Properties

Label 197.10.a.b.1.13
Level $197$
Weight $10$
Character 197.1
Self dual yes
Analytic conductor $101.462$
Analytic rank $0$
Dimension $76$
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] = [197,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(101.462059724\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
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)
 
\(f(q)\) \(=\) \(q-34.7070 q^{2} -78.4883 q^{3} +692.576 q^{4} -317.779 q^{5} +2724.09 q^{6} -4384.10 q^{7} -6267.25 q^{8} -13522.6 q^{9} +O(q^{10})\) \(q-34.7070 q^{2} -78.4883 q^{3} +692.576 q^{4} -317.779 q^{5} +2724.09 q^{6} -4384.10 q^{7} -6267.25 q^{8} -13522.6 q^{9} +11029.1 q^{10} +45420.7 q^{11} -54359.1 q^{12} -31389.1 q^{13} +152159. q^{14} +24941.9 q^{15} -137081. q^{16} +85496.2 q^{17} +469329. q^{18} -806441. q^{19} -220086. q^{20} +344101. q^{21} -1.57642e6 q^{22} +392444. q^{23} +491906. q^{24} -1.85214e6 q^{25} +1.08942e6 q^{26} +2.60625e6 q^{27} -3.03633e6 q^{28} +5.91681e6 q^{29} -865659. q^{30} +3.42516e6 q^{31} +7.96652e6 q^{32} -3.56499e6 q^{33} -2.96732e6 q^{34} +1.39317e6 q^{35} -9.36542e6 q^{36} -3.26032e6 q^{37} +2.79892e7 q^{38} +2.46368e6 q^{39} +1.99160e6 q^{40} -1.26323e7 q^{41} -1.19427e7 q^{42} -1.66489e7 q^{43} +3.14573e7 q^{44} +4.29719e6 q^{45} -1.36205e7 q^{46} -1.60830e7 q^{47} +1.07593e7 q^{48} -2.11332e7 q^{49} +6.42823e7 q^{50} -6.71045e6 q^{51} -2.17394e7 q^{52} -4.29820e7 q^{53} -9.04551e7 q^{54} -1.44337e7 q^{55} +2.74763e7 q^{56} +6.32962e7 q^{57} -2.05355e8 q^{58} -4.18933e7 q^{59} +1.72742e7 q^{60} -5.02151e7 q^{61} -1.18877e8 q^{62} +5.92844e7 q^{63} -2.06308e8 q^{64} +9.97479e6 q^{65} +1.23730e8 q^{66} -2.36103e8 q^{67} +5.92126e7 q^{68} -3.08022e7 q^{69} -4.83529e7 q^{70} -2.73121e8 q^{71} +8.47495e7 q^{72} +1.41037e7 q^{73} +1.13156e8 q^{74} +1.45371e8 q^{75} -5.58522e8 q^{76} -1.99129e8 q^{77} -8.55069e7 q^{78} -1.34719e8 q^{79} +4.35615e7 q^{80} +6.16051e7 q^{81} +4.38428e8 q^{82} -2.79556e8 q^{83} +2.38316e8 q^{84} -2.71689e7 q^{85} +5.77833e8 q^{86} -4.64400e8 q^{87} -2.84663e8 q^{88} +1.92906e8 q^{89} -1.49143e8 q^{90} +1.37613e8 q^{91} +2.71797e8 q^{92} -2.68835e8 q^{93} +5.58194e8 q^{94} +2.56270e8 q^{95} -6.25278e8 q^{96} -3.24674e8 q^{97} +7.33471e8 q^{98} -6.14206e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q + 48 q^{2} + 890 q^{3} + 20736 q^{4} + 5171 q^{5} + 2688 q^{6} + 38986 q^{7} + 36507 q^{8} + 518318 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 76 q + 48 q^{2} + 890 q^{3} + 20736 q^{4} + 5171 q^{5} + 2688 q^{6} + 38986 q^{7} + 36507 q^{8} + 518318 q^{9} + 121093 q^{10} + 120464 q^{11} + 415744 q^{12} + 480131 q^{13} + 330849 q^{14} + 544874 q^{15} + 5963776 q^{16} + 942582 q^{17} + 1483945 q^{18} + 3097319 q^{19} + 3237739 q^{20} + 889076 q^{21} + 3791921 q^{22} + 5139200 q^{23} + 1999533 q^{24} + 34080519 q^{25} + 2791454 q^{26} + 24386486 q^{27} + 20891166 q^{28} + 6886818 q^{29} + 14171083 q^{30} + 28002851 q^{31} + 16857332 q^{32} + 30921422 q^{33} + 33506194 q^{34} + 16271736 q^{35} + 151430458 q^{36} + 55950976 q^{37} + 62370882 q^{38} - 11569592 q^{39} + 129854766 q^{40} + 14990859 q^{41} + 82216531 q^{42} + 169867467 q^{43} + 41872434 q^{44} + 205007649 q^{45} + 144032301 q^{46} + 78743342 q^{47} + 156250562 q^{48} + 533861890 q^{49} + 626841163 q^{50} + 477099244 q^{51} + 560784114 q^{52} + 188670216 q^{53} + 525901687 q^{54} + 298497914 q^{55} + 56575048 q^{56} + 213972590 q^{57} + 338315251 q^{58} + 208222151 q^{59} - 615921507 q^{60} - 233556134 q^{61} - 399368105 q^{62} + 329825056 q^{63} + 876517017 q^{64} - 840557006 q^{65} - 2482481592 q^{66} + 1210808414 q^{67} - 1266757099 q^{68} + 327801786 q^{69} - 546384313 q^{70} + 345300221 q^{71} - 1549481681 q^{72} + 1192286460 q^{73} - 1471133595 q^{74} + 761630676 q^{75} - 398699826 q^{76} - 101106252 q^{77} - 2609825943 q^{78} + 955627631 q^{79} + 1059617770 q^{80} + 3387041436 q^{81} + 1062705523 q^{82} + 1538917201 q^{83} + 1394513218 q^{84} + 225481100 q^{85} + 701644810 q^{86} + 1758812842 q^{87} + 3151474875 q^{88} + 855413630 q^{89} + 6070671455 q^{90} + 4652436248 q^{91} + 8082863606 q^{92} + 3462095982 q^{93} + 2660342117 q^{94} + 1036805508 q^{95} + 12370989029 q^{96} + 6393874545 q^{97} + 7510976010 q^{98} + 8731109606 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −34.7070 −1.53385 −0.766924 0.641738i \(-0.778214\pi\)
−0.766924 + 0.641738i \(0.778214\pi\)
\(3\) −78.4883 −0.559447 −0.279724 0.960081i \(-0.590243\pi\)
−0.279724 + 0.960081i \(0.590243\pi\)
\(4\) 692.576 1.35269
\(5\) −317.779 −0.227384 −0.113692 0.993516i \(-0.536268\pi\)
−0.113692 + 0.993516i \(0.536268\pi\)
\(6\) 2724.09 0.858107
\(7\) −4384.10 −0.690144 −0.345072 0.938576i \(-0.612145\pi\)
−0.345072 + 0.938576i \(0.612145\pi\)
\(8\) −6267.25 −0.540969
\(9\) −13522.6 −0.687019
\(10\) 11029.1 0.348772
\(11\) 45420.7 0.935378 0.467689 0.883893i \(-0.345087\pi\)
0.467689 + 0.883893i \(0.345087\pi\)
\(12\) −54359.1 −0.756757
\(13\) −31389.1 −0.304813 −0.152407 0.988318i \(-0.548702\pi\)
−0.152407 + 0.988318i \(0.548702\pi\)
\(14\) 152159. 1.05858
\(15\) 24941.9 0.127209
\(16\) −137081. −0.522924
\(17\) 85496.2 0.248271 0.124136 0.992265i \(-0.460384\pi\)
0.124136 + 0.992265i \(0.460384\pi\)
\(18\) 469329. 1.05378
\(19\) −806441. −1.41965 −0.709825 0.704378i \(-0.751226\pi\)
−0.709825 + 0.704378i \(0.751226\pi\)
\(20\) −220086. −0.307579
\(21\) 344101. 0.386099
\(22\) −1.57642e6 −1.43473
\(23\) 392444. 0.292417 0.146208 0.989254i \(-0.453293\pi\)
0.146208 + 0.989254i \(0.453293\pi\)
\(24\) 491906. 0.302644
\(25\) −1.85214e6 −0.948297
\(26\) 1.08942e6 0.467537
\(27\) 2.60625e6 0.943798
\(28\) −3.03633e6 −0.933549
\(29\) 5.91681e6 1.55345 0.776724 0.629842i \(-0.216880\pi\)
0.776724 + 0.629842i \(0.216880\pi\)
\(30\) −865659. −0.195120
\(31\) 3.42516e6 0.666122 0.333061 0.942905i \(-0.391919\pi\)
0.333061 + 0.942905i \(0.391919\pi\)
\(32\) 7.96652e6 1.34305
\(33\) −3.56499e6 −0.523294
\(34\) −2.96732e6 −0.380810
\(35\) 1.39317e6 0.156928
\(36\) −9.36542e6 −0.929322
\(37\) −3.26032e6 −0.285991 −0.142996 0.989723i \(-0.545673\pi\)
−0.142996 + 0.989723i \(0.545673\pi\)
\(38\) 2.79892e7 2.17753
\(39\) 2.46368e6 0.170527
\(40\) 1.99160e6 0.123008
\(41\) −1.26323e7 −0.698159 −0.349079 0.937093i \(-0.613506\pi\)
−0.349079 + 0.937093i \(0.613506\pi\)
\(42\) −1.19427e7 −0.592217
\(43\) −1.66489e7 −0.742638 −0.371319 0.928505i \(-0.621094\pi\)
−0.371319 + 0.928505i \(0.621094\pi\)
\(44\) 3.14573e7 1.26527
\(45\) 4.29719e6 0.156217
\(46\) −1.36205e7 −0.448522
\(47\) −1.60830e7 −0.480759 −0.240380 0.970679i \(-0.577272\pi\)
−0.240380 + 0.970679i \(0.577272\pi\)
\(48\) 1.07593e7 0.292548
\(49\) −2.11332e7 −0.523701
\(50\) 6.42823e7 1.45454
\(51\) −6.71045e6 −0.138895
\(52\) −2.17394e7 −0.412317
\(53\) −4.29820e7 −0.748247 −0.374123 0.927379i \(-0.622056\pi\)
−0.374123 + 0.927379i \(0.622056\pi\)
\(54\) −9.04551e7 −1.44764
\(55\) −1.44337e7 −0.212690
\(56\) 2.74763e7 0.373346
\(57\) 6.32962e7 0.794220
\(58\) −2.05355e8 −2.38275
\(59\) −4.18933e7 −0.450102 −0.225051 0.974347i \(-0.572255\pi\)
−0.225051 + 0.974347i \(0.572255\pi\)
\(60\) 1.72742e7 0.172074
\(61\) −5.02151e7 −0.464355 −0.232178 0.972673i \(-0.574585\pi\)
−0.232178 + 0.972673i \(0.574585\pi\)
\(62\) −1.18877e8 −1.02173
\(63\) 5.92844e7 0.474142
\(64\) −2.06308e8 −1.53712
\(65\) 9.97479e6 0.0693097
\(66\) 1.23730e8 0.802654
\(67\) −2.36103e8 −1.43141 −0.715707 0.698401i \(-0.753895\pi\)
−0.715707 + 0.698401i \(0.753895\pi\)
\(68\) 5.92126e7 0.335833
\(69\) −3.08022e7 −0.163592
\(70\) −4.83529e7 −0.240703
\(71\) −2.73121e8 −1.27553 −0.637767 0.770230i \(-0.720142\pi\)
−0.637767 + 0.770230i \(0.720142\pi\)
\(72\) 8.47495e7 0.371656
\(73\) 1.41037e7 0.0581274 0.0290637 0.999578i \(-0.490747\pi\)
0.0290637 + 0.999578i \(0.490747\pi\)
\(74\) 1.13156e8 0.438667
\(75\) 1.45371e8 0.530522
\(76\) −5.58522e8 −1.92034
\(77\) −1.99129e8 −0.645545
\(78\) −8.55069e7 −0.261562
\(79\) −1.34719e8 −0.389141 −0.194570 0.980889i \(-0.562331\pi\)
−0.194570 + 0.980889i \(0.562331\pi\)
\(80\) 4.35615e7 0.118904
\(81\) 6.16051e7 0.159014
\(82\) 4.38428e8 1.07087
\(83\) −2.79556e8 −0.646573 −0.323286 0.946301i \(-0.604788\pi\)
−0.323286 + 0.946301i \(0.604788\pi\)
\(84\) 2.38316e8 0.522271
\(85\) −2.71689e7 −0.0564529
\(86\) 5.77833e8 1.13909
\(87\) −4.64400e8 −0.869072
\(88\) −2.84663e8 −0.506010
\(89\) 1.92906e8 0.325905 0.162953 0.986634i \(-0.447898\pi\)
0.162953 + 0.986634i \(0.447898\pi\)
\(90\) −1.49143e8 −0.239613
\(91\) 1.37613e8 0.210365
\(92\) 2.71797e8 0.395548
\(93\) −2.68835e8 −0.372660
\(94\) 5.58194e8 0.737411
\(95\) 2.56270e8 0.322806
\(96\) −6.25278e8 −0.751368
\(97\) −3.24674e8 −0.372370 −0.186185 0.982515i \(-0.559612\pi\)
−0.186185 + 0.982515i \(0.559612\pi\)
\(98\) 7.33471e8 0.803278
\(99\) −6.14206e8 −0.642622
\(100\) −1.28275e9 −1.28275
\(101\) −1.53343e9 −1.46629 −0.733143 0.680075i \(-0.761947\pi\)
−0.733143 + 0.680075i \(0.761947\pi\)
\(102\) 2.32899e8 0.213043
\(103\) −2.10185e8 −0.184007 −0.0920035 0.995759i \(-0.529327\pi\)
−0.0920035 + 0.995759i \(0.529327\pi\)
\(104\) 1.96724e8 0.164895
\(105\) −1.09348e8 −0.0877927
\(106\) 1.49178e9 1.14770
\(107\) 6.24581e8 0.460640 0.230320 0.973115i \(-0.426023\pi\)
0.230320 + 0.973115i \(0.426023\pi\)
\(108\) 1.80503e9 1.27666
\(109\) 1.30462e9 0.885246 0.442623 0.896708i \(-0.354048\pi\)
0.442623 + 0.896708i \(0.354048\pi\)
\(110\) 5.00952e8 0.326234
\(111\) 2.55897e8 0.159997
\(112\) 6.00979e8 0.360893
\(113\) −2.65903e9 −1.53416 −0.767080 0.641551i \(-0.778291\pi\)
−0.767080 + 0.641551i \(0.778291\pi\)
\(114\) −2.19682e9 −1.21821
\(115\) −1.24710e8 −0.0664908
\(116\) 4.09784e9 2.10133
\(117\) 4.24462e8 0.209413
\(118\) 1.45399e9 0.690388
\(119\) −3.74824e8 −0.171343
\(120\) −1.56317e8 −0.0688163
\(121\) −2.94906e8 −0.125069
\(122\) 1.74282e9 0.712250
\(123\) 9.91486e8 0.390583
\(124\) 2.37219e9 0.901054
\(125\) 1.20923e9 0.443011
\(126\) −2.05759e9 −0.727261
\(127\) 1.09075e9 0.372057 0.186029 0.982544i \(-0.440438\pi\)
0.186029 + 0.982544i \(0.440438\pi\)
\(128\) 3.08148e9 1.01465
\(129\) 1.30674e9 0.415467
\(130\) −3.46195e8 −0.106310
\(131\) 5.01286e8 0.148719 0.0743593 0.997232i \(-0.476309\pi\)
0.0743593 + 0.997232i \(0.476309\pi\)
\(132\) −2.46903e9 −0.707854
\(133\) 3.53552e9 0.979763
\(134\) 8.19443e9 2.19557
\(135\) −8.28210e8 −0.214604
\(136\) −5.35826e8 −0.134307
\(137\) 5.81288e9 1.40977 0.704886 0.709320i \(-0.250998\pi\)
0.704886 + 0.709320i \(0.250998\pi\)
\(138\) 1.06905e9 0.250925
\(139\) 5.56713e9 1.26492 0.632462 0.774591i \(-0.282044\pi\)
0.632462 + 0.774591i \(0.282044\pi\)
\(140\) 9.64879e8 0.212274
\(141\) 1.26233e9 0.268960
\(142\) 9.47920e9 1.95647
\(143\) −1.42572e9 −0.285116
\(144\) 1.85370e9 0.359259
\(145\) −1.88024e9 −0.353229
\(146\) −4.89498e8 −0.0891585
\(147\) 1.65871e9 0.292983
\(148\) −2.25802e9 −0.386857
\(149\) −6.08351e9 −1.01115 −0.505576 0.862782i \(-0.668720\pi\)
−0.505576 + 0.862782i \(0.668720\pi\)
\(150\) −5.04541e9 −0.813740
\(151\) −7.47140e8 −0.116952 −0.0584758 0.998289i \(-0.518624\pi\)
−0.0584758 + 0.998289i \(0.518624\pi\)
\(152\) 5.05417e9 0.767987
\(153\) −1.15613e9 −0.170567
\(154\) 6.91118e9 0.990168
\(155\) −1.08844e9 −0.151465
\(156\) 1.70628e9 0.230670
\(157\) −4.04825e8 −0.0531764 −0.0265882 0.999646i \(-0.508464\pi\)
−0.0265882 + 0.999646i \(0.508464\pi\)
\(158\) 4.67569e9 0.596882
\(159\) 3.37358e9 0.418605
\(160\) −2.53159e9 −0.305389
\(161\) −1.72051e9 −0.201809
\(162\) −2.13813e9 −0.243902
\(163\) 4.12010e9 0.457155 0.228577 0.973526i \(-0.426593\pi\)
0.228577 + 0.973526i \(0.426593\pi\)
\(164\) −8.74881e9 −0.944391
\(165\) 1.13288e9 0.118989
\(166\) 9.70255e9 0.991744
\(167\) −6.48009e9 −0.644699 −0.322350 0.946621i \(-0.604473\pi\)
−0.322350 + 0.946621i \(0.604473\pi\)
\(168\) −2.15657e9 −0.208868
\(169\) −9.61922e9 −0.907089
\(170\) 9.42950e8 0.0865901
\(171\) 1.09052e10 0.975327
\(172\) −1.15306e10 −1.00456
\(173\) 1.65334e10 1.40332 0.701659 0.712513i \(-0.252443\pi\)
0.701659 + 0.712513i \(0.252443\pi\)
\(174\) 1.61179e10 1.33302
\(175\) 8.11998e9 0.654461
\(176\) −6.22634e9 −0.489131
\(177\) 3.28814e9 0.251808
\(178\) −6.69520e9 −0.499889
\(179\) −1.11447e10 −0.811393 −0.405696 0.914008i \(-0.632971\pi\)
−0.405696 + 0.914008i \(0.632971\pi\)
\(180\) 2.97613e9 0.211313
\(181\) 8.09522e9 0.560629 0.280314 0.959908i \(-0.409561\pi\)
0.280314 + 0.959908i \(0.409561\pi\)
\(182\) −4.77614e9 −0.322668
\(183\) 3.94130e9 0.259782
\(184\) −2.45954e9 −0.158188
\(185\) 1.03606e9 0.0650298
\(186\) 9.33046e9 0.571603
\(187\) 3.88330e9 0.232227
\(188\) −1.11387e10 −0.650317
\(189\) −1.14261e10 −0.651356
\(190\) −8.89436e9 −0.495135
\(191\) 2.54043e10 1.38120 0.690600 0.723237i \(-0.257346\pi\)
0.690600 + 0.723237i \(0.257346\pi\)
\(192\) 1.61928e10 0.859936
\(193\) 2.69561e10 1.39846 0.699229 0.714898i \(-0.253527\pi\)
0.699229 + 0.714898i \(0.253527\pi\)
\(194\) 1.12685e10 0.571159
\(195\) −7.82904e8 −0.0387751
\(196\) −1.46364e10 −0.708404
\(197\) 1.50614e9 0.0712470
\(198\) 2.13172e10 0.985684
\(199\) −2.58701e10 −1.16939 −0.584695 0.811253i \(-0.698786\pi\)
−0.584695 + 0.811253i \(0.698786\pi\)
\(200\) 1.16078e10 0.512999
\(201\) 1.85313e10 0.800801
\(202\) 5.32209e10 2.24906
\(203\) −2.59399e10 −1.07210
\(204\) −4.64749e9 −0.187881
\(205\) 4.01427e9 0.158750
\(206\) 7.29489e9 0.282238
\(207\) −5.30685e9 −0.200896
\(208\) 4.30286e9 0.159394
\(209\) −3.66291e10 −1.32791
\(210\) 3.79514e9 0.134661
\(211\) −7.64852e9 −0.265648 −0.132824 0.991140i \(-0.542405\pi\)
−0.132824 + 0.991140i \(0.542405\pi\)
\(212\) −2.97683e10 −1.01214
\(213\) 2.14368e10 0.713594
\(214\) −2.16773e10 −0.706551
\(215\) 5.29066e9 0.168864
\(216\) −1.63340e10 −0.510565
\(217\) −1.50163e10 −0.459720
\(218\) −4.52794e10 −1.35783
\(219\) −1.10698e9 −0.0325192
\(220\) −9.99646e9 −0.287703
\(221\) −2.68365e9 −0.0756764
\(222\) −8.88142e9 −0.245411
\(223\) −1.10676e10 −0.299696 −0.149848 0.988709i \(-0.547878\pi\)
−0.149848 + 0.988709i \(0.547878\pi\)
\(224\) −3.49260e10 −0.926901
\(225\) 2.50458e10 0.651498
\(226\) 9.22871e10 2.35317
\(227\) 5.78204e9 0.144532 0.0722661 0.997385i \(-0.476977\pi\)
0.0722661 + 0.997385i \(0.476977\pi\)
\(228\) 4.38374e10 1.07433
\(229\) 3.22040e10 0.773837 0.386919 0.922114i \(-0.373539\pi\)
0.386919 + 0.922114i \(0.373539\pi\)
\(230\) 4.32832e9 0.101987
\(231\) 1.56293e10 0.361148
\(232\) −3.70821e10 −0.840366
\(233\) 5.50337e10 1.22328 0.611642 0.791135i \(-0.290510\pi\)
0.611642 + 0.791135i \(0.290510\pi\)
\(234\) −1.47318e10 −0.321207
\(235\) 5.11085e9 0.109317
\(236\) −2.90143e10 −0.608847
\(237\) 1.05739e10 0.217704
\(238\) 1.30090e10 0.262814
\(239\) 4.76937e10 0.945520 0.472760 0.881191i \(-0.343258\pi\)
0.472760 + 0.881191i \(0.343258\pi\)
\(240\) −3.41907e9 −0.0665208
\(241\) −3.33763e10 −0.637326 −0.318663 0.947868i \(-0.603234\pi\)
−0.318663 + 0.947868i \(0.603234\pi\)
\(242\) 1.02353e10 0.191836
\(243\) −5.61341e10 −1.03276
\(244\) −3.47778e10 −0.628128
\(245\) 6.71569e9 0.119081
\(246\) −3.44115e10 −0.599095
\(247\) 2.53135e10 0.432729
\(248\) −2.14664e10 −0.360351
\(249\) 2.19419e10 0.361723
\(250\) −4.19688e10 −0.679512
\(251\) −8.82721e10 −1.40376 −0.701878 0.712297i \(-0.747655\pi\)
−0.701878 + 0.712297i \(0.747655\pi\)
\(252\) 4.10590e10 0.641366
\(253\) 1.78251e10 0.273520
\(254\) −3.78568e10 −0.570679
\(255\) 2.13244e9 0.0315824
\(256\) −1.31927e9 −0.0191979
\(257\) −1.54597e9 −0.0221056 −0.0110528 0.999939i \(-0.503518\pi\)
−0.0110528 + 0.999939i \(0.503518\pi\)
\(258\) −4.53531e10 −0.637263
\(259\) 1.42936e10 0.197375
\(260\) 6.90830e9 0.0937543
\(261\) −8.00106e10 −1.06725
\(262\) −1.73982e10 −0.228112
\(263\) −9.93743e10 −1.28078 −0.640388 0.768052i \(-0.721226\pi\)
−0.640388 + 0.768052i \(0.721226\pi\)
\(264\) 2.23427e10 0.283086
\(265\) 1.36587e10 0.170139
\(266\) −1.22707e11 −1.50281
\(267\) −1.51409e10 −0.182327
\(268\) −1.63519e11 −1.93626
\(269\) 8.06972e10 0.939665 0.469833 0.882756i \(-0.344314\pi\)
0.469833 + 0.882756i \(0.344314\pi\)
\(270\) 2.87447e10 0.329170
\(271\) 7.06333e10 0.795514 0.397757 0.917491i \(-0.369789\pi\)
0.397757 + 0.917491i \(0.369789\pi\)
\(272\) −1.17199e10 −0.129827
\(273\) −1.08010e10 −0.117688
\(274\) −2.01748e11 −2.16238
\(275\) −8.41256e10 −0.887015
\(276\) −2.13329e10 −0.221288
\(277\) 5.83837e10 0.595844 0.297922 0.954590i \(-0.403706\pi\)
0.297922 + 0.954590i \(0.403706\pi\)
\(278\) −1.93218e11 −1.94020
\(279\) −4.63171e10 −0.457638
\(280\) −8.73138e9 −0.0848929
\(281\) −1.94254e10 −0.185863 −0.0929313 0.995673i \(-0.529624\pi\)
−0.0929313 + 0.995673i \(0.529624\pi\)
\(282\) −4.38117e10 −0.412543
\(283\) −1.46775e11 −1.36023 −0.680117 0.733104i \(-0.738071\pi\)
−0.680117 + 0.733104i \(0.738071\pi\)
\(284\) −1.89157e11 −1.72540
\(285\) −2.01142e10 −0.180593
\(286\) 4.94823e10 0.437324
\(287\) 5.53812e10 0.481830
\(288\) −1.07728e11 −0.922704
\(289\) −1.11278e11 −0.938361
\(290\) 6.52573e10 0.541799
\(291\) 2.54831e10 0.208321
\(292\) 9.76790e9 0.0786282
\(293\) 6.43253e10 0.509891 0.254945 0.966955i \(-0.417942\pi\)
0.254945 + 0.966955i \(0.417942\pi\)
\(294\) −5.75689e10 −0.449392
\(295\) 1.33128e10 0.102346
\(296\) 2.04333e10 0.154712
\(297\) 1.18378e11 0.882808
\(298\) 2.11140e11 1.55095
\(299\) −1.23185e10 −0.0891325
\(300\) 1.00681e11 0.717630
\(301\) 7.29905e10 0.512527
\(302\) 2.59310e10 0.179386
\(303\) 1.20357e11 0.820310
\(304\) 1.10548e11 0.742369
\(305\) 1.59573e10 0.105587
\(306\) 4.01258e10 0.261624
\(307\) −1.07620e11 −0.691468 −0.345734 0.938333i \(-0.612370\pi\)
−0.345734 + 0.938333i \(0.612370\pi\)
\(308\) −1.37912e11 −0.873221
\(309\) 1.64971e10 0.102942
\(310\) 3.77766e10 0.232325
\(311\) 2.54619e11 1.54337 0.771683 0.636007i \(-0.219415\pi\)
0.771683 + 0.636007i \(0.219415\pi\)
\(312\) −1.54405e10 −0.0922498
\(313\) −9.85166e9 −0.0580176 −0.0290088 0.999579i \(-0.509235\pi\)
−0.0290088 + 0.999579i \(0.509235\pi\)
\(314\) 1.40503e10 0.0815644
\(315\) −1.88393e10 −0.107812
\(316\) −9.33031e10 −0.526386
\(317\) 1.38869e11 0.772395 0.386198 0.922416i \(-0.373788\pi\)
0.386198 + 0.922416i \(0.373788\pi\)
\(318\) −1.17087e11 −0.642075
\(319\) 2.68746e11 1.45306
\(320\) 6.55604e10 0.349515
\(321\) −4.90223e10 −0.257704
\(322\) 5.97139e10 0.309545
\(323\) −6.89476e10 −0.352459
\(324\) 4.26662e10 0.215096
\(325\) 5.81371e10 0.289053
\(326\) −1.42996e11 −0.701205
\(327\) −1.02397e11 −0.495248
\(328\) 7.91697e10 0.377682
\(329\) 7.05097e10 0.331793
\(330\) −3.93188e10 −0.182511
\(331\) 2.16628e11 0.991948 0.495974 0.868337i \(-0.334811\pi\)
0.495974 + 0.868337i \(0.334811\pi\)
\(332\) −1.93614e11 −0.874611
\(333\) 4.40880e10 0.196481
\(334\) 2.24905e11 0.988870
\(335\) 7.50286e10 0.325480
\(336\) −4.71698e10 −0.201900
\(337\) 2.71871e11 1.14823 0.574115 0.818775i \(-0.305346\pi\)
0.574115 + 0.818775i \(0.305346\pi\)
\(338\) 3.33854e11 1.39134
\(339\) 2.08703e11 0.858282
\(340\) −1.88165e10 −0.0763631
\(341\) 1.55573e11 0.623075
\(342\) −3.78486e11 −1.49600
\(343\) 2.69565e11 1.05157
\(344\) 1.04343e11 0.401744
\(345\) 9.78829e9 0.0371981
\(346\) −5.73826e11 −2.15247
\(347\) 1.59007e11 0.588752 0.294376 0.955690i \(-0.404888\pi\)
0.294376 + 0.955690i \(0.404888\pi\)
\(348\) −3.21632e11 −1.17558
\(349\) −3.98060e11 −1.43626 −0.718132 0.695907i \(-0.755003\pi\)
−0.718132 + 0.695907i \(0.755003\pi\)
\(350\) −2.81820e11 −1.00384
\(351\) −8.18079e10 −0.287682
\(352\) 3.61845e11 1.25626
\(353\) 3.42439e10 0.117381 0.0586903 0.998276i \(-0.481308\pi\)
0.0586903 + 0.998276i \(0.481308\pi\)
\(354\) −1.14121e11 −0.386236
\(355\) 8.67919e10 0.290036
\(356\) 1.33602e11 0.440848
\(357\) 2.94193e10 0.0958573
\(358\) 3.86800e11 1.24455
\(359\) −2.17991e11 −0.692649 −0.346325 0.938115i \(-0.612570\pi\)
−0.346325 + 0.938115i \(0.612570\pi\)
\(360\) −2.69316e10 −0.0845085
\(361\) 3.27660e11 1.01541
\(362\) −2.80961e11 −0.859919
\(363\) 2.31466e10 0.0699694
\(364\) 9.53076e10 0.284558
\(365\) −4.48186e9 −0.0132172
\(366\) −1.36791e11 −0.398466
\(367\) 6.13834e11 1.76626 0.883128 0.469133i \(-0.155433\pi\)
0.883128 + 0.469133i \(0.155433\pi\)
\(368\) −5.37967e10 −0.152912
\(369\) 1.70821e11 0.479648
\(370\) −3.59586e10 −0.0997458
\(371\) 1.88437e11 0.516398
\(372\) −1.86189e11 −0.504092
\(373\) 3.88866e11 1.04018 0.520092 0.854110i \(-0.325897\pi\)
0.520092 + 0.854110i \(0.325897\pi\)
\(374\) −1.34778e11 −0.356201
\(375\) −9.49106e10 −0.247841
\(376\) 1.00796e11 0.260076
\(377\) −1.85723e11 −0.473511
\(378\) 3.96565e11 0.999081
\(379\) −4.00694e11 −0.997553 −0.498776 0.866731i \(-0.666217\pi\)
−0.498776 + 0.866731i \(0.666217\pi\)
\(380\) 1.77486e11 0.436655
\(381\) −8.56114e10 −0.208147
\(382\) −8.81706e11 −2.11855
\(383\) 5.44318e11 1.29258 0.646291 0.763091i \(-0.276319\pi\)
0.646291 + 0.763091i \(0.276319\pi\)
\(384\) −2.41860e11 −0.567642
\(385\) 6.32790e10 0.146787
\(386\) −9.35566e11 −2.14502
\(387\) 2.25136e11 0.510207
\(388\) −2.24862e11 −0.503700
\(389\) −6.24163e11 −1.38205 −0.691027 0.722829i \(-0.742841\pi\)
−0.691027 + 0.722829i \(0.742841\pi\)
\(390\) 2.71723e10 0.0594751
\(391\) 3.35524e10 0.0725986
\(392\) 1.32447e11 0.283306
\(393\) −3.93451e10 −0.0832002
\(394\) −5.22736e10 −0.109282
\(395\) 4.28108e10 0.0884843
\(396\) −4.25384e11 −0.869267
\(397\) 6.86003e10 0.138602 0.0693009 0.997596i \(-0.477923\pi\)
0.0693009 + 0.997596i \(0.477923\pi\)
\(398\) 8.97874e11 1.79367
\(399\) −2.77497e11 −0.548126
\(400\) 2.53894e11 0.495887
\(401\) −9.20590e11 −1.77794 −0.888969 0.457968i \(-0.848577\pi\)
−0.888969 + 0.457968i \(0.848577\pi\)
\(402\) −6.43167e11 −1.22831
\(403\) −1.07513e11 −0.203043
\(404\) −1.06202e12 −1.98343
\(405\) −1.95768e10 −0.0361571
\(406\) 9.00296e11 1.64444
\(407\) −1.48086e11 −0.267510
\(408\) 4.20561e10 0.0751377
\(409\) −4.77860e10 −0.0844396 −0.0422198 0.999108i \(-0.513443\pi\)
−0.0422198 + 0.999108i \(0.513443\pi\)
\(410\) −1.39323e11 −0.243498
\(411\) −4.56243e11 −0.788693
\(412\) −1.45569e11 −0.248904
\(413\) 1.83665e11 0.310635
\(414\) 1.84185e11 0.308143
\(415\) 8.88369e10 0.147020
\(416\) −2.50062e11 −0.409381
\(417\) −4.36954e11 −0.707659
\(418\) 1.27129e12 2.03681
\(419\) −1.23633e11 −0.195961 −0.0979806 0.995188i \(-0.531238\pi\)
−0.0979806 + 0.995188i \(0.531238\pi\)
\(420\) −7.57317e10 −0.118756
\(421\) −7.95967e11 −1.23488 −0.617442 0.786617i \(-0.711831\pi\)
−0.617442 + 0.786617i \(0.711831\pi\)
\(422\) 2.65457e11 0.407463
\(423\) 2.17484e11 0.330291
\(424\) 2.69379e11 0.404778
\(425\) −1.58351e11 −0.235435
\(426\) −7.44006e11 −1.09454
\(427\) 2.20148e11 0.320472
\(428\) 4.32570e11 0.623102
\(429\) 1.11902e11 0.159507
\(430\) −1.83623e11 −0.259012
\(431\) 5.29976e11 0.739790 0.369895 0.929074i \(-0.379394\pi\)
0.369895 + 0.929074i \(0.379394\pi\)
\(432\) −3.57268e11 −0.493535
\(433\) 8.23521e10 0.112585 0.0562923 0.998414i \(-0.482072\pi\)
0.0562923 + 0.998414i \(0.482072\pi\)
\(434\) 5.21170e11 0.705140
\(435\) 1.47576e11 0.197613
\(436\) 9.03547e11 1.19746
\(437\) −3.16483e11 −0.415129
\(438\) 3.84198e10 0.0498795
\(439\) 2.06575e11 0.265453 0.132727 0.991153i \(-0.457627\pi\)
0.132727 + 0.991153i \(0.457627\pi\)
\(440\) 9.04599e10 0.115059
\(441\) 2.85776e11 0.359793
\(442\) 9.31414e10 0.116076
\(443\) 3.89537e11 0.480543 0.240271 0.970706i \(-0.422764\pi\)
0.240271 + 0.970706i \(0.422764\pi\)
\(444\) 1.77228e11 0.216426
\(445\) −6.13015e10 −0.0741056
\(446\) 3.84122e11 0.459688
\(447\) 4.77484e11 0.565686
\(448\) 9.04477e11 1.06083
\(449\) −1.16999e12 −1.35854 −0.679271 0.733887i \(-0.737704\pi\)
−0.679271 + 0.733887i \(0.737704\pi\)
\(450\) −8.69263e11 −0.999298
\(451\) −5.73767e11 −0.653042
\(452\) −1.84158e12 −2.07524
\(453\) 5.86417e10 0.0654282
\(454\) −2.00677e11 −0.221690
\(455\) −4.37305e10 −0.0478336
\(456\) −3.96693e11 −0.429648
\(457\) 1.55946e12 1.67244 0.836221 0.548392i \(-0.184760\pi\)
0.836221 + 0.548392i \(0.184760\pi\)
\(458\) −1.11770e12 −1.18695
\(459\) 2.22824e11 0.234318
\(460\) −8.63713e10 −0.0899413
\(461\) 7.18326e11 0.740742 0.370371 0.928884i \(-0.379230\pi\)
0.370371 + 0.928884i \(0.379230\pi\)
\(462\) −5.42446e11 −0.553947
\(463\) 7.79371e11 0.788188 0.394094 0.919070i \(-0.371058\pi\)
0.394094 + 0.919070i \(0.371058\pi\)
\(464\) −8.11084e11 −0.812335
\(465\) 8.54300e10 0.0847369
\(466\) −1.91005e12 −1.87633
\(467\) 1.68439e12 1.63876 0.819381 0.573250i \(-0.194318\pi\)
0.819381 + 0.573250i \(0.194318\pi\)
\(468\) 2.93972e11 0.283270
\(469\) 1.03510e12 0.987882
\(470\) −1.77382e11 −0.167675
\(471\) 3.17740e10 0.0297494
\(472\) 2.62556e11 0.243491
\(473\) −7.56205e11 −0.694647
\(474\) −3.66987e11 −0.333924
\(475\) 1.49364e12 1.34625
\(476\) −2.59594e11 −0.231773
\(477\) 5.81227e11 0.514060
\(478\) −1.65531e12 −1.45028
\(479\) 4.99339e11 0.433396 0.216698 0.976239i \(-0.430471\pi\)
0.216698 + 0.976239i \(0.430471\pi\)
\(480\) 1.98700e11 0.170849
\(481\) 1.02339e11 0.0871740
\(482\) 1.15839e12 0.977560
\(483\) 1.35040e11 0.112902
\(484\) −2.04245e11 −0.169179
\(485\) 1.03175e11 0.0846710
\(486\) 1.94825e12 1.58409
\(487\) 1.78158e12 1.43524 0.717619 0.696436i \(-0.245232\pi\)
0.717619 + 0.696436i \(0.245232\pi\)
\(488\) 3.14711e11 0.251202
\(489\) −3.23379e11 −0.255754
\(490\) −2.33082e11 −0.182652
\(491\) −5.85711e11 −0.454796 −0.227398 0.973802i \(-0.573022\pi\)
−0.227398 + 0.973802i \(0.573022\pi\)
\(492\) 6.86679e11 0.528337
\(493\) 5.05864e11 0.385676
\(494\) −8.78555e11 −0.663740
\(495\) 1.95181e11 0.146122
\(496\) −4.69526e11 −0.348331
\(497\) 1.19739e12 0.880302
\(498\) −7.61537e11 −0.554828
\(499\) 3.39368e11 0.245030 0.122515 0.992467i \(-0.460904\pi\)
0.122515 + 0.992467i \(0.460904\pi\)
\(500\) 8.37485e11 0.599256
\(501\) 5.08611e11 0.360675
\(502\) 3.06366e12 2.15315
\(503\) 1.15682e12 0.805769 0.402884 0.915251i \(-0.368008\pi\)
0.402884 + 0.915251i \(0.368008\pi\)
\(504\) −3.71551e11 −0.256496
\(505\) 4.87292e11 0.333410
\(506\) −6.18655e11 −0.419538
\(507\) 7.54996e11 0.507468
\(508\) 7.55430e11 0.503277
\(509\) −2.22385e11 −0.146850 −0.0734251 0.997301i \(-0.523393\pi\)
−0.0734251 + 0.997301i \(0.523393\pi\)
\(510\) −7.40105e10 −0.0484426
\(511\) −6.18322e10 −0.0401163
\(512\) −1.53193e12 −0.985201
\(513\) −2.10179e12 −1.33986
\(514\) 5.36560e10 0.0339066
\(515\) 6.67923e10 0.0418402
\(516\) 9.05019e11 0.561997
\(517\) −7.30503e11 −0.449692
\(518\) −4.96088e11 −0.302743
\(519\) −1.29768e12 −0.785082
\(520\) −6.25145e10 −0.0374944
\(521\) 3.85547e11 0.229249 0.114625 0.993409i \(-0.463433\pi\)
0.114625 + 0.993409i \(0.463433\pi\)
\(522\) 2.77693e12 1.63699
\(523\) −1.73676e12 −1.01504 −0.507518 0.861641i \(-0.669437\pi\)
−0.507518 + 0.861641i \(0.669437\pi\)
\(524\) 3.47179e11 0.201170
\(525\) −6.37323e11 −0.366136
\(526\) 3.44898e12 1.96451
\(527\) 2.92838e11 0.165379
\(528\) 4.88694e11 0.273643
\(529\) −1.64714e12 −0.914493
\(530\) −4.74054e11 −0.260968
\(531\) 5.66507e11 0.309229
\(532\) 2.44862e12 1.32531
\(533\) 3.96516e11 0.212808
\(534\) 5.25495e11 0.279662
\(535\) −1.98478e11 −0.104742
\(536\) 1.47972e12 0.774350
\(537\) 8.74731e11 0.453932
\(538\) −2.80076e12 −1.44130
\(539\) −9.59887e11 −0.489859
\(540\) −5.73599e11 −0.290293
\(541\) 1.77719e12 0.891964 0.445982 0.895042i \(-0.352855\pi\)
0.445982 + 0.895042i \(0.352855\pi\)
\(542\) −2.45147e12 −1.22020
\(543\) −6.35380e11 −0.313642
\(544\) 6.81107e11 0.333442
\(545\) −4.14580e11 −0.201291
\(546\) 3.74871e11 0.180516
\(547\) −3.93472e12 −1.87919 −0.939595 0.342289i \(-0.888798\pi\)
−0.939595 + 0.342289i \(0.888798\pi\)
\(548\) 4.02586e12 1.90698
\(549\) 6.79039e11 0.319021
\(550\) 2.91975e12 1.36055
\(551\) −4.77156e12 −2.20535
\(552\) 1.93045e11 0.0884980
\(553\) 5.90622e11 0.268563
\(554\) −2.02632e12 −0.913934
\(555\) −8.13186e10 −0.0363807
\(556\) 3.85566e12 1.71105
\(557\) 1.79477e12 0.790061 0.395031 0.918668i \(-0.370734\pi\)
0.395031 + 0.918668i \(0.370734\pi\)
\(558\) 1.60753e12 0.701947
\(559\) 5.22594e11 0.226366
\(560\) −1.90978e11 −0.0820612
\(561\) −3.04793e11 −0.129919
\(562\) 6.74198e11 0.285085
\(563\) 1.89933e12 0.796731 0.398365 0.917227i \(-0.369578\pi\)
0.398365 + 0.917227i \(0.369578\pi\)
\(564\) 8.74260e11 0.363818
\(565\) 8.44984e11 0.348843
\(566\) 5.09413e12 2.08639
\(567\) −2.70083e11 −0.109742
\(568\) 1.71172e12 0.690024
\(569\) −2.64265e12 −1.05690 −0.528451 0.848964i \(-0.677227\pi\)
−0.528451 + 0.848964i \(0.677227\pi\)
\(570\) 6.98103e11 0.277002
\(571\) 2.85690e12 1.12469 0.562344 0.826903i \(-0.309900\pi\)
0.562344 + 0.826903i \(0.309900\pi\)
\(572\) −9.87417e11 −0.385672
\(573\) −1.99394e12 −0.772709
\(574\) −1.92212e12 −0.739054
\(575\) −7.26861e11 −0.277298
\(576\) 2.78982e12 1.05603
\(577\) 2.44238e12 0.917323 0.458662 0.888611i \(-0.348329\pi\)
0.458662 + 0.888611i \(0.348329\pi\)
\(578\) 3.86214e12 1.43930
\(579\) −2.11574e12 −0.782363
\(580\) −1.30221e12 −0.477808
\(581\) 1.22560e12 0.446228
\(582\) −8.84442e11 −0.319533
\(583\) −1.95227e12 −0.699893
\(584\) −8.83916e10 −0.0314451
\(585\) −1.34885e11 −0.0476170
\(586\) −2.23254e12 −0.782095
\(587\) −4.14506e12 −1.44098 −0.720492 0.693463i \(-0.756084\pi\)
−0.720492 + 0.693463i \(0.756084\pi\)
\(588\) 1.14878e12 0.396315
\(589\) −2.76219e12 −0.945660
\(590\) −4.62048e11 −0.156983
\(591\) −1.18214e11 −0.0398590
\(592\) 4.46929e11 0.149552
\(593\) 2.78764e12 0.925742 0.462871 0.886426i \(-0.346819\pi\)
0.462871 + 0.886426i \(0.346819\pi\)
\(594\) −4.10854e12 −1.35409
\(595\) 1.19111e11 0.0389606
\(596\) −4.21330e12 −1.36777
\(597\) 2.03050e12 0.654212
\(598\) 4.27537e11 0.136716
\(599\) 5.47204e11 0.173672 0.0868358 0.996223i \(-0.472324\pi\)
0.0868358 + 0.996223i \(0.472324\pi\)
\(600\) −9.11079e11 −0.286996
\(601\) −2.82471e12 −0.883160 −0.441580 0.897222i \(-0.645582\pi\)
−0.441580 + 0.897222i \(0.645582\pi\)
\(602\) −2.53328e12 −0.786139
\(603\) 3.19273e12 0.983408
\(604\) −5.17451e11 −0.158199
\(605\) 9.37147e10 0.0284386
\(606\) −4.17721e12 −1.25823
\(607\) −3.14194e12 −0.939397 −0.469699 0.882827i \(-0.655637\pi\)
−0.469699 + 0.882827i \(0.655637\pi\)
\(608\) −6.42453e12 −1.90667
\(609\) 2.03598e12 0.599785
\(610\) −5.53830e11 −0.161954
\(611\) 5.04832e11 0.146542
\(612\) −8.00708e11 −0.230724
\(613\) −4.73278e12 −1.35377 −0.676885 0.736089i \(-0.736670\pi\)
−0.676885 + 0.736089i \(0.736670\pi\)
\(614\) 3.73518e12 1.06061
\(615\) −3.15073e11 −0.0888123
\(616\) 1.24799e12 0.349220
\(617\) −1.65834e12 −0.460671 −0.230335 0.973111i \(-0.573982\pi\)
−0.230335 + 0.973111i \(0.573982\pi\)
\(618\) −5.72563e11 −0.157898
\(619\) 3.65341e12 1.00021 0.500105 0.865965i \(-0.333295\pi\)
0.500105 + 0.865965i \(0.333295\pi\)
\(620\) −7.53830e11 −0.204885
\(621\) 1.02281e12 0.275982
\(622\) −8.83706e12 −2.36729
\(623\) −8.45721e11 −0.224922
\(624\) −3.37724e11 −0.0891727
\(625\) 3.23320e12 0.847563
\(626\) 3.41922e11 0.0889901
\(627\) 2.87496e12 0.742895
\(628\) −2.80372e11 −0.0719310
\(629\) −2.78745e11 −0.0710034
\(630\) 6.53857e11 0.165367
\(631\) −3.13554e12 −0.787373 −0.393687 0.919245i \(-0.628800\pi\)
−0.393687 + 0.919245i \(0.628800\pi\)
\(632\) 8.44317e11 0.210513
\(633\) 6.00319e11 0.148616
\(634\) −4.81974e12 −1.18474
\(635\) −3.46618e11 −0.0845999
\(636\) 2.33646e12 0.566241
\(637\) 6.63354e11 0.159631
\(638\) −9.32736e12 −2.22877
\(639\) 3.69330e12 0.876315
\(640\) −9.79230e11 −0.230714
\(641\) 3.92823e12 0.919042 0.459521 0.888167i \(-0.348021\pi\)
0.459521 + 0.888167i \(0.348021\pi\)
\(642\) 1.70142e12 0.395278
\(643\) −2.71651e11 −0.0626702 −0.0313351 0.999509i \(-0.509976\pi\)
−0.0313351 + 0.999509i \(0.509976\pi\)
\(644\) −1.19159e12 −0.272985
\(645\) −4.15255e11 −0.0944705
\(646\) 2.39297e12 0.540618
\(647\) −4.95489e12 −1.11164 −0.555821 0.831302i \(-0.687596\pi\)
−0.555821 + 0.831302i \(0.687596\pi\)
\(648\) −3.86095e11 −0.0860214
\(649\) −1.90283e12 −0.421015
\(650\) −2.01776e12 −0.443364
\(651\) 1.17860e12 0.257189
\(652\) 2.85348e12 0.618387
\(653\) −1.24814e11 −0.0268629 −0.0134314 0.999910i \(-0.504275\pi\)
−0.0134314 + 0.999910i \(0.504275\pi\)
\(654\) 3.55390e12 0.759635
\(655\) −1.59298e11 −0.0338162
\(656\) 1.73165e12 0.365084
\(657\) −1.90719e11 −0.0399346
\(658\) −2.44718e12 −0.508920
\(659\) 6.61121e12 1.36552 0.682758 0.730645i \(-0.260780\pi\)
0.682758 + 0.730645i \(0.260780\pi\)
\(660\) 7.84605e11 0.160955
\(661\) −1.86809e12 −0.380620 −0.190310 0.981724i \(-0.560949\pi\)
−0.190310 + 0.981724i \(0.560949\pi\)
\(662\) −7.51851e12 −1.52150
\(663\) 2.10635e11 0.0423370
\(664\) 1.75205e12 0.349776
\(665\) −1.12351e12 −0.222782
\(666\) −1.53016e12 −0.301372
\(667\) 2.32201e12 0.454254
\(668\) −4.48796e12 −0.872076
\(669\) 8.68675e11 0.167664
\(670\) −2.60402e12 −0.499237
\(671\) −2.28081e12 −0.434348
\(672\) 2.74128e12 0.518552
\(673\) −7.86777e12 −1.47837 −0.739186 0.673501i \(-0.764790\pi\)
−0.739186 + 0.673501i \(0.764790\pi\)
\(674\) −9.43584e12 −1.76121
\(675\) −4.82714e12 −0.895000
\(676\) −6.66204e12 −1.22701
\(677\) −2.30065e12 −0.420922 −0.210461 0.977602i \(-0.567496\pi\)
−0.210461 + 0.977602i \(0.567496\pi\)
\(678\) −7.24346e12 −1.31647
\(679\) 1.42341e12 0.256989
\(680\) 1.70274e11 0.0305393
\(681\) −4.53822e11 −0.0808582
\(682\) −5.39948e12 −0.955702
\(683\) −1.72763e12 −0.303779 −0.151889 0.988397i \(-0.548536\pi\)
−0.151889 + 0.988397i \(0.548536\pi\)
\(684\) 7.55266e12 1.31931
\(685\) −1.84721e12 −0.320560
\(686\) −9.35578e12 −1.61295
\(687\) −2.52763e12 −0.432921
\(688\) 2.28225e12 0.388343
\(689\) 1.34917e12 0.228076
\(690\) −3.39722e11 −0.0570562
\(691\) −6.13080e12 −1.02298 −0.511489 0.859290i \(-0.670906\pi\)
−0.511489 + 0.859290i \(0.670906\pi\)
\(692\) 1.14507e13 1.89825
\(693\) 2.69274e12 0.443502
\(694\) −5.51864e12 −0.903055
\(695\) −1.76911e12 −0.287623
\(696\) 2.91051e12 0.470141
\(697\) −1.08001e12 −0.173333
\(698\) 1.38155e13 2.20301
\(699\) −4.31950e12 −0.684362
\(700\) 5.62370e12 0.885281
\(701\) 3.06258e12 0.479024 0.239512 0.970893i \(-0.423013\pi\)
0.239512 + 0.970893i \(0.423013\pi\)
\(702\) 2.83931e12 0.441261
\(703\) 2.62926e12 0.406008
\(704\) −9.37067e12 −1.43778
\(705\) −4.01142e11 −0.0611571
\(706\) −1.18850e12 −0.180044
\(707\) 6.72273e12 1.01195
\(708\) 2.27728e12 0.340618
\(709\) 7.23288e12 1.07499 0.537493 0.843268i \(-0.319371\pi\)
0.537493 + 0.843268i \(0.319371\pi\)
\(710\) −3.01229e12 −0.444871
\(711\) 1.82175e12 0.267347
\(712\) −1.20899e12 −0.176305
\(713\) 1.34418e12 0.194785
\(714\) −1.02106e12 −0.147030
\(715\) 4.53062e11 0.0648307
\(716\) −7.71858e12 −1.09756
\(717\) −3.74340e12 −0.528969
\(718\) 7.56581e12 1.06242
\(719\) 1.09651e13 1.53014 0.765072 0.643944i \(-0.222703\pi\)
0.765072 + 0.643944i \(0.222703\pi\)
\(720\) −5.89065e11 −0.0816896
\(721\) 9.21473e11 0.126991
\(722\) −1.13721e13 −1.55748
\(723\) 2.61965e12 0.356550
\(724\) 5.60656e12 0.758355
\(725\) −1.09588e13 −1.47313
\(726\) −8.03350e11 −0.107322
\(727\) 1.30404e13 1.73135 0.865676 0.500604i \(-0.166889\pi\)
0.865676 + 0.500604i \(0.166889\pi\)
\(728\) −8.62456e11 −0.113801
\(729\) 3.19329e12 0.418760
\(730\) 1.55552e11 0.0202732
\(731\) −1.42342e12 −0.184376
\(732\) 2.72965e12 0.351404
\(733\) −1.43509e12 −0.183616 −0.0918080 0.995777i \(-0.529265\pi\)
−0.0918080 + 0.995777i \(0.529265\pi\)
\(734\) −2.13043e13 −2.70917
\(735\) −5.27103e11 −0.0666197
\(736\) 3.12641e12 0.392731
\(737\) −1.07240e13 −1.33891
\(738\) −5.92869e12 −0.735707
\(739\) −9.09373e12 −1.12161 −0.560805 0.827948i \(-0.689508\pi\)
−0.560805 + 0.827948i \(0.689508\pi\)
\(740\) 7.17551e11 0.0879650
\(741\) −1.98681e12 −0.242089
\(742\) −6.54010e12 −0.792076
\(743\) −1.54607e12 −0.186115 −0.0930573 0.995661i \(-0.529664\pi\)
−0.0930573 + 0.995661i \(0.529664\pi\)
\(744\) 1.68486e12 0.201597
\(745\) 1.93321e12 0.229920
\(746\) −1.34964e13 −1.59548
\(747\) 3.78032e12 0.444208
\(748\) 2.68948e12 0.314131
\(749\) −2.73823e12 −0.317908
\(750\) 3.29406e12 0.380151
\(751\) −5.99142e12 −0.687306 −0.343653 0.939097i \(-0.611664\pi\)
−0.343653 + 0.939097i \(0.611664\pi\)
\(752\) 2.20469e12 0.251401
\(753\) 6.92832e12 0.785328
\(754\) 6.44590e12 0.726294
\(755\) 2.37425e11 0.0265929
\(756\) −7.91342e12 −0.881082
\(757\) 9.68176e12 1.07158 0.535788 0.844353i \(-0.320015\pi\)
0.535788 + 0.844353i \(0.320015\pi\)
\(758\) 1.39069e13 1.53009
\(759\) −1.39906e12 −0.153020
\(760\) −1.60611e12 −0.174628
\(761\) 9.51077e12 1.02798 0.513990 0.857796i \(-0.328167\pi\)
0.513990 + 0.857796i \(0.328167\pi\)
\(762\) 2.97131e12 0.319265
\(763\) −5.71958e12 −0.610947
\(764\) 1.75944e13 1.86833
\(765\) 3.67393e11 0.0387842
\(766\) −1.88916e13 −1.98262
\(767\) 1.31500e12 0.137197
\(768\) 1.03547e11 0.0107402
\(769\) −5.53577e12 −0.570834 −0.285417 0.958403i \(-0.592132\pi\)
−0.285417 + 0.958403i \(0.592132\pi\)
\(770\) −2.19622e12 −0.225148
\(771\) 1.21341e11 0.0123669
\(772\) 1.86692e13 1.89168
\(773\) −2.03244e12 −0.204744 −0.102372 0.994746i \(-0.532643\pi\)
−0.102372 + 0.994746i \(0.532643\pi\)
\(774\) −7.81380e12 −0.782579
\(775\) −6.34389e12 −0.631681
\(776\) 2.03481e12 0.201441
\(777\) −1.12188e12 −0.110421
\(778\) 2.16628e13 2.11986
\(779\) 1.01872e13 0.991142
\(780\) −5.42221e11 −0.0524506
\(781\) −1.24053e13 −1.19311
\(782\) −1.16450e12 −0.111355
\(783\) 1.54207e13 1.46614
\(784\) 2.89697e12 0.273856
\(785\) 1.28645e11 0.0120914
\(786\) 1.36555e12 0.127616
\(787\) 1.56177e13 1.45121 0.725607 0.688110i \(-0.241559\pi\)
0.725607 + 0.688110i \(0.241559\pi\)
\(788\) 1.04312e12 0.0963750
\(789\) 7.79971e12 0.716526
\(790\) −1.48583e12 −0.135721
\(791\) 1.16575e13 1.05879
\(792\) 3.84938e12 0.347638
\(793\) 1.57621e12 0.141542
\(794\) −2.38091e12 −0.212594
\(795\) −1.07205e12 −0.0951839
\(796\) −1.79170e13 −1.58182
\(797\) −1.65794e13 −1.45548 −0.727742 0.685851i \(-0.759430\pi\)
−0.727742 + 0.685851i \(0.759430\pi\)
\(798\) 9.63109e12 0.840741
\(799\) −1.37504e12 −0.119359
\(800\) −1.47551e13 −1.27361
\(801\) −2.60859e12 −0.223903
\(802\) 3.19509e13 2.72708
\(803\) 6.40601e11 0.0543711
\(804\) 1.28344e13 1.08323
\(805\) 5.46742e11 0.0458882
\(806\) 3.73145e12 0.311437
\(807\) −6.33378e12 −0.525693
\(808\) 9.61041e12 0.793215
\(809\) −6.70961e12 −0.550717 −0.275359 0.961342i \(-0.588797\pi\)
−0.275359 + 0.961342i \(0.588797\pi\)
\(810\) 6.79452e11 0.0554595
\(811\) −9.43966e12 −0.766236 −0.383118 0.923699i \(-0.625150\pi\)
−0.383118 + 0.923699i \(0.625150\pi\)
\(812\) −1.79654e13 −1.45022
\(813\) −5.54389e12 −0.445048
\(814\) 5.13963e12 0.410319
\(815\) −1.30928e12 −0.103950
\(816\) 9.19877e11 0.0726314
\(817\) 1.34264e13 1.05429
\(818\) 1.65851e12 0.129517
\(819\) −1.86089e12 −0.144525
\(820\) 2.78019e12 0.214739
\(821\) 5.58747e12 0.429211 0.214606 0.976701i \(-0.431153\pi\)
0.214606 + 0.976701i \(0.431153\pi\)
\(822\) 1.58348e13 1.20974
\(823\) −1.54375e12 −0.117294 −0.0586472 0.998279i \(-0.518679\pi\)
−0.0586472 + 0.998279i \(0.518679\pi\)
\(824\) 1.31728e12 0.0995420
\(825\) 6.60287e12 0.496238
\(826\) −6.37445e12 −0.476467
\(827\) 1.60862e13 1.19585 0.597926 0.801551i \(-0.295992\pi\)
0.597926 + 0.801551i \(0.295992\pi\)
\(828\) −3.67540e12 −0.271749
\(829\) 2.18581e13 1.60738 0.803689 0.595050i \(-0.202868\pi\)
0.803689 + 0.595050i \(0.202868\pi\)
\(830\) −3.08326e12 −0.225507
\(831\) −4.58244e12 −0.333343
\(832\) 6.47583e12 0.468534
\(833\) −1.80681e12 −0.130020
\(834\) 1.51654e13 1.08544
\(835\) 2.05923e12 0.146594
\(836\) −2.53685e13 −1.79625
\(837\) 8.92683e12 0.628684
\(838\) 4.29092e12 0.300574
\(839\) 1.41779e13 0.987831 0.493915 0.869510i \(-0.335565\pi\)
0.493915 + 0.869510i \(0.335565\pi\)
\(840\) 6.85311e11 0.0474931
\(841\) 2.05015e13 1.41320
\(842\) 2.76256e13 1.89412
\(843\) 1.52467e12 0.103980
\(844\) −5.29718e12 −0.359339
\(845\) 3.05678e12 0.206257
\(846\) −7.54823e12 −0.506616
\(847\) 1.29290e12 0.0863154
\(848\) 5.89203e12 0.391276
\(849\) 1.15201e13 0.760979
\(850\) 5.49589e12 0.361121
\(851\) −1.27949e12 −0.0836286
\(852\) 1.48466e13 0.965269
\(853\) 8.74262e12 0.565420 0.282710 0.959205i \(-0.408767\pi\)
0.282710 + 0.959205i \(0.408767\pi\)
\(854\) −7.64069e12 −0.491555
\(855\) −3.46543e12 −0.221774
\(856\) −3.91441e12 −0.249192
\(857\) 1.69587e13 1.07394 0.536969 0.843602i \(-0.319569\pi\)
0.536969 + 0.843602i \(0.319569\pi\)
\(858\) −3.88378e12 −0.244660
\(859\) 2.06546e13 1.29434 0.647169 0.762346i \(-0.275953\pi\)
0.647169 + 0.762346i \(0.275953\pi\)
\(860\) 3.66419e12 0.228420
\(861\) −4.34678e12 −0.269559
\(862\) −1.83939e13 −1.13473
\(863\) −1.47651e13 −0.906126 −0.453063 0.891478i \(-0.649669\pi\)
−0.453063 + 0.891478i \(0.649669\pi\)
\(864\) 2.07627e13 1.26757
\(865\) −5.25398e12 −0.319092
\(866\) −2.85819e12 −0.172688
\(867\) 8.73404e12 0.524964
\(868\) −1.03999e13 −0.621857
\(869\) −6.11903e12 −0.363993
\(870\) −5.12194e12 −0.303108
\(871\) 7.41107e12 0.436314
\(872\) −8.17637e12 −0.478890
\(873\) 4.39044e12 0.255825
\(874\) 1.09842e13 0.636745
\(875\) −5.30140e12 −0.305742
\(876\) −7.66666e11 −0.0439883
\(877\) 1.84154e13 1.05119 0.525597 0.850734i \(-0.323842\pi\)
0.525597 + 0.850734i \(0.323842\pi\)
\(878\) −7.16961e12 −0.407165
\(879\) −5.04878e12 −0.285257
\(880\) 1.97860e12 0.111221
\(881\) 4.40458e9 0.000246328 0 0.000123164 1.00000i \(-0.499961\pi\)
0.000123164 1.00000i \(0.499961\pi\)
\(882\) −9.91843e12 −0.551867
\(883\) −2.14276e13 −1.18618 −0.593090 0.805136i \(-0.702092\pi\)
−0.593090 + 0.805136i \(0.702092\pi\)
\(884\) −1.85863e12 −0.102367
\(885\) −1.04490e12 −0.0572572
\(886\) −1.35197e13 −0.737079
\(887\) −1.16483e13 −0.631836 −0.315918 0.948786i \(-0.602312\pi\)
−0.315918 + 0.948786i \(0.602312\pi\)
\(888\) −1.60377e12 −0.0865534
\(889\) −4.78198e12 −0.256773
\(890\) 2.12759e12 0.113667
\(891\) 2.79815e12 0.148738
\(892\) −7.66514e12 −0.405395
\(893\) 1.29700e13 0.682510
\(894\) −1.65721e13 −0.867676
\(895\) 3.54156e12 0.184498
\(896\) −1.35095e13 −0.700253
\(897\) 9.66855e11 0.0498649
\(898\) 4.06068e13 2.08380
\(899\) 2.02660e13 1.03478
\(900\) 1.73461e13 0.881273
\(901\) −3.67479e12 −0.185768
\(902\) 1.99137e13 1.00167
\(903\) −5.72890e12 −0.286732
\(904\) 1.66648e13 0.829933
\(905\) −2.57249e12 −0.127478
\(906\) −2.03528e12 −0.100357
\(907\) 1.69591e13 0.832088 0.416044 0.909344i \(-0.363416\pi\)
0.416044 + 0.909344i \(0.363416\pi\)
\(908\) 4.00450e12 0.195507
\(909\) 2.07360e13 1.00737
\(910\) 1.51776e12 0.0733695
\(911\) −1.52448e13 −0.733310 −0.366655 0.930357i \(-0.619497\pi\)
−0.366655 + 0.930357i \(0.619497\pi\)
\(912\) −8.67673e12 −0.415317
\(913\) −1.26976e13 −0.604790
\(914\) −5.41242e13 −2.56527
\(915\) −1.25246e12 −0.0590703
\(916\) 2.23037e13 1.04676
\(917\) −2.19769e12 −0.102637
\(918\) −7.73357e12 −0.359408
\(919\) 1.71192e13 0.791704 0.395852 0.918314i \(-0.370449\pi\)
0.395852 + 0.918314i \(0.370449\pi\)
\(920\) 7.81590e11 0.0359695
\(921\) 8.44694e12 0.386840
\(922\) −2.49309e13 −1.13619
\(923\) 8.57301e12 0.388800
\(924\) 1.08245e13 0.488521
\(925\) 6.03858e12 0.271204
\(926\) −2.70496e13 −1.20896
\(927\) 2.84225e12 0.126416
\(928\) 4.71363e13 2.08636
\(929\) −2.10069e13 −0.925318 −0.462659 0.886536i \(-0.653105\pi\)
−0.462659 + 0.886536i \(0.653105\pi\)
\(930\) −2.96502e12 −0.129973
\(931\) 1.70427e13 0.743473
\(932\) 3.81150e13 1.65472
\(933\) −1.99846e13 −0.863432
\(934\) −5.84600e13 −2.51361
\(935\) −1.23403e12 −0.0528048
\(936\) −2.66021e12 −0.113286
\(937\) 1.95899e13 0.830240 0.415120 0.909767i \(-0.363740\pi\)
0.415120 + 0.909767i \(0.363740\pi\)
\(938\) −3.59253e13 −1.51526
\(939\) 7.73240e11 0.0324578
\(940\) 3.53965e12 0.147872
\(941\) −4.14361e13 −1.72276 −0.861382 0.507957i \(-0.830401\pi\)
−0.861382 + 0.507957i \(0.830401\pi\)
\(942\) −1.10278e12 −0.0456310
\(943\) −4.95746e12 −0.204153
\(944\) 5.74280e12 0.235369
\(945\) 3.63096e12 0.148108
\(946\) 2.62456e13 1.06548
\(947\) −5.17476e12 −0.209081 −0.104541 0.994521i \(-0.533337\pi\)
−0.104541 + 0.994521i \(0.533337\pi\)
\(948\) 7.32320e12 0.294485
\(949\) −4.42703e11 −0.0177180
\(950\) −5.18399e13 −2.06494
\(951\) −1.08996e13 −0.432114
\(952\) 2.34912e12 0.0926912
\(953\) −2.26576e13 −0.889808 −0.444904 0.895578i \(-0.646762\pi\)
−0.444904 + 0.895578i \(0.646762\pi\)
\(954\) −2.01727e13 −0.788489
\(955\) −8.07294e12 −0.314063
\(956\) 3.30315e13 1.27899
\(957\) −2.10934e13 −0.812910
\(958\) −1.73305e13 −0.664764
\(959\) −2.54843e13 −0.972946
\(960\) −5.14572e12 −0.195535
\(961\) −1.47079e13 −0.556282
\(962\) −3.55187e12 −0.133712
\(963\) −8.44595e12 −0.316468
\(964\) −2.31156e13 −0.862103
\(965\) −8.56608e12 −0.317987
\(966\) −4.68684e12 −0.173174
\(967\) 4.77063e13 1.75451 0.877257 0.480021i \(-0.159371\pi\)
0.877257 + 0.480021i \(0.159371\pi\)
\(968\) 1.84825e12 0.0676583
\(969\) 5.41158e12 0.197182
\(970\) −3.58088e12 −0.129872
\(971\) −2.95828e13 −1.06796 −0.533978 0.845498i \(-0.679304\pi\)
−0.533978 + 0.845498i \(0.679304\pi\)
\(972\) −3.88771e13 −1.39700
\(973\) −2.44069e13 −0.872980
\(974\) −6.18332e13 −2.20144
\(975\) −4.56308e12 −0.161710
\(976\) 6.88356e12 0.242823
\(977\) 4.08329e13 1.43379 0.716893 0.697183i \(-0.245564\pi\)
0.716893 + 0.697183i \(0.245564\pi\)
\(978\) 1.12235e13 0.392287
\(979\) 8.76195e12 0.304845
\(980\) 4.65113e12 0.161080
\(981\) −1.76418e13 −0.608181
\(982\) 2.03283e13 0.697587
\(983\) 2.71445e13 0.927239 0.463619 0.886034i \(-0.346551\pi\)
0.463619 + 0.886034i \(0.346551\pi\)
\(984\) −6.21389e12 −0.211293
\(985\) −4.78619e11 −0.0162004
\(986\) −1.75570e13 −0.591569
\(987\) −5.53419e12 −0.185621
\(988\) 1.75315e13 0.585347
\(989\) −6.53375e12 −0.217160
\(990\) −6.77416e12 −0.224129
\(991\) 3.58312e13 1.18013 0.590064 0.807356i \(-0.299102\pi\)
0.590064 + 0.807356i \(0.299102\pi\)
\(992\) 2.72866e13 0.894637
\(993\) −1.70028e13 −0.554942
\(994\) −4.15578e13 −1.35025
\(995\) 8.22097e12 0.265901
\(996\) 1.51964e13 0.489299
\(997\) 5.01617e12 0.160785 0.0803923 0.996763i \(-0.474383\pi\)
0.0803923 + 0.996763i \(0.474383\pi\)
\(998\) −1.17785e13 −0.375838
\(999\) −8.49721e12 −0.269918
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 197.10.a.b.1.13 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
197.10.a.b.1.13 76 1.1 even 1 trivial