.06638 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.01653 q^{3} +1.00000 q^{4} -1.99229 q^{5} +2.01653 q^{6} -2.17859 q^{7} -1.00000 q^{8} +1.06638 q^{9} +1.99229 q^{10} -5.20148 q^{11} -2.01653 q^{12} +1.73089 q^{13} +2.17859 q^{14} +4.01750 q^{15} +1.00000 q^{16} -1.01764 q^{17} -1.06638 q^{18} -6.14250 q^{19} -1.99229 q^{20} +4.39319 q^{21} +5.20148 q^{22} +5.36800 q^{23} +2.01653 q^{24} -1.03080 q^{25} -1.73089 q^{26} +3.89920 q^{27} -2.17859 q^{28} +3.11531 q^{29} -4.01750 q^{30} -3.55810 q^{31} -1.00000 q^{32} +10.4889 q^{33} +1.01764 q^{34} +4.34038 q^{35} +1.06638 q^{36} -0.736872 q^{37} +6.14250 q^{38} -3.49037 q^{39} +1.99229 q^{40} +1.75227 q^{41} -4.39319 q^{42} +6.65469 q^{43} -5.20148 q^{44} -2.12452 q^{45} -5.36800 q^{46} +4.35042 q^{47} -2.01653 q^{48} -2.25372 q^{49} +1.03080 q^{50} +2.05209 q^{51} +1.73089 q^{52} -4.05057 q^{53} -3.89920 q^{54} +10.3628 q^{55} +2.17859 q^{56} +12.3865 q^{57} -3.11531 q^{58} +8.76448 q^{59} +4.01750 q^{60} +0.816696 q^{61} +3.55810 q^{62} -2.32320 q^{63} +1.00000 q^{64} -3.44842 q^{65} -10.4889 q^{66} +9.86937 q^{67} -1.01764 q^{68} -10.8247 q^{69} -4.34038 q^{70} -0.662399 q^{71} -1.06638 q^{72} +7.12644 q^{73} +0.736872 q^{74} +2.07863 q^{75} -6.14250 q^{76} +11.3319 q^{77} +3.49037 q^{78} -4.61676 q^{79} -1.99229 q^{80} -11.0620 q^{81} -1.75227 q^{82} -16.8661 q^{83} +4.39319 q^{84} +2.02742 q^{85} -6.65469 q^{86} -6.28210 q^{87} +5.20148 q^{88} +18.0780 q^{89} +2.12452 q^{90} -3.77090 q^{91} +5.36800 q^{92} +7.17501 q^{93} -4.35042 q^{94} +12.2376 q^{95} +2.01653 q^{96} +3.91078 q^{97} +2.25372 q^{98} -5.54672 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 56 q^{2} - 11 q^{3} + 56 q^{4} + 11 q^{6} - 21 q^{7} - 56 q^{8} + 53 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q - 56 q^{2} - 11 q^{3} + 56 q^{4} + 11 q^{6} - 21 q^{7} - 56 q^{8} + 53 q^{9} + 12 q^{11} - 11 q^{12} - 31 q^{13} + 21 q^{14} - 22 q^{15} + 56 q^{16} - 4 q^{17} - 53 q^{18} - 9 q^{19} + 13 q^{21} - 12 q^{22} - 39 q^{23} + 11 q^{24} + 8 q^{25} + 31 q^{26} - 44 q^{27} - 21 q^{28} + 13 q^{29} + 22 q^{30} - 35 q^{31} - 56 q^{32} - 26 q^{33} + 4 q^{34} - 7 q^{35} + 53 q^{36} - 65 q^{37} + 9 q^{38} - 27 q^{39} + 38 q^{41} - 13 q^{42} - 76 q^{43} + 12 q^{44} - 21 q^{45} + 39 q^{46} - 43 q^{47} - 11 q^{48} + 9 q^{49} - 8 q^{50} - 19 q^{51} - 31 q^{52} - 26 q^{53} + 44 q^{54} - 67 q^{55} + 21 q^{56} - 26 q^{57} - 13 q^{58} + 11 q^{59} - 22 q^{60} - 17 q^{61} + 35 q^{62} - 67 q^{63} + 56 q^{64} + 31 q^{65} + 26 q^{66} - 93 q^{67} - 4 q^{68} - 13 q^{69} + 7 q^{70} - 33 q^{71} - 53 q^{72} - 41 q^{73} + 65 q^{74} - 21 q^{75} - 9 q^{76} + 5 q^{77} + 27 q^{78} - 69 q^{79} + 36 q^{81} - 38 q^{82} + 4 q^{83} + 13 q^{84} - 40 q^{85} + 76 q^{86} - 69 q^{87} - 12 q^{88} + 40 q^{89} + 21 q^{90} - 64 q^{91} - 39 q^{92} - 57 q^{93} + 43 q^{94} - 22 q^{95} + 11 q^{96} - 71 q^{97} - 9 q^{98} + 17 q^{99}+O(q^{100}) \) 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
\(3\) −2.01653 −1.16424 −0.582121 0.813102i \(-0.697777\pi\)
−0.582121 + 0.813102i \(0.697777\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.99229 −0.890977 −0.445489 0.895288i \(-0.646970\pi\)
−0.445489 + 0.895288i \(0.646970\pi\)
\(6\) 2.01653 0.823243
\(7\) −2.17859 −0.823431 −0.411716 0.911312i \(-0.635070\pi\)
−0.411716 + 0.911312i \(0.635070\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.06638 0.355458
\(10\) 1.99229 0.630016
\(11\) −5.20148 −1.56830 −0.784152 0.620569i \(-0.786902\pi\)
−0.784152 + 0.620569i \(0.786902\pi\)
\(12\) −2.01653 −0.582121
\(13\) 1.73089 0.480061 0.240031 0.970765i \(-0.422843\pi\)
0.240031 + 0.970765i \(0.422843\pi\)
\(14\) 2.17859 0.582254
\(15\) 4.01750 1.03731
\(16\) 1.00000 0.250000
\(17\) −1.01764 −0.246813 −0.123406 0.992356i \(-0.539382\pi\)
−0.123406 + 0.992356i \(0.539382\pi\)
\(18\) −1.06638 −0.251347
\(19\) −6.14250 −1.40919 −0.704593 0.709612i \(-0.748870\pi\)
−0.704593 + 0.709612i \(0.748870\pi\)
\(20\) −1.99229 −0.445489
\(21\) 4.39319 0.958673
\(22\) 5.20148 1.10896
\(23\) 5.36800 1.11931 0.559653 0.828727i \(-0.310934\pi\)
0.559653 + 0.828727i \(0.310934\pi\)
\(24\) 2.01653 0.411622
\(25\) −1.03080 −0.206159
\(26\) −1.73089 −0.339455
\(27\) 3.89920 0.750402
\(28\) −2.17859 −0.411716
\(29\) 3.11531 0.578498 0.289249 0.957254i \(-0.406594\pi\)
0.289249 + 0.957254i \(0.406594\pi\)
\(30\) −4.01750 −0.733491
\(31\) −3.55810 −0.639054 −0.319527 0.947577i \(-0.603524\pi\)
−0.319527 + 0.947577i \(0.603524\pi\)
\(32\) −1.00000 −0.176777
\(33\) 10.4889 1.82588
\(34\) 1.01764 0.174523
\(35\) 4.34038 0.733659
\(36\) 1.06638 0.177729
\(37\) −0.736872 −0.121141 −0.0605705 0.998164i \(-0.519292\pi\)
−0.0605705 + 0.998164i \(0.519292\pi\)
\(38\) 6.14250 0.996445
\(39\) −3.49037 −0.558907
\(40\) 1.99229 0.315008
\(41\) 1.75227 0.273659 0.136829 0.990595i \(-0.456309\pi\)
0.136829 + 0.990595i \(0.456309\pi\)
\(42\) −4.39319 −0.677884
\(43\) 6.65469 1.01483 0.507415 0.861702i \(-0.330601\pi\)
0.507415 + 0.861702i \(0.330601\pi\)
\(44\) −5.20148 −0.784152
\(45\) −2.12452 −0.316705
\(46\) −5.36800 −0.791469
\(47\) 4.35042 0.634574 0.317287 0.948329i \(-0.397228\pi\)
0.317287 + 0.948329i \(0.397228\pi\)
\(48\) −2.01653 −0.291060
\(49\) −2.25372 −0.321961
\(50\) 1.03080 0.145777
\(51\) 2.05209 0.287350
\(52\) 1.73089 0.240031
\(53\) −4.05057 −0.556389 −0.278195 0.960525i \(-0.589736\pi\)
−0.278195 + 0.960525i \(0.589736\pi\)
\(54\) −3.89920 −0.530614
\(55\) 10.3628 1.39732
\(56\) 2.17859 0.291127
\(57\) 12.3865 1.64063
\(58\) −3.11531 −0.409060
\(59\) 8.76448 1.14104 0.570519 0.821284i \(-0.306742\pi\)
0.570519 + 0.821284i \(0.306742\pi\)
\(60\) 4.01750 0.518656
\(61\) 0.816696 0.104567 0.0522836 0.998632i \(-0.483350\pi\)
0.0522836 + 0.998632i \(0.483350\pi\)
\(62\) 3.55810 0.451880
\(63\) −2.32320 −0.292696
\(64\) 1.00000 0.125000
\(65\) −3.44842 −0.427724
\(66\) −10.4889 −1.29110
\(67\) 9.86937 1.20574 0.602868 0.797841i \(-0.294025\pi\)
0.602868 + 0.797841i \(0.294025\pi\)
\(68\) −1.01764 −0.123406
\(69\) −10.8247 −1.30314
\(70\) −4.34038 −0.518775
\(71\) −0.662399 −0.0786123 −0.0393061 0.999227i \(-0.512515\pi\)
−0.0393061 + 0.999227i \(0.512515\pi\)
\(72\) −1.06638 −0.125674
\(73\) 7.12644 0.834087 0.417043 0.908887i \(-0.363066\pi\)
0.417043 + 0.908887i \(0.363066\pi\)
\(74\) 0.736872 0.0856597
\(75\) 2.07863 0.240019
\(76\) −6.14250 −0.704593
\(77\) 11.3319 1.29139
\(78\) 3.49037 0.395207
\(79\) −4.61676 −0.519426 −0.259713 0.965686i \(-0.583628\pi\)
−0.259713 + 0.965686i \(0.583628\pi\)
\(80\) −1.99229 −0.222744
\(81\) −11.0620 −1.22911
\(82\) −1.75227 −0.193506
\(83\) −16.8661 −1.85129 −0.925647 0.378388i \(-0.876478\pi\)
−0.925647 + 0.378388i \(0.876478\pi\)
\(84\) 4.39319 0.479337
\(85\) 2.02742 0.219905
\(86\) −6.65469 −0.717593
\(87\) −6.28210 −0.673512
\(88\) 5.20148 0.554479
\(89\) 18.0780 1.91626 0.958131 0.286330i \(-0.0924354\pi\)
0.958131 + 0.286330i \(0.0924354\pi\)
\(90\) 2.12452 0.223944
\(91\) −3.77090 −0.395298
\(92\) 5.36800 0.559653
\(93\) 7.17501 0.744013
\(94\) −4.35042 −0.448712
\(95\) 12.2376 1.25555
\(96\) 2.01653 0.205811
\(97\) 3.91078 0.397080 0.198540 0.980093i \(-0.436380\pi\)
0.198540 + 0.980093i \(0.436380\pi\)
\(98\) 2.25372 0.227661
\(99\) −5.54672 −0.557467
\(100\) −1.03080 −0.103080
\(101\) 11.0093 1.09547 0.547735 0.836652i \(-0.315490\pi\)
0.547735 + 0.836652i \(0.315490\pi\)
\(102\) −2.05209 −0.203187
\(103\) −4.95972 −0.488696 −0.244348 0.969688i \(-0.578574\pi\)
−0.244348 + 0.969688i \(0.578574\pi\)
\(104\) −1.73089 −0.169727
\(105\) −8.75249 −0.854156
\(106\) 4.05057 0.393427
\(107\) −0.865756 −0.0836958 −0.0418479 0.999124i \(-0.513324\pi\)
−0.0418479 + 0.999124i \(0.513324\pi\)
\(108\) 3.89920 0.375201
\(109\) 11.9007 1.13988 0.569939 0.821687i \(-0.306967\pi\)
0.569939 + 0.821687i \(0.306967\pi\)
\(110\) −10.3628 −0.988057
\(111\) 1.48592 0.141037
\(112\) −2.17859 −0.205858
\(113\) −18.9162 −1.77949 −0.889744 0.456460i \(-0.849117\pi\)
−0.889744 + 0.456460i \(0.849117\pi\)
\(114\) −12.3865 −1.16010
\(115\) −10.6946 −0.997276
\(116\) 3.11531 0.289249
\(117\) 1.84577 0.170642
\(118\) −8.76448 −0.806836
\(119\) 2.21702 0.203234
\(120\) −4.01750 −0.366745
\(121\) 16.0553 1.45958
\(122\) −0.816696 −0.0739402
\(123\) −3.53350 −0.318605
\(124\) −3.55810 −0.319527
\(125\) 12.0151 1.07466
\(126\) 2.32320 0.206967
\(127\) −5.89672 −0.523249 −0.261625 0.965170i \(-0.584258\pi\)
−0.261625 + 0.965170i \(0.584258\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −13.4193 −1.18151
\(130\) 3.44842 0.302446
\(131\) 11.2970 0.987023 0.493512 0.869739i \(-0.335713\pi\)
0.493512 + 0.869739i \(0.335713\pi\)
\(132\) 10.4889 0.912942
\(133\) 13.3820 1.16037
\(134\) −9.86937 −0.852584
\(135\) −7.76833 −0.668591
\(136\) 1.01764 0.0872615
\(137\) 1.78132 0.152189 0.0760944 0.997101i \(-0.475755\pi\)
0.0760944 + 0.997101i \(0.475755\pi\)
\(138\) 10.8247 0.921461
\(139\) −17.6995 −1.50125 −0.750626 0.660728i \(-0.770248\pi\)
−0.750626 + 0.660728i \(0.770248\pi\)
\(140\) 4.34038 0.366829
\(141\) −8.77274 −0.738798
\(142\) 0.662399 0.0555873
\(143\) −9.00316 −0.752882
\(144\) 1.06638 0.0888646
\(145\) −6.20659 −0.515429
\(146\) −7.12644 −0.589788
\(147\) 4.54469 0.374840
\(148\) −0.736872 −0.0605705
\(149\) 20.1669 1.65214 0.826070 0.563567i \(-0.190572\pi\)
0.826070 + 0.563567i \(0.190572\pi\)
\(150\) −2.07863 −0.169719
\(151\) 0.805568 0.0655562 0.0327781 0.999463i \(-0.489565\pi\)
0.0327781 + 0.999463i \(0.489565\pi\)
\(152\) 6.14250 0.498222
\(153\) −1.08518 −0.0877317
\(154\) −11.3319 −0.913151
\(155\) 7.08876 0.569383
\(156\) −3.49037 −0.279454
\(157\) −15.4947 −1.23661 −0.618306 0.785937i \(-0.712181\pi\)
−0.618306 + 0.785937i \(0.712181\pi\)
\(158\) 4.61676 0.367290
\(159\) 8.16809 0.647771
\(160\) 1.99229 0.157504
\(161\) −11.6947 −0.921672
\(162\) 11.0620 0.869110
\(163\) 19.4715 1.52512 0.762562 0.646915i \(-0.223941\pi\)
0.762562 + 0.646915i \(0.223941\pi\)
\(164\) 1.75227 0.136829
\(165\) −20.8969 −1.62682
\(166\) 16.8661 1.30906
\(167\) −13.2299 −1.02376 −0.511881 0.859057i \(-0.671051\pi\)
−0.511881 + 0.859057i \(0.671051\pi\)
\(168\) −4.39319 −0.338942
\(169\) −10.0040 −0.769541
\(170\) −2.02742 −0.155496
\(171\) −6.55021 −0.500907
\(172\) 6.65469 0.507415
\(173\) 2.90149 0.220596 0.110298 0.993899i \(-0.464819\pi\)
0.110298 + 0.993899i \(0.464819\pi\)
\(174\) 6.28210 0.476245
\(175\) 2.24569 0.169758
\(176\) −5.20148 −0.392076
\(177\) −17.6738 −1.32844
\(178\) −18.0780 −1.35500
\(179\) −5.88100 −0.439566 −0.219783 0.975549i \(-0.570535\pi\)
−0.219783 + 0.975549i \(0.570535\pi\)
\(180\) −2.12452 −0.158353
\(181\) 13.7437 1.02156 0.510781 0.859711i \(-0.329356\pi\)
0.510781 + 0.859711i \(0.329356\pi\)
\(182\) 3.77090 0.279518
\(183\) −1.64689 −0.121741
\(184\) −5.36800 −0.395734
\(185\) 1.46806 0.107934
\(186\) −7.17501 −0.526097
\(187\) 5.29321 0.387078
\(188\) 4.35042 0.317287
\(189\) −8.49479 −0.617905
\(190\) −12.2376 −0.887810
\(191\) 24.6595 1.78430 0.892150 0.451740i \(-0.149197\pi\)
0.892150 + 0.451740i \(0.149197\pi\)
\(192\) −2.01653 −0.145530
\(193\) 7.24622 0.521594 0.260797 0.965394i \(-0.416015\pi\)
0.260797 + 0.965394i \(0.416015\pi\)
\(194\) −3.91078 −0.280778
\(195\) 6.95382 0.497974
\(196\) −2.25372 −0.160980
\(197\) 20.9701 1.49406 0.747030 0.664790i \(-0.231479\pi\)
0.747030 + 0.664790i \(0.231479\pi\)
\(198\) 5.54672 0.394188
\(199\) −21.2249 −1.50459 −0.752297 0.658825i \(-0.771054\pi\)
−0.752297 + 0.658825i \(0.771054\pi\)
\(200\) 1.03080 0.0728883
\(201\) −19.9018 −1.40377
\(202\) −11.0093 −0.774614
\(203\) −6.78700 −0.476354
\(204\) 2.05209 0.143675
\(205\) −3.49103 −0.243824
\(206\) 4.95972 0.345560
\(207\) 5.72430 0.397867
\(208\) 1.73089 0.120015
\(209\) 31.9501 2.21003
\(210\) 8.75249 0.603979
\(211\) −26.5375 −1.82692 −0.913459 0.406930i \(-0.866599\pi\)
−0.913459 + 0.406930i \(0.866599\pi\)
\(212\) −4.05057 −0.278195
\(213\) 1.33574 0.0915237
\(214\) 0.865756 0.0591819
\(215\) −13.2580 −0.904191
\(216\) −3.89920 −0.265307
\(217\) 7.75166 0.526217
\(218\) −11.9007 −0.806015
\(219\) −14.3706 −0.971078
\(220\) 10.3628 0.698662
\(221\) −1.76141 −0.118485
\(222\) −1.48592 −0.0997286
\(223\) 5.23517 0.350573 0.175286 0.984517i \(-0.443915\pi\)
0.175286 + 0.984517i \(0.443915\pi\)
\(224\) 2.17859 0.145563
\(225\) −1.09922 −0.0732811
\(226\) 18.9162 1.25829
\(227\) −5.70185 −0.378445 −0.189222 0.981934i \(-0.560597\pi\)
−0.189222 + 0.981934i \(0.560597\pi\)
\(228\) 12.3865 0.820316
\(229\) 10.2629 0.678192 0.339096 0.940752i \(-0.389879\pi\)
0.339096 + 0.940752i \(0.389879\pi\)
\(230\) 10.6946 0.705181
\(231\) −22.8511 −1.50349
\(232\) −3.11531 −0.204530
\(233\) 22.2019 1.45449 0.727247 0.686376i \(-0.240800\pi\)
0.727247 + 0.686376i \(0.240800\pi\)
\(234\) −1.84577 −0.120662
\(235\) −8.66729 −0.565391
\(236\) 8.76448 0.570519
\(237\) 9.30982 0.604738
\(238\) −2.21702 −0.143708
\(239\) −27.6097 −1.78592 −0.892961 0.450134i \(-0.851376\pi\)
−0.892961 + 0.450134i \(0.851376\pi\)
\(240\) 4.01750 0.259328
\(241\) 18.2226 1.17382 0.586910 0.809652i \(-0.300344\pi\)
0.586910 + 0.809652i \(0.300344\pi\)
\(242\) −16.0553 −1.03208
\(243\) 10.6091 0.680576
\(244\) 0.816696 0.0522836
\(245\) 4.49006 0.286860
\(246\) 3.53350 0.225288
\(247\) −10.6320 −0.676495
\(248\) 3.55810 0.225940
\(249\) 34.0109 2.15535
\(250\) −12.0151 −0.759900
\(251\) −4.64098 −0.292936 −0.146468 0.989215i \(-0.546790\pi\)
−0.146468 + 0.989215i \(0.546790\pi\)
\(252\) −2.32320 −0.146348
\(253\) −27.9215 −1.75541
\(254\) 5.89672 0.369993
\(255\) −4.08835 −0.256022
\(256\) 1.00000 0.0625000
\(257\) −3.10603 −0.193749 −0.0968745 0.995297i \(-0.530885\pi\)
−0.0968745 + 0.995297i \(0.530885\pi\)
\(258\) 13.4193 0.835452
\(259\) 1.60535 0.0997514
\(260\) −3.44842 −0.213862
\(261\) 3.32209 0.205632
\(262\) −11.2970 −0.697931
\(263\) −4.91173 −0.302870 −0.151435 0.988467i \(-0.548389\pi\)
−0.151435 + 0.988467i \(0.548389\pi\)
\(264\) −10.4889 −0.645548
\(265\) 8.06990 0.495730
\(266\) −13.3820 −0.820504
\(267\) −36.4547 −2.23099
\(268\) 9.86937 0.602868
\(269\) 25.0619 1.52805 0.764024 0.645187i \(-0.223221\pi\)
0.764024 + 0.645187i \(0.223221\pi\)
\(270\) 7.76833 0.472765
\(271\) 15.7737 0.958185 0.479092 0.877764i \(-0.340966\pi\)
0.479092 + 0.877764i \(0.340966\pi\)
\(272\) −1.01764 −0.0617032
\(273\) 7.60411 0.460222
\(274\) −1.78132 −0.107614
\(275\) 5.36167 0.323321
\(276\) −10.8247 −0.651571
\(277\) −21.7596 −1.30741 −0.653705 0.756749i \(-0.726786\pi\)
−0.653705 + 0.756749i \(0.726786\pi\)
\(278\) 17.6995 1.06154
\(279\) −3.79427 −0.227157
\(280\) −4.34038 −0.259388
\(281\) −2.52287 −0.150502 −0.0752509 0.997165i \(-0.523976\pi\)
−0.0752509 + 0.997165i \(0.523976\pi\)
\(282\) 8.77274 0.522409
\(283\) 11.2483 0.668640 0.334320 0.942460i \(-0.391493\pi\)
0.334320 + 0.942460i \(0.391493\pi\)
\(284\) −0.662399 −0.0393061
\(285\) −24.6775 −1.46177
\(286\) 9.00316 0.532368
\(287\) −3.81749 −0.225339
\(288\) −1.06638 −0.0628368
\(289\) −15.9644 −0.939083
\(290\) 6.20659 0.364463
\(291\) −7.88619 −0.462297
\(292\) 7.12644 0.417043
\(293\) 16.5581 0.967332 0.483666 0.875253i \(-0.339305\pi\)
0.483666 + 0.875253i \(0.339305\pi\)
\(294\) −4.54469 −0.265052
\(295\) −17.4613 −1.01664
\(296\) 0.736872 0.0428298
\(297\) −20.2816 −1.17686
\(298\) −20.1669 −1.16824
\(299\) 9.29140 0.537335
\(300\) 2.07863 0.120010
\(301\) −14.4979 −0.835643
\(302\) −0.805568 −0.0463552
\(303\) −22.2006 −1.27539
\(304\) −6.14250 −0.352296
\(305\) −1.62709 −0.0931670
\(306\) 1.08518 0.0620357
\(307\) 2.89200 0.165055 0.0825276 0.996589i \(-0.473701\pi\)
0.0825276 + 0.996589i \(0.473701\pi\)
\(308\) 11.3319 0.645695
\(309\) 10.0014 0.568960
\(310\) −7.08876 −0.402614
\(311\) −30.2618 −1.71599 −0.857994 0.513660i \(-0.828289\pi\)
−0.857994 + 0.513660i \(0.828289\pi\)
\(312\) 3.49037 0.197604
\(313\) −13.0609 −0.738245 −0.369122 0.929381i \(-0.620342\pi\)
−0.369122 + 0.929381i \(0.620342\pi\)
\(314\) 15.4947 0.874417
\(315\) 4.62848 0.260785
\(316\) −4.61676 −0.259713
\(317\) −3.91087 −0.219656 −0.109828 0.993951i \(-0.535030\pi\)
−0.109828 + 0.993951i \(0.535030\pi\)
\(318\) −8.16809 −0.458044
\(319\) −16.2042 −0.907261
\(320\) −1.99229 −0.111372
\(321\) 1.74582 0.0974421
\(322\) 11.6947 0.651720
\(323\) 6.25083 0.347805
\(324\) −11.0620 −0.614554
\(325\) −1.78419 −0.0989691
\(326\) −19.4715 −1.07843
\(327\) −23.9980 −1.32709
\(328\) −1.75227 −0.0967531
\(329\) −9.47781 −0.522529
\(330\) 20.8969 1.15034
\(331\) −32.7139 −1.79812 −0.899058 0.437829i \(-0.855748\pi\)
−0.899058 + 0.437829i \(0.855748\pi\)
\(332\) −16.8661 −0.925647
\(333\) −0.785782 −0.0430606
\(334\) 13.2299 0.723908
\(335\) −19.6626 −1.07428
\(336\) 4.39319 0.239668
\(337\) −32.1026 −1.74874 −0.874370 0.485260i \(-0.838725\pi\)
−0.874370 + 0.485260i \(0.838725\pi\)
\(338\) 10.0040 0.544148
\(339\) 38.1450 2.07175
\(340\) 2.02742 0.109952
\(341\) 18.5074 1.00223
\(342\) 6.55021 0.354195
\(343\) 20.1601 1.08854
\(344\) −6.65469 −0.358797
\(345\) 21.5659 1.16107
\(346\) −2.90149 −0.155985
\(347\) 14.6604 0.787012 0.393506 0.919322i \(-0.371262\pi\)
0.393506 + 0.919322i \(0.371262\pi\)
\(348\) −6.28210 −0.336756
\(349\) −27.2375 −1.45799 −0.728995 0.684520i \(-0.760012\pi\)
−0.728995 + 0.684520i \(0.760012\pi\)
\(350\) −2.24569 −0.120037
\(351\) 6.74908 0.360239
\(352\) 5.20148 0.277240
\(353\) −0.965735 −0.0514009 −0.0257004 0.999670i \(-0.508182\pi\)
−0.0257004 + 0.999670i \(0.508182\pi\)
\(354\) 17.6738 0.939352
\(355\) 1.31969 0.0700418
\(356\) 18.0780 0.958131
\(357\) −4.47067 −0.236613
\(358\) 5.88100 0.310820
\(359\) 5.46016 0.288176 0.144088 0.989565i \(-0.453975\pi\)
0.144088 + 0.989565i \(0.453975\pi\)
\(360\) 2.12452 0.111972
\(361\) 18.7303 0.985804
\(362\) −13.7437 −0.722353
\(363\) −32.3760 −1.69930
\(364\) −3.77090 −0.197649
\(365\) −14.1979 −0.743152
\(366\) 1.64689 0.0860842
\(367\) −4.85033 −0.253185 −0.126592 0.991955i \(-0.540404\pi\)
−0.126592 + 0.991955i \(0.540404\pi\)
\(368\) 5.36800 0.279826
\(369\) 1.86858 0.0972744
\(370\) −1.46806 −0.0763208
\(371\) 8.82456 0.458148
\(372\) 7.17501 0.372007
\(373\) 32.0711 1.66058 0.830291 0.557331i \(-0.188174\pi\)
0.830291 + 0.557331i \(0.188174\pi\)
\(374\) −5.29321 −0.273705
\(375\) −24.2287 −1.25116
\(376\) −4.35042 −0.224356
\(377\) 5.39224 0.277715
\(378\) 8.49479 0.436925
\(379\) 16.4856 0.846811 0.423405 0.905940i \(-0.360835\pi\)
0.423405 + 0.905940i \(0.360835\pi\)
\(380\) 12.2376 0.627776
\(381\) 11.8909 0.609188
\(382\) −24.6595 −1.26169
\(383\) 4.43551 0.226644 0.113322 0.993558i \(-0.463851\pi\)
0.113322 + 0.993558i \(0.463851\pi\)
\(384\) 2.01653 0.102905
\(385\) −22.5764 −1.15060
\(386\) −7.24622 −0.368823
\(387\) 7.09639 0.360730
\(388\) 3.91078 0.198540
\(389\) −2.80059 −0.141995 −0.0709977 0.997476i \(-0.522618\pi\)
−0.0709977 + 0.997476i \(0.522618\pi\)
\(390\) −6.95382 −0.352121
\(391\) −5.46267 −0.276259
\(392\) 2.25372 0.113830
\(393\) −22.7807 −1.14913
\(394\) −20.9701 −1.05646
\(395\) 9.19791 0.462797
\(396\) −5.54672 −0.278733
\(397\) 13.5015 0.677621 0.338810 0.940855i \(-0.389976\pi\)
0.338810 + 0.940855i \(0.389976\pi\)
\(398\) 21.2249 1.06391
\(399\) −26.9852 −1.35095
\(400\) −1.03080 −0.0515398
\(401\) 16.1122 0.804606 0.402303 0.915507i \(-0.368210\pi\)
0.402303 + 0.915507i \(0.368210\pi\)
\(402\) 19.9018 0.992614
\(403\) −6.15867 −0.306785
\(404\) 11.0093 0.547735
\(405\) 22.0386 1.09511
\(406\) 6.78700 0.336833
\(407\) 3.83282 0.189986
\(408\) −2.05209 −0.101594
\(409\) −39.7145 −1.96375 −0.981877 0.189519i \(-0.939307\pi\)
−0.981877 + 0.189519i \(0.939307\pi\)
\(410\) 3.49103 0.172410
\(411\) −3.59209 −0.177185
\(412\) −4.95972 −0.244348
\(413\) −19.0942 −0.939567
\(414\) −5.72430 −0.281334
\(415\) 33.6021 1.64946
\(416\) −1.73089 −0.0848636
\(417\) 35.6915 1.74782
\(418\) −31.9501 −1.56273
\(419\) 34.9604 1.70793 0.853963 0.520333i \(-0.174192\pi\)
0.853963 + 0.520333i \(0.174192\pi\)
\(420\) −8.75249 −0.427078
\(421\) 6.17265 0.300836 0.150418 0.988622i \(-0.451938\pi\)
0.150418 + 0.988622i \(0.451938\pi\)
\(422\) 26.5375 1.29183
\(423\) 4.63918 0.225565
\(424\) 4.05057 0.196713
\(425\) 1.04898 0.0508828
\(426\) −1.33574 −0.0647170
\(427\) −1.77925 −0.0861039
\(428\) −0.865756 −0.0418479
\(429\) 18.1551 0.876536
\(430\) 13.2580 0.639360
\(431\) −14.0408 −0.676322 −0.338161 0.941088i \(-0.609805\pi\)
−0.338161 + 0.941088i \(0.609805\pi\)
\(432\) 3.89920 0.187601
\(433\) −27.5736 −1.32510 −0.662550 0.749017i \(-0.730526\pi\)
−0.662550 + 0.749017i \(0.730526\pi\)
\(434\) −7.75166 −0.372092
\(435\) 12.5157 0.600084
\(436\) 11.9007 0.569939
\(437\) −32.9729 −1.57731
\(438\) 14.3706 0.686656
\(439\) 37.7467 1.80155 0.900777 0.434282i \(-0.142998\pi\)
0.900777 + 0.434282i \(0.142998\pi\)
\(440\) −10.3628 −0.494028
\(441\) −2.40332 −0.114444
\(442\) 1.76141 0.0837818
\(443\) 19.8368 0.942476 0.471238 0.882006i \(-0.343807\pi\)
0.471238 + 0.882006i \(0.343807\pi\)
\(444\) 1.48592 0.0705187
\(445\) −36.0165 −1.70735
\(446\) −5.23517 −0.247892
\(447\) −40.6671 −1.92349
\(448\) −2.17859 −0.102929
\(449\) −16.3370 −0.770990 −0.385495 0.922710i \(-0.625969\pi\)
−0.385495 + 0.922710i \(0.625969\pi\)
\(450\) 1.09922 0.0518175
\(451\) −9.11440 −0.429180
\(452\) −18.9162 −0.889744
\(453\) −1.62445 −0.0763233
\(454\) 5.70185 0.267601
\(455\) 7.51271 0.352201
\(456\) −12.3865 −0.580051
\(457\) −36.0942 −1.68842 −0.844208 0.536015i \(-0.819929\pi\)
−0.844208 + 0.536015i \(0.819929\pi\)
\(458\) −10.2629 −0.479554
\(459\) −3.96797 −0.185209
\(460\) −10.6946 −0.498638
\(461\) −4.83779 −0.225318 −0.112659 0.993634i \(-0.535937\pi\)
−0.112659 + 0.993634i \(0.535937\pi\)
\(462\) 22.8511 1.06313
\(463\) −25.1462 −1.16864 −0.584322 0.811522i \(-0.698640\pi\)
−0.584322 + 0.811522i \(0.698640\pi\)
\(464\) 3.11531 0.144625
\(465\) −14.2947 −0.662899
\(466\) −22.2019 −1.02848
\(467\) −32.1074 −1.48576 −0.742878 0.669427i \(-0.766540\pi\)
−0.742878 + 0.669427i \(0.766540\pi\)
\(468\) 1.84577 0.0853209
\(469\) −21.5014 −0.992841
\(470\) 8.66729 0.399792
\(471\) 31.2455 1.43972
\(472\) −8.76448 −0.403418
\(473\) −34.6142 −1.59156
\(474\) −9.30982 −0.427614
\(475\) 6.33167 0.290517
\(476\) 2.21702 0.101617
\(477\) −4.31943 −0.197773
\(478\) 27.6097 1.26284
\(479\) 15.5845 0.712074 0.356037 0.934472i \(-0.384128\pi\)
0.356037 + 0.934472i \(0.384128\pi\)
\(480\) −4.01750 −0.183373
\(481\) −1.27544 −0.0581551
\(482\) −18.2226 −0.830016
\(483\) 23.5827 1.07305
\(484\) 16.0553 0.729789
\(485\) −7.79140 −0.353789
\(486\) −10.6091 −0.481240
\(487\) −9.57076 −0.433693 −0.216846 0.976206i \(-0.569577\pi\)
−0.216846 + 0.976206i \(0.569577\pi\)
\(488\) −0.816696 −0.0369701
\(489\) −39.2647 −1.77561
\(490\) −4.49006 −0.202840
\(491\) −20.4555 −0.923145 −0.461573 0.887102i \(-0.652715\pi\)
−0.461573 + 0.887102i \(0.652715\pi\)
\(492\) −3.53350 −0.159303
\(493\) −3.17025 −0.142781
\(494\) 10.6320 0.478355
\(495\) 11.0507 0.496690
\(496\) −3.55810 −0.159764
\(497\) 1.44310 0.0647318
\(498\) −34.0109 −1.52407
\(499\) −8.98511 −0.402229 −0.201114 0.979568i \(-0.564456\pi\)
−0.201114 + 0.979568i \(0.564456\pi\)
\(500\) 12.0151 0.537330
\(501\) 26.6784 1.19191
\(502\) 4.64098 0.207137
\(503\) 4.45453 0.198618 0.0993089 0.995057i \(-0.468337\pi\)
0.0993089 + 0.995057i \(0.468337\pi\)
\(504\) 2.32320 0.103484
\(505\) −21.9338 −0.976039
\(506\) 27.9215 1.24126
\(507\) 20.1734 0.895932
\(508\) −5.89672 −0.261625
\(509\) 19.0643 0.845009 0.422505 0.906361i \(-0.361151\pi\)
0.422505 + 0.906361i \(0.361151\pi\)
\(510\) 4.08835 0.181035
\(511\) −15.5256 −0.686813
\(512\) −1.00000 −0.0441942
\(513\) −23.9509 −1.05746
\(514\) 3.10603 0.137001
\(515\) 9.88118 0.435417
\(516\) −13.4193 −0.590754
\(517\) −22.6286 −0.995206
\(518\) −1.60535 −0.0705349
\(519\) −5.85094 −0.256828
\(520\) 3.44842 0.151223
\(521\) −14.7804 −0.647542 −0.323771 0.946136i \(-0.604951\pi\)
−0.323771 + 0.946136i \(0.604951\pi\)
\(522\) −3.32209 −0.145404
\(523\) −12.3855 −0.541581 −0.270791 0.962638i \(-0.587285\pi\)
−0.270791 + 0.962638i \(0.587285\pi\)
\(524\) 11.2970 0.493512
\(525\) −4.52849 −0.197639
\(526\) 4.91173 0.214162
\(527\) 3.62085 0.157727
\(528\) 10.4889 0.456471
\(529\) 5.81545 0.252846
\(530\) −8.06990 −0.350534
\(531\) 9.34622 0.405591
\(532\) 13.3820 0.580184
\(533\) 3.03298 0.131373
\(534\) 36.4547 1.57755
\(535\) 1.72483 0.0745711
\(536\) −9.86937 −0.426292
\(537\) 11.8592 0.511762
\(538\) −25.0619 −1.08049
\(539\) 11.7227 0.504932
\(540\) −7.76833 −0.334296
\(541\) 15.4287 0.663332 0.331666 0.943397i \(-0.392389\pi\)
0.331666 + 0.943397i \(0.392389\pi\)
\(542\) −15.7737 −0.677539
\(543\) −27.7145 −1.18934
\(544\) 1.01764 0.0436308
\(545\) −23.7095 −1.01561
\(546\) −7.60411 −0.325426
\(547\) −29.3366 −1.25434 −0.627171 0.778881i \(-0.715787\pi\)
−0.627171 + 0.778881i \(0.715787\pi\)
\(548\) 1.78132 0.0760944
\(549\) 0.870904 0.0371693
\(550\) −5.36167 −0.228622
\(551\) −19.1358 −0.815212
\(552\) 10.8247 0.460730
\(553\) 10.0581 0.427712
\(554\) 21.7596 0.924479
\(555\) −2.96038 −0.125661
\(556\) −17.6995 −0.750626
\(557\) 10.9988 0.466036 0.233018 0.972472i \(-0.425140\pi\)
0.233018 + 0.972472i \(0.425140\pi\)
\(558\) 3.79427 0.160624
\(559\) 11.5185 0.487181
\(560\) 4.34038 0.183415
\(561\) −10.6739 −0.450652
\(562\) 2.52287 0.106421
\(563\) −9.96127 −0.419817 −0.209909 0.977721i \(-0.567317\pi\)
−0.209909 + 0.977721i \(0.567317\pi\)
\(564\) −8.77274 −0.369399
\(565\) 37.6865 1.58548
\(566\) −11.2483 −0.472800
\(567\) 24.0995 1.01209
\(568\) 0.662399 0.0277936
\(569\) 23.7346 0.995008 0.497504 0.867462i \(-0.334250\pi\)
0.497504 + 0.867462i \(0.334250\pi\)
\(570\) 24.6775 1.03362
\(571\) −42.0051 −1.75786 −0.878929 0.476953i \(-0.841741\pi\)
−0.878929 + 0.476953i \(0.841741\pi\)
\(572\) −9.00316 −0.376441
\(573\) −49.7265 −2.07736
\(574\) 3.81749 0.159339
\(575\) −5.53332 −0.230755
\(576\) 1.06638 0.0444323
\(577\) −3.76406 −0.156700 −0.0783500 0.996926i \(-0.524965\pi\)
−0.0783500 + 0.996926i \(0.524965\pi\)
\(578\) 15.9644 0.664032
\(579\) −14.6122 −0.607261
\(580\) −6.20659 −0.257714
\(581\) 36.7444 1.52441
\(582\) 7.88619 0.326893
\(583\) 21.0690 0.872587
\(584\) −7.12644 −0.294894
\(585\) −3.67731 −0.152038
\(586\) −16.5581 −0.684007
\(587\) −47.3100 −1.95269 −0.976346 0.216215i \(-0.930629\pi\)
−0.976346 + 0.216215i \(0.930629\pi\)
\(588\) 4.54469 0.187420
\(589\) 21.8556 0.900546
\(590\) 17.4613 0.718872
\(591\) −42.2868 −1.73945
\(592\) −0.736872 −0.0302853
\(593\) 19.9962 0.821144 0.410572 0.911828i \(-0.365329\pi\)
0.410572 + 0.911828i \(0.365329\pi\)
\(594\) 20.2816 0.832165
\(595\) −4.41693 −0.181076
\(596\) 20.1669 0.826070
\(597\) 42.8005 1.75171
\(598\) −9.29140 −0.379953
\(599\) 30.5204 1.24703 0.623515 0.781812i \(-0.285704\pi\)
0.623515 + 0.781812i \(0.285704\pi\)
\(600\) −2.07863 −0.0848596
\(601\) 22.8440 0.931827 0.465914 0.884830i \(-0.345726\pi\)
0.465914 + 0.884830i \(0.345726\pi\)
\(602\) 14.4979 0.590889
\(603\) 10.5245 0.428589
\(604\) 0.805568 0.0327781
\(605\) −31.9868 −1.30045
\(606\) 22.2006 0.901838
\(607\) −21.2977 −0.864448 −0.432224 0.901766i \(-0.642271\pi\)
−0.432224 + 0.901766i \(0.642271\pi\)
\(608\) 6.14250 0.249111
\(609\) 13.6862 0.554591
\(610\) 1.62709 0.0658790
\(611\) 7.53008 0.304635
\(612\) −1.08518 −0.0438659
\(613\) −26.5985 −1.07430 −0.537152 0.843486i \(-0.680500\pi\)
−0.537152 + 0.843486i \(0.680500\pi\)
\(614\) −2.89200 −0.116712
\(615\) 7.03975 0.283870
\(616\) −11.3319 −0.456576
\(617\) 5.12203 0.206205 0.103103 0.994671i \(-0.467123\pi\)
0.103103 + 0.994671i \(0.467123\pi\)
\(618\) −10.0014 −0.402315
\(619\) −0.447418 −0.0179833 −0.00899163 0.999960i \(-0.502862\pi\)
−0.00899163 + 0.999960i \(0.502862\pi\)
\(620\) 7.08876 0.284691
\(621\) 20.9309 0.839930
\(622\) 30.2618 1.21339
\(623\) −39.3846 −1.57791
\(624\) −3.49037 −0.139727
\(625\) −18.7835 −0.751339
\(626\) 13.0609 0.522018
\(627\) −64.4281 −2.57301
\(628\) −15.4947 −0.618306
\(629\) 0.749868 0.0298992
\(630\) −4.62848 −0.184403
\(631\) 16.6516 0.662888 0.331444 0.943475i \(-0.392464\pi\)
0.331444 + 0.943475i \(0.392464\pi\)
\(632\) 4.61676 0.183645
\(633\) 53.5136 2.12697
\(634\) 3.91087 0.155321
\(635\) 11.7479 0.466203
\(636\) 8.16809 0.323886
\(637\) −3.90094 −0.154561
\(638\) 16.2042 0.641531
\(639\) −0.706366 −0.0279434
\(640\) 1.99229 0.0787520
\(641\) 16.3382 0.645322 0.322661 0.946515i \(-0.395423\pi\)
0.322661 + 0.946515i \(0.395423\pi\)
\(642\) −1.74582 −0.0689020
\(643\) −39.4221 −1.55465 −0.777327 0.629097i \(-0.783425\pi\)
−0.777327 + 0.629097i \(0.783425\pi\)
\(644\) −11.6947 −0.460836
\(645\) 26.7352 1.05270
\(646\) −6.25083 −0.245935
\(647\) −46.9144 −1.84439 −0.922197 0.386720i \(-0.873608\pi\)
−0.922197 + 0.386720i \(0.873608\pi\)
\(648\) 11.0620 0.434555
\(649\) −45.5882 −1.78949
\(650\) 1.78419 0.0699817
\(651\) −15.6314 −0.612644
\(652\) 19.4715 0.762562
\(653\) −38.8508 −1.52035 −0.760174 0.649720i \(-0.774886\pi\)
−0.760174 + 0.649720i \(0.774886\pi\)
\(654\) 23.9980 0.938397
\(655\) −22.5068 −0.879415
\(656\) 1.75227 0.0684147
\(657\) 7.59946 0.296483
\(658\) 9.47781 0.369484
\(659\) −2.72442 −0.106128 −0.0530642 0.998591i \(-0.516899\pi\)
−0.0530642 + 0.998591i \(0.516899\pi\)
\(660\) −20.8969 −0.813411
\(661\) 20.1897 0.785289 0.392644 0.919690i \(-0.371560\pi\)
0.392644 + 0.919690i \(0.371560\pi\)
\(662\) 32.7139 1.27146
\(663\) 3.55193 0.137946
\(664\) 16.8661 0.654531
\(665\) −26.6608 −1.03386
\(666\) 0.785782 0.0304484
\(667\) 16.7230 0.647517
\(668\) −13.2299 −0.511881
\(669\) −10.5569 −0.408151
\(670\) 19.6626 0.759633
\(671\) −4.24802 −0.163993
\(672\) −4.39319 −0.169471
\(673\) −18.6030 −0.717095 −0.358547 0.933512i \(-0.616728\pi\)
−0.358547 + 0.933512i \(0.616728\pi\)
\(674\) 32.1026 1.23655
\(675\) −4.01929 −0.154702
\(676\) −10.0040 −0.384771
\(677\) 23.9407 0.920117 0.460059 0.887889i \(-0.347828\pi\)
0.460059 + 0.887889i \(0.347828\pi\)
\(678\) −38.1450 −1.46495
\(679\) −8.52001 −0.326968
\(680\) −2.02742 −0.0777481
\(681\) 11.4979 0.440601
\(682\) −18.5074 −0.708684
\(683\) 22.3650 0.855773 0.427887 0.903832i \(-0.359258\pi\)
0.427887 + 0.903832i \(0.359258\pi\)
\(684\) −6.55021 −0.250453
\(685\) −3.54891 −0.135597
\(686\) −20.1601 −0.769717
\(687\) −20.6954 −0.789579
\(688\) 6.65469 0.253708
\(689\) −7.01108 −0.267101
\(690\) −21.5659 −0.821001
\(691\) 4.27775 0.162733 0.0813667 0.996684i \(-0.474072\pi\)
0.0813667 + 0.996684i \(0.474072\pi\)
\(692\) 2.90149 0.110298
\(693\) 12.0841 0.459036
\(694\) −14.6604 −0.556501
\(695\) 35.2624 1.33758
\(696\) 6.28210 0.238122
\(697\) −1.78317 −0.0675426
\(698\) 27.2375 1.03095
\(699\) −44.7707 −1.69338
\(700\) 2.24569 0.0848791
\(701\) 2.85161 0.107704 0.0538519 0.998549i \(-0.482850\pi\)
0.0538519 + 0.998549i \(0.482850\pi\)
\(702\) −6.74908 −0.254727
\(703\) 4.52624 0.170710
\(704\) −5.20148 −0.196038
\(705\) 17.4778 0.658252
\(706\) 0.965735 0.0363459
\(707\) −23.9849 −0.902045
\(708\) −17.6738 −0.664222
\(709\) −0.492083 −0.0184806 −0.00924028 0.999957i \(-0.502941\pi\)
−0.00924028 + 0.999957i \(0.502941\pi\)
\(710\) −1.31969 −0.0495270
\(711\) −4.92320 −0.184634
\(712\) −18.0780 −0.677501
\(713\) −19.0999 −0.715297
\(714\) 4.47067 0.167311
\(715\) 17.9369 0.670801
\(716\) −5.88100 −0.219783
\(717\) 55.6756 2.07924
\(718\) −5.46016 −0.203772
\(719\) 21.1879 0.790175 0.395088 0.918643i \(-0.370714\pi\)
0.395088 + 0.918643i \(0.370714\pi\)
\(720\) −2.12452 −0.0791763
\(721\) 10.8052 0.402407
\(722\) −18.7303 −0.697069
\(723\) −36.7463 −1.36661
\(724\) 13.7437 0.510781
\(725\) −3.21125 −0.119263
\(726\) 32.3760 1.20159
\(727\) 32.3848 1.20108 0.600542 0.799593i \(-0.294952\pi\)
0.600542 + 0.799593i \(0.294952\pi\)
\(728\) 3.77090 0.139759
\(729\) 11.7923 0.436753
\(730\) 14.1979 0.525488
\(731\) −6.77205 −0.250473
\(732\) −1.64689 −0.0608707
\(733\) 38.9051 1.43699 0.718496 0.695531i \(-0.244831\pi\)
0.718496 + 0.695531i \(0.244831\pi\)
\(734\) 4.85033 0.179029
\(735\) −9.05433 −0.333974
\(736\) −5.36800 −0.197867
\(737\) −51.3353 −1.89096
\(738\) −1.86858 −0.0687834
\(739\) −24.1775 −0.889383 −0.444692 0.895684i \(-0.646687\pi\)
−0.444692 + 0.895684i \(0.646687\pi\)
\(740\) 1.46806 0.0539670
\(741\) 21.4396 0.787604
\(742\) −8.82456 −0.323960
\(743\) 12.4962 0.458440 0.229220 0.973375i \(-0.426382\pi\)
0.229220 + 0.973375i \(0.426382\pi\)
\(744\) −7.17501 −0.263048
\(745\) −40.1783 −1.47202
\(746\) −32.0711 −1.17421
\(747\) −17.9856 −0.658058
\(748\) 5.29321 0.193539
\(749\) 1.88613 0.0689178
\(750\) 24.2287 0.884707
\(751\) 42.2824 1.54291 0.771453 0.636286i \(-0.219530\pi\)
0.771453 + 0.636286i \(0.219530\pi\)
\(752\) 4.35042 0.158644
\(753\) 9.35864 0.341048
\(754\) −5.39224 −0.196374
\(755\) −1.60492 −0.0584091
\(756\) −8.49479 −0.308952
\(757\) −12.3443 −0.448661 −0.224331 0.974513i \(-0.572020\pi\)
−0.224331 + 0.974513i \(0.572020\pi\)
\(758\) −16.4856 −0.598785
\(759\) 56.3045 2.04372
\(760\) −12.2376 −0.443905
\(761\) −14.1956 −0.514592 −0.257296 0.966333i \(-0.582832\pi\)
−0.257296 + 0.966333i \(0.582832\pi\)
\(762\) −11.8909 −0.430761
\(763\) −25.9268 −0.938611
\(764\) 24.6595 0.892150
\(765\) 2.16199 0.0781670
\(766\) −4.43551 −0.160261
\(767\) 15.1703 0.547768
\(768\) −2.01653 −0.0727651
\(769\) −18.8012 −0.677988 −0.338994 0.940789i \(-0.610087\pi\)
−0.338994 + 0.940789i \(0.610087\pi\)
\(770\) 22.5764 0.813597
\(771\) 6.26339 0.225571
\(772\) 7.24622 0.260797
\(773\) −43.9987 −1.58252 −0.791261 0.611478i \(-0.790575\pi\)
−0.791261 + 0.611478i \(0.790575\pi\)
\(774\) −7.09639 −0.255075
\(775\) 3.66768 0.131747
\(776\) −3.91078 −0.140389
\(777\) −3.23722 −0.116135
\(778\) 2.80059 0.100406
\(779\) −10.7633 −0.385636
\(780\) 6.95382 0.248987
\(781\) 3.44545 0.123288
\(782\) 5.46267 0.195345
\(783\) 12.1472 0.434106
\(784\) −2.25372 −0.0804902
\(785\) 30.8699 1.10179
\(786\) 22.7807 0.812560
\(787\) 33.1121 1.18032 0.590160 0.807287i \(-0.299065\pi\)
0.590160 + 0.807287i \(0.299065\pi\)
\(788\) 20.9701 0.747030
\(789\) 9.90463 0.352614
\(790\) −9.19791 −0.327247
\(791\) 41.2108 1.46529
\(792\) 5.54672 0.197094
\(793\) 1.41361 0.0501986
\(794\) −13.5015 −0.479150
\(795\) −16.2732 −0.577150
\(796\) −21.2249 −0.752297
\(797\) −16.1632 −0.572531 −0.286265 0.958150i \(-0.592414\pi\)
−0.286265 + 0.958150i \(0.592414\pi\)
\(798\) 26.9852 0.955265
\(799\) −4.42715 −0.156621
\(800\) 1.03080 0.0364442
\(801\) 19.2779 0.681151
\(802\) −16.1122 −0.568942
\(803\) −37.0680 −1.30810
\(804\) −19.9018 −0.701884
\(805\) 23.2992 0.821189
\(806\) 6.15867 0.216930
\(807\) −50.5379 −1.77902
\(808\) −11.0093 −0.387307
\(809\) 14.3386 0.504117 0.252058 0.967712i \(-0.418893\pi\)
0.252058 + 0.967712i \(0.418893\pi\)
\(810\) −22.0386 −0.774358
\(811\) 29.4702 1.03484 0.517419 0.855732i \(-0.326893\pi\)
0.517419 + 0.855732i \(0.326893\pi\)
\(812\) −6.78700 −0.238177
\(813\) −31.8081 −1.11556
\(814\) −3.83282 −0.134340
\(815\) −38.7927 −1.35885
\(816\) 2.05209 0.0718375
\(817\) −40.8764 −1.43008
\(818\) 39.7145 1.38858
\(819\) −4.02119 −0.140512
\(820\) −3.49103 −0.121912
\(821\) −2.50677 −0.0874870 −0.0437435 0.999043i \(-0.513928\pi\)
−0.0437435 + 0.999043i \(0.513928\pi\)
\(822\) 3.59209 0.125288
\(823\) −35.1430 −1.22501 −0.612504 0.790467i \(-0.709838\pi\)
−0.612504 + 0.790467i \(0.709838\pi\)
\(824\) 4.95972 0.172780
\(825\) −10.8119 −0.376423
\(826\) 19.0942 0.664374
\(827\) −33.7990 −1.17531 −0.587653 0.809113i \(-0.699948\pi\)
−0.587653 + 0.809113i \(0.699948\pi\)
\(828\) 5.72430 0.198933
\(829\) 25.1882 0.874823 0.437411 0.899262i \(-0.355895\pi\)
0.437411 + 0.899262i \(0.355895\pi\)
\(830\) −33.6021 −1.16635
\(831\) 43.8789 1.52214
\(832\) 1.73089 0.0600077
\(833\) 2.29347 0.0794641
\(834\) −35.6915 −1.23589
\(835\) 26.3578 0.912148
\(836\) 31.9501 1.10502
\(837\) −13.8738 −0.479548
\(838\) −34.9604 −1.20769
\(839\) 32.8691 1.13477 0.567384 0.823453i \(-0.307956\pi\)
0.567384 + 0.823453i \(0.307956\pi\)
\(840\) 8.75249 0.301990
\(841\) −19.2948 −0.665340
\(842\) −6.17265 −0.212723
\(843\) 5.08743 0.175220
\(844\) −26.5375 −0.913459
\(845\) 19.9309 0.685644
\(846\) −4.63918 −0.159498
\(847\) −34.9781 −1.20186
\(848\) −4.05057 −0.139097
\(849\) −22.6824 −0.778459
\(850\) −1.04898 −0.0359796
\(851\) −3.95553 −0.135594
\(852\) 1.33574 0.0457618
\(853\) −2.04553 −0.0700376 −0.0350188 0.999387i \(-0.511149\pi\)
−0.0350188 + 0.999387i \(0.511149\pi\)
\(854\) 1.77925 0.0608847
\(855\) 13.0499 0.446297
\(856\) 0.865756 0.0295909
\(857\) −12.9478 −0.442289 −0.221145 0.975241i \(-0.570979\pi\)
−0.221145 + 0.975241i \(0.570979\pi\)
\(858\) −18.1551 −0.619805
\(859\) −15.8517 −0.540853 −0.270426 0.962741i \(-0.587165\pi\)
−0.270426 + 0.962741i \(0.587165\pi\)
\(860\) −13.2580 −0.452095
\(861\) 7.69807 0.262349
\(862\) 14.0408 0.478232
\(863\) −1.75669 −0.0597983 −0.0298992 0.999553i \(-0.509519\pi\)
−0.0298992 + 0.999553i \(0.509519\pi\)
\(864\) −3.89920 −0.132654
\(865\) −5.78060 −0.196546
\(866\) 27.5736 0.936988
\(867\) 32.1927 1.09332
\(868\) 7.75166 0.263109
\(869\) 24.0140 0.814618
\(870\) −12.5157 −0.424323
\(871\) 17.0828 0.578827
\(872\) −11.9007 −0.403008
\(873\) 4.17036 0.141145
\(874\) 32.9729 1.11533
\(875\) −26.1760 −0.884909
\(876\) −14.3706 −0.485539
\(877\) −18.9996 −0.641570 −0.320785 0.947152i \(-0.603947\pi\)
−0.320785 + 0.947152i \(0.603947\pi\)
\(878\) −37.7467 −1.27389
\(879\) −33.3897 −1.12621
\(880\) 10.3628 0.349331
\(881\) 25.1300 0.846652 0.423326 0.905978i \(-0.360863\pi\)
0.423326 + 0.905978i \(0.360863\pi\)
\(882\) 2.40332 0.0809239
\(883\) 13.5948 0.457500 0.228750 0.973485i \(-0.426536\pi\)
0.228750 + 0.973485i \(0.426536\pi\)
\(884\) −1.76141 −0.0592427
\(885\) 35.2113 1.18361
\(886\) −19.8368 −0.666431
\(887\) −52.5737 −1.76525 −0.882627 0.470075i \(-0.844227\pi\)
−0.882627 + 0.470075i \(0.844227\pi\)
\(888\) −1.48592 −0.0498643
\(889\) 12.8466 0.430860
\(890\) 36.0165 1.20728
\(891\) 57.5386 1.92761
\(892\) 5.23517 0.175286
\(893\) −26.7225 −0.894233
\(894\) 40.6671 1.36011
\(895\) 11.7166 0.391644
\(896\) 2.17859 0.0727817
\(897\) −18.7363 −0.625588
\(898\) 16.3370 0.545173
\(899\) −11.0846 −0.369692
\(900\) −1.09922 −0.0366405
\(901\) 4.12201 0.137324
\(902\) 9.11440 0.303476
\(903\) 29.2353 0.972891
\(904\) 18.9162 0.629144
\(905\) −27.3814 −0.910188
\(906\) 1.62445 0.0539687
\(907\) 3.92333 0.130272 0.0651360 0.997876i \(-0.479252\pi\)
0.0651360 + 0.997876i \(0.479252\pi\)
\(908\) −5.70185 −0.189222
\(909\) 11.7401 0.389394
\(910\) −7.51271 −0.249044
\(911\) −21.2137 −0.702841 −0.351420 0.936218i \(-0.614301\pi\)
−0.351420 + 0.936218i \(0.614301\pi\)
\(912\) 12.3865 0.410158
\(913\) 87.7286 2.90339
\(914\) 36.0942 1.19389
\(915\) 3.28107 0.108469
\(916\) 10.2629 0.339096
\(917\) −24.6116 −0.812746
\(918\) 3.96797 0.130963
\(919\) 38.6523 1.27502 0.637510 0.770442i \(-0.279964\pi\)
0.637510 + 0.770442i \(0.279964\pi\)
\(920\) 10.6946 0.352590
\(921\) −5.83179 −0.192164
\(922\) 4.83779 0.159324
\(923\) −1.14654 −0.0377387
\(924\) −22.8511 −0.751745
\(925\) 0.759566 0.0249744
\(926\) 25.1462 0.826356
\(927\) −5.28892 −0.173711
\(928\) −3.11531 −0.102265
\(929\) 26.7077 0.876251 0.438125 0.898914i \(-0.355643\pi\)
0.438125 + 0.898914i \(0.355643\pi\)
\(930\) 14.2947 0.468740
\(931\) 13.8435 0.453702
\(932\) 22.2019 0.727247
\(933\) 61.0236 1.99782
\(934\) 32.1074 1.05059
\(935\) −10.5456 −0.344877
\(936\) −1.84577 −0.0603310
\(937\) 51.5277 1.68334 0.841668 0.539995i \(-0.181574\pi\)
0.841668 + 0.539995i \(0.181574\pi\)
\(938\) 21.5014 0.702044
\(939\) 26.3376 0.859495
\(940\) −8.66729 −0.282696
\(941\) −10.3636 −0.337843 −0.168921 0.985630i \(-0.554028\pi\)
−0.168921 + 0.985630i \(0.554028\pi\)
\(942\) −31.2455 −1.01803
\(943\) 9.40620 0.306308
\(944\) 8.76448 0.285260
\(945\) 16.9240 0.550539
\(946\) 34.6142 1.12540
\(947\) 23.0491 0.748996 0.374498 0.927228i \(-0.377815\pi\)
0.374498 + 0.927228i \(0.377815\pi\)
\(948\) 9.30982 0.302369
\(949\) 12.3351 0.400413
\(950\) −6.33167 −0.205426
\(951\) 7.88637 0.255733
\(952\) −2.21702 −0.0718539
\(953\) 12.4672 0.403851 0.201925 0.979401i \(-0.435280\pi\)
0.201925 + 0.979401i \(0.435280\pi\)
\(954\) 4.31943 0.139847
\(955\) −49.1288 −1.58977
\(956\) −27.6097 −0.892961
\(957\) 32.6762 1.05627
\(958\) −15.5845 −0.503512
\(959\) −3.88078 −0.125317
\(960\) 4.01750 0.129664
\(961\) −18.3399 −0.591610
\(962\) 1.27544 0.0411219
\(963\) −0.923221 −0.0297504
\(964\) 18.2226 0.586910
\(965\) −14.4365 −0.464728
\(966\) −23.5827 −0.758760
\(967\) 43.1277 1.38689 0.693447 0.720508i \(-0.256091\pi\)
0.693447 + 0.720508i \(0.256091\pi\)
\(968\) −16.0553 −0.516038
\(969\) −12.6050 −0.404929
\(970\) 7.79140 0.250167
\(971\) 45.2401 1.45183 0.725913 0.687787i \(-0.241418\pi\)
0.725913 + 0.687787i \(0.241418\pi\)
\(972\) 10.6091 0.340288
\(973\) 38.5600 1.23618
\(974\) 9.57076 0.306667
\(975\) 3.59787 0.115224
\(976\) 0.816696 0.0261418
\(977\) −10.0854 −0.322659 −0.161329 0.986901i \(-0.551578\pi\)
−0.161329 + 0.986901i \(0.551578\pi\)
\(978\) 39.2647 1.25555
\(979\) −94.0322 −3.00528
\(980\) 4.49006 0.143430
\(981\) 12.6906 0.405179
\(982\) 20.4555 0.652762
\(983\) −12.7054 −0.405238 −0.202619 0.979258i \(-0.564945\pi\)
−0.202619 + 0.979258i \(0.564945\pi\)
\(984\) 3.53350 0.112644
\(985\) −41.7785 −1.33117
\(986\) 3.17025 0.100961
\(987\) 19.1122 0.608349
\(988\) −10.6320 −0.338248
\(989\) 35.7224 1.13591
\(990\) −11.0507 −0.351213
\(991\) −12.9191 −0.410388 −0.205194 0.978721i \(-0.565783\pi\)
−0.205194 + 0.978721i \(0.565783\pi\)
\(992\) 3.55810 0.112970
\(993\) 65.9684 2.09344
\(994\) −1.44310 −0.0457723
\(995\) 42.2861 1.34056
\(996\) 34.0109 1.07768
\(997\) 60.4523 1.91454 0.957271 0.289192i \(-0.0933865\pi\)
0.957271 + 0.289192i \(0.0933865\pi\)
\(998\) 8.98511 0.284419
\(999\) −2.87322 −0.0909045
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 6002.2.a.b.1.15 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6002.2.a.b.1.15 56 1.1 even 1 trivial