Properties

Label 197.14.a.a.1.18
Level $197$
Weight $14$
Character 197.1
Self dual yes
Analytic conductor $211.245$
Analytic rank $1$
Dimension $104$
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] = [197,14,Mod(1,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 197.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: \(211.244930035\)
Analytic rank: \(1\)
Dimension: \(104\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Character \(\chi\) \(=\) 197.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-135.981 q^{2} +2210.13 q^{3} +10298.9 q^{4} +24727.3 q^{5} -300537. q^{6} -333758. q^{7} -286501. q^{8} +3.29036e6 q^{9} +O(q^{10})\) \(q-135.981 q^{2} +2210.13 q^{3} +10298.9 q^{4} +24727.3 q^{5} -300537. q^{6} -333758. q^{7} -286501. q^{8} +3.29036e6 q^{9} -3.36245e6 q^{10} +2.42731e6 q^{11} +2.27620e7 q^{12} +1.74555e7 q^{13} +4.53848e7 q^{14} +5.46505e7 q^{15} -4.54099e7 q^{16} -5.12193e6 q^{17} -4.47427e8 q^{18} -2.76788e8 q^{19} +2.54664e8 q^{20} -7.37649e8 q^{21} -3.30069e8 q^{22} +3.30732e8 q^{23} -6.33205e8 q^{24} -6.09265e8 q^{25} -2.37363e9 q^{26} +3.74846e9 q^{27} -3.43734e9 q^{28} -1.77627e9 q^{29} -7.43145e9 q^{30} -4.02156e9 q^{31} +8.52192e9 q^{32} +5.36467e9 q^{33} +6.96487e8 q^{34} -8.25292e9 q^{35} +3.38871e10 q^{36} +7.48195e9 q^{37} +3.76380e10 q^{38} +3.85790e10 q^{39} -7.08439e9 q^{40} -2.60082e10 q^{41} +1.00306e11 q^{42} +3.17109e10 q^{43} +2.49986e10 q^{44} +8.13616e10 q^{45} -4.49734e10 q^{46} -9.22642e10 q^{47} -1.00362e11 q^{48} +1.45052e10 q^{49} +8.28487e10 q^{50} -1.13201e10 q^{51} +1.79773e11 q^{52} -1.70900e11 q^{53} -5.09720e11 q^{54} +6.00207e10 q^{55} +9.56219e10 q^{56} -6.11737e11 q^{57} +2.41540e11 q^{58} -3.65076e10 q^{59} +5.62841e11 q^{60} +3.48182e11 q^{61} +5.46857e11 q^{62} -1.09818e12 q^{63} -7.86823e11 q^{64} +4.31628e11 q^{65} -7.29495e11 q^{66} -5.00394e11 q^{67} -5.27503e10 q^{68} +7.30961e11 q^{69} +1.12224e12 q^{70} -2.58712e11 q^{71} -9.42690e11 q^{72} -3.41103e11 q^{73} -1.01741e12 q^{74} -1.34656e12 q^{75} -2.85061e12 q^{76} -8.10133e11 q^{77} -5.24603e12 q^{78} +4.33087e11 q^{79} -1.12286e12 q^{80} +3.03869e12 q^{81} +3.53664e12 q^{82} -3.93626e12 q^{83} -7.59698e12 q^{84} -1.26651e11 q^{85} -4.31209e12 q^{86} -3.92580e12 q^{87} -6.95426e11 q^{88} -5.61964e12 q^{89} -1.10637e13 q^{90} -5.82592e12 q^{91} +3.40618e12 q^{92} -8.88817e12 q^{93} +1.25462e13 q^{94} -6.84421e12 q^{95} +1.88346e13 q^{96} +9.80267e11 q^{97} -1.97244e12 q^{98} +7.98672e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 104 q - 128 q^{2} - 8020 q^{3} + 409600 q^{4} - 99004 q^{5} - 83328 q^{6} - 2084037 q^{7} - 2111301 q^{8} + 51549776 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 104 q - 128 q^{2} - 8020 q^{3} + 409600 q^{4} - 99004 q^{5} - 83328 q^{6} - 2084037 q^{7} - 2111301 q^{8} + 51549776 q^{9} - 9626347 q^{10} - 10688800 q^{11} - 68157440 q^{12} - 94762650 q^{13} - 52465903 q^{14} - 186128761 q^{15} + 1543503872 q^{16} - 109572251 q^{17} - 449658263 q^{18} - 1495268031 q^{19} - 1194030741 q^{20} - 832275538 q^{21} - 2695120703 q^{22} - 2632554532 q^{23} - 190649667 q^{24} + 22696779124 q^{25} - 2460454754 q^{26} - 16267542277 q^{27} - 18137835250 q^{28} + 47556769 q^{29} - 9613748117 q^{30} - 24774628288 q^{31} - 29246249180 q^{32} - 42367036666 q^{33} - 39395206606 q^{34} - 26216433864 q^{35} + 190013501626 q^{36} - 82419664050 q^{37} - 34593622142 q^{38} - 39498281801 q^{39} - 150303128514 q^{40} - 25862853768 q^{41} - 138889965149 q^{42} - 258164897781 q^{43} - 197365738094 q^{44} - 240782892021 q^{45} - 218441397971 q^{46} - 208905762731 q^{47} - 664063830814 q^{48} + 1113758315863 q^{49} - 1516068087607 q^{50} - 1019253294393 q^{51} - 1162941441840 q^{52} + 161684855900 q^{53} + 1679137315823 q^{54} - 557952233701 q^{55} + 2842561845328 q^{56} + 801593765429 q^{57} - 7249760433 q^{58} - 775487110641 q^{59} - 57598844627 q^{60} - 36786715662 q^{61} + 681529132643 q^{62} - 4349115033663 q^{63} + 4600420238797 q^{64} - 2531819073161 q^{65} - 3958922748734 q^{66} - 7314072405766 q^{67} - 10297353950393 q^{68} - 2089200206275 q^{69} - 14268943913713 q^{70} - 6055724651085 q^{71} - 26860198563805 q^{72} - 7572811533391 q^{73} - 11175265675817 q^{74} - 24431657592434 q^{75} - 26425925198106 q^{76} - 9735327037686 q^{77} - 35310017230907 q^{78} - 8981210762721 q^{79} - 25913753436330 q^{80} + 15437375349812 q^{81} - 19721330628227 q^{82} - 22209909714532 q^{83} - 18837805278768 q^{84} - 5905005171430 q^{85} - 8913876797772 q^{86} - 7341562344401 q^{87} - 35154329886441 q^{88} - 4646484034146 q^{89} + 9564902968095 q^{90} - 22752431457047 q^{91} - 13601121953458 q^{92} - 9615114240293 q^{93} + 15352521272967 q^{94} + 5258551767043 q^{95} + 87277848810881 q^{96} - 42298840804040 q^{97} + 27000102375354 q^{98} - 8666459567773 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −135.981 −1.50240 −0.751198 0.660077i \(-0.770523\pi\)
−0.751198 + 0.660077i \(0.770523\pi\)
\(3\) 2210.13 1.75037 0.875185 0.483788i \(-0.160739\pi\)
0.875185 + 0.483788i \(0.160739\pi\)
\(4\) 10298.9 1.25719
\(5\) 24727.3 0.707736 0.353868 0.935295i \(-0.384866\pi\)
0.353868 + 0.935295i \(0.384866\pi\)
\(6\) −300537. −2.62975
\(7\) −333758. −1.07225 −0.536123 0.844140i \(-0.680112\pi\)
−0.536123 + 0.844140i \(0.680112\pi\)
\(8\) −286501. −0.386404
\(9\) 3.29036e6 2.06380
\(10\) −3.36245e6 −1.06330
\(11\) 2.42731e6 0.413117 0.206558 0.978434i \(-0.433774\pi\)
0.206558 + 0.978434i \(0.433774\pi\)
\(12\) 2.27620e7 2.20055
\(13\) 1.74555e7 1.00300 0.501501 0.865157i \(-0.332781\pi\)
0.501501 + 0.865157i \(0.332781\pi\)
\(14\) 4.53848e7 1.61094
\(15\) 5.46505e7 1.23880
\(16\) −4.54099e7 −0.676661
\(17\) −5.12193e6 −0.0514654 −0.0257327 0.999669i \(-0.508192\pi\)
−0.0257327 + 0.999669i \(0.508192\pi\)
\(18\) −4.47427e8 −3.10064
\(19\) −2.76788e8 −1.34974 −0.674868 0.737939i \(-0.735799\pi\)
−0.674868 + 0.737939i \(0.735799\pi\)
\(20\) 2.54664e8 0.889760
\(21\) −7.37649e8 −1.87683
\(22\) −3.30069e8 −0.620664
\(23\) 3.30732e8 0.465849 0.232925 0.972495i \(-0.425170\pi\)
0.232925 + 0.972495i \(0.425170\pi\)
\(24\) −6.33205e8 −0.676349
\(25\) −6.09265e8 −0.499110
\(26\) −2.37363e9 −1.50690
\(27\) 3.74846e9 1.86204
\(28\) −3.43734e9 −1.34802
\(29\) −1.77627e9 −0.554527 −0.277264 0.960794i \(-0.589428\pi\)
−0.277264 + 0.960794i \(0.589428\pi\)
\(30\) −7.43145e9 −1.86117
\(31\) −4.02156e9 −0.813848 −0.406924 0.913462i \(-0.633399\pi\)
−0.406924 + 0.913462i \(0.633399\pi\)
\(32\) 8.52192e9 1.40302
\(33\) 5.36467e9 0.723107
\(34\) 6.96487e8 0.0773214
\(35\) −8.25292e9 −0.758866
\(36\) 3.38871e10 2.59459
\(37\) 7.48195e9 0.479406 0.239703 0.970846i \(-0.422950\pi\)
0.239703 + 0.970846i \(0.422950\pi\)
\(38\) 3.76380e10 2.02784
\(39\) 3.85790e10 1.75562
\(40\) −7.08439e9 −0.273472
\(41\) −2.60082e10 −0.855098 −0.427549 0.903992i \(-0.640623\pi\)
−0.427549 + 0.903992i \(0.640623\pi\)
\(42\) 1.00306e11 2.81973
\(43\) 3.17109e10 0.765005 0.382502 0.923954i \(-0.375062\pi\)
0.382502 + 0.923954i \(0.375062\pi\)
\(44\) 2.49986e10 0.519367
\(45\) 8.13616e10 1.46062
\(46\) −4.49734e10 −0.699890
\(47\) −9.22642e10 −1.24853 −0.624263 0.781215i \(-0.714600\pi\)
−0.624263 + 0.781215i \(0.714600\pi\)
\(48\) −1.00362e11 −1.18441
\(49\) 1.45052e10 0.149710
\(50\) 8.28487e10 0.749861
\(51\) −1.13201e10 −0.0900836
\(52\) 1.79773e11 1.26096
\(53\) −1.70900e11 −1.05913 −0.529564 0.848270i \(-0.677644\pi\)
−0.529564 + 0.848270i \(0.677644\pi\)
\(54\) −5.09720e11 −2.79752
\(55\) 6.00207e10 0.292377
\(56\) 9.56219e10 0.414319
\(57\) −6.11737e11 −2.36254
\(58\) 2.41540e11 0.833119
\(59\) −3.65076e10 −0.112680 −0.0563399 0.998412i \(-0.517943\pi\)
−0.0563399 + 0.998412i \(0.517943\pi\)
\(60\) 5.62841e11 1.55741
\(61\) 3.48182e11 0.865291 0.432645 0.901564i \(-0.357580\pi\)
0.432645 + 0.901564i \(0.357580\pi\)
\(62\) 5.46857e11 1.22272
\(63\) −1.09818e12 −2.21290
\(64\) −7.86823e11 −1.43122
\(65\) 4.31628e11 0.709860
\(66\) −7.29495e11 −1.08639
\(67\) −5.00394e11 −0.675811 −0.337906 0.941180i \(-0.609718\pi\)
−0.337906 + 0.941180i \(0.609718\pi\)
\(68\) −5.27503e10 −0.0647019
\(69\) 7.30961e11 0.815408
\(70\) 1.12224e12 1.14012
\(71\) −2.58712e11 −0.239683 −0.119842 0.992793i \(-0.538239\pi\)
−0.119842 + 0.992793i \(0.538239\pi\)
\(72\) −9.42690e11 −0.797458
\(73\) −3.41103e11 −0.263808 −0.131904 0.991263i \(-0.542109\pi\)
−0.131904 + 0.991263i \(0.542109\pi\)
\(74\) −1.01741e12 −0.720257
\(75\) −1.34656e12 −0.873627
\(76\) −2.85061e12 −1.69688
\(77\) −8.10133e11 −0.442962
\(78\) −5.24603e12 −2.63764
\(79\) 4.33087e11 0.200447 0.100223 0.994965i \(-0.468044\pi\)
0.100223 + 0.994965i \(0.468044\pi\)
\(80\) −1.12286e12 −0.478897
\(81\) 3.03869e12 1.19546
\(82\) 3.53664e12 1.28470
\(83\) −3.93626e12 −1.32153 −0.660763 0.750594i \(-0.729767\pi\)
−0.660763 + 0.750594i \(0.729767\pi\)
\(84\) −7.59698e12 −2.35953
\(85\) −1.26651e11 −0.0364239
\(86\) −4.31209e12 −1.14934
\(87\) −3.92580e12 −0.970628
\(88\) −6.95426e11 −0.159630
\(89\) −5.61964e12 −1.19860 −0.599299 0.800525i \(-0.704554\pi\)
−0.599299 + 0.800525i \(0.704554\pi\)
\(90\) −1.10637e13 −2.19443
\(91\) −5.82592e12 −1.07546
\(92\) 3.40618e12 0.585662
\(93\) −8.88817e12 −1.42454
\(94\) 1.25462e13 1.87578
\(95\) −6.84421e12 −0.955256
\(96\) 1.88346e13 2.45580
\(97\) 9.80267e11 0.119489 0.0597445 0.998214i \(-0.480971\pi\)
0.0597445 + 0.998214i \(0.480971\pi\)
\(98\) −1.97244e12 −0.224923
\(99\) 7.98672e12 0.852588
\(100\) −6.27477e12 −0.627477
\(101\) −1.80524e13 −1.69217 −0.846087 0.533044i \(-0.821048\pi\)
−0.846087 + 0.533044i \(0.821048\pi\)
\(102\) 1.53933e12 0.135341
\(103\) 1.05325e13 0.869137 0.434568 0.900639i \(-0.356901\pi\)
0.434568 + 0.900639i \(0.356901\pi\)
\(104\) −5.00103e12 −0.387563
\(105\) −1.82400e13 −1.32830
\(106\) 2.32392e13 1.59123
\(107\) 2.62913e13 1.69363 0.846814 0.531889i \(-0.178518\pi\)
0.846814 + 0.531889i \(0.178518\pi\)
\(108\) 3.86051e13 2.34094
\(109\) −1.71710e13 −0.980669 −0.490335 0.871534i \(-0.663125\pi\)
−0.490335 + 0.871534i \(0.663125\pi\)
\(110\) −8.16170e12 −0.439266
\(111\) 1.65361e13 0.839138
\(112\) 1.51559e13 0.725546
\(113\) 5.46644e12 0.246999 0.123499 0.992345i \(-0.460588\pi\)
0.123499 + 0.992345i \(0.460588\pi\)
\(114\) 8.31849e13 3.54946
\(115\) 8.17810e12 0.329698
\(116\) −1.82937e13 −0.697147
\(117\) 5.74350e13 2.06999
\(118\) 4.96436e12 0.169289
\(119\) 1.70948e12 0.0551836
\(120\) −1.56574e13 −0.478677
\(121\) −2.86309e13 −0.829335
\(122\) −4.73462e13 −1.30001
\(123\) −5.74816e13 −1.49674
\(124\) −4.14177e13 −1.02316
\(125\) −4.52501e13 −1.06097
\(126\) 1.49332e14 3.32464
\(127\) 3.53962e13 0.748570 0.374285 0.927314i \(-0.377888\pi\)
0.374285 + 0.927314i \(0.377888\pi\)
\(128\) 3.71817e13 0.747247
\(129\) 7.00853e13 1.33904
\(130\) −5.86933e13 −1.06649
\(131\) −1.42234e12 −0.0245889 −0.0122944 0.999924i \(-0.503914\pi\)
−0.0122944 + 0.999924i \(0.503914\pi\)
\(132\) 5.52503e13 0.909084
\(133\) 9.23801e13 1.44725
\(134\) 6.80442e13 1.01534
\(135\) 9.26892e13 1.31783
\(136\) 1.46744e12 0.0198864
\(137\) 5.44544e13 0.703638 0.351819 0.936068i \(-0.385563\pi\)
0.351819 + 0.936068i \(0.385563\pi\)
\(138\) −9.93970e13 −1.22507
\(139\) 5.74325e13 0.675401 0.337700 0.941254i \(-0.390351\pi\)
0.337700 + 0.941254i \(0.390351\pi\)
\(140\) −8.49961e13 −0.954040
\(141\) −2.03916e14 −2.18538
\(142\) 3.51800e13 0.360099
\(143\) 4.23700e13 0.414356
\(144\) −1.49415e14 −1.39649
\(145\) −4.39224e13 −0.392459
\(146\) 4.63837e13 0.396344
\(147\) 3.20585e13 0.262048
\(148\) 7.70559e13 0.602705
\(149\) −9.62237e13 −0.720396 −0.360198 0.932876i \(-0.617291\pi\)
−0.360198 + 0.932876i \(0.617291\pi\)
\(150\) 1.83106e14 1.31253
\(151\) −1.04732e14 −0.719003 −0.359501 0.933145i \(-0.617053\pi\)
−0.359501 + 0.933145i \(0.617053\pi\)
\(152\) 7.93000e13 0.521542
\(153\) −1.68530e13 −0.106214
\(154\) 1.10163e14 0.665504
\(155\) −9.94421e13 −0.575989
\(156\) 3.97322e14 2.20716
\(157\) 1.79282e14 0.955410 0.477705 0.878520i \(-0.341469\pi\)
0.477705 + 0.878520i \(0.341469\pi\)
\(158\) −5.88918e13 −0.301151
\(159\) −3.77711e14 −1.85386
\(160\) 2.10724e14 0.992964
\(161\) −1.10384e14 −0.499505
\(162\) −4.13206e14 −1.79605
\(163\) 3.97441e13 0.165979 0.0829895 0.996550i \(-0.473553\pi\)
0.0829895 + 0.996550i \(0.473553\pi\)
\(164\) −2.67857e14 −1.07502
\(165\) 1.32654e14 0.511769
\(166\) 5.35257e14 1.98546
\(167\) 9.71509e13 0.346569 0.173284 0.984872i \(-0.444562\pi\)
0.173284 + 0.984872i \(0.444562\pi\)
\(168\) 2.11337e14 0.725212
\(169\) 1.82074e12 0.00601153
\(170\) 1.72222e13 0.0547231
\(171\) −9.10731e14 −2.78558
\(172\) 3.26588e14 0.961758
\(173\) 6.78768e13 0.192496 0.0962479 0.995357i \(-0.469316\pi\)
0.0962479 + 0.995357i \(0.469316\pi\)
\(174\) 5.33835e14 1.45827
\(175\) 2.03347e14 0.535168
\(176\) −1.10224e14 −0.279540
\(177\) −8.06867e13 −0.197231
\(178\) 7.64166e14 1.80077
\(179\) 6.18015e14 1.40428 0.702141 0.712038i \(-0.252228\pi\)
0.702141 + 0.712038i \(0.252228\pi\)
\(180\) 8.37936e14 1.83628
\(181\) −4.00598e14 −0.846834 −0.423417 0.905935i \(-0.639169\pi\)
−0.423417 + 0.905935i \(0.639169\pi\)
\(182\) 7.92216e14 1.61577
\(183\) 7.69528e14 1.51458
\(184\) −9.47550e13 −0.180006
\(185\) 1.85008e14 0.339293
\(186\) 1.20862e15 2.14021
\(187\) −1.24325e13 −0.0212612
\(188\) −9.50221e14 −1.56964
\(189\) −1.25108e15 −1.99656
\(190\) 9.30684e14 1.43517
\(191\) 8.51707e12 0.0126933 0.00634663 0.999980i \(-0.497980\pi\)
0.00634663 + 0.999980i \(0.497980\pi\)
\(192\) −1.73898e15 −2.50517
\(193\) −5.88423e14 −0.819534 −0.409767 0.912190i \(-0.634390\pi\)
−0.409767 + 0.912190i \(0.634390\pi\)
\(194\) −1.33298e14 −0.179520
\(195\) 9.53954e14 1.24252
\(196\) 1.49388e14 0.188214
\(197\) −5.84517e13 −0.0712470
\(198\) −1.08604e15 −1.28092
\(199\) −1.37423e15 −1.56861 −0.784306 0.620374i \(-0.786981\pi\)
−0.784306 + 0.620374i \(0.786981\pi\)
\(200\) 1.74555e14 0.192858
\(201\) −1.10594e15 −1.18292
\(202\) 2.45478e15 2.54232
\(203\) 5.92845e14 0.594589
\(204\) −1.16585e14 −0.113252
\(205\) −6.43113e14 −0.605184
\(206\) −1.43222e15 −1.30579
\(207\) 1.08823e15 0.961418
\(208\) −7.92655e14 −0.678692
\(209\) −6.71850e14 −0.557598
\(210\) 2.48030e15 1.99563
\(211\) 6.02454e14 0.469989 0.234995 0.971997i \(-0.424493\pi\)
0.234995 + 0.971997i \(0.424493\pi\)
\(212\) −1.76008e15 −1.33153
\(213\) −5.71788e14 −0.419535
\(214\) −3.57513e15 −2.54450
\(215\) 7.84125e14 0.541421
\(216\) −1.07394e15 −0.719498
\(217\) 1.34223e15 0.872644
\(218\) 2.33493e15 1.47335
\(219\) −7.53884e14 −0.461761
\(220\) 6.18148e14 0.367574
\(221\) −8.94061e13 −0.0516199
\(222\) −2.24860e15 −1.26072
\(223\) −9.48481e14 −0.516472 −0.258236 0.966082i \(-0.583141\pi\)
−0.258236 + 0.966082i \(0.583141\pi\)
\(224\) −2.84426e15 −1.50438
\(225\) −2.00470e15 −1.03006
\(226\) −7.43333e14 −0.371090
\(227\) 3.08998e15 1.49895 0.749475 0.662032i \(-0.230306\pi\)
0.749475 + 0.662032i \(0.230306\pi\)
\(228\) −6.30023e15 −2.97016
\(229\) 1.36747e15 0.626595 0.313298 0.949655i \(-0.398566\pi\)
0.313298 + 0.949655i \(0.398566\pi\)
\(230\) −1.11207e15 −0.495337
\(231\) −1.79050e15 −0.775348
\(232\) 5.08904e14 0.214271
\(233\) −1.47789e14 −0.0605103 −0.0302551 0.999542i \(-0.509632\pi\)
−0.0302551 + 0.999542i \(0.509632\pi\)
\(234\) −7.81008e15 −3.10994
\(235\) −2.28144e15 −0.883626
\(236\) −3.75989e14 −0.141660
\(237\) 9.57180e14 0.350856
\(238\) −2.32458e14 −0.0829075
\(239\) 3.28245e15 1.13923 0.569616 0.821911i \(-0.307092\pi\)
0.569616 + 0.821911i \(0.307092\pi\)
\(240\) −2.48168e15 −0.838247
\(241\) 4.67657e15 1.53751 0.768753 0.639546i \(-0.220878\pi\)
0.768753 + 0.639546i \(0.220878\pi\)
\(242\) 3.89326e15 1.24599
\(243\) 7.39659e14 0.230458
\(244\) 3.58589e15 1.08784
\(245\) 3.58675e14 0.105955
\(246\) 7.81643e15 2.24869
\(247\) −4.83148e15 −1.35379
\(248\) 1.15218e15 0.314474
\(249\) −8.69965e15 −2.31316
\(250\) 6.15317e15 1.59400
\(251\) −6.81787e15 −1.72096 −0.860478 0.509487i \(-0.829835\pi\)
−0.860478 + 0.509487i \(0.829835\pi\)
\(252\) −1.13101e16 −2.78203
\(253\) 8.02789e14 0.192450
\(254\) −4.81322e15 −1.12465
\(255\) −2.79916e14 −0.0637554
\(256\) 1.38964e15 0.308562
\(257\) −2.01489e15 −0.436200 −0.218100 0.975926i \(-0.569986\pi\)
−0.218100 + 0.975926i \(0.569986\pi\)
\(258\) −9.53030e15 −2.01177
\(259\) −2.49716e15 −0.514041
\(260\) 4.44530e15 0.892430
\(261\) −5.84457e15 −1.14443
\(262\) 1.93411e14 0.0369422
\(263\) 3.63935e15 0.678128 0.339064 0.940763i \(-0.389890\pi\)
0.339064 + 0.940763i \(0.389890\pi\)
\(264\) −1.53698e15 −0.279411
\(265\) −4.22588e15 −0.749582
\(266\) −1.25620e16 −2.17434
\(267\) −1.24201e16 −2.09799
\(268\) −5.15351e15 −0.849624
\(269\) 1.43352e15 0.230682 0.115341 0.993326i \(-0.463204\pi\)
0.115341 + 0.993326i \(0.463204\pi\)
\(270\) −1.26040e16 −1.97990
\(271\) −6.45126e15 −0.989337 −0.494669 0.869082i \(-0.664711\pi\)
−0.494669 + 0.869082i \(0.664711\pi\)
\(272\) 2.32587e14 0.0348246
\(273\) −1.28761e16 −1.88246
\(274\) −7.40478e15 −1.05714
\(275\) −1.47887e15 −0.206191
\(276\) 7.52811e15 1.02512
\(277\) −1.08394e15 −0.144174 −0.0720870 0.997398i \(-0.522966\pi\)
−0.0720870 + 0.997398i \(0.522966\pi\)
\(278\) −7.80975e15 −1.01472
\(279\) −1.32324e16 −1.67962
\(280\) 2.36447e15 0.293229
\(281\) −4.73489e14 −0.0573745 −0.0286873 0.999588i \(-0.509133\pi\)
−0.0286873 + 0.999588i \(0.509133\pi\)
\(282\) 2.77288e16 3.28331
\(283\) 1.25676e16 1.45426 0.727128 0.686501i \(-0.240854\pi\)
0.727128 + 0.686501i \(0.240854\pi\)
\(284\) −2.66446e15 −0.301328
\(285\) −1.51266e16 −1.67205
\(286\) −5.76153e15 −0.622527
\(287\) 8.68046e15 0.916875
\(288\) 2.80402e16 2.89554
\(289\) −9.87834e15 −0.997351
\(290\) 5.97262e15 0.589628
\(291\) 2.16652e15 0.209150
\(292\) −3.51300e15 −0.331657
\(293\) −9.54568e15 −0.881389 −0.440694 0.897657i \(-0.645268\pi\)
−0.440694 + 0.897657i \(0.645268\pi\)
\(294\) −4.35935e15 −0.393699
\(295\) −9.02734e14 −0.0797475
\(296\) −2.14359e15 −0.185244
\(297\) 9.09867e15 0.769238
\(298\) 1.30846e16 1.08232
\(299\) 5.77311e15 0.467247
\(300\) −1.38681e16 −1.09832
\(301\) −1.05838e16 −0.820273
\(302\) 1.42416e16 1.08023
\(303\) −3.98981e16 −2.96193
\(304\) 1.25689e16 0.913313
\(305\) 8.60959e15 0.612397
\(306\) 2.29169e15 0.159576
\(307\) 9.39605e15 0.640539 0.320270 0.947326i \(-0.396226\pi\)
0.320270 + 0.947326i \(0.396226\pi\)
\(308\) −8.34349e15 −0.556889
\(309\) 2.32781e16 1.52131
\(310\) 1.35223e16 0.865364
\(311\) −1.95819e16 −1.22719 −0.613595 0.789621i \(-0.710277\pi\)
−0.613595 + 0.789621i \(0.710277\pi\)
\(312\) −1.10529e16 −0.678379
\(313\) −1.21714e16 −0.731648 −0.365824 0.930684i \(-0.619213\pi\)
−0.365824 + 0.930684i \(0.619213\pi\)
\(314\) −2.43790e16 −1.43540
\(315\) −2.71551e16 −1.56615
\(316\) 4.46033e15 0.252000
\(317\) 2.59948e16 1.43880 0.719401 0.694595i \(-0.244416\pi\)
0.719401 + 0.694595i \(0.244416\pi\)
\(318\) 5.13616e16 2.78524
\(319\) −4.31156e15 −0.229084
\(320\) −1.94560e16 −1.01293
\(321\) 5.81073e16 2.96448
\(322\) 1.50102e16 0.750453
\(323\) 1.41769e15 0.0694647
\(324\) 3.12953e16 1.50292
\(325\) −1.06351e16 −0.500608
\(326\) −5.40446e15 −0.249366
\(327\) −3.79501e16 −1.71653
\(328\) 7.45139e15 0.330413
\(329\) 3.07939e16 1.33873
\(330\) −1.80384e16 −0.768879
\(331\) 1.78693e16 0.746835 0.373418 0.927663i \(-0.378186\pi\)
0.373418 + 0.927663i \(0.378186\pi\)
\(332\) −4.05392e16 −1.66141
\(333\) 2.46183e16 0.989396
\(334\) −1.32107e16 −0.520683
\(335\) −1.23734e16 −0.478296
\(336\) 3.34966e16 1.26997
\(337\) −1.08602e16 −0.403873 −0.201937 0.979399i \(-0.564724\pi\)
−0.201937 + 0.979399i \(0.564724\pi\)
\(338\) −2.47587e14 −0.00903170
\(339\) 1.20815e16 0.432339
\(340\) −1.30437e15 −0.0457919
\(341\) −9.76156e15 −0.336214
\(342\) 1.23842e17 4.18504
\(343\) 2.74962e16 0.911720
\(344\) −9.08521e15 −0.295601
\(345\) 1.80747e16 0.577094
\(346\) −9.22997e15 −0.289205
\(347\) −2.33541e15 −0.0718161 −0.0359081 0.999355i \(-0.511432\pi\)
−0.0359081 + 0.999355i \(0.511432\pi\)
\(348\) −4.04314e16 −1.22027
\(349\) −1.42009e16 −0.420679 −0.210340 0.977628i \(-0.567457\pi\)
−0.210340 + 0.977628i \(0.567457\pi\)
\(350\) −2.76514e16 −0.804034
\(351\) 6.54314e16 1.86763
\(352\) 2.06853e16 0.579609
\(353\) −6.06526e16 −1.66845 −0.834227 0.551422i \(-0.814086\pi\)
−0.834227 + 0.551422i \(0.814086\pi\)
\(354\) 1.09719e16 0.296319
\(355\) −6.39725e15 −0.169633
\(356\) −5.78762e16 −1.50687
\(357\) 3.77818e15 0.0965917
\(358\) −8.40385e16 −2.10979
\(359\) −3.99343e16 −0.984537 −0.492268 0.870443i \(-0.663832\pi\)
−0.492268 + 0.870443i \(0.663832\pi\)
\(360\) −2.33102e16 −0.564390
\(361\) 3.45585e16 0.821785
\(362\) 5.44739e16 1.27228
\(363\) −6.32780e16 −1.45164
\(364\) −6.00007e16 −1.35206
\(365\) −8.43456e15 −0.186706
\(366\) −1.04641e17 −2.27550
\(367\) 4.00665e16 0.855958 0.427979 0.903789i \(-0.359226\pi\)
0.427979 + 0.903789i \(0.359226\pi\)
\(368\) −1.50185e16 −0.315222
\(369\) −8.55764e16 −1.76475
\(370\) −2.51577e16 −0.509752
\(371\) 5.70391e16 1.13564
\(372\) −9.15385e16 −1.79091
\(373\) 9.24820e16 1.77807 0.889037 0.457835i \(-0.151375\pi\)
0.889037 + 0.457835i \(0.151375\pi\)
\(374\) 1.69059e15 0.0319428
\(375\) −1.00009e17 −1.85710
\(376\) 2.64338e16 0.482435
\(377\) −3.10058e16 −0.556191
\(378\) 1.70123e17 2.99962
\(379\) −2.64927e16 −0.459168 −0.229584 0.973289i \(-0.573737\pi\)
−0.229584 + 0.973289i \(0.573737\pi\)
\(380\) −7.04879e16 −1.20094
\(381\) 7.82302e16 1.31027
\(382\) −1.15816e15 −0.0190703
\(383\) −9.07080e16 −1.46843 −0.734215 0.678917i \(-0.762450\pi\)
−0.734215 + 0.678917i \(0.762450\pi\)
\(384\) 8.21764e16 1.30796
\(385\) −2.00324e16 −0.313500
\(386\) 8.00145e16 1.23126
\(387\) 1.04340e17 1.57881
\(388\) 1.00957e16 0.150221
\(389\) 1.12227e16 0.164220 0.0821100 0.996623i \(-0.473834\pi\)
0.0821100 + 0.996623i \(0.473834\pi\)
\(390\) −1.29720e17 −1.86675
\(391\) −1.69399e15 −0.0239751
\(392\) −4.15576e15 −0.0578484
\(393\) −3.14355e15 −0.0430397
\(394\) 7.94834e15 0.107041
\(395\) 1.07091e16 0.141863
\(396\) 8.22545e16 1.07187
\(397\) −9.03879e16 −1.15870 −0.579351 0.815078i \(-0.696694\pi\)
−0.579351 + 0.815078i \(0.696694\pi\)
\(398\) 1.86870e17 2.35668
\(399\) 2.04172e17 2.53322
\(400\) 2.76667e16 0.337728
\(401\) 3.02416e16 0.363218 0.181609 0.983371i \(-0.441870\pi\)
0.181609 + 0.983371i \(0.441870\pi\)
\(402\) 1.50387e17 1.77721
\(403\) −7.01984e16 −0.816290
\(404\) −1.85920e17 −2.12739
\(405\) 7.51386e16 0.846069
\(406\) −8.06158e16 −0.893308
\(407\) 1.81610e16 0.198051
\(408\) 3.24323e15 0.0348086
\(409\) 1.03290e17 1.09108 0.545539 0.838086i \(-0.316325\pi\)
0.545539 + 0.838086i \(0.316325\pi\)
\(410\) 8.74513e16 0.909225
\(411\) 1.20351e17 1.23163
\(412\) 1.08473e17 1.09267
\(413\) 1.21847e16 0.120820
\(414\) −1.47978e17 −1.44443
\(415\) −9.73329e16 −0.935292
\(416\) 1.48755e17 1.40723
\(417\) 1.26933e17 1.18220
\(418\) 9.13590e16 0.837733
\(419\) 3.96205e15 0.0357708 0.0178854 0.999840i \(-0.494307\pi\)
0.0178854 + 0.999840i \(0.494307\pi\)
\(420\) −1.87853e17 −1.66992
\(421\) 1.33999e17 1.17292 0.586461 0.809977i \(-0.300521\pi\)
0.586461 + 0.809977i \(0.300521\pi\)
\(422\) −8.19225e16 −0.706110
\(423\) −3.03582e17 −2.57670
\(424\) 4.89629e16 0.409250
\(425\) 3.12061e15 0.0256869
\(426\) 7.77525e16 0.630307
\(427\) −1.16208e17 −0.927804
\(428\) 2.70772e17 2.12922
\(429\) 9.36432e16 0.725277
\(430\) −1.06626e17 −0.813429
\(431\) −9.09820e15 −0.0683681 −0.0341840 0.999416i \(-0.510883\pi\)
−0.0341840 + 0.999416i \(0.510883\pi\)
\(432\) −1.70217e17 −1.25997
\(433\) 8.48260e16 0.618526 0.309263 0.950977i \(-0.399918\pi\)
0.309263 + 0.950977i \(0.399918\pi\)
\(434\) −1.82518e17 −1.31106
\(435\) −9.70743e16 −0.686948
\(436\) −1.76842e17 −1.23289
\(437\) −9.15426e16 −0.628773
\(438\) 1.02514e17 0.693748
\(439\) −2.47830e17 −1.65247 −0.826235 0.563325i \(-0.809522\pi\)
−0.826235 + 0.563325i \(0.809522\pi\)
\(440\) −1.71960e16 −0.112976
\(441\) 4.77274e16 0.308971
\(442\) 1.21576e16 0.0775535
\(443\) −2.43051e17 −1.52782 −0.763910 0.645322i \(-0.776723\pi\)
−0.763910 + 0.645322i \(0.776723\pi\)
\(444\) 1.70304e17 1.05496
\(445\) −1.38958e17 −0.848290
\(446\) 1.28976e17 0.775945
\(447\) −2.12667e17 −1.26096
\(448\) 2.62608e17 1.53462
\(449\) −3.19926e17 −1.84267 −0.921337 0.388764i \(-0.872902\pi\)
−0.921337 + 0.388764i \(0.872902\pi\)
\(450\) 2.72602e17 1.54756
\(451\) −6.31301e16 −0.353255
\(452\) 5.62984e16 0.310525
\(453\) −2.31472e17 −1.25852
\(454\) −4.20179e17 −2.25202
\(455\) −1.44059e17 −0.761144
\(456\) 1.75263e17 0.912892
\(457\) −1.27474e17 −0.654584 −0.327292 0.944923i \(-0.606136\pi\)
−0.327292 + 0.944923i \(0.606136\pi\)
\(458\) −1.85950e17 −0.941394
\(459\) −1.91993e16 −0.0958306
\(460\) 8.42255e16 0.414494
\(461\) −1.02466e17 −0.497190 −0.248595 0.968607i \(-0.579969\pi\)
−0.248595 + 0.968607i \(0.579969\pi\)
\(462\) 2.43475e17 1.16488
\(463\) −2.44233e17 −1.15220 −0.576100 0.817379i \(-0.695426\pi\)
−0.576100 + 0.817379i \(0.695426\pi\)
\(464\) 8.06605e16 0.375227
\(465\) −2.19780e17 −1.00819
\(466\) 2.00966e16 0.0909104
\(467\) 1.94347e17 0.866996 0.433498 0.901155i \(-0.357279\pi\)
0.433498 + 0.901155i \(0.357279\pi\)
\(468\) 5.91518e17 2.60237
\(469\) 1.67010e17 0.724636
\(470\) 3.10234e17 1.32756
\(471\) 3.96237e17 1.67232
\(472\) 1.04595e16 0.0435398
\(473\) 7.69723e16 0.316036
\(474\) −1.30159e17 −0.527125
\(475\) 1.68637e17 0.673666
\(476\) 1.76058e16 0.0693763
\(477\) −5.62321e17 −2.18582
\(478\) −4.46352e17 −1.71158
\(479\) 1.15328e17 0.436268 0.218134 0.975919i \(-0.430003\pi\)
0.218134 + 0.975919i \(0.430003\pi\)
\(480\) 4.65727e17 1.73806
\(481\) 1.30601e17 0.480845
\(482\) −6.35926e17 −2.30994
\(483\) −2.43964e17 −0.874318
\(484\) −2.94867e17 −1.04263
\(485\) 2.42393e16 0.0845667
\(486\) −1.00580e17 −0.346238
\(487\) −1.31850e17 −0.447859 −0.223929 0.974605i \(-0.571888\pi\)
−0.223929 + 0.974605i \(0.571888\pi\)
\(488\) −9.97544e16 −0.334351
\(489\) 8.78397e16 0.290525
\(490\) −4.87731e16 −0.159186
\(491\) −4.19330e17 −1.35060 −0.675299 0.737544i \(-0.735985\pi\)
−0.675299 + 0.737544i \(0.735985\pi\)
\(492\) −5.91999e17 −1.88169
\(493\) 9.09795e15 0.0285390
\(494\) 6.56991e17 2.03392
\(495\) 1.97490e17 0.603407
\(496\) 1.82619e17 0.550699
\(497\) 8.63473e16 0.256999
\(498\) 1.18299e18 3.47528
\(499\) −1.29858e17 −0.376544 −0.188272 0.982117i \(-0.560289\pi\)
−0.188272 + 0.982117i \(0.560289\pi\)
\(500\) −4.66027e17 −1.33385
\(501\) 2.14716e17 0.606624
\(502\) 9.27103e17 2.58556
\(503\) −4.03567e17 −1.11102 −0.555512 0.831508i \(-0.687478\pi\)
−0.555512 + 0.831508i \(0.687478\pi\)
\(504\) 3.14630e17 0.855071
\(505\) −4.46386e17 −1.19761
\(506\) −1.09164e17 −0.289136
\(507\) 4.02408e15 0.0105224
\(508\) 3.64542e17 0.941096
\(509\) 7.02064e16 0.178942 0.0894708 0.995989i \(-0.471482\pi\)
0.0894708 + 0.995989i \(0.471482\pi\)
\(510\) 3.80634e16 0.0957858
\(511\) 1.13846e17 0.282867
\(512\) −4.93558e17 −1.21083
\(513\) −1.03753e18 −2.51326
\(514\) 2.73987e17 0.655345
\(515\) 2.60439e17 0.615119
\(516\) 7.21803e17 1.68343
\(517\) −2.23954e17 −0.515786
\(518\) 3.39567e17 0.772293
\(519\) 1.50017e17 0.336939
\(520\) −1.23662e17 −0.274292
\(521\) −1.64710e17 −0.360807 −0.180403 0.983593i \(-0.557740\pi\)
−0.180403 + 0.983593i \(0.557740\pi\)
\(522\) 7.94753e17 1.71939
\(523\) −8.24266e17 −1.76119 −0.880595 0.473869i \(-0.842857\pi\)
−0.880595 + 0.473869i \(0.842857\pi\)
\(524\) −1.46485e16 −0.0309130
\(525\) 4.49424e17 0.936743
\(526\) −4.94884e17 −1.01882
\(527\) 2.05981e16 0.0418850
\(528\) −2.43609e17 −0.489298
\(529\) −3.94653e17 −0.782985
\(530\) 5.74641e17 1.12617
\(531\) −1.20123e17 −0.232548
\(532\) 9.51415e17 1.81947
\(533\) −4.53988e17 −0.857665
\(534\) 1.68891e18 3.15201
\(535\) 6.50113e17 1.19864
\(536\) 1.43363e17 0.261136
\(537\) 1.36589e18 2.45801
\(538\) −1.94932e17 −0.346576
\(539\) 3.52087e16 0.0618476
\(540\) 9.54598e17 1.65677
\(541\) −7.19658e17 −1.23408 −0.617041 0.786931i \(-0.711669\pi\)
−0.617041 + 0.786931i \(0.711669\pi\)
\(542\) 8.77251e17 1.48638
\(543\) −8.85375e17 −1.48227
\(544\) −4.36487e16 −0.0722068
\(545\) −4.24591e17 −0.694055
\(546\) 1.75090e18 2.82820
\(547\) −2.44733e17 −0.390638 −0.195319 0.980740i \(-0.562574\pi\)
−0.195319 + 0.980740i \(0.562574\pi\)
\(548\) 5.60821e17 0.884608
\(549\) 1.14564e18 1.78578
\(550\) 2.01099e17 0.309780
\(551\) 4.91651e17 0.748465
\(552\) −2.09421e17 −0.315077
\(553\) −1.44546e17 −0.214928
\(554\) 1.47396e17 0.216606
\(555\) 4.08892e17 0.593888
\(556\) 5.91493e17 0.849108
\(557\) 3.20958e16 0.0455397 0.0227698 0.999741i \(-0.492752\pi\)
0.0227698 + 0.999741i \(0.492752\pi\)
\(558\) 1.79935e18 2.52345
\(559\) 5.53532e17 0.767301
\(560\) 3.74765e17 0.513495
\(561\) −2.74775e16 −0.0372150
\(562\) 6.43857e16 0.0861992
\(563\) −1.14519e18 −1.51556 −0.757778 0.652512i \(-0.773715\pi\)
−0.757778 + 0.652512i \(0.773715\pi\)
\(564\) −2.10011e18 −2.74744
\(565\) 1.35170e17 0.174810
\(566\) −1.70896e18 −2.18487
\(567\) −1.01419e18 −1.28182
\(568\) 7.41213e16 0.0926145
\(569\) 2.98861e17 0.369181 0.184590 0.982816i \(-0.440904\pi\)
0.184590 + 0.982816i \(0.440904\pi\)
\(570\) 2.05693e18 2.51208
\(571\) −7.57570e17 −0.914720 −0.457360 0.889282i \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(572\) 4.36365e17 0.520925
\(573\) 1.88238e16 0.0222179
\(574\) −1.18038e18 −1.37751
\(575\) −2.01503e17 −0.232510
\(576\) −2.58893e18 −2.95375
\(577\) 2.40451e17 0.271259 0.135629 0.990760i \(-0.456694\pi\)
0.135629 + 0.990760i \(0.456694\pi\)
\(578\) 1.34327e18 1.49842
\(579\) −1.30049e18 −1.43449
\(580\) −4.52353e17 −0.493396
\(581\) 1.31376e18 1.41700
\(582\) −2.94606e17 −0.314226
\(583\) −4.14826e17 −0.437543
\(584\) 9.77265e16 0.101936
\(585\) 1.42021e18 1.46501
\(586\) 1.29803e18 1.32419
\(587\) −9.10215e17 −0.918325 −0.459163 0.888352i \(-0.651850\pi\)
−0.459163 + 0.888352i \(0.651850\pi\)
\(588\) 3.30168e17 0.329444
\(589\) 1.11312e18 1.09848
\(590\) 1.22755e17 0.119812
\(591\) −1.29186e17 −0.124709
\(592\) −3.39755e17 −0.324395
\(593\) 1.49529e18 1.41211 0.706055 0.708157i \(-0.250473\pi\)
0.706055 + 0.708157i \(0.250473\pi\)
\(594\) −1.23725e18 −1.15570
\(595\) 4.22709e16 0.0390554
\(596\) −9.91000e17 −0.905676
\(597\) −3.03724e18 −2.74565
\(598\) −7.85034e17 −0.701990
\(599\) −1.77688e18 −1.57175 −0.785876 0.618385i \(-0.787787\pi\)
−0.785876 + 0.618385i \(0.787787\pi\)
\(600\) 3.85790e17 0.337573
\(601\) 8.12672e17 0.703447 0.351724 0.936104i \(-0.385596\pi\)
0.351724 + 0.936104i \(0.385596\pi\)
\(602\) 1.43920e18 1.23237
\(603\) −1.64647e18 −1.39474
\(604\) −1.07863e18 −0.903924
\(605\) −7.07964e17 −0.586950
\(606\) 5.42540e18 4.44999
\(607\) −1.43007e17 −0.116046 −0.0580231 0.998315i \(-0.518480\pi\)
−0.0580231 + 0.998315i \(0.518480\pi\)
\(608\) −2.35876e18 −1.89370
\(609\) 1.31027e18 1.04075
\(610\) −1.17074e18 −0.920063
\(611\) −1.61052e18 −1.25227
\(612\) −1.73567e17 −0.133532
\(613\) 1.48247e18 1.12848 0.564240 0.825611i \(-0.309169\pi\)
0.564240 + 0.825611i \(0.309169\pi\)
\(614\) −1.27769e18 −0.962343
\(615\) −1.42136e18 −1.05930
\(616\) 2.32104e17 0.171162
\(617\) 2.65142e18 1.93475 0.967375 0.253348i \(-0.0815316\pi\)
0.967375 + 0.253348i \(0.0815316\pi\)
\(618\) −3.16539e18 −2.28561
\(619\) 2.95294e16 0.0210992 0.0105496 0.999944i \(-0.496642\pi\)
0.0105496 + 0.999944i \(0.496642\pi\)
\(620\) −1.02415e18 −0.724129
\(621\) 1.23974e18 0.867428
\(622\) 2.66277e18 1.84373
\(623\) 1.87560e18 1.28519
\(624\) −1.75187e18 −1.18796
\(625\) −3.75180e17 −0.251779
\(626\) 1.65508e18 1.09922
\(627\) −1.48488e18 −0.976003
\(628\) 1.84641e18 1.20113
\(629\) −3.83220e16 −0.0246728
\(630\) 3.69258e18 2.35297
\(631\) −2.30017e18 −1.45067 −0.725336 0.688395i \(-0.758316\pi\)
−0.725336 + 0.688395i \(0.758316\pi\)
\(632\) −1.24080e17 −0.0774534
\(633\) 1.33150e18 0.822656
\(634\) −3.53481e18 −2.16165
\(635\) 8.75251e17 0.529790
\(636\) −3.89001e18 −2.33066
\(637\) 2.53197e17 0.150159
\(638\) 5.86292e17 0.344175
\(639\) −8.51256e17 −0.494658
\(640\) 9.19402e17 0.528854
\(641\) 2.55633e18 1.45559 0.727795 0.685795i \(-0.240545\pi\)
0.727795 + 0.685795i \(0.240545\pi\)
\(642\) −7.90151e18 −4.45382
\(643\) −2.36777e18 −1.32120 −0.660600 0.750738i \(-0.729698\pi\)
−0.660600 + 0.750738i \(0.729698\pi\)
\(644\) −1.13684e18 −0.627973
\(645\) 1.73302e18 0.947688
\(646\) −1.92779e17 −0.104363
\(647\) 3.11143e18 1.66756 0.833782 0.552094i \(-0.186171\pi\)
0.833782 + 0.552094i \(0.186171\pi\)
\(648\) −8.70589e17 −0.461929
\(649\) −8.86153e16 −0.0465499
\(650\) 1.44617e18 0.752111
\(651\) 2.96650e18 1.52745
\(652\) 4.09321e17 0.208668
\(653\) 1.66792e17 0.0841861 0.0420931 0.999114i \(-0.486597\pi\)
0.0420931 + 0.999114i \(0.486597\pi\)
\(654\) 5.16050e18 2.57891
\(655\) −3.51705e16 −0.0174024
\(656\) 1.18103e18 0.578612
\(657\) −1.12235e18 −0.544445
\(658\) −4.18740e18 −2.01129
\(659\) −1.38461e18 −0.658527 −0.329263 0.944238i \(-0.606800\pi\)
−0.329263 + 0.944238i \(0.606800\pi\)
\(660\) 1.36619e18 0.643391
\(661\) −1.62332e18 −0.756996 −0.378498 0.925602i \(-0.623559\pi\)
−0.378498 + 0.925602i \(0.623559\pi\)
\(662\) −2.42989e18 −1.12204
\(663\) −1.97599e17 −0.0903539
\(664\) 1.12774e18 0.510643
\(665\) 2.28431e18 1.02427
\(666\) −3.34763e18 −1.48646
\(667\) −5.87470e17 −0.258326
\(668\) 1.00055e18 0.435703
\(669\) −2.09627e18 −0.904017
\(670\) 1.68255e18 0.718590
\(671\) 8.45145e17 0.357466
\(672\) −6.28618e18 −2.63322
\(673\) −1.22676e18 −0.508933 −0.254467 0.967082i \(-0.581900\pi\)
−0.254467 + 0.967082i \(0.581900\pi\)
\(674\) 1.47679e18 0.606777
\(675\) −2.28381e18 −0.929361
\(676\) 1.87517e16 0.00755765
\(677\) 4.12504e18 1.64665 0.823325 0.567570i \(-0.192116\pi\)
0.823325 + 0.567570i \(0.192116\pi\)
\(678\) −1.64286e18 −0.649544
\(679\) −3.27172e17 −0.128122
\(680\) 3.62857e16 0.0140743
\(681\) 6.82926e18 2.62372
\(682\) 1.32739e18 0.505126
\(683\) 4.35534e18 1.64168 0.820839 0.571160i \(-0.193506\pi\)
0.820839 + 0.571160i \(0.193506\pi\)
\(684\) −9.37954e18 −3.50201
\(685\) 1.34651e18 0.497990
\(686\) −3.73897e18 −1.36976
\(687\) 3.02229e18 1.09677
\(688\) −1.43999e18 −0.517649
\(689\) −2.98315e18 −1.06231
\(690\) −2.45782e18 −0.867023
\(691\) 9.48719e17 0.331536 0.165768 0.986165i \(-0.446990\pi\)
0.165768 + 0.986165i \(0.446990\pi\)
\(692\) 6.99057e17 0.242004
\(693\) −2.66563e18 −0.914184
\(694\) 3.17572e17 0.107896
\(695\) 1.42015e18 0.478005
\(696\) 1.12474e18 0.375054
\(697\) 1.33212e17 0.0440080
\(698\) 1.93106e18 0.632027
\(699\) −3.26634e17 −0.105915
\(700\) 2.09425e18 0.672809
\(701\) 1.83506e18 0.584094 0.292047 0.956404i \(-0.405664\pi\)
0.292047 + 0.956404i \(0.405664\pi\)
\(702\) −8.89744e18 −2.80591
\(703\) −2.07091e18 −0.647071
\(704\) −1.90986e18 −0.591262
\(705\) −5.04229e18 −1.54667
\(706\) 8.24762e18 2.50668
\(707\) 6.02512e18 1.81443
\(708\) −8.30985e17 −0.247957
\(709\) −1.63548e18 −0.483552 −0.241776 0.970332i \(-0.577730\pi\)
−0.241776 + 0.970332i \(0.577730\pi\)
\(710\) 8.69907e17 0.254855
\(711\) 1.42501e18 0.413682
\(712\) 1.61003e18 0.463142
\(713\) −1.33006e18 −0.379130
\(714\) −5.13762e17 −0.145119
\(715\) 1.04769e18 0.293255
\(716\) 6.36488e18 1.76545
\(717\) 7.25465e18 1.99408
\(718\) 5.43032e18 1.47916
\(719\) −5.55267e17 −0.149887 −0.0749435 0.997188i \(-0.523878\pi\)
−0.0749435 + 0.997188i \(0.523878\pi\)
\(720\) −3.69462e18 −0.988346
\(721\) −3.51529e18 −0.931928
\(722\) −4.69931e18 −1.23465
\(723\) 1.03358e19 2.69120
\(724\) −4.12573e18 −1.06463
\(725\) 1.08222e18 0.276770
\(726\) 8.60463e18 2.18094
\(727\) 4.63573e18 1.16451 0.582257 0.813005i \(-0.302170\pi\)
0.582257 + 0.813005i \(0.302170\pi\)
\(728\) 1.66913e18 0.415563
\(729\) −3.20992e18 −0.792072
\(730\) 1.14694e18 0.280507
\(731\) −1.62421e17 −0.0393713
\(732\) 7.92530e18 1.90412
\(733\) 6.29653e18 1.49943 0.749713 0.661763i \(-0.230191\pi\)
0.749713 + 0.661763i \(0.230191\pi\)
\(734\) −5.44830e18 −1.28599
\(735\) 7.92719e17 0.185461
\(736\) 2.81847e18 0.653594
\(737\) −1.21461e18 −0.279189
\(738\) 1.16368e19 2.65135
\(739\) −5.80929e18 −1.31200 −0.656001 0.754760i \(-0.727753\pi\)
−0.656001 + 0.754760i \(0.727753\pi\)
\(740\) 1.90538e18 0.426556
\(741\) −1.06782e19 −2.36963
\(742\) −7.75625e18 −1.70619
\(743\) 3.72059e18 0.811305 0.405652 0.914027i \(-0.367044\pi\)
0.405652 + 0.914027i \(0.367044\pi\)
\(744\) 2.54647e18 0.550445
\(745\) −2.37935e18 −0.509850
\(746\) −1.25758e19 −2.67137
\(747\) −1.29517e19 −2.72736
\(748\) −1.28041e17 −0.0267294
\(749\) −8.77494e18 −1.81598
\(750\) 1.35993e19 2.79009
\(751\) −8.73904e18 −1.77748 −0.888739 0.458413i \(-0.848418\pi\)
−0.888739 + 0.458413i \(0.848418\pi\)
\(752\) 4.18971e18 0.844828
\(753\) −1.50684e19 −3.01231
\(754\) 4.21621e18 0.835619
\(755\) −2.58974e18 −0.508864
\(756\) −1.28847e19 −2.51006
\(757\) −1.18963e18 −0.229768 −0.114884 0.993379i \(-0.536650\pi\)
−0.114884 + 0.993379i \(0.536650\pi\)
\(758\) 3.60252e18 0.689852
\(759\) 1.77427e18 0.336859
\(760\) 1.96087e18 0.369114
\(761\) 5.01198e18 0.935425 0.467713 0.883881i \(-0.345078\pi\)
0.467713 + 0.883881i \(0.345078\pi\)
\(762\) −1.06378e19 −1.96855
\(763\) 5.73095e18 1.05152
\(764\) 8.77165e16 0.0159579
\(765\) −4.16728e17 −0.0751716
\(766\) 1.23346e19 2.20616
\(767\) −6.37261e17 −0.113018
\(768\) 3.07129e18 0.540098
\(769\) −5.93365e18 −1.03467 −0.517334 0.855784i \(-0.673075\pi\)
−0.517334 + 0.855784i \(0.673075\pi\)
\(770\) 2.72403e18 0.471001
\(771\) −4.45317e18 −0.763512
\(772\) −6.06012e18 −1.03031
\(773\) −1.82934e18 −0.308409 −0.154204 0.988039i \(-0.549281\pi\)
−0.154204 + 0.988039i \(0.549281\pi\)
\(774\) −1.41883e19 −2.37200
\(775\) 2.45019e18 0.406200
\(776\) −2.80847e17 −0.0461710
\(777\) −5.51905e18 −0.899762
\(778\) −1.52608e18 −0.246723
\(779\) 7.19877e18 1.15416
\(780\) 9.82469e18 1.56208
\(781\) −6.27975e17 −0.0990172
\(782\) 2.30350e17 0.0360201
\(783\) −6.65829e18 −1.03255
\(784\) −6.58682e17 −0.101303
\(785\) 4.43316e18 0.676178
\(786\) 4.27464e17 0.0646626
\(787\) 5.99750e17 0.0899776 0.0449888 0.998987i \(-0.485675\pi\)
0.0449888 + 0.998987i \(0.485675\pi\)
\(788\) −6.01989e17 −0.0895712
\(789\) 8.04345e18 1.18698
\(790\) −1.45623e18 −0.213135
\(791\) −1.82447e18 −0.264843
\(792\) −2.28820e18 −0.329443
\(793\) 6.07770e18 0.867888
\(794\) 1.22911e19 1.74083
\(795\) −9.33975e18 −1.31205
\(796\) −1.41531e19 −1.97205
\(797\) 9.62112e18 1.32968 0.664839 0.746986i \(-0.268500\pi\)
0.664839 + 0.746986i \(0.268500\pi\)
\(798\) −2.77636e19 −3.80590
\(799\) 4.72571e17 0.0642559
\(800\) −5.19211e18 −0.700259
\(801\) −1.84906e19 −2.47366
\(802\) −4.11230e18 −0.545696
\(803\) −8.27964e17 −0.108983
\(804\) −1.13899e19 −1.48716
\(805\) −2.72950e18 −0.353517
\(806\) 9.54568e18 1.22639
\(807\) 3.16827e18 0.403780
\(808\) 5.17202e18 0.653862
\(809\) 7.79142e18 0.977127 0.488563 0.872528i \(-0.337521\pi\)
0.488563 + 0.872528i \(0.337521\pi\)
\(810\) −1.02174e19 −1.27113
\(811\) −8.70719e18 −1.07459 −0.537295 0.843395i \(-0.680554\pi\)
−0.537295 + 0.843395i \(0.680554\pi\)
\(812\) 6.10566e18 0.747512
\(813\) −1.42581e19 −1.73171
\(814\) −2.46956e18 −0.297550
\(815\) 9.82763e17 0.117469
\(816\) 5.14047e17 0.0609560
\(817\) −8.77720e18 −1.03255
\(818\) −1.40455e19 −1.63923
\(819\) −1.91694e19 −2.21954
\(820\) −6.62337e18 −0.760832
\(821\) 2.21889e18 0.252874 0.126437 0.991975i \(-0.459646\pi\)
0.126437 + 0.991975i \(0.459646\pi\)
\(822\) −1.63655e19 −1.85039
\(823\) −1.24348e18 −0.139488 −0.0697442 0.997565i \(-0.522218\pi\)
−0.0697442 + 0.997565i \(0.522218\pi\)
\(824\) −3.01756e18 −0.335838
\(825\) −3.26851e18 −0.360910
\(826\) −1.65689e18 −0.181520
\(827\) −1.78746e19 −1.94290 −0.971449 0.237248i \(-0.923755\pi\)
−0.971449 + 0.237248i \(0.923755\pi\)
\(828\) 1.12076e19 1.20869
\(829\) 1.22207e19 1.30765 0.653823 0.756647i \(-0.273164\pi\)
0.653823 + 0.756647i \(0.273164\pi\)
\(830\) 1.32355e19 1.40518
\(831\) −2.39565e18 −0.252358
\(832\) −1.37344e19 −1.43552
\(833\) −7.42948e16 −0.00770488
\(834\) −1.72606e19 −1.77613
\(835\) 2.40228e18 0.245279
\(836\) −6.91932e18 −0.701008
\(837\) −1.50746e19 −1.51541
\(838\) −5.38765e17 −0.0537419
\(839\) 1.06014e19 1.04933 0.524665 0.851309i \(-0.324191\pi\)
0.524665 + 0.851309i \(0.324191\pi\)
\(840\) 5.22579e18 0.513259
\(841\) −7.10548e18 −0.692500
\(842\) −1.82214e19 −1.76219
\(843\) −1.04647e18 −0.100427
\(844\) 6.20462e18 0.590867
\(845\) 4.50220e16 0.00425458
\(846\) 4.12815e19 3.87122
\(847\) 9.55578e18 0.889250
\(848\) 7.76054e18 0.716670
\(849\) 2.77761e19 2.54549
\(850\) −4.24345e17 −0.0385919
\(851\) 2.47452e18 0.223331
\(852\) −5.88880e18 −0.527436
\(853\) 4.77561e18 0.424483 0.212242 0.977217i \(-0.431924\pi\)
0.212242 + 0.977217i \(0.431924\pi\)
\(854\) 1.58022e19 1.39393
\(855\) −2.25199e19 −1.97145
\(856\) −7.53249e18 −0.654424
\(857\) 4.15881e18 0.358587 0.179293 0.983796i \(-0.442619\pi\)
0.179293 + 0.983796i \(0.442619\pi\)
\(858\) −1.27337e19 −1.08965
\(859\) −1.40675e19 −1.19471 −0.597354 0.801978i \(-0.703781\pi\)
−0.597354 + 0.801978i \(0.703781\pi\)
\(860\) 8.07564e18 0.680670
\(861\) 1.91849e19 1.60487
\(862\) 1.23719e18 0.102716
\(863\) 1.94577e19 1.60332 0.801662 0.597777i \(-0.203949\pi\)
0.801662 + 0.597777i \(0.203949\pi\)
\(864\) 3.19441e19 2.61247
\(865\) 1.67841e18 0.136236
\(866\) −1.15347e19 −0.929270
\(867\) −2.18324e19 −1.74573
\(868\) 1.38235e19 1.09708
\(869\) 1.05124e18 0.0828080
\(870\) 1.32003e19 1.03207
\(871\) −8.73464e18 −0.677840
\(872\) 4.91950e18 0.378934
\(873\) 3.22543e18 0.246601
\(874\) 1.24481e19 0.944666
\(875\) 1.51026e19 1.13762
\(876\) −7.76418e18 −0.580522
\(877\) −3.63031e18 −0.269430 −0.134715 0.990884i \(-0.543012\pi\)
−0.134715 + 0.990884i \(0.543012\pi\)
\(878\) 3.37002e19 2.48266
\(879\) −2.10972e19 −1.54276
\(880\) −2.72554e18 −0.197840
\(881\) 5.69812e18 0.410571 0.205286 0.978702i \(-0.434188\pi\)
0.205286 + 0.978702i \(0.434188\pi\)
\(882\) −6.49004e18 −0.464196
\(883\) 1.10293e17 0.00783075 0.00391538 0.999992i \(-0.498754\pi\)
0.00391538 + 0.999992i \(0.498754\pi\)
\(884\) −9.20785e17 −0.0648961
\(885\) −1.99516e18 −0.139588
\(886\) 3.30503e19 2.29539
\(887\) 5.78303e18 0.398705 0.199353 0.979928i \(-0.436116\pi\)
0.199353 + 0.979928i \(0.436116\pi\)
\(888\) −4.73760e18 −0.324246
\(889\) −1.18138e19 −0.802651
\(890\) 1.88957e19 1.27447
\(891\) 7.37585e18 0.493864
\(892\) −9.76832e18 −0.649304
\(893\) 2.55376e19 1.68518
\(894\) 2.89187e19 1.89446
\(895\) 1.52818e19 0.993860
\(896\) −1.24097e19 −0.801233
\(897\) 1.27593e19 0.817856
\(898\) 4.35040e19 2.76843
\(899\) 7.14338e18 0.451301
\(900\) −2.06462e19 −1.29498
\(901\) 8.75336e17 0.0545084
\(902\) 8.58451e18 0.530729
\(903\) −2.33915e19 −1.43578
\(904\) −1.56614e18 −0.0954411
\(905\) −9.90570e18 −0.599335
\(906\) 3.14759e19 1.89080
\(907\) 1.57749e19 0.940848 0.470424 0.882441i \(-0.344101\pi\)
0.470424 + 0.882441i \(0.344101\pi\)
\(908\) 3.18234e19 1.88447
\(909\) −5.93987e19 −3.49230
\(910\) 1.95894e19 1.14354
\(911\) 2.45870e19 1.42507 0.712536 0.701636i \(-0.247547\pi\)
0.712536 + 0.701636i \(0.247547\pi\)
\(912\) 2.77790e19 1.59864
\(913\) −9.55451e18 −0.545945
\(914\) 1.73340e19 0.983444
\(915\) 1.90283e19 1.07192
\(916\) 1.40835e19 0.787750
\(917\) 4.74716e17 0.0263653
\(918\) 2.61075e18 0.143975
\(919\) 1.02050e19 0.558810 0.279405 0.960173i \(-0.409863\pi\)
0.279405 + 0.960173i \(0.409863\pi\)
\(920\) −2.34303e18 −0.127397
\(921\) 2.07665e19 1.12118
\(922\) 1.39334e19 0.746977
\(923\) −4.51596e18 −0.240403
\(924\) −1.84402e19 −0.974761
\(925\) −4.55849e18 −0.239276
\(926\) 3.32111e19 1.73106
\(927\) 3.46556e19 1.79372
\(928\) −1.51373e19 −0.778010
\(929\) 1.68307e19 0.859013 0.429507 0.903064i \(-0.358687\pi\)
0.429507 + 0.903064i \(0.358687\pi\)
\(930\) 2.98860e19 1.51471
\(931\) −4.01487e18 −0.202069
\(932\) −1.52207e18 −0.0760730
\(933\) −4.32785e19 −2.14804
\(934\) −2.64275e19 −1.30257
\(935\) −3.07422e17 −0.0150473
\(936\) −1.64552e19 −0.799852
\(937\) 2.03322e19 0.981469 0.490734 0.871309i \(-0.336729\pi\)
0.490734 + 0.871309i \(0.336729\pi\)
\(938\) −2.27103e19 −1.08869
\(939\) −2.69004e19 −1.28066
\(940\) −2.34964e19 −1.11089
\(941\) 2.69186e19 1.26392 0.631960 0.775001i \(-0.282251\pi\)
0.631960 + 0.775001i \(0.282251\pi\)
\(942\) −5.38809e19 −2.51249
\(943\) −8.60176e18 −0.398347
\(944\) 1.65781e18 0.0762459
\(945\) −3.09357e19 −1.41304
\(946\) −1.04668e19 −0.474811
\(947\) −2.03943e19 −0.918827 −0.459414 0.888222i \(-0.651940\pi\)
−0.459414 + 0.888222i \(0.651940\pi\)
\(948\) 9.85792e18 0.441094
\(949\) −5.95415e18 −0.264600
\(950\) −2.29315e19 −1.01211
\(951\) 5.74519e19 2.51844
\(952\) −4.89769e17 −0.0213231
\(953\) 3.03907e19 1.31412 0.657062 0.753837i \(-0.271799\pi\)
0.657062 + 0.753837i \(0.271799\pi\)
\(954\) 7.64651e19 3.28397
\(955\) 2.10604e17 0.00898347
\(956\) 3.38057e19 1.43223
\(957\) −9.52912e18 −0.400982
\(958\) −1.56824e19 −0.655447
\(959\) −1.81746e19 −0.754473
\(960\) −4.30003e19 −1.77300
\(961\) −8.24463e18 −0.337652
\(962\) −1.77594e19 −0.722419
\(963\) 8.65079e19 3.49530
\(964\) 4.81636e19 1.93294
\(965\) −1.45501e19 −0.580014
\(966\) 3.31745e19 1.31357
\(967\) −5.80632e17 −0.0228365 −0.0114182 0.999935i \(-0.503635\pi\)
−0.0114182 + 0.999935i \(0.503635\pi\)
\(968\) 8.20277e18 0.320458
\(969\) 3.13328e18 0.121589
\(970\) −3.29609e18 −0.127053
\(971\) 1.45589e19 0.557448 0.278724 0.960371i \(-0.410089\pi\)
0.278724 + 0.960371i \(0.410089\pi\)
\(972\) 7.61769e18 0.289729
\(973\) −1.91685e19 −0.724195
\(974\) 1.79291e19 0.672861
\(975\) −2.35049e19 −0.876249
\(976\) −1.58109e19 −0.585508
\(977\) −2.61387e19 −0.961546 −0.480773 0.876845i \(-0.659644\pi\)
−0.480773 + 0.876845i \(0.659644\pi\)
\(978\) −1.19446e19 −0.436483
\(979\) −1.36406e19 −0.495161
\(980\) 3.69396e18 0.133206
\(981\) −5.64986e19 −2.02390
\(982\) 5.70210e19 2.02913
\(983\) −2.23630e19 −0.790557 −0.395278 0.918561i \(-0.629352\pi\)
−0.395278 + 0.918561i \(0.629352\pi\)
\(984\) 1.64685e19 0.578345
\(985\) −1.44535e18 −0.0504241
\(986\) −1.23715e18 −0.0428768
\(987\) 6.80586e19 2.34326
\(988\) −4.97590e19 −1.70197
\(989\) 1.04878e19 0.356377
\(990\) −2.68549e19 −0.906556
\(991\) −5.93305e18 −0.198975 −0.0994877 0.995039i \(-0.531720\pi\)
−0.0994877 + 0.995039i \(0.531720\pi\)
\(992\) −3.42714e19 −1.14184
\(993\) 3.94934e19 1.30724
\(994\) −1.17416e19 −0.386115
\(995\) −3.39811e19 −1.11016
\(996\) −8.95969e19 −2.90809
\(997\) −4.98509e17 −0.0160751 −0.00803756 0.999968i \(-0.502558\pi\)
−0.00803756 + 0.999968i \(0.502558\pi\)
\(998\) 1.76583e19 0.565718
\(999\) 2.80458e19 0.892672
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 197.14.a.a.1.18 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
197.14.a.a.1.18 104 1.1 even 1 trivial