Properties

Label 8018.2.a.j.1.25
Level $8018$
Weight $2$
Character 8018.1
Self dual yes
Analytic conductor $64.024$
Analytic rank $0$
Dimension $47$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8018,2,Mod(1,8018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8018, 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("8018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8018 = 2 \cdot 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8018.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: \(64.0240523407\)
Analytic rank: \(0\)
Dimension: \(47\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.25
Character \(\chi\) \(=\) 8018.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.731229 q^{3} +1.00000 q^{4} +3.24153 q^{5} +0.731229 q^{6} +4.02612 q^{7} +1.00000 q^{8} -2.46530 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.731229 q^{3} +1.00000 q^{4} +3.24153 q^{5} +0.731229 q^{6} +4.02612 q^{7} +1.00000 q^{8} -2.46530 q^{9} +3.24153 q^{10} +1.64443 q^{11} +0.731229 q^{12} +2.18437 q^{13} +4.02612 q^{14} +2.37030 q^{15} +1.00000 q^{16} -2.11672 q^{17} -2.46530 q^{18} -1.00000 q^{19} +3.24153 q^{20} +2.94402 q^{21} +1.64443 q^{22} +7.93136 q^{23} +0.731229 q^{24} +5.50755 q^{25} +2.18437 q^{26} -3.99639 q^{27} +4.02612 q^{28} -7.04227 q^{29} +2.37030 q^{30} +2.96493 q^{31} +1.00000 q^{32} +1.20246 q^{33} -2.11672 q^{34} +13.0508 q^{35} -2.46530 q^{36} -1.09154 q^{37} -1.00000 q^{38} +1.59728 q^{39} +3.24153 q^{40} +5.86384 q^{41} +2.94402 q^{42} -10.0819 q^{43} +1.64443 q^{44} -7.99137 q^{45} +7.93136 q^{46} +7.44446 q^{47} +0.731229 q^{48} +9.20968 q^{49} +5.50755 q^{50} -1.54780 q^{51} +2.18437 q^{52} -1.80240 q^{53} -3.99639 q^{54} +5.33048 q^{55} +4.02612 q^{56} -0.731229 q^{57} -7.04227 q^{58} -8.17643 q^{59} +2.37030 q^{60} -9.55710 q^{61} +2.96493 q^{62} -9.92562 q^{63} +1.00000 q^{64} +7.08072 q^{65} +1.20246 q^{66} +10.0295 q^{67} -2.11672 q^{68} +5.79964 q^{69} +13.0508 q^{70} +2.43109 q^{71} -2.46530 q^{72} +0.129250 q^{73} -1.09154 q^{74} +4.02728 q^{75} -1.00000 q^{76} +6.62069 q^{77} +1.59728 q^{78} +2.09468 q^{79} +3.24153 q^{80} +4.47364 q^{81} +5.86384 q^{82} -1.41701 q^{83} +2.94402 q^{84} -6.86141 q^{85} -10.0819 q^{86} -5.14951 q^{87} +1.64443 q^{88} -7.58566 q^{89} -7.99137 q^{90} +8.79456 q^{91} +7.93136 q^{92} +2.16804 q^{93} +7.44446 q^{94} -3.24153 q^{95} +0.731229 q^{96} +1.83031 q^{97} +9.20968 q^{98} -4.05402 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 47 q + 47 q^{2} + 10 q^{3} + 47 q^{4} + 15 q^{5} + 10 q^{6} + 47 q^{8} + 69 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 47 q + 47 q^{2} + 10 q^{3} + 47 q^{4} + 15 q^{5} + 10 q^{6} + 47 q^{8} + 69 q^{9} + 15 q^{10} + 17 q^{11} + 10 q^{12} + 27 q^{13} + 10 q^{15} + 47 q^{16} + 16 q^{17} + 69 q^{18} - 47 q^{19} + 15 q^{20} + 26 q^{21} + 17 q^{22} + 24 q^{23} + 10 q^{24} + 86 q^{25} + 27 q^{26} + 43 q^{27} + 58 q^{29} + 10 q^{30} + 21 q^{31} + 47 q^{32} + 18 q^{33} + 16 q^{34} + 24 q^{35} + 69 q^{36} + 78 q^{37} - 47 q^{38} - 4 q^{39} + 15 q^{40} + 53 q^{41} + 26 q^{42} + 47 q^{43} + 17 q^{44} + 23 q^{45} + 24 q^{46} + 16 q^{47} + 10 q^{48} + 93 q^{49} + 86 q^{50} + 28 q^{51} + 27 q^{52} + 43 q^{53} + 43 q^{54} + 23 q^{55} - 10 q^{57} + 58 q^{58} + 3 q^{59} + 10 q^{60} + 12 q^{61} + 21 q^{62} - 5 q^{63} + 47 q^{64} + 62 q^{65} + 18 q^{66} + 68 q^{67} + 16 q^{68} + 12 q^{69} + 24 q^{70} - 13 q^{71} + 69 q^{72} + 35 q^{73} + 78 q^{74} + 22 q^{75} - 47 q^{76} + 26 q^{77} - 4 q^{78} + 21 q^{79} + 15 q^{80} + 123 q^{81} + 53 q^{82} + 23 q^{83} + 26 q^{84} + 38 q^{85} + 47 q^{86} + 39 q^{87} + 17 q^{88} + 24 q^{89} + 23 q^{90} + 49 q^{91} + 24 q^{92} + 91 q^{93} + 16 q^{94} - 15 q^{95} + 10 q^{96} + 88 q^{97} + 93 q^{98} + 26 q^{99}+O(q^{100}) \) 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
\(3\) 0.731229 0.422175 0.211088 0.977467i \(-0.432299\pi\)
0.211088 + 0.977467i \(0.432299\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.24153 1.44966 0.724829 0.688929i \(-0.241919\pi\)
0.724829 + 0.688929i \(0.241919\pi\)
\(6\) 0.731229 0.298523
\(7\) 4.02612 1.52173 0.760866 0.648909i \(-0.224774\pi\)
0.760866 + 0.648909i \(0.224774\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.46530 −0.821768
\(10\) 3.24153 1.02506
\(11\) 1.64443 0.495815 0.247907 0.968784i \(-0.420257\pi\)
0.247907 + 0.968784i \(0.420257\pi\)
\(12\) 0.731229 0.211088
\(13\) 2.18437 0.605836 0.302918 0.953017i \(-0.402039\pi\)
0.302918 + 0.953017i \(0.402039\pi\)
\(14\) 4.02612 1.07603
\(15\) 2.37030 0.612010
\(16\) 1.00000 0.250000
\(17\) −2.11672 −0.513379 −0.256690 0.966494i \(-0.582632\pi\)
−0.256690 + 0.966494i \(0.582632\pi\)
\(18\) −2.46530 −0.581078
\(19\) −1.00000 −0.229416
\(20\) 3.24153 0.724829
\(21\) 2.94402 0.642438
\(22\) 1.64443 0.350594
\(23\) 7.93136 1.65380 0.826902 0.562346i \(-0.190101\pi\)
0.826902 + 0.562346i \(0.190101\pi\)
\(24\) 0.731229 0.149261
\(25\) 5.50755 1.10151
\(26\) 2.18437 0.428391
\(27\) −3.99639 −0.769105
\(28\) 4.02612 0.760866
\(29\) −7.04227 −1.30772 −0.653858 0.756617i \(-0.726851\pi\)
−0.653858 + 0.756617i \(0.726851\pi\)
\(30\) 2.37030 0.432756
\(31\) 2.96493 0.532516 0.266258 0.963902i \(-0.414213\pi\)
0.266258 + 0.963902i \(0.414213\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.20246 0.209321
\(34\) −2.11672 −0.363014
\(35\) 13.0508 2.20599
\(36\) −2.46530 −0.410884
\(37\) −1.09154 −0.179448 −0.0897240 0.995967i \(-0.528598\pi\)
−0.0897240 + 0.995967i \(0.528598\pi\)
\(38\) −1.00000 −0.162221
\(39\) 1.59728 0.255769
\(40\) 3.24153 0.512532
\(41\) 5.86384 0.915778 0.457889 0.889009i \(-0.348606\pi\)
0.457889 + 0.889009i \(0.348606\pi\)
\(42\) 2.94402 0.454272
\(43\) −10.0819 −1.53748 −0.768741 0.639560i \(-0.779116\pi\)
−0.768741 + 0.639560i \(0.779116\pi\)
\(44\) 1.64443 0.247907
\(45\) −7.99137 −1.19128
\(46\) 7.93136 1.16942
\(47\) 7.44446 1.08589 0.542943 0.839769i \(-0.317310\pi\)
0.542943 + 0.839769i \(0.317310\pi\)
\(48\) 0.731229 0.105544
\(49\) 9.20968 1.31567
\(50\) 5.50755 0.778885
\(51\) −1.54780 −0.216736
\(52\) 2.18437 0.302918
\(53\) −1.80240 −0.247579 −0.123789 0.992309i \(-0.539505\pi\)
−0.123789 + 0.992309i \(0.539505\pi\)
\(54\) −3.99639 −0.543840
\(55\) 5.33048 0.718762
\(56\) 4.02612 0.538014
\(57\) −0.731229 −0.0968536
\(58\) −7.04227 −0.924695
\(59\) −8.17643 −1.06448 −0.532240 0.846593i \(-0.678650\pi\)
−0.532240 + 0.846593i \(0.678650\pi\)
\(60\) 2.37030 0.306005
\(61\) −9.55710 −1.22366 −0.611831 0.790989i \(-0.709567\pi\)
−0.611831 + 0.790989i \(0.709567\pi\)
\(62\) 2.96493 0.376546
\(63\) −9.92562 −1.25051
\(64\) 1.00000 0.125000
\(65\) 7.08072 0.878255
\(66\) 1.20246 0.148012
\(67\) 10.0295 1.22529 0.612647 0.790356i \(-0.290105\pi\)
0.612647 + 0.790356i \(0.290105\pi\)
\(68\) −2.11672 −0.256690
\(69\) 5.79964 0.698195
\(70\) 13.0508 1.55987
\(71\) 2.43109 0.288518 0.144259 0.989540i \(-0.453920\pi\)
0.144259 + 0.989540i \(0.453920\pi\)
\(72\) −2.46530 −0.290539
\(73\) 0.129250 0.0151276 0.00756380 0.999971i \(-0.497592\pi\)
0.00756380 + 0.999971i \(0.497592\pi\)
\(74\) −1.09154 −0.126889
\(75\) 4.02728 0.465030
\(76\) −1.00000 −0.114708
\(77\) 6.62069 0.754497
\(78\) 1.59728 0.180856
\(79\) 2.09468 0.235670 0.117835 0.993033i \(-0.462405\pi\)
0.117835 + 0.993033i \(0.462405\pi\)
\(80\) 3.24153 0.362415
\(81\) 4.47364 0.497071
\(82\) 5.86384 0.647553
\(83\) −1.41701 −0.155537 −0.0777684 0.996971i \(-0.524779\pi\)
−0.0777684 + 0.996971i \(0.524779\pi\)
\(84\) 2.94402 0.321219
\(85\) −6.86141 −0.744224
\(86\) −10.0819 −1.08716
\(87\) −5.14951 −0.552085
\(88\) 1.64443 0.175297
\(89\) −7.58566 −0.804078 −0.402039 0.915622i \(-0.631698\pi\)
−0.402039 + 0.915622i \(0.631698\pi\)
\(90\) −7.99137 −0.842364
\(91\) 8.79456 0.921920
\(92\) 7.93136 0.826902
\(93\) 2.16804 0.224815
\(94\) 7.44446 0.767838
\(95\) −3.24153 −0.332574
\(96\) 0.731229 0.0746307
\(97\) 1.83031 0.185840 0.0929201 0.995674i \(-0.470380\pi\)
0.0929201 + 0.995674i \(0.470380\pi\)
\(98\) 9.20968 0.930318
\(99\) −4.05402 −0.407445
\(100\) 5.50755 0.550755
\(101\) −17.2151 −1.71297 −0.856483 0.516175i \(-0.827356\pi\)
−0.856483 + 0.516175i \(0.827356\pi\)
\(102\) −1.54780 −0.153255
\(103\) −16.4725 −1.62309 −0.811544 0.584292i \(-0.801372\pi\)
−0.811544 + 0.584292i \(0.801372\pi\)
\(104\) 2.18437 0.214195
\(105\) 9.54314 0.931315
\(106\) −1.80240 −0.175065
\(107\) 17.0663 1.64986 0.824929 0.565236i \(-0.191215\pi\)
0.824929 + 0.565236i \(0.191215\pi\)
\(108\) −3.99639 −0.384553
\(109\) 14.2864 1.36839 0.684194 0.729300i \(-0.260154\pi\)
0.684194 + 0.729300i \(0.260154\pi\)
\(110\) 5.33048 0.508242
\(111\) −0.798165 −0.0757585
\(112\) 4.02612 0.380433
\(113\) 15.5876 1.46636 0.733178 0.680037i \(-0.238036\pi\)
0.733178 + 0.680037i \(0.238036\pi\)
\(114\) −0.731229 −0.0684859
\(115\) 25.7098 2.39745
\(116\) −7.04227 −0.653858
\(117\) −5.38514 −0.497857
\(118\) −8.17643 −0.752702
\(119\) −8.52216 −0.781226
\(120\) 2.37030 0.216378
\(121\) −8.29584 −0.754168
\(122\) −9.55710 −0.865259
\(123\) 4.28781 0.386619
\(124\) 2.96493 0.266258
\(125\) 1.64523 0.147153
\(126\) −9.92562 −0.884245
\(127\) −0.381396 −0.0338434 −0.0169217 0.999857i \(-0.505387\pi\)
−0.0169217 + 0.999857i \(0.505387\pi\)
\(128\) 1.00000 0.0883883
\(129\) −7.37221 −0.649087
\(130\) 7.08072 0.621020
\(131\) −0.745467 −0.0651318 −0.0325659 0.999470i \(-0.510368\pi\)
−0.0325659 + 0.999470i \(0.510368\pi\)
\(132\) 1.20246 0.104660
\(133\) −4.02612 −0.349109
\(134\) 10.0295 0.866414
\(135\) −12.9544 −1.11494
\(136\) −2.11672 −0.181507
\(137\) −8.61574 −0.736093 −0.368046 0.929807i \(-0.619973\pi\)
−0.368046 + 0.929807i \(0.619973\pi\)
\(138\) 5.79964 0.493698
\(139\) 10.8967 0.924244 0.462122 0.886816i \(-0.347088\pi\)
0.462122 + 0.886816i \(0.347088\pi\)
\(140\) 13.0508 1.10300
\(141\) 5.44361 0.458434
\(142\) 2.43109 0.204013
\(143\) 3.59205 0.300383
\(144\) −2.46530 −0.205442
\(145\) −22.8278 −1.89574
\(146\) 0.129250 0.0106968
\(147\) 6.73438 0.555443
\(148\) −1.09154 −0.0897240
\(149\) −7.19939 −0.589797 −0.294899 0.955529i \(-0.595286\pi\)
−0.294899 + 0.955529i \(0.595286\pi\)
\(150\) 4.02728 0.328826
\(151\) 2.39938 0.195259 0.0976294 0.995223i \(-0.468874\pi\)
0.0976294 + 0.995223i \(0.468874\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 5.21835 0.421879
\(154\) 6.62069 0.533510
\(155\) 9.61091 0.771967
\(156\) 1.59728 0.127884
\(157\) 12.9233 1.03139 0.515696 0.856772i \(-0.327533\pi\)
0.515696 + 0.856772i \(0.327533\pi\)
\(158\) 2.09468 0.166644
\(159\) −1.31797 −0.104522
\(160\) 3.24153 0.256266
\(161\) 31.9327 2.51665
\(162\) 4.47364 0.351482
\(163\) −17.2184 −1.34865 −0.674325 0.738434i \(-0.735566\pi\)
−0.674325 + 0.738434i \(0.735566\pi\)
\(164\) 5.86384 0.457889
\(165\) 3.89780 0.303444
\(166\) −1.41701 −0.109981
\(167\) 7.66193 0.592898 0.296449 0.955049i \(-0.404198\pi\)
0.296449 + 0.955049i \(0.404198\pi\)
\(168\) 2.94402 0.227136
\(169\) −8.22852 −0.632963
\(170\) −6.86141 −0.526246
\(171\) 2.46530 0.188527
\(172\) −10.0819 −0.768741
\(173\) −0.850469 −0.0646600 −0.0323300 0.999477i \(-0.510293\pi\)
−0.0323300 + 0.999477i \(0.510293\pi\)
\(174\) −5.14951 −0.390383
\(175\) 22.1741 1.67620
\(176\) 1.64443 0.123954
\(177\) −5.97884 −0.449397
\(178\) −7.58566 −0.568569
\(179\) 8.77509 0.655881 0.327940 0.944698i \(-0.393645\pi\)
0.327940 + 0.944698i \(0.393645\pi\)
\(180\) −7.99137 −0.595641
\(181\) −14.3582 −1.06724 −0.533620 0.845724i \(-0.679169\pi\)
−0.533620 + 0.845724i \(0.679169\pi\)
\(182\) 8.79456 0.651896
\(183\) −6.98843 −0.516600
\(184\) 7.93136 0.584708
\(185\) −3.53826 −0.260138
\(186\) 2.16804 0.158968
\(187\) −3.48080 −0.254541
\(188\) 7.44446 0.542943
\(189\) −16.0900 −1.17037
\(190\) −3.24153 −0.235166
\(191\) 17.3329 1.25417 0.627083 0.778953i \(-0.284249\pi\)
0.627083 + 0.778953i \(0.284249\pi\)
\(192\) 0.731229 0.0527719
\(193\) −25.1710 −1.81185 −0.905923 0.423443i \(-0.860821\pi\)
−0.905923 + 0.423443i \(0.860821\pi\)
\(194\) 1.83031 0.131409
\(195\) 5.17763 0.370778
\(196\) 9.20968 0.657834
\(197\) 2.39308 0.170500 0.0852498 0.996360i \(-0.472831\pi\)
0.0852498 + 0.996360i \(0.472831\pi\)
\(198\) −4.05402 −0.288107
\(199\) −11.6350 −0.824786 −0.412393 0.911006i \(-0.635307\pi\)
−0.412393 + 0.911006i \(0.635307\pi\)
\(200\) 5.50755 0.389442
\(201\) 7.33384 0.517289
\(202\) −17.2151 −1.21125
\(203\) −28.3530 −1.98999
\(204\) −1.54780 −0.108368
\(205\) 19.0078 1.32757
\(206\) −16.4725 −1.14770
\(207\) −19.5532 −1.35904
\(208\) 2.18437 0.151459
\(209\) −1.64443 −0.113748
\(210\) 9.54314 0.658539
\(211\) −1.00000 −0.0688428
\(212\) −1.80240 −0.123789
\(213\) 1.77768 0.121805
\(214\) 17.0663 1.16663
\(215\) −32.6810 −2.22882
\(216\) −3.99639 −0.271920
\(217\) 11.9372 0.810347
\(218\) 14.2864 0.967596
\(219\) 0.0945115 0.00638650
\(220\) 5.33048 0.359381
\(221\) −4.62370 −0.311024
\(222\) −0.798165 −0.0535694
\(223\) 12.6046 0.844063 0.422032 0.906581i \(-0.361317\pi\)
0.422032 + 0.906581i \(0.361317\pi\)
\(224\) 4.02612 0.269007
\(225\) −13.5778 −0.905185
\(226\) 15.5876 1.03687
\(227\) −0.152605 −0.0101288 −0.00506439 0.999987i \(-0.501612\pi\)
−0.00506439 + 0.999987i \(0.501612\pi\)
\(228\) −0.731229 −0.0484268
\(229\) −0.343084 −0.0226716 −0.0113358 0.999936i \(-0.503608\pi\)
−0.0113358 + 0.999936i \(0.503608\pi\)
\(230\) 25.7098 1.69525
\(231\) 4.84124 0.318530
\(232\) −7.04227 −0.462348
\(233\) −24.5919 −1.61107 −0.805533 0.592551i \(-0.798121\pi\)
−0.805533 + 0.592551i \(0.798121\pi\)
\(234\) −5.38514 −0.352038
\(235\) 24.1315 1.57416
\(236\) −8.17643 −0.532240
\(237\) 1.53169 0.0994940
\(238\) −8.52216 −0.552410
\(239\) 15.2818 0.988497 0.494249 0.869321i \(-0.335443\pi\)
0.494249 + 0.869321i \(0.335443\pi\)
\(240\) 2.37030 0.153002
\(241\) −3.76817 −0.242729 −0.121365 0.992608i \(-0.538727\pi\)
−0.121365 + 0.992608i \(0.538727\pi\)
\(242\) −8.29584 −0.533277
\(243\) 15.2604 0.978956
\(244\) −9.55710 −0.611831
\(245\) 29.8535 1.90727
\(246\) 4.28781 0.273381
\(247\) −2.18437 −0.138988
\(248\) 2.96493 0.188273
\(249\) −1.03616 −0.0656638
\(250\) 1.64523 0.104053
\(251\) 18.9149 1.19390 0.596950 0.802278i \(-0.296379\pi\)
0.596950 + 0.802278i \(0.296379\pi\)
\(252\) −9.92562 −0.625255
\(253\) 13.0426 0.819980
\(254\) −0.381396 −0.0239309
\(255\) −5.01726 −0.314193
\(256\) 1.00000 0.0625000
\(257\) −27.4598 −1.71289 −0.856447 0.516234i \(-0.827333\pi\)
−0.856447 + 0.516234i \(0.827333\pi\)
\(258\) −7.37221 −0.458974
\(259\) −4.39467 −0.273072
\(260\) 7.08072 0.439128
\(261\) 17.3613 1.07464
\(262\) −0.745467 −0.0460551
\(263\) −21.2240 −1.30873 −0.654365 0.756179i \(-0.727064\pi\)
−0.654365 + 0.756179i \(0.727064\pi\)
\(264\) 1.20246 0.0740061
\(265\) −5.84255 −0.358905
\(266\) −4.02612 −0.246858
\(267\) −5.54685 −0.339462
\(268\) 10.0295 0.612647
\(269\) 14.6404 0.892639 0.446319 0.894874i \(-0.352735\pi\)
0.446319 + 0.894874i \(0.352735\pi\)
\(270\) −12.9544 −0.788382
\(271\) −8.05587 −0.489359 −0.244680 0.969604i \(-0.578683\pi\)
−0.244680 + 0.969604i \(0.578683\pi\)
\(272\) −2.11672 −0.128345
\(273\) 6.43083 0.389212
\(274\) −8.61574 −0.520496
\(275\) 9.05678 0.546145
\(276\) 5.79964 0.349098
\(277\) 9.22890 0.554511 0.277255 0.960796i \(-0.410575\pi\)
0.277255 + 0.960796i \(0.410575\pi\)
\(278\) 10.8967 0.653539
\(279\) −7.30944 −0.437605
\(280\) 13.0508 0.779936
\(281\) −23.5180 −1.40297 −0.701483 0.712686i \(-0.747478\pi\)
−0.701483 + 0.712686i \(0.747478\pi\)
\(282\) 5.44361 0.324162
\(283\) 11.7905 0.700871 0.350435 0.936587i \(-0.386034\pi\)
0.350435 + 0.936587i \(0.386034\pi\)
\(284\) 2.43109 0.144259
\(285\) −2.37030 −0.140405
\(286\) 3.59205 0.212403
\(287\) 23.6086 1.39357
\(288\) −2.46530 −0.145269
\(289\) −12.5195 −0.736442
\(290\) −22.8278 −1.34049
\(291\) 1.33838 0.0784571
\(292\) 0.129250 0.00756380
\(293\) −5.50119 −0.321383 −0.160691 0.987005i \(-0.551372\pi\)
−0.160691 + 0.987005i \(0.551372\pi\)
\(294\) 6.73438 0.392757
\(295\) −26.5042 −1.54313
\(296\) −1.09154 −0.0634445
\(297\) −6.57179 −0.381334
\(298\) −7.19939 −0.417049
\(299\) 17.3251 1.00193
\(300\) 4.02728 0.232515
\(301\) −40.5912 −2.33964
\(302\) 2.39938 0.138069
\(303\) −12.5882 −0.723172
\(304\) −1.00000 −0.0573539
\(305\) −30.9797 −1.77389
\(306\) 5.21835 0.298313
\(307\) −3.32632 −0.189843 −0.0949215 0.995485i \(-0.530260\pi\)
−0.0949215 + 0.995485i \(0.530260\pi\)
\(308\) 6.62069 0.377249
\(309\) −12.0452 −0.685227
\(310\) 9.61091 0.545863
\(311\) −18.7637 −1.06399 −0.531995 0.846748i \(-0.678557\pi\)
−0.531995 + 0.846748i \(0.678557\pi\)
\(312\) 1.59728 0.0904280
\(313\) −6.41053 −0.362345 −0.181172 0.983451i \(-0.557989\pi\)
−0.181172 + 0.983451i \(0.557989\pi\)
\(314\) 12.9233 0.729304
\(315\) −32.1742 −1.81281
\(316\) 2.09468 0.117835
\(317\) 30.6899 1.72372 0.861858 0.507150i \(-0.169301\pi\)
0.861858 + 0.507150i \(0.169301\pi\)
\(318\) −1.31797 −0.0739080
\(319\) −11.5805 −0.648385
\(320\) 3.24153 0.181207
\(321\) 12.4793 0.696529
\(322\) 31.9327 1.77954
\(323\) 2.11672 0.117777
\(324\) 4.47364 0.248535
\(325\) 12.0305 0.667334
\(326\) −17.2184 −0.953640
\(327\) 10.4466 0.577699
\(328\) 5.86384 0.323777
\(329\) 29.9723 1.65243
\(330\) 3.89780 0.214567
\(331\) 1.32964 0.0730837 0.0365419 0.999332i \(-0.488366\pi\)
0.0365419 + 0.999332i \(0.488366\pi\)
\(332\) −1.41701 −0.0777684
\(333\) 2.69098 0.147465
\(334\) 7.66193 0.419242
\(335\) 32.5109 1.77626
\(336\) 2.94402 0.160609
\(337\) −3.74910 −0.204226 −0.102113 0.994773i \(-0.532560\pi\)
−0.102113 + 0.994773i \(0.532560\pi\)
\(338\) −8.22852 −0.447572
\(339\) 11.3981 0.619059
\(340\) −6.86141 −0.372112
\(341\) 4.87562 0.264030
\(342\) 2.46530 0.133308
\(343\) 8.89644 0.480363
\(344\) −10.0819 −0.543582
\(345\) 18.7997 1.01214
\(346\) −0.850469 −0.0457215
\(347\) −29.8884 −1.60449 −0.802246 0.596994i \(-0.796362\pi\)
−0.802246 + 0.596994i \(0.796362\pi\)
\(348\) −5.14951 −0.276043
\(349\) 11.4539 0.613114 0.306557 0.951852i \(-0.400823\pi\)
0.306557 + 0.951852i \(0.400823\pi\)
\(350\) 22.1741 1.18525
\(351\) −8.72960 −0.465952
\(352\) 1.64443 0.0876485
\(353\) −32.0552 −1.70613 −0.853063 0.521809i \(-0.825257\pi\)
−0.853063 + 0.521809i \(0.825257\pi\)
\(354\) −5.97884 −0.317772
\(355\) 7.88047 0.418252
\(356\) −7.58566 −0.402039
\(357\) −6.23165 −0.329814
\(358\) 8.77509 0.463778
\(359\) −19.8681 −1.04860 −0.524299 0.851534i \(-0.675673\pi\)
−0.524299 + 0.851534i \(0.675673\pi\)
\(360\) −7.99137 −0.421182
\(361\) 1.00000 0.0526316
\(362\) −14.3582 −0.754652
\(363\) −6.06616 −0.318391
\(364\) 8.79456 0.460960
\(365\) 0.418969 0.0219299
\(366\) −6.98843 −0.365291
\(367\) 8.49405 0.443386 0.221693 0.975117i \(-0.428842\pi\)
0.221693 + 0.975117i \(0.428842\pi\)
\(368\) 7.93136 0.413451
\(369\) −14.4562 −0.752557
\(370\) −3.53826 −0.183946
\(371\) −7.25670 −0.376749
\(372\) 2.16804 0.112408
\(373\) 23.7991 1.23227 0.616136 0.787640i \(-0.288697\pi\)
0.616136 + 0.787640i \(0.288697\pi\)
\(374\) −3.48080 −0.179988
\(375\) 1.20304 0.0621245
\(376\) 7.44446 0.383919
\(377\) −15.3829 −0.792262
\(378\) −16.0900 −0.827578
\(379\) 29.6559 1.52332 0.761660 0.647977i \(-0.224385\pi\)
0.761660 + 0.647977i \(0.224385\pi\)
\(380\) −3.24153 −0.166287
\(381\) −0.278888 −0.0142878
\(382\) 17.3329 0.886829
\(383\) −2.71447 −0.138703 −0.0693514 0.997592i \(-0.522093\pi\)
−0.0693514 + 0.997592i \(0.522093\pi\)
\(384\) 0.731229 0.0373154
\(385\) 21.4612 1.09376
\(386\) −25.1710 −1.28117
\(387\) 24.8551 1.26345
\(388\) 1.83031 0.0929201
\(389\) 30.6576 1.55440 0.777202 0.629251i \(-0.216638\pi\)
0.777202 + 0.629251i \(0.216638\pi\)
\(390\) 5.17763 0.262179
\(391\) −16.7884 −0.849028
\(392\) 9.20968 0.465159
\(393\) −0.545107 −0.0274970
\(394\) 2.39308 0.120561
\(395\) 6.78997 0.341641
\(396\) −4.05402 −0.203722
\(397\) 2.17022 0.108920 0.0544600 0.998516i \(-0.482656\pi\)
0.0544600 + 0.998516i \(0.482656\pi\)
\(398\) −11.6350 −0.583212
\(399\) −2.94402 −0.147385
\(400\) 5.50755 0.275377
\(401\) 18.6126 0.929468 0.464734 0.885450i \(-0.346150\pi\)
0.464734 + 0.885450i \(0.346150\pi\)
\(402\) 7.33384 0.365779
\(403\) 6.47650 0.322618
\(404\) −17.2151 −0.856483
\(405\) 14.5014 0.720583
\(406\) −28.3530 −1.40714
\(407\) −1.79496 −0.0889730
\(408\) −1.54780 −0.0766277
\(409\) 12.3470 0.610522 0.305261 0.952269i \(-0.401256\pi\)
0.305261 + 0.952269i \(0.401256\pi\)
\(410\) 19.0078 0.938731
\(411\) −6.30008 −0.310760
\(412\) −16.4725 −0.811544
\(413\) −32.9193 −1.61985
\(414\) −19.5532 −0.960989
\(415\) −4.59328 −0.225475
\(416\) 2.18437 0.107098
\(417\) 7.96796 0.390193
\(418\) −1.64443 −0.0804318
\(419\) −34.3591 −1.67855 −0.839275 0.543706i \(-0.817020\pi\)
−0.839275 + 0.543706i \(0.817020\pi\)
\(420\) 9.54314 0.465657
\(421\) −31.1637 −1.51883 −0.759413 0.650609i \(-0.774514\pi\)
−0.759413 + 0.650609i \(0.774514\pi\)
\(422\) −1.00000 −0.0486792
\(423\) −18.3529 −0.892347
\(424\) −1.80240 −0.0875324
\(425\) −11.6579 −0.565492
\(426\) 1.77768 0.0861291
\(427\) −38.4781 −1.86208
\(428\) 17.0663 0.824929
\(429\) 2.62661 0.126814
\(430\) −32.6810 −1.57602
\(431\) −15.7166 −0.757044 −0.378522 0.925592i \(-0.623568\pi\)
−0.378522 + 0.925592i \(0.623568\pi\)
\(432\) −3.99639 −0.192276
\(433\) 35.8497 1.72282 0.861412 0.507906i \(-0.169580\pi\)
0.861412 + 0.507906i \(0.169580\pi\)
\(434\) 11.9372 0.573002
\(435\) −16.6923 −0.800335
\(436\) 14.2864 0.684194
\(437\) −7.93136 −0.379409
\(438\) 0.0945115 0.00451594
\(439\) 8.77697 0.418902 0.209451 0.977819i \(-0.432832\pi\)
0.209451 + 0.977819i \(0.432832\pi\)
\(440\) 5.33048 0.254121
\(441\) −22.7047 −1.08117
\(442\) −4.62370 −0.219927
\(443\) 13.3669 0.635080 0.317540 0.948245i \(-0.397143\pi\)
0.317540 + 0.948245i \(0.397143\pi\)
\(444\) −0.798165 −0.0378793
\(445\) −24.5892 −1.16564
\(446\) 12.6046 0.596843
\(447\) −5.26440 −0.248998
\(448\) 4.02612 0.190217
\(449\) −2.35505 −0.111142 −0.0555708 0.998455i \(-0.517698\pi\)
−0.0555708 + 0.998455i \(0.517698\pi\)
\(450\) −13.5778 −0.640062
\(451\) 9.64269 0.454056
\(452\) 15.5876 0.733178
\(453\) 1.75450 0.0824335
\(454\) −0.152605 −0.00716213
\(455\) 28.5079 1.33647
\(456\) −0.731229 −0.0342429
\(457\) −2.75822 −0.129024 −0.0645121 0.997917i \(-0.520549\pi\)
−0.0645121 + 0.997917i \(0.520549\pi\)
\(458\) −0.343084 −0.0160312
\(459\) 8.45922 0.394843
\(460\) 25.7098 1.19873
\(461\) −32.4649 −1.51204 −0.756020 0.654549i \(-0.772859\pi\)
−0.756020 + 0.654549i \(0.772859\pi\)
\(462\) 4.84124 0.225235
\(463\) 33.8729 1.57421 0.787104 0.616821i \(-0.211580\pi\)
0.787104 + 0.616821i \(0.211580\pi\)
\(464\) −7.04227 −0.326929
\(465\) 7.02778 0.325905
\(466\) −24.5919 −1.13920
\(467\) −28.8638 −1.33566 −0.667828 0.744315i \(-0.732776\pi\)
−0.667828 + 0.744315i \(0.732776\pi\)
\(468\) −5.38514 −0.248928
\(469\) 40.3799 1.86457
\(470\) 24.1315 1.11310
\(471\) 9.44990 0.435428
\(472\) −8.17643 −0.376351
\(473\) −16.5791 −0.762306
\(474\) 1.53169 0.0703529
\(475\) −5.50755 −0.252704
\(476\) −8.52216 −0.390613
\(477\) 4.44347 0.203453
\(478\) 15.2818 0.698973
\(479\) 12.6792 0.579327 0.289663 0.957129i \(-0.406457\pi\)
0.289663 + 0.957129i \(0.406457\pi\)
\(480\) 2.37030 0.108189
\(481\) −2.38433 −0.108716
\(482\) −3.76817 −0.171636
\(483\) 23.3501 1.06247
\(484\) −8.29584 −0.377084
\(485\) 5.93302 0.269405
\(486\) 15.2604 0.692227
\(487\) 18.3562 0.831798 0.415899 0.909411i \(-0.363467\pi\)
0.415899 + 0.909411i \(0.363467\pi\)
\(488\) −9.55710 −0.432630
\(489\) −12.5906 −0.569367
\(490\) 29.8535 1.34864
\(491\) −7.73203 −0.348942 −0.174471 0.984662i \(-0.555821\pi\)
−0.174471 + 0.984662i \(0.555821\pi\)
\(492\) 4.28781 0.193309
\(493\) 14.9065 0.671354
\(494\) −2.18437 −0.0982796
\(495\) −13.1413 −0.590656
\(496\) 2.96493 0.133129
\(497\) 9.78788 0.439046
\(498\) −1.03616 −0.0464313
\(499\) −7.68285 −0.343932 −0.171966 0.985103i \(-0.555012\pi\)
−0.171966 + 0.985103i \(0.555012\pi\)
\(500\) 1.64523 0.0735767
\(501\) 5.60263 0.250307
\(502\) 18.9149 0.844215
\(503\) −2.08396 −0.0929190 −0.0464595 0.998920i \(-0.514794\pi\)
−0.0464595 + 0.998920i \(0.514794\pi\)
\(504\) −9.92562 −0.442122
\(505\) −55.8033 −2.48322
\(506\) 13.0426 0.579814
\(507\) −6.01693 −0.267221
\(508\) −0.381396 −0.0169217
\(509\) 31.7306 1.40643 0.703217 0.710975i \(-0.251746\pi\)
0.703217 + 0.710975i \(0.251746\pi\)
\(510\) −5.01726 −0.222168
\(511\) 0.520378 0.0230202
\(512\) 1.00000 0.0441942
\(513\) 3.99639 0.176445
\(514\) −27.4598 −1.21120
\(515\) −53.3963 −2.35292
\(516\) −7.37221 −0.324543
\(517\) 12.2419 0.538399
\(518\) −4.39467 −0.193091
\(519\) −0.621888 −0.0272978
\(520\) 7.08072 0.310510
\(521\) −16.9267 −0.741572 −0.370786 0.928718i \(-0.620912\pi\)
−0.370786 + 0.928718i \(0.620912\pi\)
\(522\) 17.3613 0.759885
\(523\) 9.27600 0.405611 0.202806 0.979219i \(-0.434994\pi\)
0.202806 + 0.979219i \(0.434994\pi\)
\(524\) −0.745467 −0.0325659
\(525\) 16.2143 0.707651
\(526\) −21.2240 −0.925412
\(527\) −6.27591 −0.273383
\(528\) 1.20246 0.0523302
\(529\) 39.9065 1.73507
\(530\) −5.84255 −0.253784
\(531\) 20.1574 0.874756
\(532\) −4.02612 −0.174555
\(533\) 12.8088 0.554811
\(534\) −5.54685 −0.240036
\(535\) 55.3209 2.39173
\(536\) 10.0295 0.433207
\(537\) 6.41660 0.276897
\(538\) 14.6404 0.631191
\(539\) 15.1447 0.652328
\(540\) −12.9544 −0.557470
\(541\) 24.7005 1.06196 0.530979 0.847385i \(-0.321824\pi\)
0.530979 + 0.847385i \(0.321824\pi\)
\(542\) −8.05587 −0.346029
\(543\) −10.4992 −0.450562
\(544\) −2.11672 −0.0907535
\(545\) 46.3098 1.98369
\(546\) 6.43083 0.275214
\(547\) −15.5522 −0.664964 −0.332482 0.943110i \(-0.607886\pi\)
−0.332482 + 0.943110i \(0.607886\pi\)
\(548\) −8.61574 −0.368046
\(549\) 23.5612 1.00557
\(550\) 9.05678 0.386183
\(551\) 7.04227 0.300011
\(552\) 5.79964 0.246849
\(553\) 8.43344 0.358626
\(554\) 9.22890 0.392098
\(555\) −2.58728 −0.109824
\(556\) 10.8967 0.462122
\(557\) 45.7360 1.93790 0.968949 0.247262i \(-0.0795307\pi\)
0.968949 + 0.247262i \(0.0795307\pi\)
\(558\) −7.30944 −0.309433
\(559\) −22.0227 −0.931462
\(560\) 13.0508 0.551498
\(561\) −2.54526 −0.107461
\(562\) −23.5180 −0.992047
\(563\) 0.429964 0.0181208 0.00906040 0.999959i \(-0.497116\pi\)
0.00906040 + 0.999959i \(0.497116\pi\)
\(564\) 5.44361 0.229217
\(565\) 50.5277 2.12571
\(566\) 11.7905 0.495590
\(567\) 18.0114 0.756409
\(568\) 2.43109 0.102006
\(569\) 18.6866 0.783385 0.391693 0.920096i \(-0.371890\pi\)
0.391693 + 0.920096i \(0.371890\pi\)
\(570\) −2.37030 −0.0992811
\(571\) −31.7284 −1.32779 −0.663897 0.747824i \(-0.731099\pi\)
−0.663897 + 0.747824i \(0.731099\pi\)
\(572\) 3.59205 0.150191
\(573\) 12.6743 0.529477
\(574\) 23.6086 0.985402
\(575\) 43.6823 1.82168
\(576\) −2.46530 −0.102721
\(577\) −9.91499 −0.412766 −0.206383 0.978471i \(-0.566169\pi\)
−0.206383 + 0.978471i \(0.566169\pi\)
\(578\) −12.5195 −0.520743
\(579\) −18.4057 −0.764916
\(580\) −22.8278 −0.947871
\(581\) −5.70505 −0.236685
\(582\) 1.33838 0.0554776
\(583\) −2.96393 −0.122753
\(584\) 0.129250 0.00534841
\(585\) −17.4561 −0.721722
\(586\) −5.50119 −0.227252
\(587\) −12.4239 −0.512790 −0.256395 0.966572i \(-0.582535\pi\)
−0.256395 + 0.966572i \(0.582535\pi\)
\(588\) 6.73438 0.277721
\(589\) −2.96493 −0.122168
\(590\) −26.5042 −1.09116
\(591\) 1.74989 0.0719807
\(592\) −1.09154 −0.0448620
\(593\) −45.8510 −1.88288 −0.941438 0.337186i \(-0.890525\pi\)
−0.941438 + 0.337186i \(0.890525\pi\)
\(594\) −6.57179 −0.269644
\(595\) −27.6249 −1.13251
\(596\) −7.19939 −0.294899
\(597\) −8.50788 −0.348204
\(598\) 17.3251 0.708474
\(599\) 11.2392 0.459220 0.229610 0.973283i \(-0.426255\pi\)
0.229610 + 0.973283i \(0.426255\pi\)
\(600\) 4.02728 0.164413
\(601\) −27.9883 −1.14167 −0.570833 0.821066i \(-0.693380\pi\)
−0.570833 + 0.821066i \(0.693380\pi\)
\(602\) −40.5912 −1.65437
\(603\) −24.7257 −1.00691
\(604\) 2.39938 0.0976294
\(605\) −26.8913 −1.09329
\(606\) −12.5882 −0.511360
\(607\) 39.7790 1.61458 0.807289 0.590156i \(-0.200934\pi\)
0.807289 + 0.590156i \(0.200934\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −20.7326 −0.840126
\(610\) −30.9797 −1.25433
\(611\) 16.2615 0.657869
\(612\) 5.21835 0.210939
\(613\) 1.31277 0.0530223 0.0265111 0.999649i \(-0.491560\pi\)
0.0265111 + 0.999649i \(0.491560\pi\)
\(614\) −3.32632 −0.134239
\(615\) 13.8991 0.560465
\(616\) 6.62069 0.266755
\(617\) 28.8387 1.16100 0.580501 0.814260i \(-0.302857\pi\)
0.580501 + 0.814260i \(0.302857\pi\)
\(618\) −12.0452 −0.484529
\(619\) 31.8027 1.27826 0.639129 0.769100i \(-0.279295\pi\)
0.639129 + 0.769100i \(0.279295\pi\)
\(620\) 9.61091 0.385983
\(621\) −31.6968 −1.27195
\(622\) −18.7637 −0.752354
\(623\) −30.5408 −1.22359
\(624\) 1.59728 0.0639422
\(625\) −22.2047 −0.888187
\(626\) −6.41053 −0.256216
\(627\) −1.20246 −0.0480215
\(628\) 12.9233 0.515696
\(629\) 2.31048 0.0921249
\(630\) −32.1742 −1.28185
\(631\) −31.9630 −1.27243 −0.636214 0.771512i \(-0.719501\pi\)
−0.636214 + 0.771512i \(0.719501\pi\)
\(632\) 2.09468 0.0833219
\(633\) −0.731229 −0.0290637
\(634\) 30.6899 1.21885
\(635\) −1.23631 −0.0490614
\(636\) −1.31797 −0.0522609
\(637\) 20.1174 0.797079
\(638\) −11.5805 −0.458478
\(639\) −5.99338 −0.237094
\(640\) 3.24153 0.128133
\(641\) −3.51583 −0.138867 −0.0694335 0.997587i \(-0.522119\pi\)
−0.0694335 + 0.997587i \(0.522119\pi\)
\(642\) 12.4793 0.492521
\(643\) 12.0885 0.476722 0.238361 0.971177i \(-0.423390\pi\)
0.238361 + 0.971177i \(0.423390\pi\)
\(644\) 31.9327 1.25832
\(645\) −23.8973 −0.940954
\(646\) 2.11672 0.0832811
\(647\) −19.6176 −0.771249 −0.385624 0.922656i \(-0.626014\pi\)
−0.385624 + 0.922656i \(0.626014\pi\)
\(648\) 4.47364 0.175741
\(649\) −13.4456 −0.527785
\(650\) 12.0305 0.471876
\(651\) 8.72880 0.342109
\(652\) −17.2184 −0.674325
\(653\) 21.3444 0.835273 0.417636 0.908614i \(-0.362859\pi\)
0.417636 + 0.908614i \(0.362859\pi\)
\(654\) 10.4466 0.408495
\(655\) −2.41646 −0.0944188
\(656\) 5.86384 0.228945
\(657\) −0.318641 −0.0124314
\(658\) 29.9723 1.16844
\(659\) −6.63518 −0.258470 −0.129235 0.991614i \(-0.541252\pi\)
−0.129235 + 0.991614i \(0.541252\pi\)
\(660\) 3.89780 0.151722
\(661\) −5.44306 −0.211710 −0.105855 0.994382i \(-0.533758\pi\)
−0.105855 + 0.994382i \(0.533758\pi\)
\(662\) 1.32964 0.0516780
\(663\) −3.38098 −0.131306
\(664\) −1.41701 −0.0549906
\(665\) −13.0508 −0.506089
\(666\) 2.69098 0.104273
\(667\) −55.8548 −2.16271
\(668\) 7.66193 0.296449
\(669\) 9.21681 0.356343
\(670\) 32.5109 1.25600
\(671\) −15.7160 −0.606710
\(672\) 2.94402 0.113568
\(673\) −28.0842 −1.08257 −0.541283 0.840840i \(-0.682061\pi\)
−0.541283 + 0.840840i \(0.682061\pi\)
\(674\) −3.74910 −0.144410
\(675\) −22.0103 −0.847177
\(676\) −8.22852 −0.316481
\(677\) 30.7617 1.18227 0.591134 0.806574i \(-0.298681\pi\)
0.591134 + 0.806574i \(0.298681\pi\)
\(678\) 11.3981 0.437741
\(679\) 7.36907 0.282799
\(680\) −6.86141 −0.263123
\(681\) −0.111590 −0.00427612
\(682\) 4.87562 0.186697
\(683\) 37.1331 1.42086 0.710429 0.703769i \(-0.248501\pi\)
0.710429 + 0.703769i \(0.248501\pi\)
\(684\) 2.46530 0.0942633
\(685\) −27.9282 −1.06708
\(686\) 8.89644 0.339668
\(687\) −0.250873 −0.00957139
\(688\) −10.0819 −0.384370
\(689\) −3.93712 −0.149992
\(690\) 18.7997 0.715694
\(691\) 27.4658 1.04485 0.522424 0.852686i \(-0.325028\pi\)
0.522424 + 0.852686i \(0.325028\pi\)
\(692\) −0.850469 −0.0323300
\(693\) −16.3220 −0.620022
\(694\) −29.8884 −1.13455
\(695\) 35.3219 1.33984
\(696\) −5.14951 −0.195192
\(697\) −12.4121 −0.470141
\(698\) 11.4539 0.433537
\(699\) −17.9823 −0.680152
\(700\) 22.1741 0.838101
\(701\) −11.2595 −0.425267 −0.212633 0.977132i \(-0.568204\pi\)
−0.212633 + 0.977132i \(0.568204\pi\)
\(702\) −8.72960 −0.329478
\(703\) 1.09154 0.0411682
\(704\) 1.64443 0.0619769
\(705\) 17.6456 0.664573
\(706\) −32.0552 −1.20641
\(707\) −69.3101 −2.60668
\(708\) −5.97884 −0.224699
\(709\) −20.5576 −0.772058 −0.386029 0.922487i \(-0.626154\pi\)
−0.386029 + 0.922487i \(0.626154\pi\)
\(710\) 7.88047 0.295749
\(711\) −5.16402 −0.193666
\(712\) −7.58566 −0.284285
\(713\) 23.5159 0.880678
\(714\) −6.23165 −0.233214
\(715\) 11.6438 0.435452
\(716\) 8.77509 0.327940
\(717\) 11.1745 0.417319
\(718\) −19.8681 −0.741471
\(719\) −27.4192 −1.02256 −0.511281 0.859413i \(-0.670829\pi\)
−0.511281 + 0.859413i \(0.670829\pi\)
\(720\) −7.99137 −0.297821
\(721\) −66.3205 −2.46990
\(722\) 1.00000 0.0372161
\(723\) −2.75540 −0.102474
\(724\) −14.3582 −0.533620
\(725\) −38.7856 −1.44046
\(726\) −6.06616 −0.225136
\(727\) 25.9929 0.964024 0.482012 0.876165i \(-0.339906\pi\)
0.482012 + 0.876165i \(0.339906\pi\)
\(728\) 8.79456 0.325948
\(729\) −2.26205 −0.0837796
\(730\) 0.418969 0.0155067
\(731\) 21.3406 0.789311
\(732\) −6.98843 −0.258300
\(733\) −28.0599 −1.03642 −0.518209 0.855254i \(-0.673401\pi\)
−0.518209 + 0.855254i \(0.673401\pi\)
\(734\) 8.49405 0.313521
\(735\) 21.8297 0.805202
\(736\) 7.93136 0.292354
\(737\) 16.4928 0.607519
\(738\) −14.4562 −0.532138
\(739\) −7.29777 −0.268453 −0.134226 0.990951i \(-0.542855\pi\)
−0.134226 + 0.990951i \(0.542855\pi\)
\(740\) −3.53826 −0.130069
\(741\) −1.59728 −0.0586774
\(742\) −7.25670 −0.266402
\(743\) −32.4781 −1.19151 −0.595753 0.803168i \(-0.703146\pi\)
−0.595753 + 0.803168i \(0.703146\pi\)
\(744\) 2.16804 0.0794842
\(745\) −23.3371 −0.855004
\(746\) 23.7991 0.871348
\(747\) 3.49336 0.127815
\(748\) −3.48080 −0.127271
\(749\) 68.7109 2.51064
\(750\) 1.20304 0.0439287
\(751\) 51.0438 1.86261 0.931307 0.364234i \(-0.118669\pi\)
0.931307 + 0.364234i \(0.118669\pi\)
\(752\) 7.44446 0.271472
\(753\) 13.8311 0.504035
\(754\) −15.3829 −0.560214
\(755\) 7.77768 0.283059
\(756\) −16.0900 −0.585186
\(757\) −34.6026 −1.25765 −0.628827 0.777545i \(-0.716465\pi\)
−0.628827 + 0.777545i \(0.716465\pi\)
\(758\) 29.6559 1.07715
\(759\) 9.53712 0.346175
\(760\) −3.24153 −0.117583
\(761\) 22.2985 0.808321 0.404160 0.914688i \(-0.367564\pi\)
0.404160 + 0.914688i \(0.367564\pi\)
\(762\) −0.278888 −0.0101030
\(763\) 57.5188 2.08232
\(764\) 17.3329 0.627083
\(765\) 16.9155 0.611580
\(766\) −2.71447 −0.0980777
\(767\) −17.8604 −0.644901
\(768\) 0.731229 0.0263860
\(769\) −16.4702 −0.593932 −0.296966 0.954888i \(-0.595975\pi\)
−0.296966 + 0.954888i \(0.595975\pi\)
\(770\) 21.4612 0.773407
\(771\) −20.0794 −0.723142
\(772\) −25.1710 −0.905923
\(773\) −7.44134 −0.267646 −0.133823 0.991005i \(-0.542725\pi\)
−0.133823 + 0.991005i \(0.542725\pi\)
\(774\) 24.8551 0.893396
\(775\) 16.3295 0.586572
\(776\) 1.83031 0.0657044
\(777\) −3.21351 −0.115284
\(778\) 30.6576 1.09913
\(779\) −5.86384 −0.210094
\(780\) 5.17763 0.185389
\(781\) 3.99777 0.143051
\(782\) −16.7884 −0.600354
\(783\) 28.1436 1.00577
\(784\) 9.20968 0.328917
\(785\) 41.8913 1.49517
\(786\) −0.545107 −0.0194433
\(787\) 1.16450 0.0415099 0.0207549 0.999785i \(-0.493393\pi\)
0.0207549 + 0.999785i \(0.493393\pi\)
\(788\) 2.39308 0.0852498
\(789\) −15.5196 −0.552513
\(790\) 6.78997 0.241576
\(791\) 62.7575 2.23140
\(792\) −4.05402 −0.144054
\(793\) −20.8763 −0.741338
\(794\) 2.17022 0.0770181
\(795\) −4.27224 −0.151521
\(796\) −11.6350 −0.412393
\(797\) 27.2896 0.966647 0.483324 0.875442i \(-0.339429\pi\)
0.483324 + 0.875442i \(0.339429\pi\)
\(798\) −2.94402 −0.104217
\(799\) −15.7578 −0.557471
\(800\) 5.50755 0.194721
\(801\) 18.7010 0.660766
\(802\) 18.6126 0.657233
\(803\) 0.212543 0.00750049
\(804\) 7.33384 0.258645
\(805\) 103.511 3.64828
\(806\) 6.47650 0.228125
\(807\) 10.7055 0.376850
\(808\) −17.2151 −0.605625
\(809\) 9.36235 0.329163 0.164581 0.986364i \(-0.447373\pi\)
0.164581 + 0.986364i \(0.447373\pi\)
\(810\) 14.5014 0.509529
\(811\) −41.7318 −1.46540 −0.732701 0.680550i \(-0.761741\pi\)
−0.732701 + 0.680550i \(0.761741\pi\)
\(812\) −28.3530 −0.994997
\(813\) −5.89069 −0.206595
\(814\) −1.79496 −0.0629134
\(815\) −55.8141 −1.95508
\(816\) −1.54780 −0.0541840
\(817\) 10.0819 0.352723
\(818\) 12.3470 0.431704
\(819\) −21.6813 −0.757604
\(820\) 19.0078 0.663783
\(821\) 55.5967 1.94034 0.970169 0.242431i \(-0.0779449\pi\)
0.970169 + 0.242431i \(0.0779449\pi\)
\(822\) −6.30008 −0.219741
\(823\) 15.5036 0.540420 0.270210 0.962801i \(-0.412907\pi\)
0.270210 + 0.962801i \(0.412907\pi\)
\(824\) −16.4725 −0.573848
\(825\) 6.62258 0.230569
\(826\) −32.9193 −1.14541
\(827\) −11.3875 −0.395984 −0.197992 0.980204i \(-0.563442\pi\)
−0.197992 + 0.980204i \(0.563442\pi\)
\(828\) −19.5532 −0.679522
\(829\) 46.7508 1.62372 0.811861 0.583851i \(-0.198455\pi\)
0.811861 + 0.583851i \(0.198455\pi\)
\(830\) −4.59328 −0.159435
\(831\) 6.74844 0.234101
\(832\) 2.18437 0.0757295
\(833\) −19.4943 −0.675437
\(834\) 7.96796 0.275908
\(835\) 24.8364 0.859500
\(836\) −1.64443 −0.0568739
\(837\) −11.8490 −0.409561
\(838\) −34.3591 −1.18691
\(839\) 3.90182 0.134706 0.0673528 0.997729i \(-0.478545\pi\)
0.0673528 + 0.997729i \(0.478545\pi\)
\(840\) 9.54314 0.329270
\(841\) 20.5935 0.710122
\(842\) −31.1637 −1.07397
\(843\) −17.1970 −0.592298
\(844\) −1.00000 −0.0344214
\(845\) −26.6730 −0.917580
\(846\) −18.3529 −0.630984
\(847\) −33.4001 −1.14764
\(848\) −1.80240 −0.0618947
\(849\) 8.62153 0.295890
\(850\) −11.6579 −0.399863
\(851\) −8.65740 −0.296772
\(852\) 1.77768 0.0609025
\(853\) −6.80197 −0.232895 −0.116448 0.993197i \(-0.537151\pi\)
−0.116448 + 0.993197i \(0.537151\pi\)
\(854\) −38.4781 −1.31669
\(855\) 7.99137 0.273299
\(856\) 17.0663 0.583313
\(857\) −7.46206 −0.254899 −0.127450 0.991845i \(-0.540679\pi\)
−0.127450 + 0.991845i \(0.540679\pi\)
\(858\) 2.62661 0.0896711
\(859\) 40.6587 1.38726 0.693629 0.720332i \(-0.256011\pi\)
0.693629 + 0.720332i \(0.256011\pi\)
\(860\) −32.6810 −1.11441
\(861\) 17.2633 0.588330
\(862\) −15.7166 −0.535311
\(863\) −19.9143 −0.677890 −0.338945 0.940806i \(-0.610070\pi\)
−0.338945 + 0.940806i \(0.610070\pi\)
\(864\) −3.99639 −0.135960
\(865\) −2.75682 −0.0937349
\(866\) 35.8497 1.21822
\(867\) −9.15463 −0.310908
\(868\) 11.9372 0.405174
\(869\) 3.44456 0.116849
\(870\) −16.6923 −0.565923
\(871\) 21.9081 0.742328
\(872\) 14.2864 0.483798
\(873\) −4.51228 −0.152718
\(874\) −7.93136 −0.268282
\(875\) 6.62388 0.223928
\(876\) 0.0945115 0.00319325
\(877\) 25.5496 0.862749 0.431375 0.902173i \(-0.358029\pi\)
0.431375 + 0.902173i \(0.358029\pi\)
\(878\) 8.77697 0.296209
\(879\) −4.02263 −0.135680
\(880\) 5.33048 0.179691
\(881\) 47.4187 1.59758 0.798788 0.601613i \(-0.205475\pi\)
0.798788 + 0.601613i \(0.205475\pi\)
\(882\) −22.7047 −0.764506
\(883\) −45.8201 −1.54197 −0.770985 0.636854i \(-0.780236\pi\)
−0.770985 + 0.636854i \(0.780236\pi\)
\(884\) −4.62370 −0.155512
\(885\) −19.3806 −0.651473
\(886\) 13.3669 0.449069
\(887\) −25.2259 −0.847002 −0.423501 0.905896i \(-0.639199\pi\)
−0.423501 + 0.905896i \(0.639199\pi\)
\(888\) −0.798165 −0.0267847
\(889\) −1.53555 −0.0515006
\(890\) −24.5892 −0.824231
\(891\) 7.35659 0.246455
\(892\) 12.6046 0.422032
\(893\) −7.44446 −0.249119
\(894\) −5.26440 −0.176068
\(895\) 28.4447 0.950803
\(896\) 4.02612 0.134503
\(897\) 12.6686 0.422992
\(898\) −2.35505 −0.0785890
\(899\) −20.8798 −0.696380
\(900\) −13.5778 −0.452592
\(901\) 3.81517 0.127102
\(902\) 9.64269 0.321066
\(903\) −29.6814 −0.987736
\(904\) 15.5876 0.518435
\(905\) −46.5427 −1.54713
\(906\) 1.75450 0.0582893
\(907\) −11.4744 −0.381000 −0.190500 0.981687i \(-0.561011\pi\)
−0.190500 + 0.981687i \(0.561011\pi\)
\(908\) −0.152605 −0.00506439
\(909\) 42.4405 1.40766
\(910\) 28.5079 0.945026
\(911\) 15.4063 0.510432 0.255216 0.966884i \(-0.417853\pi\)
0.255216 + 0.966884i \(0.417853\pi\)
\(912\) −0.731229 −0.0242134
\(913\) −2.33017 −0.0771175
\(914\) −2.75822 −0.0912338
\(915\) −22.6532 −0.748893
\(916\) −0.343084 −0.0113358
\(917\) −3.00134 −0.0991131
\(918\) 8.45922 0.279196
\(919\) 2.74438 0.0905286 0.0452643 0.998975i \(-0.485587\pi\)
0.0452643 + 0.998975i \(0.485587\pi\)
\(920\) 25.7098 0.847627
\(921\) −2.43230 −0.0801470
\(922\) −32.4649 −1.06917
\(923\) 5.31041 0.174794
\(924\) 4.84124 0.159265
\(925\) −6.01170 −0.197664
\(926\) 33.8729 1.11313
\(927\) 40.6098 1.33380
\(928\) −7.04227 −0.231174
\(929\) 40.7179 1.33591 0.667955 0.744201i \(-0.267170\pi\)
0.667955 + 0.744201i \(0.267170\pi\)
\(930\) 7.02778 0.230450
\(931\) −9.20968 −0.301835
\(932\) −24.5919 −0.805533
\(933\) −13.7205 −0.449190
\(934\) −28.8638 −0.944452
\(935\) −11.2831 −0.368998
\(936\) −5.38514 −0.176019
\(937\) −3.78336 −0.123597 −0.0617985 0.998089i \(-0.519684\pi\)
−0.0617985 + 0.998089i \(0.519684\pi\)
\(938\) 40.3799 1.31845
\(939\) −4.68757 −0.152973
\(940\) 24.1315 0.787082
\(941\) 33.5344 1.09319 0.546594 0.837398i \(-0.315924\pi\)
0.546594 + 0.837398i \(0.315924\pi\)
\(942\) 9.44990 0.307894
\(943\) 46.5083 1.51452
\(944\) −8.17643 −0.266120
\(945\) −52.1562 −1.69664
\(946\) −16.5791 −0.539032
\(947\) 11.7566 0.382038 0.191019 0.981586i \(-0.438821\pi\)
0.191019 + 0.981586i \(0.438821\pi\)
\(948\) 1.53169 0.0497470
\(949\) 0.282331 0.00916485
\(950\) −5.50755 −0.178688
\(951\) 22.4413 0.727710
\(952\) −8.52216 −0.276205
\(953\) 27.4805 0.890181 0.445090 0.895486i \(-0.353172\pi\)
0.445090 + 0.895486i \(0.353172\pi\)
\(954\) 4.44347 0.143863
\(955\) 56.1852 1.81811
\(956\) 15.2818 0.494249
\(957\) −8.46802 −0.273732
\(958\) 12.6792 0.409646
\(959\) −34.6881 −1.12014
\(960\) 2.37030 0.0765012
\(961\) −22.2092 −0.716426
\(962\) −2.38433 −0.0768739
\(963\) −42.0735 −1.35580
\(964\) −3.76817 −0.121365
\(965\) −81.5925 −2.62656
\(966\) 23.3501 0.751277
\(967\) 38.4447 1.23630 0.618148 0.786061i \(-0.287883\pi\)
0.618148 + 0.786061i \(0.287883\pi\)
\(968\) −8.29584 −0.266639
\(969\) 1.54780 0.0497226
\(970\) 5.93302 0.190498
\(971\) 52.6254 1.68883 0.844414 0.535691i \(-0.179949\pi\)
0.844414 + 0.535691i \(0.179949\pi\)
\(972\) 15.2604 0.489478
\(973\) 43.8714 1.40645
\(974\) 18.3562 0.588170
\(975\) 8.79707 0.281732
\(976\) −9.55710 −0.305915
\(977\) −38.2740 −1.22449 −0.612247 0.790667i \(-0.709734\pi\)
−0.612247 + 0.790667i \(0.709734\pi\)
\(978\) −12.5906 −0.402603
\(979\) −12.4741 −0.398674
\(980\) 29.8535 0.953635
\(981\) −35.2203 −1.12450
\(982\) −7.73203 −0.246739
\(983\) −35.6313 −1.13646 −0.568230 0.822869i \(-0.692372\pi\)
−0.568230 + 0.822869i \(0.692372\pi\)
\(984\) 4.28781 0.136690
\(985\) 7.75724 0.247166
\(986\) 14.9065 0.474719
\(987\) 21.9166 0.697614
\(988\) −2.18437 −0.0694942
\(989\) −79.9636 −2.54269
\(990\) −13.1413 −0.417657
\(991\) 7.57125 0.240509 0.120254 0.992743i \(-0.461629\pi\)
0.120254 + 0.992743i \(0.461629\pi\)
\(992\) 2.96493 0.0941365
\(993\) 0.972273 0.0308541
\(994\) 9.78788 0.310453
\(995\) −37.7154 −1.19566
\(996\) −1.03616 −0.0328319
\(997\) 23.4970 0.744158 0.372079 0.928201i \(-0.378645\pi\)
0.372079 + 0.928201i \(0.378645\pi\)
\(998\) −7.68285 −0.243196
\(999\) 4.36222 0.138014
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 8018.2.a.j.1.25 47
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8018.2.a.j.1.25 47 1.1 even 1 trivial