Properties

Label 4009.2.a.e.1.10
Level $4009$
Weight $2$
Character 4009.1
Self dual yes
Analytic conductor $32.012$
Analytic rank $0$
Dimension $82$
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] = [4009,2,Mod(1,4009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4009, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4009 = 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4009.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.0120261703\)
Analytic rank: \(0\)
Dimension: \(82\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Character \(\chi\) \(=\) 4009.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.21501 q^{2} +1.93892 q^{3} +2.90626 q^{4} +3.78369 q^{5} -4.29473 q^{6} +0.133386 q^{7} -2.00737 q^{8} +0.759427 q^{9} +O(q^{10})\) \(q-2.21501 q^{2} +1.93892 q^{3} +2.90626 q^{4} +3.78369 q^{5} -4.29473 q^{6} +0.133386 q^{7} -2.00737 q^{8} +0.759427 q^{9} -8.38090 q^{10} -1.09409 q^{11} +5.63502 q^{12} +3.29925 q^{13} -0.295450 q^{14} +7.33629 q^{15} -1.36617 q^{16} +1.86199 q^{17} -1.68214 q^{18} +1.00000 q^{19} +10.9964 q^{20} +0.258625 q^{21} +2.42341 q^{22} +7.74684 q^{23} -3.89214 q^{24} +9.31631 q^{25} -7.30786 q^{26} -4.34430 q^{27} +0.387653 q^{28} -3.39199 q^{29} -16.2499 q^{30} +2.03974 q^{31} +7.04083 q^{32} -2.12135 q^{33} -4.12432 q^{34} +0.504690 q^{35} +2.20709 q^{36} +8.03020 q^{37} -2.21501 q^{38} +6.39699 q^{39} -7.59528 q^{40} +5.20858 q^{41} -0.572856 q^{42} -3.74804 q^{43} -3.17970 q^{44} +2.87343 q^{45} -17.1593 q^{46} -4.19531 q^{47} -2.64891 q^{48} -6.98221 q^{49} -20.6357 q^{50} +3.61025 q^{51} +9.58847 q^{52} -9.76739 q^{53} +9.62266 q^{54} -4.13969 q^{55} -0.267755 q^{56} +1.93892 q^{57} +7.51329 q^{58} -0.931634 q^{59} +21.3212 q^{60} +7.87186 q^{61} -4.51803 q^{62} +0.101297 q^{63} -12.8631 q^{64} +12.4833 q^{65} +4.69881 q^{66} +3.01933 q^{67} +5.41142 q^{68} +15.0205 q^{69} -1.11789 q^{70} -1.26895 q^{71} -1.52445 q^{72} -0.126954 q^{73} -17.7869 q^{74} +18.0636 q^{75} +2.90626 q^{76} -0.145936 q^{77} -14.1694 q^{78} -0.541512 q^{79} -5.16918 q^{80} -10.7016 q^{81} -11.5370 q^{82} +2.31165 q^{83} +0.751631 q^{84} +7.04518 q^{85} +8.30194 q^{86} -6.57682 q^{87} +2.19624 q^{88} -3.44189 q^{89} -6.36468 q^{90} +0.440072 q^{91} +22.5143 q^{92} +3.95489 q^{93} +9.29264 q^{94} +3.78369 q^{95} +13.6516 q^{96} -17.7453 q^{97} +15.4656 q^{98} -0.830879 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.21501 −1.56625 −0.783124 0.621866i \(-0.786375\pi\)
−0.783124 + 0.621866i \(0.786375\pi\)
\(3\) 1.93892 1.11944 0.559719 0.828682i \(-0.310909\pi\)
0.559719 + 0.828682i \(0.310909\pi\)
\(4\) 2.90626 1.45313
\(5\) 3.78369 1.69212 0.846059 0.533090i \(-0.178969\pi\)
0.846059 + 0.533090i \(0.178969\pi\)
\(6\) −4.29473 −1.75332
\(7\) 0.133386 0.0504150 0.0252075 0.999682i \(-0.491975\pi\)
0.0252075 + 0.999682i \(0.491975\pi\)
\(8\) −2.00737 −0.709713
\(9\) 0.759427 0.253142
\(10\) −8.38090 −2.65027
\(11\) −1.09409 −0.329880 −0.164940 0.986304i \(-0.552743\pi\)
−0.164940 + 0.986304i \(0.552743\pi\)
\(12\) 5.63502 1.62669
\(13\) 3.29925 0.915046 0.457523 0.889198i \(-0.348737\pi\)
0.457523 + 0.889198i \(0.348737\pi\)
\(14\) −0.295450 −0.0789624
\(15\) 7.33629 1.89422
\(16\) −1.36617 −0.341543
\(17\) 1.86199 0.451598 0.225799 0.974174i \(-0.427501\pi\)
0.225799 + 0.974174i \(0.427501\pi\)
\(18\) −1.68214 −0.396483
\(19\) 1.00000 0.229416
\(20\) 10.9964 2.45887
\(21\) 0.258625 0.0564365
\(22\) 2.42341 0.516673
\(23\) 7.74684 1.61533 0.807664 0.589642i \(-0.200731\pi\)
0.807664 + 0.589642i \(0.200731\pi\)
\(24\) −3.89214 −0.794480
\(25\) 9.31631 1.86326
\(26\) −7.30786 −1.43319
\(27\) −4.34430 −0.836061
\(28\) 0.387653 0.0732596
\(29\) −3.39199 −0.629877 −0.314939 0.949112i \(-0.601984\pi\)
−0.314939 + 0.949112i \(0.601984\pi\)
\(30\) −16.2499 −2.96682
\(31\) 2.03974 0.366347 0.183174 0.983081i \(-0.441363\pi\)
0.183174 + 0.983081i \(0.441363\pi\)
\(32\) 7.04083 1.24465
\(33\) −2.12135 −0.369280
\(34\) −4.12432 −0.707314
\(35\) 0.504690 0.0853082
\(36\) 2.20709 0.367848
\(37\) 8.03020 1.32016 0.660078 0.751197i \(-0.270523\pi\)
0.660078 + 0.751197i \(0.270523\pi\)
\(38\) −2.21501 −0.359322
\(39\) 6.39699 1.02434
\(40\) −7.59528 −1.20092
\(41\) 5.20858 0.813444 0.406722 0.913552i \(-0.366672\pi\)
0.406722 + 0.913552i \(0.366672\pi\)
\(42\) −0.572856 −0.0883936
\(43\) −3.74804 −0.571571 −0.285785 0.958294i \(-0.592254\pi\)
−0.285785 + 0.958294i \(0.592254\pi\)
\(44\) −3.17970 −0.479358
\(45\) 2.87343 0.428346
\(46\) −17.1593 −2.53000
\(47\) −4.19531 −0.611948 −0.305974 0.952040i \(-0.598982\pi\)
−0.305974 + 0.952040i \(0.598982\pi\)
\(48\) −2.64891 −0.382337
\(49\) −6.98221 −0.997458
\(50\) −20.6357 −2.91833
\(51\) 3.61025 0.505536
\(52\) 9.58847 1.32968
\(53\) −9.76739 −1.34165 −0.670827 0.741614i \(-0.734061\pi\)
−0.670827 + 0.741614i \(0.734061\pi\)
\(54\) 9.62266 1.30948
\(55\) −4.13969 −0.558195
\(56\) −0.267755 −0.0357802
\(57\) 1.93892 0.256817
\(58\) 7.51329 0.986543
\(59\) −0.931634 −0.121288 −0.0606442 0.998159i \(-0.519315\pi\)
−0.0606442 + 0.998159i \(0.519315\pi\)
\(60\) 21.3212 2.75255
\(61\) 7.87186 1.00789 0.503944 0.863736i \(-0.331882\pi\)
0.503944 + 0.863736i \(0.331882\pi\)
\(62\) −4.51803 −0.573790
\(63\) 0.101297 0.0127622
\(64\) −12.8631 −1.60789
\(65\) 12.4833 1.54837
\(66\) 4.69881 0.578384
\(67\) 3.01933 0.368869 0.184435 0.982845i \(-0.440955\pi\)
0.184435 + 0.982845i \(0.440955\pi\)
\(68\) 5.41142 0.656231
\(69\) 15.0205 1.80826
\(70\) −1.11789 −0.133614
\(71\) −1.26895 −0.150597 −0.0752986 0.997161i \(-0.523991\pi\)
−0.0752986 + 0.997161i \(0.523991\pi\)
\(72\) −1.52445 −0.179658
\(73\) −0.126954 −0.0148588 −0.00742942 0.999972i \(-0.502365\pi\)
−0.00742942 + 0.999972i \(0.502365\pi\)
\(74\) −17.7869 −2.06769
\(75\) 18.0636 2.08581
\(76\) 2.90626 0.333371
\(77\) −0.145936 −0.0166309
\(78\) −14.1694 −1.60437
\(79\) −0.541512 −0.0609248 −0.0304624 0.999536i \(-0.509698\pi\)
−0.0304624 + 0.999536i \(0.509698\pi\)
\(80\) −5.16918 −0.577932
\(81\) −10.7016 −1.18906
\(82\) −11.5370 −1.27405
\(83\) 2.31165 0.253736 0.126868 0.991920i \(-0.459508\pi\)
0.126868 + 0.991920i \(0.459508\pi\)
\(84\) 0.751631 0.0820096
\(85\) 7.04518 0.764157
\(86\) 8.30194 0.895221
\(87\) −6.57682 −0.705109
\(88\) 2.19624 0.234120
\(89\) −3.44189 −0.364839 −0.182420 0.983221i \(-0.558393\pi\)
−0.182420 + 0.983221i \(0.558393\pi\)
\(90\) −6.36468 −0.670896
\(91\) 0.440072 0.0461321
\(92\) 22.5143 2.34728
\(93\) 3.95489 0.410103
\(94\) 9.29264 0.958462
\(95\) 3.78369 0.388198
\(96\) 13.6516 1.39331
\(97\) −17.7453 −1.80176 −0.900879 0.434070i \(-0.857077\pi\)
−0.900879 + 0.434070i \(0.857077\pi\)
\(98\) 15.4656 1.56227
\(99\) −0.830879 −0.0835065
\(100\) 27.0756 2.70756
\(101\) −7.49718 −0.745997 −0.372999 0.927832i \(-0.621670\pi\)
−0.372999 + 0.927832i \(0.621670\pi\)
\(102\) −7.99674 −0.791795
\(103\) 16.5721 1.63289 0.816447 0.577420i \(-0.195940\pi\)
0.816447 + 0.577420i \(0.195940\pi\)
\(104\) −6.62282 −0.649420
\(105\) 0.978556 0.0954972
\(106\) 21.6348 2.10136
\(107\) 19.0770 1.84424 0.922121 0.386901i \(-0.126454\pi\)
0.922121 + 0.386901i \(0.126454\pi\)
\(108\) −12.6257 −1.21491
\(109\) 3.55901 0.340892 0.170446 0.985367i \(-0.445479\pi\)
0.170446 + 0.985367i \(0.445479\pi\)
\(110\) 9.16944 0.874272
\(111\) 15.5699 1.47783
\(112\) −0.182228 −0.0172189
\(113\) 12.9011 1.21363 0.606816 0.794842i \(-0.292447\pi\)
0.606816 + 0.794842i \(0.292447\pi\)
\(114\) −4.29473 −0.402239
\(115\) 29.3117 2.73333
\(116\) −9.85801 −0.915293
\(117\) 2.50553 0.231637
\(118\) 2.06358 0.189968
\(119\) 0.248362 0.0227673
\(120\) −14.7267 −1.34435
\(121\) −9.80297 −0.891179
\(122\) −17.4362 −1.57860
\(123\) 10.0990 0.910600
\(124\) 5.92800 0.532350
\(125\) 16.3316 1.46074
\(126\) −0.224373 −0.0199887
\(127\) 12.6975 1.12672 0.563362 0.826210i \(-0.309508\pi\)
0.563362 + 0.826210i \(0.309508\pi\)
\(128\) 14.4103 1.27370
\(129\) −7.26717 −0.639838
\(130\) −27.6507 −2.42512
\(131\) 12.7292 1.11216 0.556080 0.831129i \(-0.312305\pi\)
0.556080 + 0.831129i \(0.312305\pi\)
\(132\) −6.16520 −0.536612
\(133\) 0.133386 0.0115660
\(134\) −6.68783 −0.577741
\(135\) −16.4375 −1.41471
\(136\) −3.73770 −0.320505
\(137\) 1.95373 0.166919 0.0834594 0.996511i \(-0.473403\pi\)
0.0834594 + 0.996511i \(0.473403\pi\)
\(138\) −33.2706 −2.83218
\(139\) −5.31613 −0.450908 −0.225454 0.974254i \(-0.572387\pi\)
−0.225454 + 0.974254i \(0.572387\pi\)
\(140\) 1.46676 0.123964
\(141\) −8.13438 −0.685039
\(142\) 2.81074 0.235872
\(143\) −3.60966 −0.301855
\(144\) −1.03751 −0.0864590
\(145\) −12.8342 −1.06583
\(146\) 0.281204 0.0232726
\(147\) −13.5380 −1.11659
\(148\) 23.3378 1.91836
\(149\) −11.3748 −0.931863 −0.465932 0.884821i \(-0.654281\pi\)
−0.465932 + 0.884821i \(0.654281\pi\)
\(150\) −40.0110 −3.26689
\(151\) −8.26595 −0.672673 −0.336337 0.941742i \(-0.609188\pi\)
−0.336337 + 0.941742i \(0.609188\pi\)
\(152\) −2.00737 −0.162819
\(153\) 1.41404 0.114319
\(154\) 0.323249 0.0260481
\(155\) 7.71773 0.619903
\(156\) 18.5913 1.48850
\(157\) 8.44089 0.673656 0.336828 0.941566i \(-0.390646\pi\)
0.336828 + 0.941566i \(0.390646\pi\)
\(158\) 1.19945 0.0954234
\(159\) −18.9382 −1.50190
\(160\) 26.6403 2.10610
\(161\) 1.03332 0.0814369
\(162\) 23.7040 1.86236
\(163\) 4.73495 0.370870 0.185435 0.982657i \(-0.440631\pi\)
0.185435 + 0.982657i \(0.440631\pi\)
\(164\) 15.1375 1.18204
\(165\) −8.02654 −0.624865
\(166\) −5.12032 −0.397414
\(167\) −14.0935 −1.09059 −0.545294 0.838245i \(-0.683582\pi\)
−0.545294 + 0.838245i \(0.683582\pi\)
\(168\) −0.519156 −0.0400538
\(169\) −2.11498 −0.162691
\(170\) −15.6051 −1.19686
\(171\) 0.759427 0.0580748
\(172\) −10.8928 −0.830567
\(173\) 20.9623 1.59374 0.796868 0.604153i \(-0.206489\pi\)
0.796868 + 0.604153i \(0.206489\pi\)
\(174\) 14.5677 1.10437
\(175\) 1.24266 0.0939364
\(176\) 1.49471 0.112668
\(177\) −1.80637 −0.135775
\(178\) 7.62380 0.571428
\(179\) −7.19946 −0.538113 −0.269056 0.963124i \(-0.586712\pi\)
−0.269056 + 0.963124i \(0.586712\pi\)
\(180\) 8.35095 0.622443
\(181\) −18.1059 −1.34580 −0.672901 0.739733i \(-0.734952\pi\)
−0.672901 + 0.739733i \(0.734952\pi\)
\(182\) −0.974763 −0.0722543
\(183\) 15.2629 1.12827
\(184\) −15.5508 −1.14642
\(185\) 30.3838 2.23386
\(186\) −8.76012 −0.642323
\(187\) −2.03718 −0.148973
\(188\) −12.1927 −0.889241
\(189\) −0.579468 −0.0421501
\(190\) −8.38090 −0.608015
\(191\) −9.94658 −0.719709 −0.359855 0.933008i \(-0.617174\pi\)
−0.359855 + 0.933008i \(0.617174\pi\)
\(192\) −24.9407 −1.79994
\(193\) 7.40629 0.533116 0.266558 0.963819i \(-0.414114\pi\)
0.266558 + 0.963819i \(0.414114\pi\)
\(194\) 39.3059 2.82200
\(195\) 24.2042 1.73330
\(196\) −20.2921 −1.44944
\(197\) −9.21642 −0.656642 −0.328321 0.944566i \(-0.606483\pi\)
−0.328321 + 0.944566i \(0.606483\pi\)
\(198\) 1.84040 0.130792
\(199\) 3.37710 0.239396 0.119698 0.992810i \(-0.461807\pi\)
0.119698 + 0.992810i \(0.461807\pi\)
\(200\) −18.7013 −1.32238
\(201\) 5.85424 0.412926
\(202\) 16.6063 1.16842
\(203\) −0.452443 −0.0317553
\(204\) 10.4923 0.734610
\(205\) 19.7077 1.37644
\(206\) −36.7073 −2.55752
\(207\) 5.88316 0.408908
\(208\) −4.50734 −0.312528
\(209\) −1.09409 −0.0756796
\(210\) −2.16751 −0.149572
\(211\) −1.00000 −0.0688428
\(212\) −28.3866 −1.94960
\(213\) −2.46041 −0.168584
\(214\) −42.2557 −2.88854
\(215\) −14.1814 −0.967165
\(216\) 8.72063 0.593364
\(217\) 0.272071 0.0184694
\(218\) −7.88324 −0.533921
\(219\) −0.246154 −0.0166336
\(220\) −12.0310 −0.811130
\(221\) 6.14315 0.413233
\(222\) −34.4875 −2.31465
\(223\) −7.59468 −0.508577 −0.254289 0.967128i \(-0.581841\pi\)
−0.254289 + 0.967128i \(0.581841\pi\)
\(224\) 0.939146 0.0627493
\(225\) 7.07505 0.471670
\(226\) −28.5760 −1.90085
\(227\) 26.1549 1.73596 0.867982 0.496596i \(-0.165417\pi\)
0.867982 + 0.496596i \(0.165417\pi\)
\(228\) 5.63502 0.373188
\(229\) −25.7038 −1.69855 −0.849277 0.527948i \(-0.822961\pi\)
−0.849277 + 0.527948i \(0.822961\pi\)
\(230\) −64.9256 −4.28106
\(231\) −0.282958 −0.0186173
\(232\) 6.80899 0.447032
\(233\) 10.0026 0.655293 0.327647 0.944800i \(-0.393744\pi\)
0.327647 + 0.944800i \(0.393744\pi\)
\(234\) −5.54978 −0.362800
\(235\) −15.8737 −1.03549
\(236\) −2.70757 −0.176248
\(237\) −1.04995 −0.0682016
\(238\) −0.550125 −0.0356593
\(239\) −15.1150 −0.977711 −0.488855 0.872365i \(-0.662585\pi\)
−0.488855 + 0.872365i \(0.662585\pi\)
\(240\) −10.0226 −0.646959
\(241\) 18.9626 1.22149 0.610745 0.791827i \(-0.290870\pi\)
0.610745 + 0.791827i \(0.290870\pi\)
\(242\) 21.7137 1.39581
\(243\) −7.71659 −0.495019
\(244\) 22.8777 1.46459
\(245\) −26.4185 −1.68782
\(246\) −22.3695 −1.42622
\(247\) 3.29925 0.209926
\(248\) −4.09451 −0.260002
\(249\) 4.48211 0.284042
\(250\) −36.1746 −2.28788
\(251\) 3.41922 0.215819 0.107910 0.994161i \(-0.465584\pi\)
0.107910 + 0.994161i \(0.465584\pi\)
\(252\) 0.294394 0.0185451
\(253\) −8.47573 −0.532864
\(254\) −28.1251 −1.76473
\(255\) 13.6601 0.855427
\(256\) −6.19266 −0.387041
\(257\) −23.9928 −1.49663 −0.748314 0.663345i \(-0.769136\pi\)
−0.748314 + 0.663345i \(0.769136\pi\)
\(258\) 16.0968 1.00214
\(259\) 1.07111 0.0665557
\(260\) 36.2798 2.24998
\(261\) −2.57597 −0.159448
\(262\) −28.1954 −1.74192
\(263\) 25.7428 1.58737 0.793684 0.608330i \(-0.208160\pi\)
0.793684 + 0.608330i \(0.208160\pi\)
\(264\) 4.25835 0.262083
\(265\) −36.9568 −2.27024
\(266\) −0.295450 −0.0181152
\(267\) −6.67355 −0.408415
\(268\) 8.77495 0.536015
\(269\) −1.47273 −0.0897942 −0.0448971 0.998992i \(-0.514296\pi\)
−0.0448971 + 0.998992i \(0.514296\pi\)
\(270\) 36.4092 2.21579
\(271\) −1.69114 −0.102729 −0.0513646 0.998680i \(-0.516357\pi\)
−0.0513646 + 0.998680i \(0.516357\pi\)
\(272\) −2.54380 −0.154240
\(273\) 0.853266 0.0516420
\(274\) −4.32754 −0.261436
\(275\) −10.1929 −0.614652
\(276\) 43.6536 2.62764
\(277\) −17.0367 −1.02364 −0.511819 0.859093i \(-0.671028\pi\)
−0.511819 + 0.859093i \(0.671028\pi\)
\(278\) 11.7753 0.706233
\(279\) 1.54903 0.0927379
\(280\) −1.01310 −0.0605444
\(281\) −5.36336 −0.319951 −0.159976 0.987121i \(-0.551142\pi\)
−0.159976 + 0.987121i \(0.551142\pi\)
\(282\) 18.0177 1.07294
\(283\) −13.8625 −0.824043 −0.412021 0.911174i \(-0.635177\pi\)
−0.412021 + 0.911174i \(0.635177\pi\)
\(284\) −3.68791 −0.218837
\(285\) 7.33629 0.434564
\(286\) 7.99543 0.472780
\(287\) 0.694750 0.0410098
\(288\) 5.34699 0.315075
\(289\) −13.5330 −0.796059
\(290\) 28.4280 1.66935
\(291\) −34.4067 −2.01696
\(292\) −0.368961 −0.0215918
\(293\) −16.2038 −0.946639 −0.473319 0.880891i \(-0.656944\pi\)
−0.473319 + 0.880891i \(0.656944\pi\)
\(294\) 29.9867 1.74886
\(295\) −3.52501 −0.205234
\(296\) −16.1196 −0.936933
\(297\) 4.75305 0.275800
\(298\) 25.1954 1.45953
\(299\) 25.5587 1.47810
\(300\) 52.4976 3.03095
\(301\) −0.499935 −0.0288158
\(302\) 18.3091 1.05357
\(303\) −14.5365 −0.835098
\(304\) −1.36617 −0.0783554
\(305\) 29.7847 1.70546
\(306\) −3.13211 −0.179051
\(307\) −10.7359 −0.612729 −0.306365 0.951914i \(-0.599113\pi\)
−0.306365 + 0.951914i \(0.599113\pi\)
\(308\) −0.424127 −0.0241669
\(309\) 32.1320 1.82793
\(310\) −17.0948 −0.970921
\(311\) −3.11213 −0.176473 −0.0882365 0.996100i \(-0.528123\pi\)
−0.0882365 + 0.996100i \(0.528123\pi\)
\(312\) −12.8411 −0.726986
\(313\) 24.4746 1.38339 0.691693 0.722191i \(-0.256865\pi\)
0.691693 + 0.722191i \(0.256865\pi\)
\(314\) −18.6966 −1.05511
\(315\) 0.383275 0.0215951
\(316\) −1.57377 −0.0885317
\(317\) 9.11430 0.511910 0.255955 0.966689i \(-0.417610\pi\)
0.255955 + 0.966689i \(0.417610\pi\)
\(318\) 41.9483 2.35234
\(319\) 3.71114 0.207784
\(320\) −48.6702 −2.72074
\(321\) 36.9888 2.06452
\(322\) −2.28881 −0.127550
\(323\) 1.86199 0.103604
\(324\) −31.1015 −1.72786
\(325\) 30.7368 1.70497
\(326\) −10.4880 −0.580874
\(327\) 6.90066 0.381607
\(328\) −10.4556 −0.577312
\(329\) −0.559594 −0.0308514
\(330\) 17.7789 0.978693
\(331\) −5.22913 −0.287419 −0.143709 0.989620i \(-0.545903\pi\)
−0.143709 + 0.989620i \(0.545903\pi\)
\(332\) 6.71825 0.368712
\(333\) 6.09834 0.334187
\(334\) 31.2172 1.70813
\(335\) 11.4242 0.624170
\(336\) −0.353326 −0.0192755
\(337\) 13.1489 0.716268 0.358134 0.933670i \(-0.383413\pi\)
0.358134 + 0.933670i \(0.383413\pi\)
\(338\) 4.68469 0.254814
\(339\) 25.0142 1.35859
\(340\) 20.4751 1.11042
\(341\) −2.23165 −0.120851
\(342\) −1.68214 −0.0909595
\(343\) −1.86503 −0.100702
\(344\) 7.52371 0.405651
\(345\) 56.8331 3.05979
\(346\) −46.4317 −2.49618
\(347\) 20.3548 1.09270 0.546351 0.837556i \(-0.316016\pi\)
0.546351 + 0.837556i \(0.316016\pi\)
\(348\) −19.1139 −1.02461
\(349\) 3.24662 0.173788 0.0868938 0.996218i \(-0.472306\pi\)
0.0868938 + 0.996218i \(0.472306\pi\)
\(350\) −2.75251 −0.147128
\(351\) −14.3329 −0.765035
\(352\) −7.70328 −0.410586
\(353\) 17.0312 0.906478 0.453239 0.891389i \(-0.350268\pi\)
0.453239 + 0.891389i \(0.350268\pi\)
\(354\) 4.00112 0.212657
\(355\) −4.80133 −0.254828
\(356\) −10.0030 −0.530159
\(357\) 0.481556 0.0254866
\(358\) 15.9469 0.842817
\(359\) 17.4891 0.923040 0.461520 0.887130i \(-0.347304\pi\)
0.461520 + 0.887130i \(0.347304\pi\)
\(360\) −5.76805 −0.304003
\(361\) 1.00000 0.0526316
\(362\) 40.1047 2.10786
\(363\) −19.0072 −0.997620
\(364\) 1.27896 0.0670359
\(365\) −0.480355 −0.0251429
\(366\) −33.8075 −1.76715
\(367\) −6.71410 −0.350473 −0.175237 0.984526i \(-0.556069\pi\)
−0.175237 + 0.984526i \(0.556069\pi\)
\(368\) −10.5835 −0.551705
\(369\) 3.95554 0.205917
\(370\) −67.3003 −3.49878
\(371\) −1.30283 −0.0676395
\(372\) 11.4939 0.595933
\(373\) −10.1350 −0.524773 −0.262386 0.964963i \(-0.584510\pi\)
−0.262386 + 0.964963i \(0.584510\pi\)
\(374\) 4.51236 0.233329
\(375\) 31.6657 1.63521
\(376\) 8.42154 0.434308
\(377\) −11.1910 −0.576367
\(378\) 1.28353 0.0660174
\(379\) −3.79265 −0.194815 −0.0974077 0.995245i \(-0.531055\pi\)
−0.0974077 + 0.995245i \(0.531055\pi\)
\(380\) 10.9964 0.564103
\(381\) 24.6195 1.26130
\(382\) 22.0318 1.12724
\(383\) 9.68916 0.495093 0.247547 0.968876i \(-0.420376\pi\)
0.247547 + 0.968876i \(0.420376\pi\)
\(384\) 27.9405 1.42583
\(385\) −0.552175 −0.0281414
\(386\) −16.4050 −0.834992
\(387\) −2.84636 −0.144689
\(388\) −51.5723 −2.61819
\(389\) −18.9737 −0.962004 −0.481002 0.876719i \(-0.659727\pi\)
−0.481002 + 0.876719i \(0.659727\pi\)
\(390\) −53.6125 −2.71478
\(391\) 14.4245 0.729480
\(392\) 14.0159 0.707910
\(393\) 24.6810 1.24499
\(394\) 20.4144 1.02846
\(395\) −2.04891 −0.103092
\(396\) −2.41475 −0.121346
\(397\) −29.5411 −1.48263 −0.741314 0.671159i \(-0.765797\pi\)
−0.741314 + 0.671159i \(0.765797\pi\)
\(398\) −7.48030 −0.374954
\(399\) 0.258625 0.0129474
\(400\) −12.7277 −0.636385
\(401\) 6.85782 0.342463 0.171232 0.985231i \(-0.445225\pi\)
0.171232 + 0.985231i \(0.445225\pi\)
\(402\) −12.9672 −0.646745
\(403\) 6.72959 0.335225
\(404\) −21.7888 −1.08403
\(405\) −40.4913 −2.01203
\(406\) 1.00217 0.0497366
\(407\) −8.78574 −0.435493
\(408\) −7.24712 −0.358786
\(409\) −5.98969 −0.296171 −0.148086 0.988975i \(-0.547311\pi\)
−0.148086 + 0.988975i \(0.547311\pi\)
\(410\) −43.6526 −2.15585
\(411\) 3.78814 0.186855
\(412\) 48.1627 2.37281
\(413\) −0.124267 −0.00611476
\(414\) −13.0312 −0.640451
\(415\) 8.74656 0.429352
\(416\) 23.2294 1.13892
\(417\) −10.3076 −0.504764
\(418\) 2.42341 0.118533
\(419\) 1.61323 0.0788116 0.0394058 0.999223i \(-0.487454\pi\)
0.0394058 + 0.999223i \(0.487454\pi\)
\(420\) 2.84394 0.138770
\(421\) −19.9700 −0.973279 −0.486639 0.873603i \(-0.661777\pi\)
−0.486639 + 0.873603i \(0.661777\pi\)
\(422\) 2.21501 0.107825
\(423\) −3.18603 −0.154910
\(424\) 19.6068 0.952190
\(425\) 17.3468 0.841446
\(426\) 5.44982 0.264045
\(427\) 1.04999 0.0508127
\(428\) 55.4427 2.67992
\(429\) −6.99886 −0.337908
\(430\) 31.4120 1.51482
\(431\) 37.7877 1.82017 0.910084 0.414425i \(-0.136017\pi\)
0.910084 + 0.414425i \(0.136017\pi\)
\(432\) 5.93507 0.285551
\(433\) −15.6294 −0.751101 −0.375550 0.926802i \(-0.622546\pi\)
−0.375550 + 0.926802i \(0.622546\pi\)
\(434\) −0.602640 −0.0289277
\(435\) −24.8846 −1.19313
\(436\) 10.3434 0.495360
\(437\) 7.74684 0.370582
\(438\) 0.545233 0.0260523
\(439\) 2.83564 0.135338 0.0676688 0.997708i \(-0.478444\pi\)
0.0676688 + 0.997708i \(0.478444\pi\)
\(440\) 8.30990 0.396159
\(441\) −5.30247 −0.252499
\(442\) −13.6071 −0.647225
\(443\) −17.0073 −0.808041 −0.404020 0.914750i \(-0.632388\pi\)
−0.404020 + 0.914750i \(0.632388\pi\)
\(444\) 45.2503 2.14748
\(445\) −13.0230 −0.617351
\(446\) 16.8223 0.796557
\(447\) −22.0550 −1.04316
\(448\) −1.71576 −0.0810620
\(449\) 19.8904 0.938685 0.469343 0.883016i \(-0.344491\pi\)
0.469343 + 0.883016i \(0.344491\pi\)
\(450\) −15.6713 −0.738752
\(451\) −5.69864 −0.268339
\(452\) 37.4939 1.76356
\(453\) −16.0270 −0.753016
\(454\) −57.9334 −2.71895
\(455\) 1.66510 0.0780609
\(456\) −3.89214 −0.182266
\(457\) 10.0041 0.467974 0.233987 0.972240i \(-0.424823\pi\)
0.233987 + 0.972240i \(0.424823\pi\)
\(458\) 56.9341 2.66035
\(459\) −8.08903 −0.377564
\(460\) 85.1873 3.97188
\(461\) 23.4614 1.09270 0.546352 0.837555i \(-0.316016\pi\)
0.546352 + 0.837555i \(0.316016\pi\)
\(462\) 0.626754 0.0291592
\(463\) 39.2805 1.82552 0.912760 0.408497i \(-0.133947\pi\)
0.912760 + 0.408497i \(0.133947\pi\)
\(464\) 4.63405 0.215130
\(465\) 14.9641 0.693943
\(466\) −22.1559 −1.02635
\(467\) 11.9742 0.554102 0.277051 0.960855i \(-0.410643\pi\)
0.277051 + 0.960855i \(0.410643\pi\)
\(468\) 7.28173 0.336598
\(469\) 0.402735 0.0185966
\(470\) 35.1605 1.62183
\(471\) 16.3662 0.754117
\(472\) 1.87014 0.0860800
\(473\) 4.10068 0.188550
\(474\) 2.32565 0.106821
\(475\) 9.31631 0.427462
\(476\) 0.721806 0.0330839
\(477\) −7.41761 −0.339629
\(478\) 33.4799 1.53134
\(479\) −20.4965 −0.936511 −0.468256 0.883593i \(-0.655117\pi\)
−0.468256 + 0.883593i \(0.655117\pi\)
\(480\) 51.6536 2.35765
\(481\) 26.4936 1.20800
\(482\) −42.0024 −1.91316
\(483\) 2.00353 0.0911636
\(484\) −28.4900 −1.29500
\(485\) −67.1426 −3.04879
\(486\) 17.0923 0.775323
\(487\) −42.8780 −1.94299 −0.971494 0.237065i \(-0.923815\pi\)
−0.971494 + 0.237065i \(0.923815\pi\)
\(488\) −15.8017 −0.715312
\(489\) 9.18071 0.415166
\(490\) 58.5172 2.64354
\(491\) 25.9281 1.17012 0.585060 0.810990i \(-0.301071\pi\)
0.585060 + 0.810990i \(0.301071\pi\)
\(492\) 29.3504 1.32322
\(493\) −6.31585 −0.284451
\(494\) −7.30786 −0.328796
\(495\) −3.14379 −0.141303
\(496\) −2.78663 −0.125123
\(497\) −0.169260 −0.00759236
\(498\) −9.92790 −0.444880
\(499\) −15.3463 −0.686995 −0.343498 0.939154i \(-0.611612\pi\)
−0.343498 + 0.939154i \(0.611612\pi\)
\(500\) 47.4638 2.12265
\(501\) −27.3262 −1.22085
\(502\) −7.57359 −0.338026
\(503\) −3.16345 −0.141051 −0.0705256 0.997510i \(-0.522468\pi\)
−0.0705256 + 0.997510i \(0.522468\pi\)
\(504\) −0.203340 −0.00905749
\(505\) −28.3670 −1.26232
\(506\) 18.7738 0.834597
\(507\) −4.10078 −0.182122
\(508\) 36.9023 1.63728
\(509\) 31.2968 1.38721 0.693603 0.720357i \(-0.256022\pi\)
0.693603 + 0.720357i \(0.256022\pi\)
\(510\) −30.2572 −1.33981
\(511\) −0.0169338 −0.000749109 0
\(512\) −15.1038 −0.667502
\(513\) −4.34430 −0.191806
\(514\) 53.1442 2.34409
\(515\) 62.7036 2.76305
\(516\) −21.1203 −0.929768
\(517\) 4.59003 0.201869
\(518\) −2.37252 −0.104243
\(519\) 40.6444 1.78409
\(520\) −25.0587 −1.09890
\(521\) −13.4515 −0.589323 −0.294661 0.955602i \(-0.595207\pi\)
−0.294661 + 0.955602i \(0.595207\pi\)
\(522\) 5.70579 0.249736
\(523\) −16.8159 −0.735307 −0.367653 0.929963i \(-0.619839\pi\)
−0.367653 + 0.929963i \(0.619839\pi\)
\(524\) 36.9945 1.61611
\(525\) 2.40943 0.105156
\(526\) −57.0205 −2.48621
\(527\) 3.79796 0.165442
\(528\) 2.89814 0.126125
\(529\) 37.0136 1.60929
\(530\) 81.8595 3.55575
\(531\) −0.707507 −0.0307032
\(532\) 0.387653 0.0168069
\(533\) 17.1844 0.744339
\(534\) 14.7820 0.639679
\(535\) 72.1814 3.12068
\(536\) −6.06091 −0.261792
\(537\) −13.9592 −0.602384
\(538\) 3.26212 0.140640
\(539\) 7.63915 0.329041
\(540\) −47.7716 −2.05576
\(541\) −30.1432 −1.29596 −0.647979 0.761658i \(-0.724386\pi\)
−0.647979 + 0.761658i \(0.724386\pi\)
\(542\) 3.74588 0.160899
\(543\) −35.1060 −1.50654
\(544\) 13.1099 0.562084
\(545\) 13.4662 0.576829
\(546\) −1.88999 −0.0808842
\(547\) 17.4722 0.747058 0.373529 0.927619i \(-0.378148\pi\)
0.373529 + 0.927619i \(0.378148\pi\)
\(548\) 5.67806 0.242555
\(549\) 5.97810 0.255139
\(550\) 22.5773 0.962698
\(551\) −3.39199 −0.144504
\(552\) −30.1518 −1.28335
\(553\) −0.0722299 −0.00307153
\(554\) 37.7365 1.60327
\(555\) 58.9118 2.50067
\(556\) −15.4500 −0.655228
\(557\) −2.69500 −0.114191 −0.0570954 0.998369i \(-0.518184\pi\)
−0.0570954 + 0.998369i \(0.518184\pi\)
\(558\) −3.43111 −0.145251
\(559\) −12.3657 −0.523014
\(560\) −0.689494 −0.0291364
\(561\) −3.94993 −0.166766
\(562\) 11.8799 0.501123
\(563\) −21.0355 −0.886540 −0.443270 0.896388i \(-0.646182\pi\)
−0.443270 + 0.896388i \(0.646182\pi\)
\(564\) −23.6406 −0.995450
\(565\) 48.8137 2.05361
\(566\) 30.7056 1.29065
\(567\) −1.42743 −0.0599466
\(568\) 2.54726 0.106881
\(569\) −23.9589 −1.00441 −0.502204 0.864749i \(-0.667477\pi\)
−0.502204 + 0.864749i \(0.667477\pi\)
\(570\) −16.2499 −0.680635
\(571\) −7.47725 −0.312913 −0.156457 0.987685i \(-0.550007\pi\)
−0.156457 + 0.987685i \(0.550007\pi\)
\(572\) −10.4906 −0.438635
\(573\) −19.2857 −0.805670
\(574\) −1.53888 −0.0642315
\(575\) 72.1720 3.00978
\(576\) −9.76862 −0.407026
\(577\) −16.9353 −0.705025 −0.352512 0.935807i \(-0.614673\pi\)
−0.352512 + 0.935807i \(0.614673\pi\)
\(578\) 29.9757 1.24683
\(579\) 14.3602 0.596791
\(580\) −37.2997 −1.54878
\(581\) 0.308341 0.0127921
\(582\) 76.2112 3.15905
\(583\) 10.6864 0.442585
\(584\) 0.254844 0.0105455
\(585\) 9.48017 0.391957
\(586\) 35.8916 1.48267
\(587\) −14.3523 −0.592383 −0.296192 0.955129i \(-0.595717\pi\)
−0.296192 + 0.955129i \(0.595717\pi\)
\(588\) −39.3449 −1.62255
\(589\) 2.03974 0.0840458
\(590\) 7.80793 0.321447
\(591\) −17.8699 −0.735071
\(592\) −10.9706 −0.450891
\(593\) −26.6774 −1.09551 −0.547755 0.836639i \(-0.684517\pi\)
−0.547755 + 0.836639i \(0.684517\pi\)
\(594\) −10.5280 −0.431971
\(595\) 0.939726 0.0385250
\(596\) −33.0582 −1.35412
\(597\) 6.54794 0.267989
\(598\) −56.6128 −2.31507
\(599\) 48.7765 1.99295 0.996477 0.0838624i \(-0.0267256\pi\)
0.996477 + 0.0838624i \(0.0267256\pi\)
\(600\) −36.2604 −1.48032
\(601\) −40.4504 −1.65001 −0.825003 0.565129i \(-0.808826\pi\)
−0.825003 + 0.565129i \(0.808826\pi\)
\(602\) 1.10736 0.0451326
\(603\) 2.29296 0.0933764
\(604\) −24.0230 −0.977482
\(605\) −37.0914 −1.50798
\(606\) 32.1984 1.30797
\(607\) 39.5777 1.60641 0.803204 0.595704i \(-0.203127\pi\)
0.803204 + 0.595704i \(0.203127\pi\)
\(608\) 7.04083 0.285543
\(609\) −0.877253 −0.0355481
\(610\) −65.9733 −2.67118
\(611\) −13.8413 −0.559961
\(612\) 4.10957 0.166120
\(613\) −34.8981 −1.40952 −0.704761 0.709445i \(-0.748946\pi\)
−0.704761 + 0.709445i \(0.748946\pi\)
\(614\) 23.7801 0.959685
\(615\) 38.2117 1.54084
\(616\) 0.292947 0.0118032
\(617\) −5.60470 −0.225637 −0.112818 0.993616i \(-0.535988\pi\)
−0.112818 + 0.993616i \(0.535988\pi\)
\(618\) −71.1726 −2.86298
\(619\) 3.95757 0.159068 0.0795342 0.996832i \(-0.474657\pi\)
0.0795342 + 0.996832i \(0.474657\pi\)
\(620\) 22.4297 0.900799
\(621\) −33.6546 −1.35051
\(622\) 6.89340 0.276400
\(623\) −0.459098 −0.0183934
\(624\) −8.73939 −0.349856
\(625\) 15.2121 0.608482
\(626\) −54.2115 −2.16673
\(627\) −2.12135 −0.0847187
\(628\) 24.5314 0.978910
\(629\) 14.9521 0.596180
\(630\) −0.848957 −0.0338233
\(631\) 22.0798 0.878982 0.439491 0.898247i \(-0.355159\pi\)
0.439491 + 0.898247i \(0.355159\pi\)
\(632\) 1.08702 0.0432392
\(633\) −1.93892 −0.0770653
\(634\) −20.1883 −0.801778
\(635\) 48.0435 1.90655
\(636\) −55.0394 −2.18245
\(637\) −23.0360 −0.912720
\(638\) −8.22020 −0.325441
\(639\) −0.963678 −0.0381225
\(640\) 54.5241 2.15526
\(641\) 19.9360 0.787423 0.393711 0.919234i \(-0.371191\pi\)
0.393711 + 0.919234i \(0.371191\pi\)
\(642\) −81.9306 −3.23354
\(643\) −11.3083 −0.445956 −0.222978 0.974823i \(-0.571578\pi\)
−0.222978 + 0.974823i \(0.571578\pi\)
\(644\) 3.00309 0.118338
\(645\) −27.4967 −1.08268
\(646\) −4.12432 −0.162269
\(647\) −45.7563 −1.79887 −0.899433 0.437058i \(-0.856021\pi\)
−0.899433 + 0.437058i \(0.856021\pi\)
\(648\) 21.4820 0.843893
\(649\) 1.01929 0.0400106
\(650\) −68.0822 −2.67040
\(651\) 0.527526 0.0206754
\(652\) 13.7610 0.538922
\(653\) 12.2945 0.481119 0.240560 0.970634i \(-0.422669\pi\)
0.240560 + 0.970634i \(0.422669\pi\)
\(654\) −15.2850 −0.597691
\(655\) 48.1635 1.88190
\(656\) −7.11583 −0.277826
\(657\) −0.0964122 −0.00376140
\(658\) 1.23950 0.0483209
\(659\) 30.1860 1.17588 0.587940 0.808905i \(-0.299939\pi\)
0.587940 + 0.808905i \(0.299939\pi\)
\(660\) −23.3272 −0.908010
\(661\) 47.8970 1.86298 0.931489 0.363770i \(-0.118511\pi\)
0.931489 + 0.363770i \(0.118511\pi\)
\(662\) 11.5826 0.450169
\(663\) 11.9111 0.462589
\(664\) −4.64034 −0.180080
\(665\) 0.504690 0.0195710
\(666\) −13.5079 −0.523420
\(667\) −26.2772 −1.01746
\(668\) −40.9594 −1.58477
\(669\) −14.7255 −0.569321
\(670\) −25.3047 −0.977605
\(671\) −8.61250 −0.332482
\(672\) 1.82093 0.0702440
\(673\) 13.8506 0.533900 0.266950 0.963710i \(-0.413984\pi\)
0.266950 + 0.963710i \(0.413984\pi\)
\(674\) −29.1250 −1.12185
\(675\) −40.4729 −1.55780
\(676\) −6.14667 −0.236411
\(677\) 35.7238 1.37298 0.686488 0.727141i \(-0.259151\pi\)
0.686488 + 0.727141i \(0.259151\pi\)
\(678\) −55.4067 −2.12788
\(679\) −2.36696 −0.0908357
\(680\) −14.1423 −0.542333
\(681\) 50.7124 1.94330
\(682\) 4.94312 0.189282
\(683\) 17.5216 0.670446 0.335223 0.942139i \(-0.391188\pi\)
0.335223 + 0.942139i \(0.391188\pi\)
\(684\) 2.20709 0.0843902
\(685\) 7.39233 0.282446
\(686\) 4.13105 0.157724
\(687\) −49.8377 −1.90143
\(688\) 5.12047 0.195216
\(689\) −32.2250 −1.22768
\(690\) −125.886 −4.79239
\(691\) 28.7634 1.09421 0.547105 0.837064i \(-0.315730\pi\)
0.547105 + 0.837064i \(0.315730\pi\)
\(692\) 60.9220 2.31591
\(693\) −0.110827 −0.00420998
\(694\) −45.0860 −1.71144
\(695\) −20.1146 −0.762989
\(696\) 13.2021 0.500425
\(697\) 9.69831 0.367350
\(698\) −7.19129 −0.272194
\(699\) 19.3943 0.733560
\(700\) 3.61150 0.136502
\(701\) 11.8718 0.448393 0.224196 0.974544i \(-0.428024\pi\)
0.224196 + 0.974544i \(0.428024\pi\)
\(702\) 31.7475 1.19823
\(703\) 8.03020 0.302865
\(704\) 14.0734 0.530412
\(705\) −30.7780 −1.15917
\(706\) −37.7242 −1.41977
\(707\) −1.00002 −0.0376095
\(708\) −5.24977 −0.197299
\(709\) 49.2027 1.84785 0.923923 0.382579i \(-0.124964\pi\)
0.923923 + 0.382579i \(0.124964\pi\)
\(710\) 10.6350 0.399124
\(711\) −0.411238 −0.0154226
\(712\) 6.90915 0.258931
\(713\) 15.8015 0.591771
\(714\) −1.06665 −0.0399184
\(715\) −13.6578 −0.510775
\(716\) −20.9235 −0.781947
\(717\) −29.3069 −1.09449
\(718\) −38.7385 −1.44571
\(719\) −3.00601 −0.112105 −0.0560525 0.998428i \(-0.517851\pi\)
−0.0560525 + 0.998428i \(0.517851\pi\)
\(720\) −3.92561 −0.146299
\(721\) 2.21048 0.0823225
\(722\) −2.21501 −0.0824341
\(723\) 36.7671 1.36738
\(724\) −52.6205 −1.95562
\(725\) −31.6009 −1.17363
\(726\) 42.1011 1.56252
\(727\) −43.0194 −1.59550 −0.797751 0.602987i \(-0.793977\pi\)
−0.797751 + 0.602987i \(0.793977\pi\)
\(728\) −0.883389 −0.0327406
\(729\) 17.1428 0.634918
\(730\) 1.06399 0.0393800
\(731\) −6.97880 −0.258120
\(732\) 44.3580 1.63952
\(733\) −47.7648 −1.76423 −0.882116 0.471032i \(-0.843882\pi\)
−0.882116 + 0.471032i \(0.843882\pi\)
\(734\) 14.8718 0.548928
\(735\) −51.2235 −1.88941
\(736\) 54.5442 2.01053
\(737\) −3.30341 −0.121683
\(738\) −8.76154 −0.322517
\(739\) −9.86435 −0.362866 −0.181433 0.983403i \(-0.558074\pi\)
−0.181433 + 0.983403i \(0.558074\pi\)
\(740\) 88.3031 3.24609
\(741\) 6.39699 0.234999
\(742\) 2.88578 0.105940
\(743\) 32.8173 1.20395 0.601975 0.798515i \(-0.294381\pi\)
0.601975 + 0.798515i \(0.294381\pi\)
\(744\) −7.93894 −0.291056
\(745\) −43.0389 −1.57682
\(746\) 22.4492 0.821924
\(747\) 1.75553 0.0642313
\(748\) −5.92057 −0.216477
\(749\) 2.54460 0.0929776
\(750\) −70.1397 −2.56114
\(751\) −40.2147 −1.46745 −0.733727 0.679444i \(-0.762221\pi\)
−0.733727 + 0.679444i \(0.762221\pi\)
\(752\) 5.73152 0.209007
\(753\) 6.62960 0.241596
\(754\) 24.7882 0.902733
\(755\) −31.2758 −1.13824
\(756\) −1.68408 −0.0612495
\(757\) −48.1991 −1.75183 −0.875914 0.482467i \(-0.839741\pi\)
−0.875914 + 0.482467i \(0.839741\pi\)
\(758\) 8.40076 0.305129
\(759\) −16.4338 −0.596509
\(760\) −7.59528 −0.275510
\(761\) −5.42956 −0.196822 −0.0984108 0.995146i \(-0.531376\pi\)
−0.0984108 + 0.995146i \(0.531376\pi\)
\(762\) −54.5325 −1.97550
\(763\) 0.474721 0.0171861
\(764\) −28.9073 −1.04583
\(765\) 5.35030 0.193440
\(766\) −21.4616 −0.775438
\(767\) −3.07369 −0.110984
\(768\) −12.0071 −0.433269
\(769\) 28.3638 1.02282 0.511412 0.859336i \(-0.329123\pi\)
0.511412 + 0.859336i \(0.329123\pi\)
\(770\) 1.22307 0.0440765
\(771\) −46.5202 −1.67538
\(772\) 21.5246 0.774687
\(773\) 15.9944 0.575279 0.287639 0.957739i \(-0.407130\pi\)
0.287639 + 0.957739i \(0.407130\pi\)
\(774\) 6.30471 0.226618
\(775\) 19.0028 0.682601
\(776\) 35.6214 1.27873
\(777\) 2.07681 0.0745050
\(778\) 42.0269 1.50674
\(779\) 5.20858 0.186617
\(780\) 70.3437 2.51871
\(781\) 1.38835 0.0496790
\(782\) −31.9504 −1.14255
\(783\) 14.7358 0.526616
\(784\) 9.53891 0.340675
\(785\) 31.9377 1.13991
\(786\) −54.6687 −1.94997
\(787\) −30.5297 −1.08827 −0.544133 0.838999i \(-0.683141\pi\)
−0.544133 + 0.838999i \(0.683141\pi\)
\(788\) −26.7853 −0.954187
\(789\) 49.9133 1.77696
\(790\) 4.53836 0.161468
\(791\) 1.72082 0.0611853
\(792\) 1.66788 0.0592657
\(793\) 25.9712 0.922264
\(794\) 65.4339 2.32216
\(795\) −71.6564 −2.54139
\(796\) 9.81473 0.347874
\(797\) −26.6068 −0.942463 −0.471231 0.882010i \(-0.656190\pi\)
−0.471231 + 0.882010i \(0.656190\pi\)
\(798\) −0.572856 −0.0202789
\(799\) −7.81161 −0.276355
\(800\) 65.5945 2.31912
\(801\) −2.61386 −0.0923562
\(802\) −15.1901 −0.536382
\(803\) 0.138899 0.00490163
\(804\) 17.0140 0.600036
\(805\) 3.90976 0.137801
\(806\) −14.9061 −0.525045
\(807\) −2.85552 −0.100519
\(808\) 15.0496 0.529444
\(809\) 13.8396 0.486575 0.243287 0.969954i \(-0.421774\pi\)
0.243287 + 0.969954i \(0.421774\pi\)
\(810\) 89.6887 3.15134
\(811\) −19.0621 −0.669362 −0.334681 0.942331i \(-0.608629\pi\)
−0.334681 + 0.942331i \(0.608629\pi\)
\(812\) −1.31492 −0.0461446
\(813\) −3.27899 −0.114999
\(814\) 19.4605 0.682089
\(815\) 17.9156 0.627556
\(816\) −4.93223 −0.172663
\(817\) −3.74804 −0.131127
\(818\) 13.2672 0.463877
\(819\) 0.334202 0.0116780
\(820\) 57.2756 2.00015
\(821\) −18.7960 −0.655985 −0.327993 0.944680i \(-0.606372\pi\)
−0.327993 + 0.944680i \(0.606372\pi\)
\(822\) −8.39077 −0.292662
\(823\) −36.8738 −1.28534 −0.642670 0.766143i \(-0.722173\pi\)
−0.642670 + 0.766143i \(0.722173\pi\)
\(824\) −33.2663 −1.15889
\(825\) −19.7632 −0.688065
\(826\) 0.275251 0.00957722
\(827\) −7.32533 −0.254727 −0.127363 0.991856i \(-0.540651\pi\)
−0.127363 + 0.991856i \(0.540651\pi\)
\(828\) 17.0980 0.594196
\(829\) −5.97441 −0.207500 −0.103750 0.994603i \(-0.533084\pi\)
−0.103750 + 0.994603i \(0.533084\pi\)
\(830\) −19.3737 −0.672471
\(831\) −33.0329 −1.14590
\(832\) −42.4387 −1.47130
\(833\) −13.0008 −0.450450
\(834\) 22.8313 0.790585
\(835\) −53.3255 −1.84540
\(836\) −3.17970 −0.109972
\(837\) −8.86123 −0.306289
\(838\) −3.57332 −0.123438
\(839\) −0.781041 −0.0269645 −0.0134823 0.999909i \(-0.504292\pi\)
−0.0134823 + 0.999909i \(0.504292\pi\)
\(840\) −1.96433 −0.0677757
\(841\) −17.4944 −0.603255
\(842\) 44.2337 1.52440
\(843\) −10.3992 −0.358166
\(844\) −2.90626 −0.100038
\(845\) −8.00242 −0.275292
\(846\) 7.05708 0.242627
\(847\) −1.30758 −0.0449288
\(848\) 13.3439 0.458233
\(849\) −26.8784 −0.922465
\(850\) −38.4234 −1.31791
\(851\) 62.2087 2.13249
\(852\) −7.15058 −0.244975
\(853\) −9.63724 −0.329973 −0.164986 0.986296i \(-0.552758\pi\)
−0.164986 + 0.986296i \(0.552758\pi\)
\(854\) −2.32574 −0.0795853
\(855\) 2.87343 0.0982694
\(856\) −38.2946 −1.30888
\(857\) −57.7223 −1.97176 −0.985878 0.167462i \(-0.946443\pi\)
−0.985878 + 0.167462i \(0.946443\pi\)
\(858\) 15.5025 0.529248
\(859\) −41.7538 −1.42462 −0.712310 0.701865i \(-0.752351\pi\)
−0.712310 + 0.701865i \(0.752351\pi\)
\(860\) −41.2149 −1.40542
\(861\) 1.34707 0.0459079
\(862\) −83.6999 −2.85083
\(863\) 25.5299 0.869048 0.434524 0.900660i \(-0.356917\pi\)
0.434524 + 0.900660i \(0.356917\pi\)
\(864\) −30.5875 −1.04061
\(865\) 79.3149 2.69679
\(866\) 34.6192 1.17641
\(867\) −26.2395 −0.891139
\(868\) 0.790710 0.0268385
\(869\) 0.592461 0.0200979
\(870\) 55.1197 1.86873
\(871\) 9.96150 0.337532
\(872\) −7.14427 −0.241935
\(873\) −13.4762 −0.456101
\(874\) −17.1593 −0.580423
\(875\) 2.17840 0.0736433
\(876\) −0.715388 −0.0241707
\(877\) 47.3418 1.59862 0.799309 0.600921i \(-0.205199\pi\)
0.799309 + 0.600921i \(0.205199\pi\)
\(878\) −6.28096 −0.211972
\(879\) −31.4180 −1.05970
\(880\) 5.65553 0.190648
\(881\) 31.9711 1.07714 0.538568 0.842582i \(-0.318966\pi\)
0.538568 + 0.842582i \(0.318966\pi\)
\(882\) 11.7450 0.395475
\(883\) 15.2184 0.512140 0.256070 0.966658i \(-0.417572\pi\)
0.256070 + 0.966658i \(0.417572\pi\)
\(884\) 17.8536 0.600481
\(885\) −6.83473 −0.229747
\(886\) 37.6713 1.26559
\(887\) −27.2918 −0.916369 −0.458184 0.888857i \(-0.651500\pi\)
−0.458184 + 0.888857i \(0.651500\pi\)
\(888\) −31.2547 −1.04884
\(889\) 1.69367 0.0568038
\(890\) 28.8461 0.966924
\(891\) 11.7084 0.392247
\(892\) −22.0721 −0.739029
\(893\) −4.19531 −0.140391
\(894\) 48.8519 1.63385
\(895\) −27.2405 −0.910550
\(896\) 1.92213 0.0642138
\(897\) 49.5565 1.65464
\(898\) −44.0574 −1.47021
\(899\) −6.91877 −0.230754
\(900\) 20.5619 0.685398
\(901\) −18.1868 −0.605889
\(902\) 12.6225 0.420285
\(903\) −0.969336 −0.0322575
\(904\) −25.8973 −0.861331
\(905\) −68.5071 −2.27725
\(906\) 35.5000 1.17941
\(907\) −22.4136 −0.744231 −0.372115 0.928187i \(-0.621367\pi\)
−0.372115 + 0.928187i \(0.621367\pi\)
\(908\) 76.0130 2.52258
\(909\) −5.69356 −0.188843
\(910\) −3.68820 −0.122263
\(911\) 17.5758 0.582311 0.291156 0.956676i \(-0.405960\pi\)
0.291156 + 0.956676i \(0.405960\pi\)
\(912\) −2.64891 −0.0877141
\(913\) −2.52914 −0.0837025
\(914\) −22.1592 −0.732962
\(915\) 57.7502 1.90916
\(916\) −74.7018 −2.46822
\(917\) 1.69790 0.0560696
\(918\) 17.9173 0.591358
\(919\) 2.54400 0.0839190 0.0419595 0.999119i \(-0.486640\pi\)
0.0419595 + 0.999119i \(0.486640\pi\)
\(920\) −58.8394 −1.93988
\(921\) −20.8161 −0.685913
\(922\) −51.9671 −1.71145
\(923\) −4.18659 −0.137803
\(924\) −0.822350 −0.0270533
\(925\) 74.8118 2.45980
\(926\) −87.0066 −2.85921
\(927\) 12.5853 0.413355
\(928\) −23.8824 −0.783980
\(929\) 4.75504 0.156008 0.0780040 0.996953i \(-0.475145\pi\)
0.0780040 + 0.996953i \(0.475145\pi\)
\(930\) −33.1456 −1.08689
\(931\) −6.98221 −0.228833
\(932\) 29.0702 0.952226
\(933\) −6.03419 −0.197551
\(934\) −26.5230 −0.867860
\(935\) −7.70805 −0.252080
\(936\) −5.02954 −0.164396
\(937\) −50.9314 −1.66386 −0.831929 0.554882i \(-0.812763\pi\)
−0.831929 + 0.554882i \(0.812763\pi\)
\(938\) −0.892061 −0.0291268
\(939\) 47.4544 1.54862
\(940\) −46.1332 −1.50470
\(941\) −45.9920 −1.49930 −0.749648 0.661837i \(-0.769777\pi\)
−0.749648 + 0.661837i \(0.769777\pi\)
\(942\) −36.2514 −1.18113
\(943\) 40.3501 1.31398
\(944\) 1.27277 0.0414252
\(945\) −2.19253 −0.0713229
\(946\) −9.08305 −0.295315
\(947\) −21.9572 −0.713514 −0.356757 0.934197i \(-0.616118\pi\)
−0.356757 + 0.934197i \(0.616118\pi\)
\(948\) −3.05143 −0.0991058
\(949\) −0.418852 −0.0135965
\(950\) −20.6357 −0.669510
\(951\) 17.6719 0.573052
\(952\) −0.498556 −0.0161583
\(953\) −7.15874 −0.231894 −0.115947 0.993255i \(-0.536990\pi\)
−0.115947 + 0.993255i \(0.536990\pi\)
\(954\) 16.4301 0.531943
\(955\) −37.6348 −1.21783
\(956\) −43.9282 −1.42074
\(957\) 7.19561 0.232601
\(958\) 45.4000 1.46681
\(959\) 0.260600 0.00841522
\(960\) −94.3677 −3.04571
\(961\) −26.8395 −0.865790
\(962\) −58.6835 −1.89203
\(963\) 14.4876 0.466856
\(964\) 55.1104 1.77499
\(965\) 28.0231 0.902095
\(966\) −4.43782 −0.142785
\(967\) −15.7504 −0.506499 −0.253249 0.967401i \(-0.581499\pi\)
−0.253249 + 0.967401i \(0.581499\pi\)
\(968\) 19.6782 0.632482
\(969\) 3.61025 0.115978
\(970\) 148.721 4.77515
\(971\) 27.6116 0.886100 0.443050 0.896497i \(-0.353896\pi\)
0.443050 + 0.896497i \(0.353896\pi\)
\(972\) −22.4264 −0.719328
\(973\) −0.709095 −0.0227325
\(974\) 94.9751 3.04320
\(975\) 59.5963 1.90861
\(976\) −10.7543 −0.344237
\(977\) −6.07091 −0.194226 −0.0971129 0.995273i \(-0.530961\pi\)
−0.0971129 + 0.995273i \(0.530961\pi\)
\(978\) −20.3353 −0.650253
\(979\) 3.76572 0.120353
\(980\) −76.7791 −2.45262
\(981\) 2.70281 0.0862941
\(982\) −57.4310 −1.83270
\(983\) 45.2595 1.44355 0.721777 0.692125i \(-0.243326\pi\)
0.721777 + 0.692125i \(0.243326\pi\)
\(984\) −20.2725 −0.646265
\(985\) −34.8721 −1.11112
\(986\) 13.9897 0.445521
\(987\) −1.08501 −0.0345362
\(988\) 9.58847 0.305050
\(989\) −29.0355 −0.923275
\(990\) 6.96352 0.221315
\(991\) 35.1589 1.11686 0.558429 0.829552i \(-0.311404\pi\)
0.558429 + 0.829552i \(0.311404\pi\)
\(992\) 14.3614 0.455976
\(993\) −10.1389 −0.321748
\(994\) 0.374913 0.0118915
\(995\) 12.7779 0.405086
\(996\) 13.0262 0.412750
\(997\) −14.8559 −0.470490 −0.235245 0.971936i \(-0.575589\pi\)
−0.235245 + 0.971936i \(0.575589\pi\)
\(998\) 33.9922 1.07600
\(999\) −34.8856 −1.10373
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 4009.2.a.e.1.10 82
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4009.2.a.e.1.10 82 1.1 even 1 trivial