Properties

Label 4006.2.a.i.1.20
Level $4006$
Weight $2$
Character 4006.1
Self dual yes
Analytic conductor $31.988$
Analytic rank $0$
Dimension $46$
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] = [4006,2,Mod(1,4006)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4006, 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("4006.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4006.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: \(31.9880710497\)
Analytic rank: \(0\)
Dimension: \(46\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 4006.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.0490190 q^{3} +1.00000 q^{4} +0.547018 q^{5} +0.0490190 q^{6} -4.51424 q^{7} +1.00000 q^{8} -2.99760 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.0490190 q^{3} +1.00000 q^{4} +0.547018 q^{5} +0.0490190 q^{6} -4.51424 q^{7} +1.00000 q^{8} -2.99760 q^{9} +0.547018 q^{10} +0.850349 q^{11} +0.0490190 q^{12} +5.33838 q^{13} -4.51424 q^{14} +0.0268143 q^{15} +1.00000 q^{16} -2.19834 q^{17} -2.99760 q^{18} -0.355291 q^{19} +0.547018 q^{20} -0.221284 q^{21} +0.850349 q^{22} +4.91805 q^{23} +0.0490190 q^{24} -4.70077 q^{25} +5.33838 q^{26} -0.293996 q^{27} -4.51424 q^{28} -1.79123 q^{29} +0.0268143 q^{30} +10.6418 q^{31} +1.00000 q^{32} +0.0416832 q^{33} -2.19834 q^{34} -2.46937 q^{35} -2.99760 q^{36} +3.67532 q^{37} -0.355291 q^{38} +0.261682 q^{39} +0.547018 q^{40} +5.45355 q^{41} -0.221284 q^{42} -4.54506 q^{43} +0.850349 q^{44} -1.63974 q^{45} +4.91805 q^{46} +5.16120 q^{47} +0.0490190 q^{48} +13.3784 q^{49} -4.70077 q^{50} -0.107760 q^{51} +5.33838 q^{52} +3.22422 q^{53} -0.293996 q^{54} +0.465156 q^{55} -4.51424 q^{56} -0.0174160 q^{57} -1.79123 q^{58} -1.10040 q^{59} +0.0268143 q^{60} -5.65294 q^{61} +10.6418 q^{62} +13.5319 q^{63} +1.00000 q^{64} +2.92019 q^{65} +0.0416832 q^{66} +7.10303 q^{67} -2.19834 q^{68} +0.241078 q^{69} -2.46937 q^{70} +1.00310 q^{71} -2.99760 q^{72} -6.80012 q^{73} +3.67532 q^{74} -0.230427 q^{75} -0.355291 q^{76} -3.83868 q^{77} +0.261682 q^{78} +5.87430 q^{79} +0.547018 q^{80} +8.97838 q^{81} +5.45355 q^{82} -10.5420 q^{83} -0.221284 q^{84} -1.20253 q^{85} -4.54506 q^{86} -0.0878042 q^{87} +0.850349 q^{88} +8.57054 q^{89} -1.63974 q^{90} -24.0987 q^{91} +4.91805 q^{92} +0.521651 q^{93} +5.16120 q^{94} -0.194351 q^{95} +0.0490190 q^{96} +1.16982 q^{97} +13.3784 q^{98} -2.54900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46 q + 46 q^{2} + 21 q^{3} + 46 q^{4} + 23 q^{5} + 21 q^{6} + 26 q^{7} + 46 q^{8} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 46 q + 46 q^{2} + 21 q^{3} + 46 q^{4} + 23 q^{5} + 21 q^{6} + 26 q^{7} + 46 q^{8} + 59 q^{9} + 23 q^{10} + 39 q^{11} + 21 q^{12} + 8 q^{13} + 26 q^{14} + 14 q^{15} + 46 q^{16} + 36 q^{17} + 59 q^{18} + 37 q^{19} + 23 q^{20} + 20 q^{21} + 39 q^{22} + 38 q^{23} + 21 q^{24} + 57 q^{25} + 8 q^{26} + 63 q^{27} + 26 q^{28} + 23 q^{29} + 14 q^{30} + 44 q^{31} + 46 q^{32} + 25 q^{33} + 36 q^{34} + 26 q^{35} + 59 q^{36} + 9 q^{37} + 37 q^{38} - 2 q^{39} + 23 q^{40} + 50 q^{41} + 20 q^{42} + 46 q^{43} + 39 q^{44} + 30 q^{45} + 38 q^{46} + 57 q^{47} + 21 q^{48} + 62 q^{49} + 57 q^{50} + 5 q^{51} + 8 q^{52} + 21 q^{53} + 63 q^{54} + 40 q^{55} + 26 q^{56} + 3 q^{57} + 23 q^{58} + 68 q^{59} + 14 q^{60} - q^{61} + 44 q^{62} + 40 q^{63} + 46 q^{64} + 18 q^{65} + 25 q^{66} + 42 q^{67} + 36 q^{68} - 7 q^{69} + 26 q^{70} + 67 q^{71} + 59 q^{72} + 48 q^{73} + 9 q^{74} + 71 q^{75} + 37 q^{76} - 5 q^{77} - 2 q^{78} + 40 q^{79} + 23 q^{80} + 82 q^{81} + 50 q^{82} + 48 q^{83} + 20 q^{84} - 68 q^{85} + 46 q^{86} + 18 q^{87} + 39 q^{88} + 90 q^{89} + 30 q^{90} + 9 q^{91} + 38 q^{92} - 42 q^{93} + 57 q^{94} + 6 q^{95} + 21 q^{96} + 46 q^{97} + 62 q^{98} + 61 q^{99}+O(q^{100}) \) 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
\(3\) 0.0490190 0.0283011 0.0141506 0.999900i \(-0.495496\pi\)
0.0141506 + 0.999900i \(0.495496\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.547018 0.244634 0.122317 0.992491i \(-0.460968\pi\)
0.122317 + 0.992491i \(0.460968\pi\)
\(6\) 0.0490190 0.0200119
\(7\) −4.51424 −1.70622 −0.853112 0.521728i \(-0.825288\pi\)
−0.853112 + 0.521728i \(0.825288\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.99760 −0.999199
\(10\) 0.547018 0.172982
\(11\) 0.850349 0.256390 0.128195 0.991749i \(-0.459082\pi\)
0.128195 + 0.991749i \(0.459082\pi\)
\(12\) 0.0490190 0.0141506
\(13\) 5.33838 1.48060 0.740300 0.672277i \(-0.234683\pi\)
0.740300 + 0.672277i \(0.234683\pi\)
\(14\) −4.51424 −1.20648
\(15\) 0.0268143 0.00692342
\(16\) 1.00000 0.250000
\(17\) −2.19834 −0.533175 −0.266587 0.963811i \(-0.585896\pi\)
−0.266587 + 0.963811i \(0.585896\pi\)
\(18\) −2.99760 −0.706540
\(19\) −0.355291 −0.0815094 −0.0407547 0.999169i \(-0.512976\pi\)
−0.0407547 + 0.999169i \(0.512976\pi\)
\(20\) 0.547018 0.122317
\(21\) −0.221284 −0.0482880
\(22\) 0.850349 0.181295
\(23\) 4.91805 1.02548 0.512742 0.858542i \(-0.328630\pi\)
0.512742 + 0.858542i \(0.328630\pi\)
\(24\) 0.0490190 0.0100060
\(25\) −4.70077 −0.940154
\(26\) 5.33838 1.04694
\(27\) −0.293996 −0.0565796
\(28\) −4.51424 −0.853112
\(29\) −1.79123 −0.332623 −0.166311 0.986073i \(-0.553186\pi\)
−0.166311 + 0.986073i \(0.553186\pi\)
\(30\) 0.0268143 0.00489559
\(31\) 10.6418 1.91133 0.955663 0.294464i \(-0.0951411\pi\)
0.955663 + 0.294464i \(0.0951411\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.0416832 0.00725612
\(34\) −2.19834 −0.377011
\(35\) −2.46937 −0.417400
\(36\) −2.99760 −0.499600
\(37\) 3.67532 0.604218 0.302109 0.953273i \(-0.402309\pi\)
0.302109 + 0.953273i \(0.402309\pi\)
\(38\) −0.355291 −0.0576359
\(39\) 0.261682 0.0419026
\(40\) 0.547018 0.0864912
\(41\) 5.45355 0.851702 0.425851 0.904793i \(-0.359975\pi\)
0.425851 + 0.904793i \(0.359975\pi\)
\(42\) −0.221284 −0.0341448
\(43\) −4.54506 −0.693114 −0.346557 0.938029i \(-0.612649\pi\)
−0.346557 + 0.938029i \(0.612649\pi\)
\(44\) 0.850349 0.128195
\(45\) −1.63974 −0.244438
\(46\) 4.91805 0.725127
\(47\) 5.16120 0.752838 0.376419 0.926450i \(-0.377155\pi\)
0.376419 + 0.926450i \(0.377155\pi\)
\(48\) 0.0490190 0.00707528
\(49\) 13.3784 1.91120
\(50\) −4.70077 −0.664789
\(51\) −0.107760 −0.0150894
\(52\) 5.33838 0.740300
\(53\) 3.22422 0.442881 0.221441 0.975174i \(-0.428924\pi\)
0.221441 + 0.975174i \(0.428924\pi\)
\(54\) −0.293996 −0.0400078
\(55\) 0.465156 0.0627217
\(56\) −4.51424 −0.603241
\(57\) −0.0174160 −0.00230681
\(58\) −1.79123 −0.235200
\(59\) −1.10040 −0.143261 −0.0716303 0.997431i \(-0.522820\pi\)
−0.0716303 + 0.997431i \(0.522820\pi\)
\(60\) 0.0268143 0.00346171
\(61\) −5.65294 −0.723785 −0.361892 0.932220i \(-0.617869\pi\)
−0.361892 + 0.932220i \(0.617869\pi\)
\(62\) 10.6418 1.35151
\(63\) 13.5319 1.70486
\(64\) 1.00000 0.125000
\(65\) 2.92019 0.362205
\(66\) 0.0416832 0.00513085
\(67\) 7.10303 0.867773 0.433886 0.900968i \(-0.357142\pi\)
0.433886 + 0.900968i \(0.357142\pi\)
\(68\) −2.19834 −0.266587
\(69\) 0.241078 0.0290224
\(70\) −2.46937 −0.295147
\(71\) 1.00310 0.119047 0.0595233 0.998227i \(-0.481042\pi\)
0.0595233 + 0.998227i \(0.481042\pi\)
\(72\) −2.99760 −0.353270
\(73\) −6.80012 −0.795894 −0.397947 0.917408i \(-0.630277\pi\)
−0.397947 + 0.917408i \(0.630277\pi\)
\(74\) 3.67532 0.427247
\(75\) −0.230427 −0.0266074
\(76\) −0.355291 −0.0407547
\(77\) −3.83868 −0.437458
\(78\) 0.261682 0.0296296
\(79\) 5.87430 0.660910 0.330455 0.943822i \(-0.392798\pi\)
0.330455 + 0.943822i \(0.392798\pi\)
\(80\) 0.547018 0.0611585
\(81\) 8.97838 0.997598
\(82\) 5.45355 0.602244
\(83\) −10.5420 −1.15713 −0.578565 0.815636i \(-0.696387\pi\)
−0.578565 + 0.815636i \(0.696387\pi\)
\(84\) −0.221284 −0.0241440
\(85\) −1.20253 −0.130433
\(86\) −4.54506 −0.490106
\(87\) −0.0878042 −0.00941360
\(88\) 0.850349 0.0906475
\(89\) 8.57054 0.908475 0.454238 0.890881i \(-0.349912\pi\)
0.454238 + 0.890881i \(0.349912\pi\)
\(90\) −1.63974 −0.172844
\(91\) −24.0987 −2.52623
\(92\) 4.91805 0.512742
\(93\) 0.521651 0.0540926
\(94\) 5.16120 0.532337
\(95\) −0.194351 −0.0199400
\(96\) 0.0490190 0.00500298
\(97\) 1.16982 0.118777 0.0593885 0.998235i \(-0.481085\pi\)
0.0593885 + 0.998235i \(0.481085\pi\)
\(98\) 13.3784 1.35142
\(99\) −2.54900 −0.256184
\(100\) −4.70077 −0.470077
\(101\) 18.0610 1.79714 0.898568 0.438833i \(-0.144608\pi\)
0.898568 + 0.438833i \(0.144608\pi\)
\(102\) −0.107760 −0.0106698
\(103\) 14.5396 1.43263 0.716314 0.697778i \(-0.245828\pi\)
0.716314 + 0.697778i \(0.245828\pi\)
\(104\) 5.33838 0.523471
\(105\) −0.121046 −0.0118129
\(106\) 3.22422 0.313164
\(107\) 15.7386 1.52151 0.760756 0.649038i \(-0.224828\pi\)
0.760756 + 0.649038i \(0.224828\pi\)
\(108\) −0.293996 −0.0282898
\(109\) 16.9779 1.62619 0.813094 0.582132i \(-0.197782\pi\)
0.813094 + 0.582132i \(0.197782\pi\)
\(110\) 0.465156 0.0443509
\(111\) 0.180160 0.0171001
\(112\) −4.51424 −0.426556
\(113\) 13.4910 1.26912 0.634561 0.772873i \(-0.281181\pi\)
0.634561 + 0.772873i \(0.281181\pi\)
\(114\) −0.0174160 −0.00163116
\(115\) 2.69026 0.250868
\(116\) −1.79123 −0.166311
\(117\) −16.0023 −1.47941
\(118\) −1.10040 −0.101300
\(119\) 9.92382 0.909715
\(120\) 0.0268143 0.00244780
\(121\) −10.2769 −0.934264
\(122\) −5.65294 −0.511793
\(123\) 0.267327 0.0241041
\(124\) 10.6418 0.955663
\(125\) −5.30650 −0.474628
\(126\) 13.5319 1.20552
\(127\) 3.46526 0.307492 0.153746 0.988110i \(-0.450866\pi\)
0.153746 + 0.988110i \(0.450866\pi\)
\(128\) 1.00000 0.0883883
\(129\) −0.222794 −0.0196159
\(130\) 2.92019 0.256118
\(131\) −14.4600 −1.26338 −0.631688 0.775222i \(-0.717638\pi\)
−0.631688 + 0.775222i \(0.717638\pi\)
\(132\) 0.0416832 0.00362806
\(133\) 1.60387 0.139073
\(134\) 7.10303 0.613608
\(135\) −0.160821 −0.0138413
\(136\) −2.19834 −0.188506
\(137\) −0.325206 −0.0277842 −0.0138921 0.999903i \(-0.504422\pi\)
−0.0138921 + 0.999903i \(0.504422\pi\)
\(138\) 0.241078 0.0205219
\(139\) −1.59207 −0.135038 −0.0675190 0.997718i \(-0.521508\pi\)
−0.0675190 + 0.997718i \(0.521508\pi\)
\(140\) −2.46937 −0.208700
\(141\) 0.252997 0.0213062
\(142\) 1.00310 0.0841787
\(143\) 4.53948 0.379611
\(144\) −2.99760 −0.249800
\(145\) −0.979835 −0.0813709
\(146\) −6.80012 −0.562782
\(147\) 0.655795 0.0540891
\(148\) 3.67532 0.302109
\(149\) −15.9572 −1.30726 −0.653630 0.756814i \(-0.726755\pi\)
−0.653630 + 0.756814i \(0.726755\pi\)
\(150\) −0.230427 −0.0188143
\(151\) 9.08255 0.739127 0.369564 0.929205i \(-0.379507\pi\)
0.369564 + 0.929205i \(0.379507\pi\)
\(152\) −0.355291 −0.0288179
\(153\) 6.58972 0.532748
\(154\) −3.83868 −0.309330
\(155\) 5.82126 0.467575
\(156\) 0.261682 0.0209513
\(157\) −18.4756 −1.47451 −0.737257 0.675613i \(-0.763879\pi\)
−0.737257 + 0.675613i \(0.763879\pi\)
\(158\) 5.87430 0.467334
\(159\) 0.158048 0.0125340
\(160\) 0.547018 0.0432456
\(161\) −22.2013 −1.74971
\(162\) 8.97838 0.705408
\(163\) 19.2850 1.51052 0.755260 0.655426i \(-0.227511\pi\)
0.755260 + 0.655426i \(0.227511\pi\)
\(164\) 5.45355 0.425851
\(165\) 0.0228015 0.00177509
\(166\) −10.5420 −0.818214
\(167\) −1.56342 −0.120981 −0.0604904 0.998169i \(-0.519266\pi\)
−0.0604904 + 0.998169i \(0.519266\pi\)
\(168\) −0.221284 −0.0170724
\(169\) 15.4983 1.19218
\(170\) −1.20253 −0.0922298
\(171\) 1.06502 0.0814441
\(172\) −4.54506 −0.346557
\(173\) 5.22013 0.396879 0.198440 0.980113i \(-0.436413\pi\)
0.198440 + 0.980113i \(0.436413\pi\)
\(174\) −0.0878042 −0.00665642
\(175\) 21.2204 1.60411
\(176\) 0.850349 0.0640974
\(177\) −0.0539407 −0.00405443
\(178\) 8.57054 0.642389
\(179\) 14.1956 1.06103 0.530515 0.847675i \(-0.321999\pi\)
0.530515 + 0.847675i \(0.321999\pi\)
\(180\) −1.63974 −0.122219
\(181\) 17.6045 1.30853 0.654266 0.756265i \(-0.272978\pi\)
0.654266 + 0.756265i \(0.272978\pi\)
\(182\) −24.0987 −1.78632
\(183\) −0.277101 −0.0204839
\(184\) 4.91805 0.362564
\(185\) 2.01047 0.147812
\(186\) 0.521651 0.0382493
\(187\) −1.86935 −0.136700
\(188\) 5.16120 0.376419
\(189\) 1.32717 0.0965374
\(190\) −0.194351 −0.0140997
\(191\) 0.544050 0.0393661 0.0196830 0.999806i \(-0.493734\pi\)
0.0196830 + 0.999806i \(0.493734\pi\)
\(192\) 0.0490190 0.00353764
\(193\) −18.0871 −1.30194 −0.650969 0.759105i \(-0.725637\pi\)
−0.650969 + 0.759105i \(0.725637\pi\)
\(194\) 1.16982 0.0839881
\(195\) 0.143145 0.0102508
\(196\) 13.3784 0.955599
\(197\) −8.52995 −0.607734 −0.303867 0.952715i \(-0.598278\pi\)
−0.303867 + 0.952715i \(0.598278\pi\)
\(198\) −2.54900 −0.181150
\(199\) −18.0997 −1.28305 −0.641525 0.767102i \(-0.721698\pi\)
−0.641525 + 0.767102i \(0.721698\pi\)
\(200\) −4.70077 −0.332395
\(201\) 0.348183 0.0245589
\(202\) 18.0610 1.27077
\(203\) 8.08604 0.567529
\(204\) −0.107760 −0.00754472
\(205\) 2.98319 0.208355
\(206\) 14.5396 1.01302
\(207\) −14.7423 −1.02466
\(208\) 5.33838 0.370150
\(209\) −0.302122 −0.0208982
\(210\) −0.121046 −0.00835298
\(211\) 1.60334 0.110379 0.0551894 0.998476i \(-0.482424\pi\)
0.0551894 + 0.998476i \(0.482424\pi\)
\(212\) 3.22422 0.221441
\(213\) 0.0491712 0.00336915
\(214\) 15.7386 1.07587
\(215\) −2.48623 −0.169559
\(216\) −0.293996 −0.0200039
\(217\) −48.0397 −3.26115
\(218\) 16.9779 1.14989
\(219\) −0.333335 −0.0225247
\(220\) 0.465156 0.0313608
\(221\) −11.7355 −0.789418
\(222\) 0.180160 0.0120916
\(223\) 16.8365 1.12746 0.563728 0.825960i \(-0.309367\pi\)
0.563728 + 0.825960i \(0.309367\pi\)
\(224\) −4.51424 −0.301621
\(225\) 14.0910 0.939401
\(226\) 13.4910 0.897405
\(227\) −18.7675 −1.24564 −0.622821 0.782365i \(-0.714013\pi\)
−0.622821 + 0.782365i \(0.714013\pi\)
\(228\) −0.0174160 −0.00115340
\(229\) 9.78439 0.646571 0.323285 0.946302i \(-0.395213\pi\)
0.323285 + 0.946302i \(0.395213\pi\)
\(230\) 2.69026 0.177391
\(231\) −0.188168 −0.0123806
\(232\) −1.79123 −0.117600
\(233\) −29.9794 −1.96402 −0.982008 0.188841i \(-0.939527\pi\)
−0.982008 + 0.188841i \(0.939527\pi\)
\(234\) −16.0023 −1.04610
\(235\) 2.82327 0.184170
\(236\) −1.10040 −0.0716303
\(237\) 0.287952 0.0187045
\(238\) 9.92382 0.643266
\(239\) −2.66434 −0.172342 −0.0861709 0.996280i \(-0.527463\pi\)
−0.0861709 + 0.996280i \(0.527463\pi\)
\(240\) 0.0268143 0.00173085
\(241\) −27.4438 −1.76781 −0.883907 0.467663i \(-0.845096\pi\)
−0.883907 + 0.467663i \(0.845096\pi\)
\(242\) −10.2769 −0.660625
\(243\) 1.32210 0.0848127
\(244\) −5.65294 −0.361892
\(245\) 7.31822 0.467544
\(246\) 0.267327 0.0170442
\(247\) −1.89668 −0.120683
\(248\) 10.6418 0.675756
\(249\) −0.516756 −0.0327481
\(250\) −5.30650 −0.335612
\(251\) 26.0953 1.64712 0.823560 0.567230i \(-0.191985\pi\)
0.823560 + 0.567230i \(0.191985\pi\)
\(252\) 13.5319 0.852428
\(253\) 4.18206 0.262924
\(254\) 3.46526 0.217430
\(255\) −0.0589468 −0.00369139
\(256\) 1.00000 0.0625000
\(257\) −13.5155 −0.843074 −0.421537 0.906811i \(-0.638509\pi\)
−0.421537 + 0.906811i \(0.638509\pi\)
\(258\) −0.222794 −0.0138705
\(259\) −16.5913 −1.03093
\(260\) 2.92019 0.181103
\(261\) 5.36938 0.332356
\(262\) −14.4600 −0.893342
\(263\) 18.1005 1.11612 0.558061 0.829800i \(-0.311545\pi\)
0.558061 + 0.829800i \(0.311545\pi\)
\(264\) 0.0416832 0.00256542
\(265\) 1.76371 0.108344
\(266\) 1.60387 0.0983397
\(267\) 0.420119 0.0257109
\(268\) 7.10303 0.433886
\(269\) 2.39550 0.146056 0.0730282 0.997330i \(-0.476734\pi\)
0.0730282 + 0.997330i \(0.476734\pi\)
\(270\) −0.160821 −0.00978727
\(271\) −23.5917 −1.43309 −0.716546 0.697540i \(-0.754278\pi\)
−0.716546 + 0.697540i \(0.754278\pi\)
\(272\) −2.19834 −0.133294
\(273\) −1.18130 −0.0714953
\(274\) −0.325206 −0.0196464
\(275\) −3.99729 −0.241046
\(276\) 0.241078 0.0145112
\(277\) 22.1683 1.33196 0.665982 0.745968i \(-0.268013\pi\)
0.665982 + 0.745968i \(0.268013\pi\)
\(278\) −1.59207 −0.0954863
\(279\) −31.8999 −1.90979
\(280\) −2.46937 −0.147573
\(281\) 9.29990 0.554785 0.277393 0.960757i \(-0.410530\pi\)
0.277393 + 0.960757i \(0.410530\pi\)
\(282\) 0.252997 0.0150657
\(283\) 23.5009 1.39698 0.698490 0.715620i \(-0.253856\pi\)
0.698490 + 0.715620i \(0.253856\pi\)
\(284\) 1.00310 0.0595233
\(285\) −0.00952688 −0.000564324 0
\(286\) 4.53948 0.268425
\(287\) −24.6187 −1.45319
\(288\) −2.99760 −0.176635
\(289\) −12.1673 −0.715725
\(290\) −0.979835 −0.0575379
\(291\) 0.0573433 0.00336152
\(292\) −6.80012 −0.397947
\(293\) −11.1623 −0.652106 −0.326053 0.945351i \(-0.605719\pi\)
−0.326053 + 0.945351i \(0.605719\pi\)
\(294\) 0.655795 0.0382467
\(295\) −0.601942 −0.0350464
\(296\) 3.67532 0.213623
\(297\) −0.249999 −0.0145064
\(298\) −15.9572 −0.924373
\(299\) 26.2544 1.51833
\(300\) −0.230427 −0.0133037
\(301\) 20.5175 1.18261
\(302\) 9.08255 0.522642
\(303\) 0.885332 0.0508610
\(304\) −0.355291 −0.0203774
\(305\) −3.09226 −0.177062
\(306\) 6.58972 0.376709
\(307\) 3.01939 0.172326 0.0861629 0.996281i \(-0.472539\pi\)
0.0861629 + 0.996281i \(0.472539\pi\)
\(308\) −3.83868 −0.218729
\(309\) 0.712716 0.0405450
\(310\) 5.82126 0.330626
\(311\) −20.1603 −1.14319 −0.571593 0.820537i \(-0.693674\pi\)
−0.571593 + 0.820537i \(0.693674\pi\)
\(312\) 0.261682 0.0148148
\(313\) 0.175370 0.00991250 0.00495625 0.999988i \(-0.498422\pi\)
0.00495625 + 0.999988i \(0.498422\pi\)
\(314\) −18.4756 −1.04264
\(315\) 7.40219 0.417066
\(316\) 5.87430 0.330455
\(317\) −23.9922 −1.34754 −0.673769 0.738942i \(-0.735326\pi\)
−0.673769 + 0.738942i \(0.735326\pi\)
\(318\) 0.158048 0.00886290
\(319\) −1.52317 −0.0852811
\(320\) 0.547018 0.0305793
\(321\) 0.771492 0.0430605
\(322\) −22.2013 −1.23723
\(323\) 0.781050 0.0434588
\(324\) 8.97838 0.498799
\(325\) −25.0945 −1.39199
\(326\) 19.2850 1.06810
\(327\) 0.832239 0.0460229
\(328\) 5.45355 0.301122
\(329\) −23.2989 −1.28451
\(330\) 0.0228015 0.00125518
\(331\) −2.13517 −0.117359 −0.0586797 0.998277i \(-0.518689\pi\)
−0.0586797 + 0.998277i \(0.518689\pi\)
\(332\) −10.5420 −0.578565
\(333\) −11.0171 −0.603735
\(334\) −1.56342 −0.0855463
\(335\) 3.88549 0.212287
\(336\) −0.221284 −0.0120720
\(337\) −25.4435 −1.38600 −0.692999 0.720939i \(-0.743711\pi\)
−0.692999 + 0.720939i \(0.743711\pi\)
\(338\) 15.4983 0.842996
\(339\) 0.661313 0.0359176
\(340\) −1.20253 −0.0652163
\(341\) 9.04925 0.490044
\(342\) 1.06502 0.0575897
\(343\) −28.7936 −1.55471
\(344\) −4.54506 −0.245053
\(345\) 0.131874 0.00709986
\(346\) 5.22013 0.280636
\(347\) −23.1490 −1.24270 −0.621352 0.783531i \(-0.713416\pi\)
−0.621352 + 0.783531i \(0.713416\pi\)
\(348\) −0.0878042 −0.00470680
\(349\) −13.4485 −0.719884 −0.359942 0.932975i \(-0.617204\pi\)
−0.359942 + 0.932975i \(0.617204\pi\)
\(350\) 21.2204 1.13428
\(351\) −1.56946 −0.0837717
\(352\) 0.850349 0.0453237
\(353\) −2.93296 −0.156106 −0.0780529 0.996949i \(-0.524870\pi\)
−0.0780529 + 0.996949i \(0.524870\pi\)
\(354\) −0.0539407 −0.00286692
\(355\) 0.548717 0.0291229
\(356\) 8.57054 0.454238
\(357\) 0.486455 0.0257460
\(358\) 14.1956 0.750262
\(359\) 13.6695 0.721449 0.360725 0.932672i \(-0.382529\pi\)
0.360725 + 0.932672i \(0.382529\pi\)
\(360\) −1.63974 −0.0864219
\(361\) −18.8738 −0.993356
\(362\) 17.6045 0.925272
\(363\) −0.503763 −0.0264407
\(364\) −24.0987 −1.26312
\(365\) −3.71979 −0.194703
\(366\) −0.277101 −0.0144843
\(367\) 11.5125 0.600949 0.300474 0.953790i \(-0.402855\pi\)
0.300474 + 0.953790i \(0.402855\pi\)
\(368\) 4.91805 0.256371
\(369\) −16.3476 −0.851019
\(370\) 2.01047 0.104519
\(371\) −14.5549 −0.755654
\(372\) 0.521651 0.0270463
\(373\) −1.32606 −0.0686607 −0.0343303 0.999411i \(-0.510930\pi\)
−0.0343303 + 0.999411i \(0.510930\pi\)
\(374\) −1.86935 −0.0966618
\(375\) −0.260119 −0.0134325
\(376\) 5.16120 0.266168
\(377\) −9.56226 −0.492481
\(378\) 1.32717 0.0682622
\(379\) −19.0090 −0.976426 −0.488213 0.872724i \(-0.662351\pi\)
−0.488213 + 0.872724i \(0.662351\pi\)
\(380\) −0.194351 −0.00996999
\(381\) 0.169864 0.00870237
\(382\) 0.544050 0.0278360
\(383\) 11.7329 0.599521 0.299760 0.954015i \(-0.403093\pi\)
0.299760 + 0.954015i \(0.403093\pi\)
\(384\) 0.0490190 0.00250149
\(385\) −2.09983 −0.107017
\(386\) −18.0871 −0.920609
\(387\) 13.6242 0.692559
\(388\) 1.16982 0.0593885
\(389\) 24.3102 1.23257 0.616287 0.787522i \(-0.288636\pi\)
0.616287 + 0.787522i \(0.288636\pi\)
\(390\) 0.143145 0.00724842
\(391\) −10.8115 −0.546762
\(392\) 13.3784 0.675711
\(393\) −0.708815 −0.0357550
\(394\) −8.52995 −0.429732
\(395\) 3.21335 0.161681
\(396\) −2.54900 −0.128092
\(397\) 2.85651 0.143364 0.0716822 0.997428i \(-0.477163\pi\)
0.0716822 + 0.997428i \(0.477163\pi\)
\(398\) −18.0997 −0.907254
\(399\) 0.0786201 0.00393593
\(400\) −4.70077 −0.235039
\(401\) 12.4445 0.621448 0.310724 0.950500i \(-0.399428\pi\)
0.310724 + 0.950500i \(0.399428\pi\)
\(402\) 0.348183 0.0173658
\(403\) 56.8100 2.82991
\(404\) 18.0610 0.898568
\(405\) 4.91134 0.244046
\(406\) 8.08604 0.401304
\(407\) 3.12530 0.154915
\(408\) −0.107760 −0.00533492
\(409\) −13.0403 −0.644804 −0.322402 0.946603i \(-0.604490\pi\)
−0.322402 + 0.946603i \(0.604490\pi\)
\(410\) 2.98319 0.147329
\(411\) −0.0159413 −0.000786325 0
\(412\) 14.5396 0.716314
\(413\) 4.96750 0.244434
\(414\) −14.7423 −0.724546
\(415\) −5.76664 −0.283073
\(416\) 5.33838 0.261736
\(417\) −0.0780419 −0.00382173
\(418\) −0.302122 −0.0147772
\(419\) −7.28880 −0.356081 −0.178041 0.984023i \(-0.556976\pi\)
−0.178041 + 0.984023i \(0.556976\pi\)
\(420\) −0.121046 −0.00590645
\(421\) 29.7670 1.45075 0.725377 0.688351i \(-0.241665\pi\)
0.725377 + 0.688351i \(0.241665\pi\)
\(422\) 1.60334 0.0780496
\(423\) −15.4712 −0.752235
\(424\) 3.22422 0.156582
\(425\) 10.3339 0.501266
\(426\) 0.0491712 0.00238235
\(427\) 25.5187 1.23494
\(428\) 15.7386 0.760756
\(429\) 0.222521 0.0107434
\(430\) −2.48623 −0.119897
\(431\) 14.2404 0.685935 0.342968 0.939347i \(-0.388568\pi\)
0.342968 + 0.939347i \(0.388568\pi\)
\(432\) −0.293996 −0.0141449
\(433\) −0.908372 −0.0436536 −0.0218268 0.999762i \(-0.506948\pi\)
−0.0218268 + 0.999762i \(0.506948\pi\)
\(434\) −48.0397 −2.30598
\(435\) −0.0480305 −0.00230289
\(436\) 16.9779 0.813094
\(437\) −1.74734 −0.0835867
\(438\) −0.333335 −0.0159274
\(439\) 24.0351 1.14713 0.573567 0.819159i \(-0.305559\pi\)
0.573567 + 0.819159i \(0.305559\pi\)
\(440\) 0.465156 0.0221755
\(441\) −40.1030 −1.90967
\(442\) −11.7355 −0.558203
\(443\) 7.12066 0.338313 0.169156 0.985589i \(-0.445896\pi\)
0.169156 + 0.985589i \(0.445896\pi\)
\(444\) 0.180160 0.00855003
\(445\) 4.68824 0.222244
\(446\) 16.8365 0.797232
\(447\) −0.782203 −0.0369969
\(448\) −4.51424 −0.213278
\(449\) 12.7532 0.601862 0.300931 0.953646i \(-0.402703\pi\)
0.300931 + 0.953646i \(0.402703\pi\)
\(450\) 14.0910 0.664257
\(451\) 4.63742 0.218368
\(452\) 13.4910 0.634561
\(453\) 0.445217 0.0209181
\(454\) −18.7675 −0.880801
\(455\) −13.1825 −0.618003
\(456\) −0.0174160 −0.000815580 0
\(457\) 24.8661 1.16319 0.581593 0.813480i \(-0.302430\pi\)
0.581593 + 0.813480i \(0.302430\pi\)
\(458\) 9.78439 0.457194
\(459\) 0.646302 0.0301668
\(460\) 2.69026 0.125434
\(461\) −32.8411 −1.52956 −0.764781 0.644291i \(-0.777153\pi\)
−0.764781 + 0.644291i \(0.777153\pi\)
\(462\) −0.188168 −0.00875438
\(463\) 21.3501 0.992225 0.496113 0.868258i \(-0.334760\pi\)
0.496113 + 0.868258i \(0.334760\pi\)
\(464\) −1.79123 −0.0831557
\(465\) 0.285352 0.0132329
\(466\) −29.9794 −1.38877
\(467\) −31.2761 −1.44728 −0.723641 0.690176i \(-0.757533\pi\)
−0.723641 + 0.690176i \(0.757533\pi\)
\(468\) −16.0023 −0.739707
\(469\) −32.0648 −1.48061
\(470\) 2.82327 0.130228
\(471\) −0.905655 −0.0417304
\(472\) −1.10040 −0.0506502
\(473\) −3.86488 −0.177707
\(474\) 0.287952 0.0132261
\(475\) 1.67014 0.0766314
\(476\) 9.92382 0.454857
\(477\) −9.66493 −0.442527
\(478\) −2.66434 −0.121864
\(479\) 30.3216 1.38543 0.692715 0.721211i \(-0.256414\pi\)
0.692715 + 0.721211i \(0.256414\pi\)
\(480\) 0.0268143 0.00122390
\(481\) 19.6202 0.894606
\(482\) −27.4438 −1.25003
\(483\) −1.08828 −0.0495186
\(484\) −10.2769 −0.467132
\(485\) 0.639912 0.0290569
\(486\) 1.32210 0.0599716
\(487\) 23.4994 1.06486 0.532429 0.846474i \(-0.321279\pi\)
0.532429 + 0.846474i \(0.321279\pi\)
\(488\) −5.65294 −0.255897
\(489\) 0.945332 0.0427494
\(490\) 7.31822 0.330604
\(491\) −16.6852 −0.752994 −0.376497 0.926418i \(-0.622871\pi\)
−0.376497 + 0.926418i \(0.622871\pi\)
\(492\) 0.267327 0.0120521
\(493\) 3.93772 0.177346
\(494\) −1.89668 −0.0853357
\(495\) −1.39435 −0.0626714
\(496\) 10.6418 0.477831
\(497\) −4.52826 −0.203120
\(498\) −0.516756 −0.0231564
\(499\) 20.2918 0.908386 0.454193 0.890903i \(-0.349928\pi\)
0.454193 + 0.890903i \(0.349928\pi\)
\(500\) −5.30650 −0.237314
\(501\) −0.0766370 −0.00342389
\(502\) 26.0953 1.16469
\(503\) −3.68169 −0.164159 −0.0820793 0.996626i \(-0.526156\pi\)
−0.0820793 + 0.996626i \(0.526156\pi\)
\(504\) 13.5319 0.602758
\(505\) 9.87970 0.439641
\(506\) 4.18206 0.185915
\(507\) 0.759711 0.0337399
\(508\) 3.46526 0.153746
\(509\) 22.8932 1.01473 0.507363 0.861733i \(-0.330620\pi\)
0.507363 + 0.861733i \(0.330620\pi\)
\(510\) −0.0589468 −0.00261021
\(511\) 30.6974 1.35797
\(512\) 1.00000 0.0441942
\(513\) 0.104454 0.00461177
\(514\) −13.5155 −0.596143
\(515\) 7.95342 0.350470
\(516\) −0.222794 −0.00980796
\(517\) 4.38882 0.193020
\(518\) −16.5913 −0.728979
\(519\) 0.255885 0.0112321
\(520\) 2.92019 0.128059
\(521\) −2.33914 −0.102480 −0.0512398 0.998686i \(-0.516317\pi\)
−0.0512398 + 0.998686i \(0.516317\pi\)
\(522\) 5.36938 0.235011
\(523\) 31.1293 1.36119 0.680596 0.732659i \(-0.261721\pi\)
0.680596 + 0.732659i \(0.261721\pi\)
\(524\) −14.4600 −0.631688
\(525\) 1.04020 0.0453982
\(526\) 18.1005 0.789218
\(527\) −23.3943 −1.01907
\(528\) 0.0416832 0.00181403
\(529\) 1.18724 0.0516191
\(530\) 1.76371 0.0766107
\(531\) 3.29857 0.143146
\(532\) 1.60387 0.0695367
\(533\) 29.1131 1.26103
\(534\) 0.420119 0.0181803
\(535\) 8.60932 0.372214
\(536\) 7.10303 0.306804
\(537\) 0.695855 0.0300283
\(538\) 2.39550 0.103278
\(539\) 11.3763 0.490012
\(540\) −0.160821 −0.00692064
\(541\) −9.67810 −0.416094 −0.208047 0.978119i \(-0.566711\pi\)
−0.208047 + 0.978119i \(0.566711\pi\)
\(542\) −23.5917 −1.01335
\(543\) 0.862954 0.0370329
\(544\) −2.19834 −0.0942528
\(545\) 9.28722 0.397821
\(546\) −1.18130 −0.0505548
\(547\) −29.3785 −1.25614 −0.628068 0.778159i \(-0.716154\pi\)
−0.628068 + 0.778159i \(0.716154\pi\)
\(548\) −0.325206 −0.0138921
\(549\) 16.9452 0.723205
\(550\) −3.99729 −0.170445
\(551\) 0.636408 0.0271119
\(552\) 0.241078 0.0102610
\(553\) −26.5180 −1.12766
\(554\) 22.1683 0.941841
\(555\) 0.0985510 0.00418326
\(556\) −1.59207 −0.0675190
\(557\) −31.4834 −1.33400 −0.666998 0.745060i \(-0.732421\pi\)
−0.666998 + 0.745060i \(0.732421\pi\)
\(558\) −31.8999 −1.35043
\(559\) −24.2632 −1.02623
\(560\) −2.46937 −0.104350
\(561\) −0.0916337 −0.00386878
\(562\) 9.29990 0.392292
\(563\) 33.2445 1.40109 0.700543 0.713610i \(-0.252941\pi\)
0.700543 + 0.713610i \(0.252941\pi\)
\(564\) 0.252997 0.0106531
\(565\) 7.37980 0.310470
\(566\) 23.5009 0.987814
\(567\) −40.5306 −1.70212
\(568\) 1.00310 0.0420893
\(569\) 3.44296 0.144336 0.0721681 0.997392i \(-0.477008\pi\)
0.0721681 + 0.997392i \(0.477008\pi\)
\(570\) −0.00952688 −0.000399037 0
\(571\) 28.2755 1.18329 0.591646 0.806198i \(-0.298478\pi\)
0.591646 + 0.806198i \(0.298478\pi\)
\(572\) 4.53948 0.189805
\(573\) 0.0266688 0.00111410
\(574\) −24.6187 −1.02756
\(575\) −23.1186 −0.964114
\(576\) −2.99760 −0.124900
\(577\) −28.0531 −1.16786 −0.583932 0.811802i \(-0.698487\pi\)
−0.583932 + 0.811802i \(0.698487\pi\)
\(578\) −12.1673 −0.506094
\(579\) −0.886611 −0.0368463
\(580\) −0.979835 −0.0406854
\(581\) 47.5889 1.97432
\(582\) 0.0573433 0.00237696
\(583\) 2.74171 0.113550
\(584\) −6.80012 −0.281391
\(585\) −8.75356 −0.361915
\(586\) −11.1623 −0.461109
\(587\) 6.14993 0.253835 0.126917 0.991913i \(-0.459492\pi\)
0.126917 + 0.991913i \(0.459492\pi\)
\(588\) 0.655795 0.0270445
\(589\) −3.78094 −0.155791
\(590\) −0.601942 −0.0247815
\(591\) −0.418129 −0.0171995
\(592\) 3.67532 0.151055
\(593\) −8.74886 −0.359273 −0.179636 0.983733i \(-0.557492\pi\)
−0.179636 + 0.983733i \(0.557492\pi\)
\(594\) −0.249999 −0.0102576
\(595\) 5.42851 0.222547
\(596\) −15.9572 −0.653630
\(597\) −0.887227 −0.0363118
\(598\) 26.2544 1.07362
\(599\) −24.5956 −1.00495 −0.502475 0.864592i \(-0.667577\pi\)
−0.502475 + 0.864592i \(0.667577\pi\)
\(600\) −0.230427 −0.00940714
\(601\) −0.679717 −0.0277262 −0.0138631 0.999904i \(-0.504413\pi\)
−0.0138631 + 0.999904i \(0.504413\pi\)
\(602\) 20.5175 0.836230
\(603\) −21.2920 −0.867078
\(604\) 9.08255 0.369564
\(605\) −5.62166 −0.228553
\(606\) 0.885332 0.0359641
\(607\) 34.1855 1.38755 0.693773 0.720193i \(-0.255947\pi\)
0.693773 + 0.720193i \(0.255947\pi\)
\(608\) −0.355291 −0.0144090
\(609\) 0.396369 0.0160617
\(610\) −3.09226 −0.125202
\(611\) 27.5524 1.11465
\(612\) 6.58972 0.266374
\(613\) −32.7836 −1.32412 −0.662058 0.749452i \(-0.730317\pi\)
−0.662058 + 0.749452i \(0.730317\pi\)
\(614\) 3.01939 0.121853
\(615\) 0.146233 0.00589669
\(616\) −3.83868 −0.154665
\(617\) 16.5536 0.666423 0.333212 0.942852i \(-0.391868\pi\)
0.333212 + 0.942852i \(0.391868\pi\)
\(618\) 0.712716 0.0286696
\(619\) 25.7728 1.03590 0.517948 0.855412i \(-0.326696\pi\)
0.517948 + 0.855412i \(0.326696\pi\)
\(620\) 5.82126 0.233788
\(621\) −1.44589 −0.0580215
\(622\) −20.1603 −0.808355
\(623\) −38.6895 −1.55006
\(624\) 0.261682 0.0104757
\(625\) 20.6011 0.824044
\(626\) 0.175370 0.00700919
\(627\) −0.0148097 −0.000591442 0
\(628\) −18.4756 −0.737257
\(629\) −8.07958 −0.322154
\(630\) 7.40219 0.294910
\(631\) −14.1508 −0.563334 −0.281667 0.959512i \(-0.590887\pi\)
−0.281667 + 0.959512i \(0.590887\pi\)
\(632\) 5.87430 0.233667
\(633\) 0.0785943 0.00312384
\(634\) −23.9922 −0.952854
\(635\) 1.89556 0.0752230
\(636\) 0.158048 0.00626702
\(637\) 71.4189 2.82972
\(638\) −1.52317 −0.0603028
\(639\) −3.00690 −0.118951
\(640\) 0.547018 0.0216228
\(641\) −34.8017 −1.37458 −0.687292 0.726381i \(-0.741201\pi\)
−0.687292 + 0.726381i \(0.741201\pi\)
\(642\) 0.771492 0.0304484
\(643\) −28.4031 −1.12011 −0.560054 0.828456i \(-0.689219\pi\)
−0.560054 + 0.828456i \(0.689219\pi\)
\(644\) −22.2013 −0.874853
\(645\) −0.121872 −0.00479872
\(646\) 0.781050 0.0307300
\(647\) −37.8389 −1.48760 −0.743801 0.668401i \(-0.766979\pi\)
−0.743801 + 0.668401i \(0.766979\pi\)
\(648\) 8.97838 0.352704
\(649\) −0.935728 −0.0367305
\(650\) −25.0945 −0.984287
\(651\) −2.35486 −0.0922941
\(652\) 19.2850 0.755260
\(653\) 33.0285 1.29250 0.646252 0.763124i \(-0.276335\pi\)
0.646252 + 0.763124i \(0.276335\pi\)
\(654\) 0.832239 0.0325431
\(655\) −7.90989 −0.309065
\(656\) 5.45355 0.212925
\(657\) 20.3840 0.795257
\(658\) −23.2989 −0.908286
\(659\) −1.72785 −0.0673075 −0.0336537 0.999434i \(-0.510714\pi\)
−0.0336537 + 0.999434i \(0.510714\pi\)
\(660\) 0.0228015 0.000887547 0
\(661\) 7.06851 0.274933 0.137467 0.990506i \(-0.456104\pi\)
0.137467 + 0.990506i \(0.456104\pi\)
\(662\) −2.13517 −0.0829856
\(663\) −0.575264 −0.0223414
\(664\) −10.5420 −0.409107
\(665\) 0.877347 0.0340221
\(666\) −11.0171 −0.426905
\(667\) −8.80936 −0.341100
\(668\) −1.56342 −0.0604904
\(669\) 0.825309 0.0319083
\(670\) 3.88549 0.150109
\(671\) −4.80697 −0.185571
\(672\) −0.221284 −0.00853620
\(673\) 29.5820 1.14030 0.570151 0.821540i \(-0.306885\pi\)
0.570151 + 0.821540i \(0.306885\pi\)
\(674\) −25.4435 −0.980048
\(675\) 1.38201 0.0531935
\(676\) 15.4983 0.596088
\(677\) 38.0283 1.46154 0.730772 0.682621i \(-0.239160\pi\)
0.730772 + 0.682621i \(0.239160\pi\)
\(678\) 0.661313 0.0253976
\(679\) −5.28084 −0.202660
\(680\) −1.20253 −0.0461149
\(681\) −0.919963 −0.0352530
\(682\) 9.04925 0.346514
\(683\) 20.0533 0.767318 0.383659 0.923475i \(-0.374664\pi\)
0.383659 + 0.923475i \(0.374664\pi\)
\(684\) 1.06502 0.0407221
\(685\) −0.177894 −0.00679697
\(686\) −28.7936 −1.09934
\(687\) 0.479621 0.0182987
\(688\) −4.54506 −0.173279
\(689\) 17.2121 0.655730
\(690\) 0.131874 0.00502036
\(691\) −20.1682 −0.767235 −0.383618 0.923492i \(-0.625322\pi\)
−0.383618 + 0.923492i \(0.625322\pi\)
\(692\) 5.22013 0.198440
\(693\) 11.5068 0.437108
\(694\) −23.1490 −0.878725
\(695\) −0.870894 −0.0330349
\(696\) −0.0878042 −0.00332821
\(697\) −11.9887 −0.454106
\(698\) −13.4485 −0.509035
\(699\) −1.46956 −0.0555838
\(700\) 21.2204 0.802057
\(701\) −12.5956 −0.475728 −0.237864 0.971298i \(-0.576447\pi\)
−0.237864 + 0.971298i \(0.576447\pi\)
\(702\) −1.56946 −0.0592355
\(703\) −1.30581 −0.0492495
\(704\) 0.850349 0.0320487
\(705\) 0.138394 0.00521221
\(706\) −2.93296 −0.110384
\(707\) −81.5318 −3.06632
\(708\) −0.0539407 −0.00202722
\(709\) −13.0028 −0.488332 −0.244166 0.969733i \(-0.578514\pi\)
−0.244166 + 0.969733i \(0.578514\pi\)
\(710\) 0.548717 0.0205930
\(711\) −17.6088 −0.660381
\(712\) 8.57054 0.321195
\(713\) 52.3370 1.96003
\(714\) 0.486455 0.0182051
\(715\) 2.48318 0.0928657
\(716\) 14.1956 0.530515
\(717\) −0.130603 −0.00487746
\(718\) 13.6695 0.510142
\(719\) −52.2726 −1.94944 −0.974719 0.223435i \(-0.928273\pi\)
−0.974719 + 0.223435i \(0.928273\pi\)
\(720\) −1.63974 −0.0611095
\(721\) −65.6352 −2.44438
\(722\) −18.8738 −0.702409
\(723\) −1.34527 −0.0500311
\(724\) 17.6045 0.654266
\(725\) 8.42016 0.312717
\(726\) −0.503763 −0.0186964
\(727\) 12.3230 0.457033 0.228517 0.973540i \(-0.426612\pi\)
0.228517 + 0.973540i \(0.426612\pi\)
\(728\) −24.0987 −0.893159
\(729\) −26.8703 −0.995197
\(730\) −3.71979 −0.137676
\(731\) 9.99155 0.369551
\(732\) −0.277101 −0.0102420
\(733\) −6.57385 −0.242811 −0.121405 0.992603i \(-0.538740\pi\)
−0.121405 + 0.992603i \(0.538740\pi\)
\(734\) 11.5125 0.424935
\(735\) 0.358732 0.0132320
\(736\) 4.91805 0.181282
\(737\) 6.04005 0.222488
\(738\) −16.3476 −0.601762
\(739\) −15.3370 −0.564180 −0.282090 0.959388i \(-0.591028\pi\)
−0.282090 + 0.959388i \(0.591028\pi\)
\(740\) 2.01047 0.0739062
\(741\) −0.0929733 −0.00341546
\(742\) −14.5549 −0.534328
\(743\) −11.8513 −0.434781 −0.217390 0.976085i \(-0.569754\pi\)
−0.217390 + 0.976085i \(0.569754\pi\)
\(744\) 0.521651 0.0191246
\(745\) −8.72886 −0.319801
\(746\) −1.32606 −0.0485504
\(747\) 31.6005 1.15620
\(748\) −1.86935 −0.0683502
\(749\) −71.0480 −2.59604
\(750\) −0.260119 −0.00949821
\(751\) −16.9962 −0.620200 −0.310100 0.950704i \(-0.600363\pi\)
−0.310100 + 0.950704i \(0.600363\pi\)
\(752\) 5.16120 0.188209
\(753\) 1.27916 0.0466153
\(754\) −9.56226 −0.348237
\(755\) 4.96832 0.180816
\(756\) 1.32717 0.0482687
\(757\) −35.2526 −1.28128 −0.640639 0.767842i \(-0.721331\pi\)
−0.640639 + 0.767842i \(0.721331\pi\)
\(758\) −19.0090 −0.690437
\(759\) 0.205000 0.00744104
\(760\) −0.194351 −0.00704985
\(761\) 8.11829 0.294288 0.147144 0.989115i \(-0.452992\pi\)
0.147144 + 0.989115i \(0.452992\pi\)
\(762\) 0.169864 0.00615351
\(763\) −76.6424 −2.77464
\(764\) 0.544050 0.0196830
\(765\) 3.60470 0.130328
\(766\) 11.7329 0.423925
\(767\) −5.87438 −0.212112
\(768\) 0.0490190 0.00176882
\(769\) 29.6433 1.06896 0.534482 0.845180i \(-0.320507\pi\)
0.534482 + 0.845180i \(0.320507\pi\)
\(770\) −2.09983 −0.0756726
\(771\) −0.662516 −0.0238599
\(772\) −18.0871 −0.650969
\(773\) 45.5742 1.63919 0.819595 0.572944i \(-0.194199\pi\)
0.819595 + 0.572944i \(0.194199\pi\)
\(774\) 13.6242 0.489713
\(775\) −50.0247 −1.79694
\(776\) 1.16982 0.0419940
\(777\) −0.813287 −0.0291765
\(778\) 24.3102 0.871561
\(779\) −1.93760 −0.0694217
\(780\) 0.143145 0.00512541
\(781\) 0.852989 0.0305223
\(782\) −10.8115 −0.386619
\(783\) 0.526614 0.0188197
\(784\) 13.3784 0.477800
\(785\) −10.1065 −0.360716
\(786\) −0.708815 −0.0252826
\(787\) −26.8828 −0.958267 −0.479133 0.877742i \(-0.659049\pi\)
−0.479133 + 0.877742i \(0.659049\pi\)
\(788\) −8.52995 −0.303867
\(789\) 0.887266 0.0315875
\(790\) 3.21335 0.114326
\(791\) −60.9014 −2.16541
\(792\) −2.54900 −0.0905749
\(793\) −30.1775 −1.07164
\(794\) 2.85651 0.101374
\(795\) 0.0864552 0.00306625
\(796\) −18.0997 −0.641525
\(797\) −24.0564 −0.852123 −0.426061 0.904694i \(-0.640099\pi\)
−0.426061 + 0.904694i \(0.640099\pi\)
\(798\) 0.0786201 0.00278312
\(799\) −11.3460 −0.401394
\(800\) −4.70077 −0.166197
\(801\) −25.6910 −0.907748
\(802\) 12.4445 0.439430
\(803\) −5.78247 −0.204059
\(804\) 0.348183 0.0122795
\(805\) −12.1445 −0.428038
\(806\) 56.8100 2.00105
\(807\) 0.117425 0.00413356
\(808\) 18.0610 0.635384
\(809\) −36.0928 −1.26896 −0.634478 0.772941i \(-0.718785\pi\)
−0.634478 + 0.772941i \(0.718785\pi\)
\(810\) 4.91134 0.172567
\(811\) −30.4154 −1.06803 −0.534015 0.845475i \(-0.679317\pi\)
−0.534015 + 0.845475i \(0.679317\pi\)
\(812\) 8.08604 0.283764
\(813\) −1.15644 −0.0405581
\(814\) 3.12530 0.109542
\(815\) 10.5493 0.369524
\(816\) −0.107760 −0.00377236
\(817\) 1.61482 0.0564954
\(818\) −13.0403 −0.455945
\(819\) 72.2383 2.52421
\(820\) 2.98319 0.104178
\(821\) 12.8725 0.449253 0.224626 0.974445i \(-0.427884\pi\)
0.224626 + 0.974445i \(0.427884\pi\)
\(822\) −0.0159413 −0.000556015 0
\(823\) −3.25035 −0.113300 −0.0566501 0.998394i \(-0.518042\pi\)
−0.0566501 + 0.998394i \(0.518042\pi\)
\(824\) 14.5396 0.506510
\(825\) −0.195943 −0.00682187
\(826\) 4.96750 0.172841
\(827\) −17.6675 −0.614359 −0.307180 0.951652i \(-0.599385\pi\)
−0.307180 + 0.951652i \(0.599385\pi\)
\(828\) −14.7423 −0.512332
\(829\) 31.1367 1.08142 0.540711 0.841209i \(-0.318155\pi\)
0.540711 + 0.841209i \(0.318155\pi\)
\(830\) −5.76664 −0.200163
\(831\) 1.08667 0.0376961
\(832\) 5.33838 0.185075
\(833\) −29.4102 −1.01900
\(834\) −0.0780419 −0.00270237
\(835\) −0.855217 −0.0295960
\(836\) −0.302122 −0.0104491
\(837\) −3.12865 −0.108142
\(838\) −7.28880 −0.251788
\(839\) 23.1978 0.800878 0.400439 0.916323i \(-0.368858\pi\)
0.400439 + 0.916323i \(0.368858\pi\)
\(840\) −0.121046 −0.00417649
\(841\) −25.7915 −0.889362
\(842\) 29.7670 1.02584
\(843\) 0.455871 0.0157010
\(844\) 1.60334 0.0551894
\(845\) 8.47785 0.291647
\(846\) −15.4712 −0.531910
\(847\) 46.3925 1.59406
\(848\) 3.22422 0.110720
\(849\) 1.15199 0.0395361
\(850\) 10.3339 0.354449
\(851\) 18.0754 0.619617
\(852\) 0.0491712 0.00168458
\(853\) −0.972998 −0.0333148 −0.0166574 0.999861i \(-0.505302\pi\)
−0.0166574 + 0.999861i \(0.505302\pi\)
\(854\) 25.5187 0.873233
\(855\) 0.582586 0.0199240
\(856\) 15.7386 0.537936
\(857\) −2.17457 −0.0742820 −0.0371410 0.999310i \(-0.511825\pi\)
−0.0371410 + 0.999310i \(0.511825\pi\)
\(858\) 0.222521 0.00759674
\(859\) 24.5912 0.839041 0.419521 0.907746i \(-0.362198\pi\)
0.419521 + 0.907746i \(0.362198\pi\)
\(860\) −2.48623 −0.0847797
\(861\) −1.20678 −0.0411270
\(862\) 14.2404 0.485029
\(863\) 4.24342 0.144448 0.0722239 0.997388i \(-0.476990\pi\)
0.0722239 + 0.997388i \(0.476990\pi\)
\(864\) −0.293996 −0.0100019
\(865\) 2.85551 0.0970902
\(866\) −0.908372 −0.0308677
\(867\) −0.596430 −0.0202558
\(868\) −48.0397 −1.63057
\(869\) 4.99520 0.169451
\(870\) −0.0480305 −0.00162839
\(871\) 37.9186 1.28482
\(872\) 16.9779 0.574944
\(873\) −3.50664 −0.118682
\(874\) −1.74734 −0.0591047
\(875\) 23.9548 0.809821
\(876\) −0.333335 −0.0112623
\(877\) −7.86293 −0.265512 −0.132756 0.991149i \(-0.542383\pi\)
−0.132756 + 0.991149i \(0.542383\pi\)
\(878\) 24.0351 0.811146
\(879\) −0.547162 −0.0184553
\(880\) 0.465156 0.0156804
\(881\) 7.18705 0.242138 0.121069 0.992644i \(-0.461368\pi\)
0.121069 + 0.992644i \(0.461368\pi\)
\(882\) −40.1030 −1.35034
\(883\) 20.9108 0.703705 0.351852 0.936055i \(-0.385552\pi\)
0.351852 + 0.936055i \(0.385552\pi\)
\(884\) −11.7355 −0.394709
\(885\) −0.0295066 −0.000991852 0
\(886\) 7.12066 0.239223
\(887\) −2.21558 −0.0743920 −0.0371960 0.999308i \(-0.511843\pi\)
−0.0371960 + 0.999308i \(0.511843\pi\)
\(888\) 0.180160 0.00604578
\(889\) −15.6430 −0.524650
\(890\) 4.68824 0.157150
\(891\) 7.63475 0.255774
\(892\) 16.8365 0.563728
\(893\) −1.83373 −0.0613634
\(894\) −0.782203 −0.0261608
\(895\) 7.76526 0.259564
\(896\) −4.51424 −0.150810
\(897\) 1.28697 0.0429705
\(898\) 12.7532 0.425580
\(899\) −19.0619 −0.635750
\(900\) 14.0910 0.469701
\(901\) −7.08793 −0.236133
\(902\) 4.63742 0.154409
\(903\) 1.00575 0.0334691
\(904\) 13.4910 0.448702
\(905\) 9.62998 0.320111
\(906\) 0.445217 0.0147914
\(907\) 29.7125 0.986587 0.493294 0.869863i \(-0.335793\pi\)
0.493294 + 0.869863i \(0.335793\pi\)
\(908\) −18.7675 −0.622821
\(909\) −54.1396 −1.79570
\(910\) −13.1825 −0.436994
\(911\) 29.8920 0.990367 0.495183 0.868789i \(-0.335101\pi\)
0.495183 + 0.868789i \(0.335101\pi\)
\(912\) −0.0174160 −0.000576702 0
\(913\) −8.96434 −0.296676
\(914\) 24.8661 0.822497
\(915\) −0.151579 −0.00501106
\(916\) 9.78439 0.323285
\(917\) 65.2760 2.15560
\(918\) 0.646302 0.0213311
\(919\) −9.64806 −0.318260 −0.159130 0.987258i \(-0.550869\pi\)
−0.159130 + 0.987258i \(0.550869\pi\)
\(920\) 2.69026 0.0886954
\(921\) 0.148007 0.00487701
\(922\) −32.8411 −1.08156
\(923\) 5.35495 0.176260
\(924\) −0.188168 −0.00619028
\(925\) −17.2768 −0.568059
\(926\) 21.3501 0.701609
\(927\) −43.5838 −1.43148
\(928\) −1.79123 −0.0588000
\(929\) 33.5306 1.10010 0.550051 0.835131i \(-0.314608\pi\)
0.550051 + 0.835131i \(0.314608\pi\)
\(930\) 0.285352 0.00935707
\(931\) −4.75323 −0.155781
\(932\) −29.9794 −0.982008
\(933\) −0.988238 −0.0323534
\(934\) −31.2761 −1.02338
\(935\) −1.02257 −0.0334416
\(936\) −16.0023 −0.523052
\(937\) −31.5401 −1.03037 −0.515185 0.857079i \(-0.672277\pi\)
−0.515185 + 0.857079i \(0.672277\pi\)
\(938\) −32.0648 −1.04695
\(939\) 0.00859646 0.000280535 0
\(940\) 2.82327 0.0920849
\(941\) 2.60195 0.0848212 0.0424106 0.999100i \(-0.486496\pi\)
0.0424106 + 0.999100i \(0.486496\pi\)
\(942\) −0.905655 −0.0295078
\(943\) 26.8209 0.873407
\(944\) −1.10040 −0.0358151
\(945\) 0.725986 0.0236163
\(946\) −3.86488 −0.125658
\(947\) 11.9055 0.386876 0.193438 0.981113i \(-0.438036\pi\)
0.193438 + 0.981113i \(0.438036\pi\)
\(948\) 0.287952 0.00935225
\(949\) −36.3016 −1.17840
\(950\) 1.67014 0.0541866
\(951\) −1.17608 −0.0381368
\(952\) 9.92382 0.321633
\(953\) −24.3663 −0.789303 −0.394652 0.918831i \(-0.629135\pi\)
−0.394652 + 0.918831i \(0.629135\pi\)
\(954\) −9.66493 −0.312914
\(955\) 0.297605 0.00963028
\(956\) −2.66434 −0.0861709
\(957\) −0.0746642 −0.00241355
\(958\) 30.3216 0.979647
\(959\) 1.46806 0.0474061
\(960\) 0.0268143 0.000865427 0
\(961\) 82.2481 2.65316
\(962\) 19.6202 0.632582
\(963\) −47.1781 −1.52029
\(964\) −27.4438 −0.883907
\(965\) −9.89397 −0.318498
\(966\) −1.08828 −0.0350150
\(967\) 1.54217 0.0495928 0.0247964 0.999693i \(-0.492106\pi\)
0.0247964 + 0.999693i \(0.492106\pi\)
\(968\) −10.2769 −0.330312
\(969\) 0.0382862 0.00122993
\(970\) 0.639912 0.0205463
\(971\) 20.9905 0.673616 0.336808 0.941573i \(-0.390653\pi\)
0.336808 + 0.941573i \(0.390653\pi\)
\(972\) 1.32210 0.0424064
\(973\) 7.18701 0.230405
\(974\) 23.4994 0.752969
\(975\) −1.23011 −0.0393949
\(976\) −5.65294 −0.180946
\(977\) −20.6989 −0.662217 −0.331109 0.943593i \(-0.607423\pi\)
−0.331109 + 0.943593i \(0.607423\pi\)
\(978\) 0.945332 0.0302284
\(979\) 7.28795 0.232924
\(980\) 7.31822 0.233772
\(981\) −50.8929 −1.62489
\(982\) −16.6852 −0.532447
\(983\) −48.6672 −1.55224 −0.776122 0.630583i \(-0.782816\pi\)
−0.776122 + 0.630583i \(0.782816\pi\)
\(984\) 0.267327 0.00852209
\(985\) −4.66604 −0.148672
\(986\) 3.93772 0.125403
\(987\) −1.14209 −0.0363531
\(988\) −1.89668 −0.0603414
\(989\) −22.3528 −0.710778
\(990\) −1.39435 −0.0443154
\(991\) −43.1229 −1.36984 −0.684922 0.728616i \(-0.740164\pi\)
−0.684922 + 0.728616i \(0.740164\pi\)
\(992\) 10.6418 0.337878
\(993\) −0.104664 −0.00332140
\(994\) −4.52826 −0.143628
\(995\) −9.90085 −0.313878
\(996\) −0.516756 −0.0163740
\(997\) −4.15228 −0.131504 −0.0657520 0.997836i \(-0.520945\pi\)
−0.0657520 + 0.997836i \(0.520945\pi\)
\(998\) 20.2918 0.642326
\(999\) −1.08053 −0.0341864
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 4006.2.a.i.1.20 46
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4006.2.a.i.1.20 46 1.1 even 1 trivial